Produced by Marius Masi, Don Kretz and the Online
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Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
      printed in subscript. Letter subscripts are preceded by an
      underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective
      paragraphs.

(4) Macrons and breves above letters and dots below letters were not
      inserted.

(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek
      letters, [oo] for infinity and [dP] for partial differential
      symbol.

(6) The following typographical errors have been corrected:

    ARTICLE MAURITIUS: "... in 1893 a great part of Port Louis was
      destroyed by fire." 'a' added.

    ARTICLE MAXIMA AND MINIMA: "If d²u/dx² vanishes, then there is no
      maximum or minimum unless d²u/dx² vanishes ..." 'minimum' amended
      from 'minimun.'

    ARTICLE MAYOR: "Any female servant or slave in the household of a
      barbarian, whose business it was to overlook other female servants
      or slaves, would be quite naturally called a majorissa."
      'household' amended from 'houselold'.

    ARTICLE MAZANDARAN: "They speak a marked Persian dialect, but a
      Turki idiom closely akin to the Turkoman is still current amongst
      the tribes, although they have mostly already passed from the nomad
      to the settled state." 'idiom' amended from 'idion'.

    ARTICLE MAZARIN, JULES: "But he began to wish for a wider sphere
      than papal negotiations, and, seeing that he had no chance of
      becoming a cardinal except by the aid of some great power ..."
      'sphere' amended from 'shpere'.

    ARTICLE MAZZINI, GIUSEPPE: "he did not actually hinder more than he
      helped the course of events by which the realization of so much of
      the great dream of his life was at last brought about." 'hinder'
      amended from 'binder'.

    ARTICLE MEAUX: "The building, which is 275 ft. long and 105 ft.
      high, consists of a short nave, with aisles, a fine transept, a
      choir and a sanctuary." 'sanctuary' amended from 'sanctury'.

    ARTICLE MECHANICS: "The simplest case is that of a frame of three
      bars, when the three joints A, B, C fall into a straight [**
      amended from straght] line ..." 'straight' amended from 'straght'.

    ARTICLE MECHANICS: "... a determinate series of quantities having
      to one another the above-mentioned ratios, whilst the constants C
      ..." 'quantities' amended from 'quantites'.

    ARTICLE MECHANICS: "Then assuming that the acceleration of one
      point of a particular link of the mechanism is known together with
      the corresponding configuration of the mechanism ..." 'particular'
      amended from 'particuar'.

    ARTICLE MECKLENBURG: "... were succeeded in the 6th century by some
      Slavonic tribes, one of these being the Obotrites, whose chief
      fortress was Michilenburg ..." 'Slavonic' amended from 'Salvonic'.




          ENCYCLOPAEDIA BRITANNICA

  A DICTIONARY OF ARTS, SCIENCES, LITERATURE
           AND GENERAL INFORMATION

              ELEVENTH EDITION


           VOLUME XVII, SLICE VIII

            Matter to Mecklenburg




ARTICLES IN THIS SLICE:


  MATTER                            MAX MÜLLER, FRIEDRICH
  MATTERHORN                        MAXWELL
  MATTEUCCI, CARLO                  MAXWELL, JAMES CLERK
  MATTHEW, ST                       MAXWELLTOWN
  MATTHEW, TOBIAS                   MAY, PHIL
  MATTHEW, GOSPEL OF ST             MAY, THOMAS
  MATTHEW CANTACUZENUS              MAY, WILLIAM
  MATTHEW OF PARIS                  MAY (month)
  MATTHEW OF WESTMINSTER            MAY, ISLE OF
  MATTHEWS, STANLEY                 MAYA
  MATTHIAE, AUGUST HEINRICH         MAYAGUEZ
  MATTHIAS (disciple)               MAYAVARAM
  MATTHIAS (Roman emperor)          MAYBOLE
  MATTHIAS I., HUNYADI              MAYEN
  MATTHISSON, FRIEDRICH VON         MAYENNE, CHARLES OF LORRAINE
  MATTING                           MAYENNE (department of France)
  MATTOCK                           MAYENNE (town of France)
  MATTO GROSSO                      MAYER, JOHANN TOBIAS
  MATTOON                           MAYER, JULIUS ROBERT
  MATTRESS                          MAYFLOWER
  MATURIN, CHARLES ROBERT           MAY-FLY
  MATVYEEV, ARTAMON SERGYEEVICH     MAYHEM
  MAUBEUGE                          MAYHEW, HENRY
  MAUCH CHUNK                       MAYHEW, JONATHAN
  MAUCHLINE                         MAYHEW, THOMAS
  MAUDE, CYRIL                      MAYMYO
  MAULE                             MAYNARD, FRANÇOIS DE
  MAULÉON, SAVARI DE                MAYNE, JASPER
  MAULSTICK                         MAYNOOTH
  MAUNDY THURSDAY                   MAYO, RICHARD SOUTHWELL BOURKE
  MAUPASSANT, HENRI GUY DE          MAYO
  MAUPEOU, RENÉ NICOLAS AUGUSTIN    MAYOR, JOHN EYTON BICKERSTETH
  MAUPERTUIS, PIERRE MOREAU DE      MAYOR
  MAU RANIPUR                       MAYOR OF THE PALACE
  MAUREL, ABDIAS                    MAYORUNA
  MAUREL, VICTOR                    MAYO-SMITH, RICHMOND
  MAURENBRECHER, KARL WILHELM       MAYOTTE
  MAUREPAS, JEAN PHÉLYPEAUX         MAYOW, JOHN
  MAURER, GEORG LUDWIG VON          MAYSVILLE
  MAURETANIA                        MAZAGAN
  MAURIAC                           MAZAMET
  MAURICE, ST                       MAZANDARAN
  MAURICE (Roman emperor)           MAZARIN, JULES
  MAURICE (elector of Saxony)       MAZAR-I-SHARIF
  MAURICE, JOHN FREDERICK DENISON   MAZARRÓN
  MAURICE OF NASSAU                 MAZATLÁN
  MAURISTS                          MAZE
  MAURITIUS                         MAZEPA-KOLEDINSKY, IVAN STEPANOVICH
  MAURY, JEAN SIFFREIN              MAZER
  MAURY, LOUIS FERDINAND ALFRED     MAZURKA
  MAURY, MATTHEW FONTAINE           MAZZARA DEL VALLO
  MAUSOLEUM                         MAZZINI, GIUSEPPE
  MAUSOLUS                          MAZZONI, GIACOMO
  MAUVE, ANTON                      MAZZONI, GUIDO
  MAVROCORDATO                      MEAD, LARKIN GOLDSMITH
  MAWKMAI                           MEAD, RICHARD
  MAXENTIUS, MARCUS VALERIUS        MEAD
  MAXIM, SIR HIRAM STEVENS          MEADE, GEORGE GORDON
  MAXIMA AND MINIMA                 MEADE, WILLIAM
  MAXIMIANUS                        MEADVILLE
  MAXIMIANUS, MARCUS VALERIUS       MEAGHER, THOMAS FRANCIS
  MAXIMILIAN I. (elector of Bavaria) MEAL
  MAXIMILIAN I. (king of Bavaria)   MEALIE
  MAXIMILIAN II. (king of Bavaria)  MEAN
  MAXIMILIAN I. (Roman emperor)     MEASLES
  MAXIMILIAN II. (Roman emperor)    MEAT
  MAXIMILIAN (emperor of Mexico)    MEATH
  MAXIMINUS, GAIUS JULIUS VERUS     MEAUX
  MAXIMINUS, GALERIUS VALERIUS      MECCA
  MAXIMS, LEGAL                     MECHANICS
  MAXIMUS                           MECHANICVILLE
  MAXIMUS, ST                       MECHITHARISTS
  MAXIMUS OF SMYRNA                 MECKLENBURG
  MAXIMUS OF TYRE




MATTER. Our conceptions of the nature and structure of matter have been
profoundly influenced in recent years by investigations on the
Conduction of Electricity through Gases (see CONDUCTION, ELECTRIC) and
on Radio-activity (q.v.). These researches and the ideas which they have
suggested have already thrown much light on some of the most fundamental
questions connected with matter; they have, too, furnished us with far
more powerful methods for investigating many problems connected with the
structure of matter than those hitherto available. There is thus every
reason to believe that our knowledge of the structure of matter will
soon become far more precise and complete than it is at present, for now
we have the means of settling by testing directly many points which are
still doubtful, but which formerly seemed far beyond the reach of
experiment.

The Molecular Theory of Matter--the only theory ever seriously
advocated--supposes that all visible forms of matter are collocations of
simpler and smaller portions. There has been a continuous tendency as
science has advanced to reduce further and further the number of the
different kinds of things of which all matter is supposed to be built
up. First came the molecular theory teaching us to regard matter as made
up of an enormous number of small particles, each kind of matter having
its characteristic particle, thus the particles of water were supposed
to be different from those of air and indeed from those of any other
substance. Then came Dalton's Atomic Theory which taught that these
molecules, in spite of their almost infinite variety, were all built up
of still smaller bodies, the atoms of the chemical elements, and that
the number of different types of these smaller bodies was limited to the
sixty or seventy types which represent the atoms of the substance
regarded by chemists as elements.

In 1815 Prout suggested that the atoms of the heavier chemical elements
were themselves composite and that they were all built up of atoms of
the lightest element, hydrogen, so that all the different forms of
matter are edifices built of the same material--the atom of hydrogen. If
the atoms of hydrogen do not alter in weight when they combine to form
atoms of other elements the atomic weights of all elements would be
multiples of that of hydrogen; though the number of elements whose
atomic weights are multiples or very nearly so of hydrogen is very
striking, there are several which are universally admitted to have
atomic weights differing largely from whole numbers. We do not know
enough about gravity to say whether this is due to the change of weight
of the hydrogen atoms when they combine to form other atoms, or whether
the primordial form from which all matter is built up is something other
than the hydrogen atom. Whatever may be the nature of this primordial
form, the tendency of all recent discoveries has been to emphasize the
truth of the conception of a common basis of matter of all kinds. That
the atoms of the different elements have a common basis, that they
behave as if they consisted of different numbers of small particles of
the same kind, is proved to most minds by the Periodic Law of Mendeléeff
and Newlands (see ELEMENT). This law shows that the physical and
chemical properties of the different elements are determined by their
atomic weights, or to use the language of mathematics, the properties of
an element are functions of its atomic weight. Now if we constructed
models of the atoms out of different materials, the atomic weight would
be but one factor out of many which would influence the physical and
chemical properties of the model, we should require to know more than
the atomic weight to fix its behaviour. If we were to plot a curve
representing the variation of some property of the substance with the
atomic weight we should not expect the curve to be a smooth one, for
instance two atoms might have the same atomic weight and yet if they
were made of different materials have no other property in common. The
influence of the atomic weight on the properties of the elements is
nowhere more strikingly shown than in the recent developments of physics
connected with the discharge of electricity through gases and with
radio-activity. The transparency of bodies to Röntgen rays, to cathode
rays, to the rays emitted by radio-active substances, the quality of the
secondary radiation emitted by the different elements are all determined
by the atomic weight of the element. So much is this the case that the
behaviour of the element with respect to these rays has been used to
determine its atomic weight, when as in the case of Indium, uncertainty
as to the valency of the element makes the result of ordinary chemical
methods ambiguous.

The radio-active elements indeed furnish us with direct evidence of this
unity of composition of matter, for not only does one element uranium,
produce another, radium, but all the radio-active substances give rise
to helium, so that the substance of the atoms of this gas must be
contained in the atoms of the radio-active elements.

It is not radio-active atoms alone that contain a common constituent,
for it has been found that all bodies can by suitable treatment, such as
raising them to incandescence or exposing them to ultra-violet light, be
made to emit negatively electrified particles, and that these particles
are the same from whatever source they may be derived. These particles
all carry the same charge of negative electricity and all have the same
mass, this mass is exceedingly small even when compared with the mass of
an atom of hydrogen, which until the discovery of these particles was
the smallest mass known to science. These particles are called
corpuscles or electrons; their mass according to the most recent
determinations is only about 1/1700 of that of an atom of hydrogen, and
their radius is only about one hundred-thousandth part of the radius of
the hydrogen atom. As corpuscles of this kind can be obtained from all
substances, we infer that they form a constituent of the atoms of all
bodies. The atoms of the different elements do not all contain the same
number of corpuscles--there are more corpuscles in the atoms of the
heavier elements than in the atoms of the lighter ones; in fact, many
different considerations point to the conclusion that the number of
corpuscles in the atom of any element is proportional to the atomic
weight of the element. Different methods of estimating the exact number
of corpuscles in the atom have all led to the conclusion that this
number is of the same order as the atomic weight; that, for instance,
the number of corpuscles in the atom of oxygen is not a large multiple
of 16. Some methods indicate that the number of corpuscles in the atom
is equal to the atomic weight, while the maximum value obtained by any
method is only about four times the atomic weight. This is one of the
points on which further experiments will enable us to speak with greater
precision. Thus one of the constituents of all atoms is the negatively
charged corpuscle; since the atoms are electrically neutral, this
negative charge must be accompanied by an equal positive one, so that on
this view the atoms must contain a charge of positive electricity
proportional to the atomic weight; the way in which this positive
electricity is arranged is a matter of great importance in the
consideration of the constitution of matter. The question naturally
arises, is the positive electricity done up into definite units like the
negative, or does it merely indicate a property acquired by an atom when
one or more corpuscles leave it? It is very remarkable that we have up
to the present (1910), in spite of many investigations on this point, no
direct evidence of the existence of positively charged particles with a
mass comparable with that of a corpuscle; the smallest positive particle
of which we have any direct indication has a mass equal to the mass of
an atom of hydrogen, and it is a most remarkable fact that we get
positively charged particles having this mass when we send the electric
discharge through gases at low pressures, whatever be the kind of gas.
It is no doubt exceedingly difficult to get rid of traces of hydrogen in
vessels containing gases at low pressures through which an electric
discharge is passing, but the circumstances under which the positively
electrified particles just alluded to appear, and the way in which they
remain unaltered in spite of all efforts to clear out any traces of
hydrogen, all seem to indicate that these positively electrified
particles, whose mass is equal to that of an atom of hydrogen, do not
come from minute traces of hydrogen present as an impurity but from the
oxygen, nitrogen, or helium, or whatever may be the gas through which
the discharge passes. If this is so, then the most natural conclusion we
can come to is that these positively electrified particles with the mass
of the atom of hydrogen are the natural units of positive electricity,
just as the corpuscles are those of negative, and that these positive
particles form a part of all atoms.

Thus in this way we are led to an electrical view of the constitution of
the atom. We regard the atom as built up of units of negative
electricity and of an equal number of units of positive electricity;
these two units are of very different mass, the mass of the negative
unit being only 1/1700 of that of the positive. The number of units of
either kind is proportional to the atomic weight of the element and of
the same order as this quantity. Whether this is anything besides the
positive and negative electricity in the atom we do not know. In the
present state of our knowledge of the properties of matter it is
unnecessary to postulate the existence of anything besides these
positive and negative units.

The atom of a chemical element on this view of the constitution of
matter is a system formed by n corpuscles and n units of positive
electricity which is in equilibrium or in a state of steady motion under
the electrical forces which the charged 2n constituents exert upon each
other. Sir J. J. Thomson (_Phil. Mag._, March 1904, "Corpuscular Theory
of Matter") has investigated the systems in steady motion which can be
formed by various numbers of negatively electrified particles immersed
in a sphere of uniform positive electrification, a case, which in
consequence of the enormous volume of the units of positive electricity
in comparison with that of the negative has much in common with the
problem under consideration, and has shown that some of the properties
of n systems of corpuscles vary in a periodic way suggestive of the
Periodic Law in Chemistry as n is continually increased.

_Mass on the Electrical Theory of Matter._--One of the most
characteristic things about matter is the possession of mass. When we
take the electrical theory of matter the idea of mass takes new and
interesting forms. This point may be illustrated by the case of a single
electrified particle; when this moves it produces in the region around
it a magnetic field, the magnetic force being proportional to the
velocity of the electrified particle.[1] In a magnetic field, however,
there is energy, and the amount of energy per unit volume at any place
is proportional to the square of the magnetic force at that place. Thus
there will be energy distributed through the space around the moving
particle, and when the velocity of the particle is small compared with
that of light we can easily show that the energy in the region around
the charged particle is ([mu]e²)/(3a), when v is the velocity of the
particle, e its charge, a its radius, and [mu] the magnetic permeability
of the region round the particle. If m is the ordinary mass of the
particle, the part of the kinetic energy due to the motion of this mass
is ½mv², thus the total kinetic energy is ½[m + (2/3)[mu]e²/a]. Thus the
electric charge on the particle makes it behave as if its mass were
increased by (2/3)[mu]e²/a. Since this increase in mass is due to the
energy in the region outside the charged particle, it is natural to look
to that region for this additional mass. This region is traversed by the
tubes of force which start from the electrified body and move with it,
and a very simple calculation shows that we should get the increase in
the mass which is due to the electrification if we suppose that these
tubes of force as they move carry with them a certain amount of the
ether, and that this ether had mass. The mass of ether thus carried
along must be such that the amount of it in unit volume at any part of
the field is such that if this were to move with the velocity of light
its kinetic energy would be equal to the potential energy of the
electric field in the unit volume under consideration. When a tube moves
this mass of ether only participates in the motion at right angles to
the tube, it is not set in motion by a movement of the tube along its
length. We may compare the mass which a charged body acquires in virtue
of its charge with the additional mass which a ball apparently acquires
when it is placed in water; a ball placed in water behaves as if its
mass were greater than its mass when moving in vacuo; we can easily
understand why this should be the case, because when the ball in the
water moves the water around it must move as well; so that when a force
acting on the ball sets it in motion it has to move some of the water as
well as the ball, and thus the ball behaves as if its mass were
increased. Similarly in the case of the electrified particle, which when
it moves carries with it its lines of force, which grip the ether and
carry some of it along with them. When the electrified particle is moved
a mass of ether has to be moved too, and thus the apparent mass of the
particle is increased. The mass of the electrified particle is thus
resident in every part of space reached by its lines of force; in this
sense an electrified body may be said to extend to an infinite distance;
the amount of the mass of the ether attached to the particle diminishes
so rapidly as we recede from it that the contributions of regions remote
from the particle are quite insignificant, and in the case of a
particle as small as a corpuscle not one millionth part of its mass will
be farther away from it than the radius of an atom.

The increase in the mass of a particle due to given charges varies as we
have seen inversely as the radius of the particle; thus the smaller the
particle the greater the increase in the mass. For bodies of appreciable
size or even for those as small as ordinary atoms the effect of any
realizable electric charge is quite insignificant, on the other hand for
the smallest bodies known, the corpuscle, there is evidence that the
whole of the mass is due to the electric charge. This result has been
deduced by the help of an extremely interesting property of the mass due
to a charge of electricity, which is that this mass is not constant but
varies with the velocity. This comes about in the following way. When
the charged particle, which for simplicity we shall suppose to be
spherical, is at rest or moving very slowly the lines of electric force
are distributed uniformly around it in all directions; when the sphere
moves, however, magnetic forces are produced in the region around it,
while these, in consequence of electro-magnetic induction in a moving
magnetic field, give rise to electric forces which displace the tubes of
electric force in such a way as to make them set themselves so as to be
more at right angles to the direction in which they are moving than they
were before. Thus if the charged sphere were moving along the line AB,
the tubes of force would, when the sphere was in motion, tend to leave
the region near AB and crowd towards a plane through the centre of the
sphere and at right angles to AB, where they would be moving more nearly
at right angles to themselves. This crowding of the lines of force
increases, however, the potential energy of the electric field, and
since the mass of the ether carried along by the lines of force is
proportional to the potential energy, the mass of the charged particle
will also be increased. The amount of variation of the mass with the
velocity depends to some extent on the assumptions we make as to the
shape of the corpuscle and the way in which it is electrified. The
simplest expression connecting the mass with the velocity is that when
the velocity is v the mass is equal to (2/3)[mu]e²/a [1/(1 - v²/c²)^½]
where c is the velocity of light. We see from this that the variation of
mass with velocity is very small unless the velocity of the body
approaches that of light, but when, as in the case of the [beta]
particles emitted by radium, the velocity is only a few per cent less
than that of light, the effect of velocity on the mass becomes very
considerable; the formula indicates that if the particles were moving
with a velocity equal to that of light they would behave as if their
mass were infinite. By observing the variation in the mass of a
corpuscle as its velocity changes we can determine how much of the mass
depends upon the electric charge and how much is independent of it. For
since the latter part of the mass is independent of the velocity, if it
predominates the variation with velocity of the mass of a corpuscle will
be small; if on the other hand it is negligible the variation in mass
with velocity will be that indicated by theory given above. The
experiment of Kaufmann (_Göttingen Nach._, Nov. 8, 1901), Bucherer
(_Ann. der Physik._, xxviii. 513, 1909) on the masses of the [beta]
particles shot out by radium, as well as those by Hupka (_Berichte der
deutsch. physik. Gesell._, 1909, p. 249) on the masses of the corpuscle
in cathode rays are in agreement with the view that the _whole_ of the
mass of these particles is due to their electric charge.

The alteration in the mass of a moving charge with its velocity is
primarily due to the increase in the potential energy which accompanies
the increase in velocity. The connexion between potential energy and
mass is general and holds for any arrangement of electrified particles;
thus if we assume the electrical constitution of matter, there will be a
part of the mass of any system dependent upon the potential energy and
in fact proportional to it. Thus every change in potential energy, such
for example as occurs when two elements combine with evolution or
absorption of heat, must be attended by a change in mass. The amount of
this change can be calculated by the rule that if a mass equal to the
change in mass were to move with the velocity of light its kinetic
energy would equal the change in the potential energy. If we apply this
result to the case of the combination of hydrogen and oxygen, where the
evolution of heat, about 1.6 × 10^11 ergs per gramme of water, is
greater than in any other known case of chemical combination, we see
that the change in mass would only amount to one part in 3000 million,
which is far beyond the reach of experiment. The evolution of energy by
radio-active substances is enormously larger than in ordinary chemical
transformations; thus one gramme of radium emits per day about as much
energy as is evolved in the formation of one gramme of water, and goes
on doing this for thousands of years. We see, however, that even in this
case it would require hundreds of years before the changes in mass
became appreciable.

The evolution of energy from the gaseous emanation given off by radium
is more rapid than that from radium itself, since according to the
experiments of Rutherford (Rutherford, _Radio-activity_, p. 432) a
gramme of the emanation would evolve about 2.1 × 10^16 ergs in four
days; this by the rule given above would diminish the mass by about one
part in 20,000; but since only very small quantities of the emanation
could be used the detection of the change of mass does not seem feasible
even in this case.

On the view we have been discussing the existence of potential energy
due to an electric field is always associated with mass; wherever there
is potential energy there is mass. On the electro-magnetic theory of
light, however, a wave of light is accompanied by electric forces, and
therefore by potential energy; thus waves of light must behave as if
they possessed mass. It may be shown that it follows from the same
principles that they must also possess momentum, the direction of the
momentum being the direction along which the light is travelling; when
the light is absorbed by an opaque substance the momentum in the light
is communicated to the substance, which therefore behaves as if the
light pressed upon it. The pressure exerted by light was shown by
Maxwell (_Electricity and Magnetism_, 3rd ed., p. 440) to be a
consequence of his electro-magnetic theory, its existence has been
established by the experiment of Lebedew, of Nichols and Hull, and of
Poynting.


  Weight.

We have hitherto been considering mass from the point of view that the
constitution of matter is electrical; we shall proceed to consider the
question of weight from the same point of view. The relation between
mass and weight is, while the simplest in expression, perhaps the most
fundamental and mysterious property possessed by matter. The weight of a
body is proportional to its mass, that is if the weights of a number of
substances are equal the masses will be equal, whatever the substances
may be. This result was verified to a considerable degree of
approximation by Newton by means of experiments with pendulums; later,
in 1830 Bessel by a very extensive and accurate series of experiments,
also made on pendulums, showed that the ratio of mass to weight was
certainly to one part in 60,000 the same for all the substances examined
by him, these included brass, silver, iron, lead, copper, ivory, water.

The constancy of this ratio acquires new interest when looked at from
the point of view of the electrical constitution of matter. We have seen
that the atoms of all bodies contain corpuscles, that the mass of a
corpuscle is only 1/1700 of the mass of an atom of hydrogen, that it
carries a constant charge of negative electricity, and that its mass is
entirely due to this charge, and can be regarded as arising from ether
gripped by the lines of force starting from the electrical charge. The
question at once suggests itself, Is this kind of mass ponderable? does
it add to the weight of the body? and, if so, is the proportion between
mass and weight the same as for ordinary bodies? Let us suppose for a
moment that this mass is not ponderable, so that the corpuscles increase
the mass but not the weight of an atom. Then, since the mass of a
corpuscle is 1/1700 that of an atom of hydrogen, the addition or removal
of one corpuscle would in the case of an atom of atomic weight x alter
the mass by one part in 1700 x, without altering the weight, this would
produce an effect of the same magnitude on the ratio of mass to weight
and would in the case of the atoms of the lighter elements be easily
measurable in experiments of the same order of accuracy as those made by
Bessel. If the number of corpuscles in the atom were proportional to the
atomic weight, then the ratio of mass to weight would be constant
whether the corpuscles were ponderable or not. If the number were not
proportional there would be greater discrepancies in the ratio of mass
to weight than is consistent with Bessel's experiments if the corpuscles
had no weight. We have seen there are other grounds for concluding that
the number of corpuscles in an atom is proportional to the atom weight,
so that the constancy of the ratio of mass to weight for a large number
of substances does not enable us to determine whether or not mass due to
charges of electricity is ponderable or not.

There seems some hope that the determination of this ratio for
radio-active substances may throw some light on this point. The enormous
amount of heat evolved by these bodies may indicate that they possess
much greater stores of potential energy than other substances. If we
suppose that the heat developed by one gramme of a radio-active
substance in the transformations which it undergoes before it reaches
the non-radio-active stage is a measure of the excess of the potential
energy in a gramme of this substance above that in a gramme of
non-radio-active substance, it would follow that a larger part of the
mass was due to electric charges in radio-active than in
non-radio-active substances; in the case of uranium this difference
would amount to at least one part in 20,000 of the total mass. If this
extra mass had no weight the ratio of mass to weight for uranium would
differ from the normal amount by more than one part in 20,000, a
quantity quite within the range of pendulum experiments. It thus appears
very desirable to make experiments on the ratio of mass to weight for
radio-active substances. Sir J. J. Thomson, by swinging a small pendulum
whose bob was made of radium bromide, has shown that this ratio for
radium does not differ from the normal by one part in 2000. The small
quantity of radium available prevented the attainment of greater
accuracy. Experiments just completed (1910) by Southerns at the
Cavendish Laboratory on this ratio for uranium show that it is normal to
an accuracy of one part in 200,000; indicating that in non-radio-active,
as in radio-active, substances the electrical mass is proportional to
the atomic weight.

Though but few experiments have been made in recent years on the value
of the ratio of mass to weight, many important investigations have been
made on the effect of alterations in the chemical and physical
conditions on the weight of bodies. These have all led to the conclusion
that no change which can be detected by our present means of
investigation occurs in the weight of a body in consequence of any
physical or chemical changes yet investigated. Thus Landolt, who devoted
a great number of years to the question whether any change in weight
occurs during chemical combination, came finally to the conclusion that
in no case out of the many he investigated did any measurable change of
weight occur during chemical combination. Poynting and Phillips (_Proc.
Roy. Soc._, 76, p. 445), as well as Southerns (78, p. 392), have shown
that change in temperature produces no change in the weight of a body;
and Poynting has also shown that neither the weight of a crystal nor the
attraction between two crystals depends at all upon the direction in
which the axis of the crystal points. The result of these laborious and
very carefully made experiments has been to strengthen the conviction
that the weight of a given portion of matter is absolutely independent
of its physical condition or state of chemical combinations. It should,
however, be noticed that we have as yet no accurate investigation as to
whether or not any changes of weight occur during radio-active
transformations, such for example as the emanation from radium undergoes
when the atoms themselves of the substance are disrupted.

It is a matter of some interest in connexion with a discussion of any
views of the constitution of matter to consider the theories of
gravitation which have been put forward to explain that apparently
invariable property of matter--its weight. It would be impossible to
consider in detail the numerous theories which have been put forward to
account for gravitation; a concise summary of many of these has been
given by Drude (Wied. _Ann._ 62, p. 1);[2] there is no dearth of
theories as to the cause of gravitation, what is lacking is the means of
putting any of them to a decisive test.

There are, however, two theories of gravitation, both old, which seem to
be especially closely connected with the idea of the electrical
constitution of matter. The first of these is the theory, associated
with the two fluid theory of electricity, that gravity is a kind of
residual electrical effect, due to the attraction between the units of
positive and negative electricity being a little greater than the
repulsion between the units of electricity of the same kind. Thus on
this view two charges of equal magnitude, but of opposite sign, would
exert an attraction varying inversely as the square of the distance on a
charge of electricity of either sign, and therefore an attraction on a
system consisting of two charges equal in magnitude but opposite in sign
forming an electrically neutral system. Thus if we had two neutral
systems, A and B, A consisting of m positive units of electricity and an
equal number of negative, while B has n units of each kind, then the
gravitational attraction between A and B would be inversely proportional
to the square of the distance and proportional to n m. The connexion
between this view of gravity and that of the electrical constitution of
matter is evidently very close, for if gravity arose in this way the
weight of a body would only depend upon the number of units of
electricity in the body. On the view that the constitution of matter is
electrical, the fundamental units which build up matter are the units of
electric charge, and as the magnitude of these charges does not change,
whatever chemical or physical vicissitudes matter, the weight of matter
ought not to be affected by such changes. There is one result of this
theory which might possibly afford a means of testing it: since the
charge on a corpuscle is equal to that on a positive unit, the weights
of the two are equal; but the mass of the corpuscle is only 1/1700 of
that of the positive unit, so that the acceleration of the corpuscle
under gravity will be 1700 times that of the positive unit, which we
should expect to be the same as that for ponderable matter or 981.

The acceleration of the corpuscle under gravity on this view would be
1.6 × 10^6. It does not seem altogether impossible that with methods
slightly more powerful than those we now possess we might measure the
effect of gravity on a corpuscle if the acceleration were as large as
this.

The other theory of gravitation to which we call attention is that due
to Le Sage of Geneva and published in 1818. Le Sage supposed that the
universe was thronged with exceedingly small particles moving with very
great velocities. These particles he called ultra-mundane corpuscles,
because they came to us from regions far beyond the solar system. He
assumed that these were so penetrating that they could pass through
masses as large as the sun or the earth without being absorbed to more
than a very small extent. There is, however, some absorption, and if
bodies are made up of the same kind of atoms, whose dimensions are small
compared with the distances between them, the absorption will be
proportional to the mass of the body. So that as the ultra-mundane
corpuscles stream through the body a small fraction, proportional to the
mass of the body, of their momentum is communicated to it. If the
direction of the ultra-mundane corpuscles passing through the body were
uniformly distributed, the momentum communicated by them to the body
would not tend to move it in one direction rather than in another, so
that a body, A, alone in the universe and exposed to bombardment by the
ultra-mundane corpuscles would remain at rest. If, however, there were a
second body, B, in the neighbourhood of A, B will shield A from some of
the corpuscles moving in the direction BA; thus A will not receive as
much momentum in this direction as when it was alone; but in this case
it only received just enough to keep it in equilibrium, so that when B
is present the momentum in the opposite direction will get the upper
hand and A will move in the direction AB, and will thus be attracted by
B. Similarly, we see that B will be attracted by A. Le Sage proved that
the rate at which momentum was being communicated to A or B by the
passage through them of his corpuscles was proportional to the product
of the masses of A and B, and if the distance between A and B was large
compared with their dimensions, inversely proportional to the square of
the distance between them; in fact, that the forces acting on them would
obey the same laws as the gravitational attraction between them. Clerk
Maxwell (article "ATOM," _Ency. Brit._, 9th ed.) pointed out that this
transference of momentum from the ultra-mundane corpuscles to the body
through which they passed involved the loss of kinetic energy by the
corpuscles, and if the loss of momentum were large enough to account for
the gravitational attraction, the loss of kinetic energy would be so
large that if converted into heat it would be sufficient to keep the
body white hot. We need not, however, suppose that this energy is
converted into heat; it might, as in the case where Röntgen rays are
produced by the passage of electrified corpuscles through matter, be
transformed into the energy of a still more penetrating form of
radiation, which might escape from the gravitating body without heating
it. It is a very interesting result of recent discoveries that the
machinery which Le Sage introduced for the purpose of his theory has a
very close analogy with things for which we have now direct experimental
evidence. We know that small particles moving with very high speeds do
exist, that they possess considerable powers of penetrating solids,
though not, as far as we know at present, to an extent comparable with
that postulated by Le Sage; and we know that the energy lost by them as
they pass through a solid is to a large extent converted into a still
more penetrating form of radiation, Röntgen rays. In Le Sage's theory
the only function of the corpuscles is to act as carriers of momentum,
any systems which possessed momentum, moved with a high velocity and had
the power of penetrating solids, might be substituted for them; now
waves of electric and magnetic force, such as light waves or Röntgen
rays, possess momentum, move with a high velocity, and the latter at any
rate possess considerable powers of penetration; so that we might
formulate a theory in which penetrating Röntgen rays replaced Le Sage's
corpuscles. Röntgen rays, however, when absorbed do not, as far as we
know, give rise to more penetrating Röntgen rays as they should to
explain attraction, but either to less penetrating rays or to rays of
the same kind.

We have confined our attention in this article to the view that the
constitution of matter is electrical; we have done so because this view
is more closely in touch with experiment than any other yet advanced.
The units of which matter is built up on this theory have been isolated
and detected in the laboratory, and we may hope to discover more and
more of their properties. By seeing whether the properties of matter are
or are not such as would arise from a collection of units having these
properties, we can apply to this theory tests of a much more definite
and rigorous character than we can apply to any other theory of matter.
     (J. J. T.)


FOOTNOTES:

  [1] We may measure this velocity with reference to any axes, provided
    we refer the motion of all the bodies which come into consideration
    to the same axes.

  [2] A theory published after Drude's paper in that of Professor
    Osborne Reynolds, given in his Rede lecture "On an Inversion of Ideas
    as to the Structure of the Universe."




MATTERHORN, one of the best known mountains (14,782 ft.) in the Alps. It
rises S.W. of the village of Zermatt, and on the frontier between
Switzerland (canton of the Valais) and Italy. Though on the Swiss side
it appears to be an isolated obelisk, it is really but the butt end of a
ridge, while the Swiss slope is not nearly as steep or difficult as the
grand terraced walls of the Italian slope. It was first conquered, after
a number of attempts chiefly on the Italian side, on the 14th of July
1865, by Mr E. Whymper's party, three members of which (Lord Francis
Douglas, the Rev. C. Hudson and Mr Hadow) with the guide, Michel Croz,
perished by a slip on the descent. Three days later it was scaled from
the Italian side by a party of men from Val Tournanche. Nowadays it is
frequently ascended in summer, especially from Zermatt.




MATTEUCCI, CARLO (1811-1868), Italian physicist, was born at Forlì on
the 20th of June 1811. After attending the École Polytechnique at
Paris, he became professor of physics successively at Bologna (1832),
Ravenna (1837) and Pisa (1840). From 1847 he took an active part in
politics, and in 1860 was chosen an Italian senator, at the same time
becoming inspector-general of the Italian telegraph lines. Two years
later he was minister of education. He died near Leghorn on the 25th of
June 1868.

  He was the author of four scientific treatises: _Lezioni di fisica_ (2
  vols., Pisa, 1841), _Lezioni sui fenomeni fisicochimici dei corpi
  viventi_ (Pisa, 1844), _Manuale di telegrafia elettrica_ (Pisa, 1850)
  and _Cours spécial sur l'induction, le magnetisme de rotation_, &c.
  (Paris, 1854). His numerous papers were published in the _Annales de
  chimie et de physique_ (1829-1858); and most of them also appeared at
  the time in the Italian scientific journals. They relate almost
  entirely to electrical phenomena, such as the magnetic rotation of
  light, the action of gas batteries, the effects of torsion on
  magnetism, the polarization of electrodes, &c., sufficiently complete
  accounts of which are given in Wiedemann's _Galvanismus_. Nine
  memoirs, entitled "Electro-Physiological Researches," were published
  in the _Philosophical Transactions_, 1845-1860. See Bianchi's _Carlo
  Matteucci e l'Italia del suo tempo_ (Rome, 1874).




MATTHEW, ST ([Greek: Maththaios] or [Greek: Matthaios], probably a
shortened form of the Hebrew equivalent to Theodorus), one of the twelve
apostles, and the traditional author of the First Gospel, where he is
described as having been a tax-gatherer or customs-officer ([Greek:
telônês], x. 3), in the service of the tetrarch Herod. The circumstances
of his call to become a follower of Jesus, received as he sat in the
"customs house" in one of the towns by the Sea of Galilee--apparently
Capernaum (Mark ii. 1, 13), are briefly related in ix. 9. We should
gather from the parallel narrative in Mark ii. 14, Luke v. 27, that he
was at the time known as "Levi the son of Alphaeus" (compare Simon
Cephas, Joseph Barnabas): if so, "James the son of Alphaeus" may have
been his brother. Possibly "Matthew" (Yahweh's gift) was his Christian
surname, since two native names, neither being a patronymic, is contrary
to Jewish usage. It must be noted, however, that Matthew and Levi were
sometimes distinguished in early times, as by Heracleon (c. 170 A.D.),
and more dubiously by Origen (c. _Celsum_, i. 62), also apparently in
the Syriac _Didascalia_ (sec. iii.), V. xiv. 14. It has generally been
supposed, on the strength of Luke's account (v. 29), that Matthew gave a
feast in Jesus' honour (like Zacchaeus, Luke xix. 6 seq.). But Mark (ii.
15), followed by Matthew (ix. 10), may mean that the meal in question
was one in Jesus' own home at Capernaum (cf. v. 1). In the lists of the
Apostles given in the Synoptic Gospels and in Acts, Matthew ranks third
or fourth in the second group of four--a fair index of his relative
importance in the apostolic age. The only other facts related of Matthew
on good authority concern him as Evangelist. Eusebius (_H.E._ iii. 24)
says that he, like John, wrote only at the spur of necessity. "For
Matthew, after preaching to Hebrews, when about to go also to others,
committed to writing in his native tongue the Gospel that bears his
name; and so by his writing supplied, for those whom he was leaving, the
loss of his presence." The value of this tradition, which may be based
on Papias, who certainly reported that "Matthew compiled the Oracles (of
the Lord) in Hebrew," can be estimated only in connexion with the study
of the Gospel itself (see below). No historical use can be made of the
artificial story, in _Sanhedrin_ 43a, that Matthew was condemned to
death by a Jewish court (see Laihle, _Christ in the Talmud_, 71 seq.).
According to the Gnostic Heracleon, quoted by Clement of Alexandria
(_Strom._ iv. 9), Matthew died a natural death. The tradition as to his
ascetic diet (in Clem. Alex. _Paedag._ ii. 16) maybe due to confusion
with Matthias (cf. _Mart. Matthaei_, i.). The earliest legend as to his
later labours, one of Syrian origin, places them in the Parthian
kingdom, where it represents him as dying a natural death at Hierapolis
(= Mabog on the Euphrates). This agrees with his legend as known to
Ambrose and Paulinus of Nola, and is the most probable in itself. The
legends which make him work with Andrew among the Anthropophagi near the
Black Sea, or again in Ethiopia (Rufinus, and Socrates, _H.E._ i. 19),
are due to confusion with Matthias, who from the first was associated in
his Acts with Andrew (see M. Bonnet, _Acta Apost. apocr._, 1808, II. i.
65). Another legend, his _Martyrium_, makes him labour and suffer in
Mysore. He is commemorated as a martyr by the Greek Church on the 16th
of November, and by the Roman on the 21st of September, the scene of his
martyrdom being placed in Ethiopia. The Latin Breviary also affirms that
his body was afterwards translated to Salerno, where it is said to lie
in the church built by Robert Guiscard. In Christian art (following
Jerome) the Evangelist Matthew is generally symbolized by the "man" in
the imagery of Ezek. i. 10, Rev. iv. 7.

  For the historical Matthew, see _Ency. Bibl._ and Zahn, _Introd. to
  New Test._, ii. 506 seq., 522 seq. For his legends, as under MARK.
       (J. V. B.)




MATTHEW, TOBIAS, or TOBIE (1546-1628), archbishop of York, was the son
of Sir John Matthew of Ross in Herefordshire, and of his wife Eleanor
Crofton of Ludlow. He was born at Bristol in 1546. He was educated at
Wells, and then in succession at University College and Christ Church,
Oxford. He proceeded B.A. in 1564, and M.A. in 1566. He attracted the
favourable notice of Queen Elizabeth, and his rise was steady though not
very rapid. He was public orator in 1569, president of St John's
College, Oxford, in 1572, dean of Christ Church in 1576, vice-chancellor
of the university in 1579, dean of Durham in 1583, bishop of Durham in
1595, and archbishop of York in 1606. In 1581 he had a controversy with
the Jesuit Edmund Campion, and published at Oxford his arguments in 1638
under the title, _Piissimi et eminentissimi viri Tobiae Matthew,
archiepiscopi olim Eboracencis concio apologetica adversus Campianam_.
While in the north he was active in forcing the recusants to conform to
the Church of England, preaching hundreds of sermons and carrying out
thorough visitations. During his later years he was to some extent in
opposition to the administration of James I. He was exempted from
attendance in the parliament of 1625 on the ground of age and
infirmities, and died on the 29th of March 1628. His wife, Frances, was
the daughter of William Barlow, bishop of Chichester.

His son, SIR TOBIAS, or TOBIE, MATTHEW (1577-1655), is remembered as the
correspondent and friend of Francis Bacon. He was educated at Christ
Church, and was early attached to the court, serving in the embassy at
Paris. His debts and dissipations were a great source of sorrow to his
father, from whom he is known to have received at different times
£14,000, the modern equivalent of which is much larger. He was chosen
member for Newport in Cornwall in the parliament of 1601, and member for
St Albans in 1604. Before this time he had become the intimate friend of
Bacon, whom he replaced as member for St Albans. When peace was made
with Spain, on the accession of James I., he wished to travel abroad.
His family, who feared his conversion to Roman Catholicism, opposed his
wish, but he promised not to go beyond France. When once safe out of
England he broke his word and went to Italy. The persuasion of some of
his countrymen in Florence, one of whom is said to have been the Jesuit
Robert Parsons, and a story he heard of the miraculous liquefaction of
the blood of San Januarius at Naples, led to his conversion in 1606.
When he returned to England he was imprisoned, and many efforts were
made to obtain his reconversion without success. He would not take the
oath of allegiance to the king. In 1608 he was exiled, and remained out
of England for ten years, mostly in Flanders and Spain. He returned in
1617, but went abroad again in 1619. His friends obtained his leave to
return in 1621. At home he was known as the intimate friend of Gondomar,
the Spanish ambassador. In 1623 he was sent to join Prince Charles,
afterwards Charles I., at Madrid, and was knighted on the 23rd of
October of that year. He remained in England till 1640, when he was
finally driven abroad by the parliament, which looked upon him as an
agent of the pope. He died in the English college in Ghent on the 13th
of October 1655. In 1618 he published an Italian translation of Bacon's
essays. The "Essay on Friendship" was written for him. He was also the
author of a translation of _The Confessions of the Incomparable Doctor
St Augustine_, which led him into controversy. His correspondence was
published in London in 1660.

  For the father, see John Le Neve's _Fasti ecclesiae anglicanae_
  (London, 1716), and Anthony Wood's _Athenae oxonienses_. For the son,
  the notice in _Athenae oxonienses_, an abridgment of his
  autobiographical _Historical Relation_ of his own life, published by
  Alban Butler in 1795, and A. H. Matthew and A. Calthrop, _Life of Sir
  Tobie Matthew_ (London, 1907).




MATTHEW, GOSPEL OF ST, the first of the four canonical Gospels of the
Christian Church. The indications of the use of this Gospel in the two
or three generations following the Apostolic Age (see GOSPEL) are more
plentiful than of any of the others. Throughout the history of the
Church, also, it has held a place second to none of the Gospels alike in
public instruction and in the private reading of Christians. The reasons
for its having impressed itself in this way and become thus familiar are
in large part to be found in the characteristics noticed below. But in
addition there has been from an early time the belief that it was the
work of one of those publicans whose heart Jesus touched and of whose
call to follow Him the three Synoptics contain an interesting account,
but who is identified as Matthew (q.v.) only in this one (Matt. ix. 9-13
= Mark ii. 13-17 = Luke v. 27-32).

1. _The Connexion of our Greek Gospel of Matthew with the Apostle whose
name it bears._--The earliest reference to a writing by Matthew occurs
in a fragment taken by Eusebius from the same work of Papias from which
he has given an account of the composition of a record by Mark (Euseb.
_Hist. Eccl._ iii. 39; see MARK, GOSPEL OF ST). The statement about
Matthew is much briefer and is harder to interpret. In spite of much
controversy, the same measure of agreement as to its meaning cannot be
said to have been attained. This is the fragment: "Matthew, however, put
together and wrote down the Oracles ([Greek: ta logia synegrapsen]) in
the Hebrew language, and each man interpreted them as he was able."
Whether "the elder" referred to in the passage on Mark, or some other
like authority, was the source of this statement also does not appear;
but it is probable that this was the case from the context in which
Eusebius gives it. Conservative writers on the Gospels have frequently
maintained that the writing here referred to was virtually the Hebrew
original of our Greek Gospel which bears his name. And it is indeed
likely that Papias himself closely associated the latter with the Hebrew
(or Aramaic) work by Matthew, of which he had been told, since the
traditional connexion of this Greek Gospel with Matthew can hardly have
begun later than this time. It is reasonable also to suppose that there
was some ground for it. The description, however, of what Matthew did
suits better the making of a collection of Christ's discourses and
sayings than the composition of a work corresponding in form and
character to our Gospel of Matthew.

The next reference in Christian literature to a Gospel-record by Matthew
is that of Irenaeus in his famous passage on the four Gospels (_Adv.
haer._ iii. i. r). He says that it was written in Hebrew; but in all
probability he regarded the Greek Gospel, which stood first in his, as
it does in our, enumeration, as in the strict sense a translation of the
Apostle's work; and this was the view of it universally taken till the
16th century, when some of the scholars of the Reformation maintained
that the Greek Gospel itself was by Matthew.

The actual phenomena, however, of this Gospel, and of its relation to
sources that have been used in it, cannot be explained consistently with
either of the two views just mentioned. It is a composite work in which
two chief sources, known in Greek to the author of our present Gospel,
have, together with some other matter, been combined. It is
inconceivable that one of the Twelve should have proceeded in this way
in giving an account of Christ's ministry. One of the chief documents,
however, here referred to seems to correspond in character with the
description given in Papias' fragment of a record of the compilation of
"the divine utterances" made by Matthew; and the use made of it in our
first Gospel may explain the connexion of this Apostle's name with it.
In the Gospel of Luke also, it is true, this same source has been used
for the teaching of Jesus. But the original Aramaic Logian document may
have been more largely reproduced in our Greek Matthew. Indeed, in the
case of one important passage (v. 17-48) this is suggested by a
comparison with Luke itself, and there are one or two others where from
the character of the matter it seems not improbable, especially vi. 1-18
and xxiii. 1-5, 7b-10, 15-22. On the whole, as will be seen below, what
appears to be a Palestinian form of the Gospel-tradition is most fully
represented in this Gospel; but in many instances at least this may well
be due to some other cause than the use of the original Logian document.

2. _The Plan on which the Contents is arranged._--In two respects the
arrangement of the book itself is significant.

  (a) As to the general outline in the first half of the account of the
  Galilean ministry (iv. 23-xi. 30). Immediately after relating the call
  of the first four disciples (iv. 18-22) the evangelist gives in iv. 23
  a comprehensive summary of Christ's work in Galilee under its two
  chief aspects, teaching and healing. In the sequel both these are
  illustrated. First, he gives in the Sermon on the Mount (v.-vii.) a
  considerable body of teaching, of the kind required by the disciples
  of Jesus generally, and a large portion of which probably also stood
  not far from the beginning of the Logian document. After this he turns
  to the other aspect. Up to this point he has mentioned no miracle. He
  now describes a number in succession, introducing all but the first of
  those told between Mark i. 23 and ii. 12, and also four specially
  remarkable ones, which occurred a good deal later according to Mark's
  order (Matt. viii. 23-34 = Mark iv. 35-v. 20; Matt. ix. 18-26 = Mark
  v. 21-43); and he also adds some derived from another source, or other
  sources (viii. 5-13; ix. 27-34). Then, after another general
  description at ix. 35, similar to that at iv. 23, he brings strikingly
  before us the needs of the masses of the people and Christ's
  compassion for them, and so introduces the mission of the Twelve
  (which again occurs later according to Mark's order, viz. at vi. 7
  seq.), whereby the ministry both of teaching and of healing was
  further extended (ix. 36-x. 42). Finally, the message of John the
  Baptist, and the reply of Jesus, and the reflections that follow
  (xi.), bring out the significance of the preceding narrative. It
  should be observed that examples have been given of every kind of
  mighty work referred to in the reply of Jesus to the messengers of the
  Baptist; and that in the discourse which follows their departure the
  perversity and unbelief of the people generally are condemned, and the
  faith of the humble-minded is contrasted therewith. The greater part
  of the matter from ix. 37 to end of xi. is taken from the Logian
  document. After this point, i.e. from xii. 1 onwards, the first
  evangelist follows Mark almost step by step down to the point (Mark
  xvi. 8), after which Mark's Gospel breaks off, and another ending has
  been supplied; and gives in substance almost the whole of Mark's
  contents, with the exception that he passes over the few narratives
  that he has (as we have seen) placed earlier. At the same time he
  brings in additional matter in connexion with most of the Marcan
  sections.

  (b) With the accounts of the words of Jesus spoken on certain
  occasions, which our first evangelist found given in one or another of
  his sources, he has combined other pieces, taken from other parts of
  the same source or from different sources, which seemed to him
  connected in subject, e.g. into the discourse spoken on a mountain,
  when crowds from all parts were present, given in the Logian document,
  he has introduced some pieces which, as we infer from Luke, stood
  separately in that document (cf. Matt. vi. 19-21 with Luke xii. 33,
  34; Matt. vi. 22, 23 with Luke xi. 34-36; Matt. vi. 24 with Luke xvi.
  13; Matt. vi. 25-34 with Luke xii. 22-32; Matt. vii. 7-11 with Luke
  xi. 9-13). Again, the address to the Twelve in Mark vi. 7-11, which in
  Matthew is combined with an address to disciples, from the Logian
  document, is connected by Luke with the sending out of seventy
  disciples (Luke x. 1-16). Our first evangelist has also added here
  various other sayings (Matt. x. 17-39, 42). Again, with the Marcan
  account of the charge of collusion with Satan and Christ's reply (Mark
  iii. 22-30), the first evangelist (xii. 24-45) combines the parallel
  account in the Logian document and adds Christ's reply to another
  attack (Luke xi. 14-16, 17-26, 29-32). These are some examples. He has
  in all in this manner constructed eight discourses or collections of
  sayings, into which the greater part of Christ's teaching is gathered:
  (1) On the character of the heirs of the kingdom (v.-vii.); (2) The
  Mission address (x.); (3) Teaching suggested by the message of John
  the Baptist (xi.); (4) The reply to an accusation and a challenge
  (xii. 22-45); (5) The teaching by parables (xiii.); (6) On offences
  (xviii.); (7) Concerning the Scribes and Pharisees (xxiii.); (8) On
  the Last Things (xxiv., xxv.). In this arrangement of his material the
  writer has in many instances disregarded chronological considerations.
  But his documents also gave only very imperfect indications of the
  occasions of many of the utterances; and the result of his method of
  procedure has been to give us an exceedingly effective representation
  of the teaching of Jesus.

  In the concluding verses of the Gospel, where the original Marcan
  parallel is wanting, the evangelist may still have followed in part
  that document while making additions as before. The account of the
  silencing of the Roman guard by the chief priests is the sequel to the
  setting of this guard and their presence at the Resurrection, which at
  an earlier point arc peculiar to Matthew (xxvii. 62-66, xxviii. 4).
  And, further, this matter seems to belong to the same cycle of
  tradition as the story of Pilate's wife and his throwing the guilt of
  the Crucifixion of Jesus upon the Jews, and the testimony borne by
  the Roman guard (as well as the centurion) who kept watch by the cross
  (xxvii. 15-26, 54), all which also are peculiar to this Gospel. It
  cannot but seem probable that these are legendary additions which had
  arisen through the desire to commend the Gospel to the Romans.

  On the other hand, the meeting of Jesus with the disciples in Galilee
  (Matt. xxviii. 16 seq.) is the natural sequel to the message to them
  related in Mark xvi. 7, as well as in Matt, xxviii. 7. Again, the
  commission to them to preach throughout the world is supported by Luke
  xxiv. 47, and by the present ending of Mark (xvi. 15), though neither
  of these mention Galilee as the place where it was given. The
  baptismal formula in Matt. xxviii. 19, is, however, peculiar, and in
  view of its non-occurrence in the Acts and Epistles of the New
  Testament must be regarded as probably an addition in accordance with
  Church usage at the time the Gospel was written.

3. _The Palestinian Element._--Teaching is preserved in this Gospel
which would have peculiar interest and be specially required in the home
of Judaism. The best examples of this are the passages already referred
to near end of § 1, as probably derived from the Logian document. There
are, besides, a good many turns of expression and sayings peculiar to
this Gospel which have a Semitic cast, or which suggest a point of view
that would be natural to Palestinian Christians, e.g. "kingdom of
heaven" frequently for "kingdom of God"; xiii. 52 ("every scribe");
xxiv. 20 ("neither on a Sabbath"). See also v. 35 and xix. 9; x. 5, 23.
Again, several of the quotations which are peculiar to this Gospel are
not taken from the LXX., as those in the other Gospels and in the
corresponding contexts in this Gospel commonly are, but are wholly or
partly independent renderings from the Hebrew (ii. 6, 15, 18; viii. 17,
xii. 17-21, &c.). Once more, there is somewhat more parallelism between
the fragments of the Gospel according to the Hebrews and this Gospel
than is the case with Luke, not to say Mark.

4. _Doctrinal Character._--In this Gospel, more decidedly than in either
of the other two Synoptics, there is a doctrinal point of view from
which the whole history is regarded. Certain aspects which are of
profound significance are dwelt upon, and this without there being any
great difference between this Gospel and the two other Synoptics in
respect to the facts recorded or the beliefs implied. The effect is
produced partly by the comments of the evangelist, which especially take
the form of citations from the Old Testament; partly by the frequency
with which certain expressions are used, and the prominence that is
given in this and other ways to particular traits and topics.

He sets forth the restriction of the mission of Jesus during His life on
earth to the people of Israel in a way which suggests at first sight a
spirit of Jewish exclusiveness. But there are various indications that
this is not the true explanation. In particular the evangelist brings
out more strongly than either Mark or Luke the national rejection of
Jesus, while the Gospel ends with the commission of Jesus to His
disciples after His resurrection to "make disciples of all the peoples."
One may divine in all this an intention to "justify the ways of God" to
the Jew, by proving that God in His faithfulness to His ancient people
had given them the first opportunity of salvation through Christ, but
that now their national privilege had been rightly forfeited. He was
also specially concerned to show that prophecy is fulfilled in the life
and work of Jesus, but the conception of this fulfilment which is
presented to us is a large one; it is to be seen not merely in
particular events or features of Christ's ministry, but in the whole new
dispensation, new relations between God and men, and new rules of
conduct which Christ has introduced. The divine meaning of the work of
Jesus is thus made apparent, while of the majesty and glory of His
person a peculiarly strong impression is conveyed.

Some illustrations in detail of these points are subjoined. Where there
are parallels in the other Gospels they should be compared and the words
in Matthew noted which in many instances serve to emphasize the points
in question.

  (a) _The Ministry of Jesus among the Jewish People as their promised
  Messiah, their rejection of Him, and the extension of the Gospel to
  the Gentiles._ The mission to Israel: Matt. i. 21; iv. 23 (note in
  these passages the use of [Greek: ho laos], which here, as generally
  in Matthew, denotes the chosen nation), ix. 33, 35, xv. 31. For the
  rule limiting the work of Jesus while on earth see xv. 24 (and note
  [Greek: ixelthousa] in verse 22, which implies that Jesus had not
  himself entered the heathen borders), and for a similar rule
  prescribed to the disciples, x. 5, 6 and 23.

  The rejection of Jesus by the people in Galilee, xi. 21; xiii. 13-15,
  and by the heads of "the nation," xxvi. 3, 47 and by "the whole
  nation," xxvii. 25; their condemnation xxiii. 38.

  Mercy to the Gentiles and the punishment of "the sons of the kingdom"
  is foretold viii. 11, 12. The commission to go and convert Gentile
  peoples ([Greek: ethnê]) is given after Christ's resurrection (xxviii.
  19).

  (b) _The Fulfilment of Prophecy._--In the birth and childhood of
  Jesus, i. 23; ii. 6, 15, 18, 23. By these citations attention is drawn
  to the lowliness of the beginnings of the Saviour's life, the
  unexpected and secret manner of His appearing, the dangers to which
  from the first He was exposed and from which He escaped.

  The ministry of Christ's forerunner, iii. 3. (The same prophecy, Isa.
  xl. 3, is also quoted in the other Gospels.)

  The ministry of Jesus. The quotations serve to bring out the
  significance of important events, especially such as were
  turning-points, and also to mark the broad features of Christ's life
  and work, iv. 15, 16; viii. 17; xii. 18 seq.; xiii. 35; xxi. 5; xxvii.
  9.

  (c) _The Teaching on the Kingdom of God._--Note the collection of
  parables "of the Kingdom" in xiii.; also the use of [Greek: hê
  basileia] ("the Kingdom") without further definition as a term the
  reference of which could not be misunderstood, especially in the
  following phrases peculiar to this Gospel: [Greek: to euangelion tês
  basileias] ("the Gospel of the Kingdom") iv. 23, ix. 35, xxiv. 14; and
  [Greek: ho logos tês basileias] ("the word of the kingdom") xiii. 19.
  The following descriptions of the kingdom, peculiar to this Gospel,
  are also interesting [Greek: hê basileia tou patros autôn] ("the
  kingdom of their father") xiii. 43 and [Greek: tou patros mou]("of my
  father") xxvi. 29.

  (d) _The Relation of the New Law to the Old._--Verses 17-48, cf. also,
  addition at xxii. 40 and xix. 19b. Further, his use of [Greek:
  dikaiosynê] ("righteousness") and [Greek: dikaios]("righteous")
  (specially frequent in this Gospel) is such as to connect the New with
  the Old; the standard in mind is the law which "fulfilled" that
  previously given.

  (e) _The Christian Ecclesia._--Chap. xvi. 18, xviii. 17.

  (f) _The Messianic Dignity and Glory of Jesus._--The narrative in i.
  and ii. show the royalty of the new-born child. The title "Son of
  David" occurs with special frequency in this Gospel. The following
  instances are without parallels in the other Gospels: ix. 27; xii. 23;
  xv. 22; xxi. 9; xxi. 15. The title "Son of God" is also used with
  somewhat greater frequency than in Mark and Luke: ii. 15; xiv. 33;
  xvi. 16; xxii. 2 seq. (where it is implied); xxvii. 40, 43.

  The thought of the future coming of Christ, and in particular of the
  judgment to be executed by Him then, is much more prominent in this
  Gospel than in the others. Some of the following predictions are
  peculiar to it, while in several others there are additional touches:
  vii. 22, 23; x. 23, 32, 33; xiii. 39-43; xvi. 27, 28; xix. 28; xxiv.
  3, 27, 30, 31, 37, 39; xxv. 31-46; xxvi. 64.

  The majesty of Christ is also impressed upon us by the signs at His
  crucifixion, some of which are related only in this Gospel, xxvii.
  51-53, and by the sublime vision of the Risen Christ at the close,
  xxviii. 16-20.

(5) _Time of Composition and Readers addressed._--The signs of dogmatic
reflection in this Gospel point to its having been composed somewhat
late in the 1st century, probably after Luke's Gospel, and this is in
accord with the conclusion that some insertions had been made in the
Marcan document used by this evangelist which were not in that used by
Luke (see LUKE, GOSPEL OF ST). We may assign A.D. 80-100 as a probable
time for the composition.

The author was in all probability a Jew by race, and he would seem to
have addressed himself especially to Jewish readers; but they were Jews
of the Dispersion. For although he was in specially close touch with
Palestine, either personally or through the sources at his command, or
both, his book was composed in Greek by the aid of Greek documents.

  See commentaries by Th. Zahn (1903) and W. C. Allen (in the series of
  International Critical Commentaries, 1907); also books on the Four
  Gospels or the Synoptic Gospels cited at the end of GOSPEL.
       (V. H. S.)




MATTHEW CANTACUZENUS, Byzantine emperor, was the son of John VI.
Cantacuzenus (q.v.). In return for the support he gave to his father
during his struggle with John V. he was allowed to annex part of Thrace
under his own dominion and in 1353 was proclaimed joint emperor. From
his Thracian principality he levied several wars against the Servians.
An attack which he prepared in 1350 was frustrated by the defection of
his Turkish auxiliaries. In 1357 he was captured by his enemies, who
delivered him to the rival emperor, John V. Compelled to abdicate, he
withdrew to a monastery, where he busied himself with writing
commentaries on the Scriptures.




MATTHEW OF PARIS (d. 1259), English monk and chronicler known to us only
through his voluminous writings. In spite of his surname, and of his
knowledge of the French language, his attitude towards foreigners
attests that he was of English birth. He may have studied at Paris in
his youth, but the earliest fact which he records of himself is his
admission as a monk at St Albans in the year 1217. His life was mainly
spent in this religious house. In 1248, however, he was sent to Norway
as the bearer of a message from Louis IX. of France to Haakon VI.; he
made himself so agreeable to the Norwegian sovereign that he was
invited, a little later, to superintend the reformation of the
Benedictine monastery of St Benet Holme at Trondhjem. Apart from these
missions, his activities were devoted to the composition of history, a
pursuit for which the monks of St Albans had long been famous. Matthew
edited anew the works of Abbot John de Cella and Roger of Wendover,
which in their altered form constitute the first part of his most
important work, the _Chronica majora_. From 1235, the point at which
Wendover dropped his pen, Matthew continued the history on the plan
which his predecessors had followed. He derived much of his information
from the letters of important personages, which he sometimes inserts,
but much more from conversation with the eye-witnesses of events. Among
his informants were Earl Richard of Cornwall and Henry III. With the
latter he appears to have been on terms of intimacy. The king knew that
Matthew was writing a history, and showed some anxiety that it should be
as exact as possible. In 1257, in the course of a week's visit to St
Albans, Henry kept the chronicler beside him night and day, "and guided
my pen," says Paris, "with much good will and diligence." It is
therefore curious that the _Chronica majora_ should give so unfavourable
an account of the king's policy. Luard supposes that Matthew never
intended his work to see the light in its present form, and many
passages of the autograph have against them the note _offendiculum_,
which shows that the writer understood the danger which he ran. On the
other hand, unexpurgated copies were made in Matthew's lifetime; though
the offending passages are duly omitted or softened in his abridgment of
his longer work, the _Historia Anglorum_ (written about 1253), the real
sentiments of the author must have been an open secret. In any case
there is no ground for the old theory that he was an official
historiographer.

  Matthew Paris was unfortunate in living at a time when English
  politics were peculiarly involved and tedious. His talent is for
  narrative and description. Though he took a keen interest in the
  personal side of politics he has no claim to be considered a judge of
  character. His appreciations of his contemporaries throw more light on
  his own prejudices than on their aims and ideas. His work is always
  vigorous, but he imputes motives in the spirit of a partisan who never
  pauses to weigh the evidence or to take a comprehensive view of the
  situation. His redeeming feature is his generous admiration for
  strength of character, even when it goes along with a policy of which
  he disapproves. Thus he praises Grosseteste, while he denounces
  Grosseteste's scheme of monastic reform. Matthew is a vehement
  supporter of the monastic orders against their rivals, the secular
  clergy and the mendicant friars. He is violently opposed to the court
  and the foreign favourites. He despises the king as a statesman,
  though for the man he has some kindly feeling. The frankness with
  which he attacks the court of Rome for its exactions is remarkable;
  so, too, is the intense nationalism which he displays in dealing with
  this topic. His faults of presentment are more often due to
  carelessness and narrow views than to deliberate purpose. But he is
  sometimes guilty of inserting rhetorical speeches which are not only
  fictitious, but also misleading as an account of the speaker's
  sentiments. In other cases he tampers with the documents which he
  inserts (as, for instance, with the text of Magna Carta). His
  chronology is, for a contemporary, inexact; and he occasionally
  inserts duplicate versions of the same incident in different places.
  Hence he must always be rigorously checked where other authorities
  exist and used with caution where he is our sole informant. None the
  less, he gives a more vivid impression of his age than any other
  English chronicler; and it is a matter for regret that his great
  history breaks off in 1259, on the eve of the crowning struggle
  between Henry III and the baronage.

  AUTHORITIES.--The relation of Matthew Paris's work to those of John de
  Cella and Roger of Wendover may best be studied in H. R. Luard's
  edition of the _Chronica majora_ (7 vols., Rolls series, 1872-1883),
  which contains valuable prefaces. The _Historia_ _Anglorum sive
  historia minor_ (1067-1253) has been edited by F. Madden (3 vols.,
  Rolls series, 1866-1869). Matthew Paris is often confused with
  "Matthew of Westminster," the reputed author of the _Flores
  historiarum_ edited by H. R. Luard (3 vols., Rolls series, 1890). This
  work, compiled by various hands, is an edition of Matthew Paris, with
  continuations extending to 1326. Matthew Paris also wrote a life of
  Edmund Rich (q.v.), which is probably the work printed in W. Wallace's
  _St Edmund of Canterbury_ (London, 1893) pp. 543-588, though this is
  attributed by the editor to the monk Eustace; _Vitae abbatum S Albani_
  (up to 1225) which have been edited by W. Watts (1640, &c.); and
  (possibly) the _Abbreviatio chronicorum_ (1000-1255), edited by F.
  Madden, in the third volume of the _Historia Anglorum_. On the value
  of Matthew as an historian see F. Liebermann in G. H. Pertz's
  _Scriptores_ xxviii. pp. 74-106; A. Jessopp's _Studies by a Recluse_
  (London, 1893); H. Plehn's _Politische Character Matheus Parisiensis_
  (Leipzig, 1897).     (H. W. C. D.)




MATTHEW OF WESTMINSTER, the name of an imaginary person who was long
regarded as the author of the _Flores Historiarum_. The error was first
discovered in 1826 by Sir F. Palgrave, who said that Matthew was "a
phantom who never existed," and later the truth of this statement was
completely proved by H. R. Luard. The name appears to have been taken
from that of Matthew of Paris, from whose _Chronica majora_ the earlier
part of the work was mainly copied, and from Westminster, the abbey in
which the work was partially written.

  The _Flores historiarum_ is a Latin chronicle dealing with English
  history from the creation to 1326, although some of the earlier
  manuscripts end at 1306; it was compiled by various persons, and
  written partly at St Albans and partly at Westminster. The part from
  1306 to 1326 was written by Robert of Reading (d. 1325) and another
  Westminster monk. Except for parts dealing with the reign of Edward I.
  its value is not great. It was first printed by Matthew Parker,
  archbishop of Canterbury, in 1567, and the best edition is the one
  edited with introduction by H. R. Luard for the Rolls series (London,
  1890). It has been translated into English by C. D. Yonge (London,
  1853). See Luard's introduction, and C. Bémont in the _Revue critique
  d'histoire_ (Paris, 1891).




MATTHEWS, STANLEY (1824-1889), American jurist, was born in Cincinnati,
Ohio, on the 21st of July 1824. He graduated from Kenyon College in
1840, studied law, and in 1842 was admitted to the bar of Maury county,
Tennessee. In 1844 he became assistant prosecuting attorney of Hamilton
county, Ohio; and in 1846-1849 edited a short-lived anti-slavery paper,
the _Cincinnati Herald_. He was clerk of the Ohio House of
Representatives in 1848-1849, a judge of common pleas of Hamilton county
in 1850-1853, state senator in 1856-1858, and U.S. district-attorney for
the southern district of Ohio in 1858-1861. First a Whig and then a
Free-Soiler, he joined the Republican party in 1861. After the outbreak
of the Civil War he was commissioned a lieutenant of the 23rd Ohio, of
which Rutherford B. Hayes was major; but saw service only with the 57th
Ohio, of which he was colonel, and with a brigade which he commanded in
the Army of the Cumberland. He resigned from the army in 1863, and was
judge of the Cincinnati superior court in 1863-1864. He was a Republican
presidential elector in 1864 and 1868. In 1872 he joined the Liberal
Republican movement, and was temporary chairman of the Cincinnati
convention which nominated Horace Greeley for the presidency, but in the
campaign he supported Grant. In 1877, as counsel before the Electoral
Commission, he opened the argument for the Republican electors of
Florida and made the principal argument for the Republican electors of
Oregon. In March of the same year he succeeded John Sherman as senator
from Ohio, and served until March 1879. In 1881 President Hayes
nominated him as associate justice of the Supreme Court, to succeed Noah
H. Swayne; there was much opposition, especially in the press, to this
appointment, because Matthews had been a prominent railway and
corporation lawyer and had been one of the Republican "visiting
statesmen" who witnessed the canvass of the vote of Louisiana[1] in
1876; and the nomination had not been approved when the session of
Congress expired. Matthews was renominated by President Garfield on the
15th of March, and the nomination was confirmed by the Senate (22 for,
21 against) on the 12th of May. He was an honest, impartial and
conscientious judge. He died in Washington, on the 22nd of March 1889.


FOOTNOTE:

  [1] It seems certain that Matthews and Charles Foster of Ohio gave
    their written promise that Hayes, if elected, would recognize the
    Democratic governors in Louisiana and South Carolina.




MATTHIAE, AUGUST HEINRICH (1769-1835), German classical scholar, was
born at Göttingen, on the 25th of December 1769, and educated at the
university. He then spent some years as a tutor in Amsterdam. In 1798 he
returned to Germany, and in 1802 was appointed director of the
Friedrichsgymnasium at Altenburg, which post he held till his death, on
the 6th of January 1835. Of his numerous important works the best-known
are his _Greek Grammar_ (3rd ed., 1835), translated into English by E.
V. Blomfield (5th ed., by J. Kenrick, 1832), his edition of _Euripides_
(9 vols., 1813-1829), _Grundriss der Geschichte der griechischen und
römischen Litteratur_ (3rd ed., 1834, Eng. trans., Oxford, 1841)
_Lehrbuch für den ersten Unterricht in der Philosophie_ (3rd ed., 1833),
_Encyklopädie und Methodologie der Philologie_ (1835). His _Life_ was
written by his son Constantin (1845).

His brother, FRIEDRICH CHRISTIAN MATTHIAE (1763-1822), rector of the
Frankfort gymnasium, published valuable editions of Seneca's _Letters_,
Aratus, and Dionysius Periegetes.




MATTHIAS, the disciple elected by the primitive Christian community to
fill the place in the Twelve vacated by Judas Iscariot (Acts i. 21-26).
Nothing further is recorded of him in the New Testament. Eusebius
(_Hist. Eccl._, I. xii.) says he was, like his competitor, Barsabas
Justus, one of the seventy, and the Syriac version of Eusebius calls him
throughout not Matthias but Tolmai, i.e. Bartholomew, without confusing
him with the Bartholomew who was originally one of the Twelve, and is
often identified with the Nathanael mentioned in the Fourth Gospel
(_Expository Times_, ix. 566). Clement of Alexandria says some
identified him with Zacchaeus, the Clementine _Recognitions_ identify
him with Barnabas, Hilgenfeld thinks he is the same as Nathanael.

  Various works--a Gospel, Traditions and Apocryphal Words--were
  ascribed to him; and there is also extant _The Acts of Andrew and
  Matthias_, which places his activity in "the city of the cannibals" in
  Ethiopia. Clement of Alexandria quotes two sayings from the
  Traditions: (1) Wonder at the things before you (suggesting, like
  Plato, that wonder is the first step to new knowledge); (2) If an
  elect man's neighbour sin, the elect man has sinned.




MATTHIAS (1557-1619), Roman emperor, son of the emperor Maximilian II.
and Maria, daughter of the emperor Charles V., was born in Vienna, on
the 24th of February 1557. Educated by the diplomatist O. G. de Busbecq,
he began his public life in 1577, soon after his father's death, when he
was invited to assume the governorship of the Netherlands, then in the
midst of the long struggle with Spain. He eagerly accepted this
invitation, although it involved a definite breach with his Spanish
kinsman, Philip II., and entering Brussels in January 1578 was named
governor-general; but he was merely a cipher, and only held the position
for about three years, returning to Germany in October 1581. Matthias
was appointed governor of Austria in 1593 by his brother, the emperor
Rudolph II.; and two years later, when another brother, the archduke
Ernest, died, he became a person of more importance as the eldest
surviving brother of the unmarried emperor. As governor of Austria
Matthias continued the policy of crushing the Protestants, although
personally he appears to have been inclined to religious tolerance; and
he dealt with the rising of the peasants in 1595, in addition to
representing Rudolph at the imperial diets, and gaining some fame as a
soldier during the Turkish War. A few years later the discontent felt by
the members of the Habsburg family at the incompetence of the emperor
became very acute, and the lead was taken by Matthias. Obtaining in May
1605 a reluctant consent from his brother, he took over the conduct of
affairs in Hungary, where a revolt had broken out, and was formally
recognized by the Habsburgs as their head in April 1606, and was
promised the succession to the Empire. In June 1606 he concluded the
peace of Vienna with the rebellious Hungarians, and was thus in a better
position to treat with the sultan, with whom peace was made in November.
This pacific policy was displeasing to Rudolph, who prepared to renew
the Turkish War; but having secured the support of the national party in
Hungary and gathered an army, Matthias forced his brother to cede to him
this kingdom, together with Austria and Moravia, both of which had
thrown in their lot with Hungary (1608). The king of Hungary, as
Matthias now became, was reluctantly compelled to grant religious
liberty to the inhabitants of Austria. The strained relations which had
arisen between Rudolph and Matthias as a result of these proceedings
were temporarily improved, and a formal reconciliation took place in
1610; but affairs in Bohemia soon destroyed this fraternal peace. In
spite of the letter of majesty (_Majestätsbrief_) which the Bohemians
had extorted from Rudolph, they were very dissatisfied with their ruler,
whose troops were ravaging their land; and in 1611 they invited Matthias
to come to their aid. Accepting this invitation, he inflicted another
humiliation upon his brother, and was crowned king of Bohemia in May
1611. Rudolph, however, was successful in preventing the election of
Matthias as German king, or king of the Romans, and when he died, in
January 1612, no provision had been made for a successor. Already king
of Hungary and Bohemia, however, Matthias obtained the remaining
hereditary dominions of the Habsburgs, and in June 1612 was crowned
emperor, although the ecclesiastical electors favoured his younger
brother, the archduke Albert (1559-1621).

The short reign of the new emperor was troubled by the religious
dissensions of Germany. His health became impaired and his indolence
increased, and he fell completely under the influence of Melchior Klesl
(q.v.), who practically conducted the imperial business. By Klesl's
advice he took up an attitude of moderation and sought to reconcile the
contending religious parties; but the proceedings at the diet of
Regensburg in 1613 proved the hopelessness of these attempts, while
their author was regarded with general distrust. Meanwhile the younger
Habsburgs, led by the emperor's brother, the archduke Maximilian, and
his cousin, Ferdinand, archduke of Styria, afterwards the emperor
Ferdinand II., disliking the peaceful policy of Klesl, had allied
themselves with the unyielding Roman Catholics, while the question of
the imperial succession was forcing its way to the front. In 1611
Matthias had married his cousin Anna (d. 1618), daughter of the archduke
Ferdinand (d. 1595), but he was old and childless and the Habsburgs were
anxious to retain his extensive possessions in the family. Klesl, on the
one hand, wished the settlement of the religious difficulties to precede
any arrangement about the imperial succession; the Habsburgs, on the
other, regarded the question of the succession as urgent and vital.
Meanwhile the disputed succession to the duchies of Cleves and Jülich
again threatened a European war; the imperial commands were flouted in
Cologne and Aix-la-Chapelle, and the Bohemians were again becoming
troublesome. Having decided that Ferdinand should succeed Matthias as
emperor, the Habsburgs had secured his election as king of Bohemia in
June 1617, but were unable to stem the rising tide of disorder in that
country. Matthias and Klesl were in favour of concessions, but Ferdinand
and Maximilian met this move by seizing and imprisoning Klesl. Ferdinand
had just secured his coronation as king of Hungary when there broke out
in Bohemia those struggles which heralded the Thirty Years' War; and on
the 20th of March 1619 the emperor died at Vienna.

  For the life and reign of Matthias the following works may be
  consulted: J. Heling, _Die Wahl des römischen Königs Matthias_
  (Belgrade, 1892); A. Gindely, _Rudolf II. und seine Zeit_ (Prague,
  1862-1868); F. Stieve, _Die Verhandlungen über die Nachfolge Kaisers
  Rudolf II._ (Munich, 1880); P. von Chlumecky, _Karl von Zierotin und
  seine Zeit_ (Brünn, 1862-1879); A. Kerschbaumer, _Kardinal Klesel_
  (Vienna, 1865); M. Ritter, _Quellenbeiträge zur Geschichte des Kaisers
  Rudolf II._ (Munich, 1872); _Deutsche Geschichte im Zeitalter der
  Gegenreformation und des dreissigjährigen Krieges_ (Stuttgart, 1887,
  seq.); and the article on Matthias in the _Allgemeine deutsche
  Biographie_, Bd. XX. (Leipzig, 1884); L. von Ranke, _Zur deutschen
  Geschichte vom Religionsfrieden bis zum 30-jährigen Kriege_ (Leipzig,
  1888); and J. Janssen, _Geschichte des deutschen Volks seit dem
  Ausgang des Mittelalters_ (Freiburg, 1878 seq.), Eng. trans. by M. A.
  Mitchell and A. M. Christie (London, 1896, seq.).




MATTHIAS I., HUNYADI (1440-1490), king of Hungary, also known as
Matthias Corvinus, a surname which he received from the raven (_corvus_)
on his escutcheon, second son of János Hunyadi and Elizabeth Szilágyi,
was born at Kolozsvár, probably on

the 23rd of February 1440. His tutors were the learned János Vitéz,
bishop of Nagyvárad, whom he subsequently raised to the primacy, and the
Polish humanist Gregory Sanocki. The precocious lad quickly mastered the
German, Latin and principal Slavonic languages, frequently acting as his
father's interpreter at the reception of ambassadors. His military
training proceeded under the eye of his father, whom he began to follow
on his campaigns when only twelve years of age. In 1453 he was created
count of Bistercze, and was knighted at the siege of Belgrade in 1454.
The same care for his welfare led his father to choose him a bride in
the powerful Cilli family, but the young Elizabeth died before the
marriage was consummated, leaving Matthias a widower at the age of
fifteen. On the death of his father he was inveigled to Buda by the
enemies of his house, and, on the pretext of being concerned in a purely
imaginary conspiracy against Ladislaus V., was condemned to
decapitation, but was spared on account of his youth, and on the king's
death fell into the hands of George Podebrad, governor of Bohemia, the
friend of the Hunyadis, in whose interests it was that a national king
should sit on the Magyar throne. Podebrad treated Matthias hospitably
and affianced him with his daughter Catherine, but still detained him,
for safety's sake, in Prague, even after a Magyar deputation had
hastened thither to offer the youth the crown. Matthias was the elect of
the Hungarian people, gratefully mindful of his father's services to the
state and inimical to all foreign candidates; and though an influential
section of the magnates, headed by the palatine László Garai and the
voivode of Transylvania, Miklós Ujlaki, who had been concerned in the
judicial murder of Matthias's brother László, and hated the Hunyadis as
semi-foreign upstarts, were fiercely opposed to Matthias's election,
they were not strong enough to resist the manifest wish of the nation,
supported as it was by Matthias's uncle Mihály Szilágyi at the head of
15,000 veterans. On the 24th of January 1458, 40,000 Hungarian noblemen,
assembled on the ice of the frozen Danube, unanimously elected Matthias
Hunyadi king of Hungary, and on the 14th of February the new king made
his state entry into Buda.

The realm at this time was environed by perils. The Turks and the
Venetians threatened it from the south, the emperor Frederick III. from
the west, and Casimir IV. of Poland from the north, both Frederick and
Casimir claiming the throne. The Czech mercenaries under Giszkra held
the northern counties and from thence plundered those in the centre.
Meanwhile Matthias's friends had only pacified the hostile dignitaries
by engaging to marry the daughter of the palatine Garai to their
nominee, whereas Matthias not unnaturally refused to marry into the
family of one of his brother's murderers, and on the 9th of February
confirmed his previous nuptial contract with the daughter of George
Podebrad, who shortly afterwards was elected king of Bohemia (March 2,
1458). Throughout 1458 the struggle between the young king and the
magnates, reinforced by Matthias's own uncle and guardian Szilágyi, was
acute. But Matthias, who began by deposing Garai and dismissing
Szilágyi, and then proceeded to levy a tax, without the consent of the
Diet, in order to hire mercenaries, easily prevailed. Nor did these
complications prevent him from recovering the fortress of Galamboc from
the Turks, successfully invading Servia, and reasserting the suzerainty
of the Hungarian crown over Bosnia. In the following year there was a
fresh rebellion, when the emperor Frederick was actually crowned king by
the malcontents at Vienna-Neustadt (March 4, 1459); but Matthias drove
him out, and Pope Pius II. intervened so as to leave Matthias free to
engage in a projected crusade against the Turks, which subsequent
political complications, however, rendered impossible. From 1461 to 1465
the career of Matthias was a perpetual struggle punctuated by truces.
Having come to an understanding with his father-in-law Podebrad, he was
able to turn his arms against the emperor Frederick, and in April 1462
Frederick restored the holy crown for 60,000 ducats and was allowed to
retain certain Hungarian counties with the title of king; in return for
which concessions, extorted from Matthias by the necessity of coping
with a simultaneous rebellion of the Magyar noble in league with
Podebrad's son Victorinus, the emperor recognized Matthias as the actual
sovereign of Hungary. Only now was Matthias able to turn against the
Turks, who were again threatening the southern provinces. He began by
defeating Ali Pasha, and then penetrated into Bosnia, and captured the
newly built fortress of Jajce after a long and obstinate defence (Dec.
1463). On returning home he was crowned with the holy crown on the 29th
of March 1464, and, after driving the Czechs out of his northern
counties, turned southwards again, this time recovering all the parts of
Bosnia which still remained in Turkish hands.

A political event of the first importance now riveted his attention upon
the north. Podebrad, who had gained the throne of Bohemia with the aid
of the Hussites and Utraquists, had long been in ill odour at Rome, and
in 1465 Pope Paul II. determined to depose the semi-Catholic monarch.
All the neighbouring princes, the emperor, Casimir IV. of Poland and
Matthias, were commanded in turn to execute the papal decree of
deposition, and Matthias gladly placed his army at the disposal of the
Holy See. The war began on the 31st of May 1468, but, as early as the
27th of February 1469, Matthias anticipated an alliance between George
and Frederick by himself concluding an armistice with the former. On the
3rd of May the Czech Catholics elected Matthias king of Bohemia, but
this was contrary to the wishes of both pope and emperor, who preferred
to partition Bohemia. But now George discomfited all his enemies by
suddenly excluding his own son from the throne in favour of Ladislaus,
the eldest son of Casimir IV., thus skilfully enlisting Poland on his
side. The sudden death of Podebrad on the 22nd of March 1471 led to
fresh complications. At the very moment when Matthias was about to
profit by the disappearance of his most capable rival, another dangerous
rebellion, headed by the primate and the chief dignitaries of the state,
with the object of placing Casimir, son of Casimir IV., on the throne,
paralysed Matthias's foreign policy during the critical years 1470-1471.
He suppressed this domestic rebellion indeed, but in the meantime the
Poles had invaded the Bohemian domains with 60,000 men, and when in 1474
Matthias was at last able to take the field against them in order to
raise the siege of Breslau, he was obliged to fortify himself in an
entrenched camp, whence he so skilfully harried the enemy that the
Poles, impatient to return to their own country, made peace at Breslau
(Feb. 1475) on an _uti possidetis_ basis, a peace subsequently confirmed
by the congress of Olmütz (July 1479). During the interval between these
peaces, Matthias, in self-defence, again made war on the emperor,
reducing Frederick to such extremities that he was glad to accept peace
on any terms. By the final arrangement made between the contending
princes, Matthias recognized Ladislaus as king of Bohemia proper in
return for the surrender of Moravia, Silesia and Upper and Lower
Lusatia, hitherto component parts of the Czech monarchy, till he should
have redeemed them for 400,000 florins. The emperor promised to pay
Matthias 100,000 florins as a war indemnity, and recognized him as the
legitimate king of Hungary on the understanding that he should succeed
him if he died without male issue, a contingency at this time somewhat
improbable, as Matthias, only three years previously (Dec. 15, 1476),
had married his third wife, Beatrice of Naples, daughter of Ferdinand of
Aragon.

The endless tergiversations and depredations of the emperor speedily
induced Matthias to declare war against him for the third time (1481),
the Magyar king conquering all the fortresses in Frederick's hereditary
domains. Finally, on the 1st of June 1485, at the head of 8000 veterans,
he made his triumphal entry into Vienna, which he henceforth made his
capital. Styria, Carinthia and Carniola were next subdued, and Trieste
was only saved by the intervention of the Venetians. Matthias
consolidated his position by alliances with the dukes of Saxony and
Bavaria, with the Swiss Confederation, and the archbishop of Salzburg,
and was henceforth the greatest potentate in central Europe. His
far-reaching hand even extended to Italy. Thus, in 1480, when a Turkish
fleet seized Otranto, Matthias, at the earnest solicitation of the pope,
sent Balasz Magyar to recover the fortress, which surrendered to him on
the 10th of May 1481. Again in 1488, Matthias took Ancona under his
protection for a time and occupied it with a Hungarian garrison.

Though Matthias's policy was so predominantly occidental that he soon
abandoned his youthful idea of driving the Turks out of Europe, he at
least succeeded in making them respect Hungarian territory. Thus in 1479
a huge Turkish army, on its return home from ravaging Transylvania, was
annihilated at Szászváros (Oct. 13), and in 1480 Matthias recaptured
Jajce, drove the Turks from Servia and erected two new military banates,
Jajce and Srebernik, out of reconquered Bosnian territory. On the death
of Mahommed II. in 1481, a unique opportunity for the intervention of
Europe in Turkish affairs presented itself. A civil war ensued in Turkey
between his sons Bayezid and Jem, and the latter, being worsted, fled to
the knights of Rhodes, by whom he was kept in custody in France (see
BAYEZID II.). Matthias, as the next-door neighbour of the Turks, claimed
the custody of so valuable a hostage, and would have used him as a means
of extorting concessions from Bayezid. But neither the pope nor the
Venetians would hear of such a transfer, and the negotiations on this
subject greatly embittered Matthias against the Curia. The last days of
Matthias were occupied in endeavouring to secure the succession to the
throne for his illegitimate son János (see CORVINUS, JÁNOS); but Queen
Beatrice, though childless, fiercely and openly opposed the idea and the
matter was still pending when Matthias, who had long been crippled by
gout, expired very suddenly on Palm Sunday, the 4th of April 1490.

Matthias Hunyadi was indisputably the greatest man of his day, and one
of the greatest monarchs who ever reigned. The precocity and
universality of his genius impress one the most. Like Napoleon, with
whom he has often been compared, he was equally illustrious as a
soldier, a statesman, an orator, a legislator and an administrator. But
in all moral qualities the brilliant adventurer of the 15th was
infinitely superior to the brilliant adventurer of the 19th century.
Though naturally passionate, Matthias's self-control was almost
superhuman, and throughout his stormy life, with his innumerable
experiences of ingratitude and treachery, he never was guilty of a
single cruel or vindictive action. His capacity for work was
inexhaustible. Frequently half his nights were spent in reading, after
the labour of his most strenuous days. There was no branch of knowledge
in which he did not take an absorbing interest, no polite art which he
did not cultivate and encourage. His camp was a school of chivalry, his
court a nursery of poets and artists. Matthias was a middle-sized,
broad-shouldered man of martial bearing, with a large fleshy nose, hair
reaching to his heels, and the clean-shaven, heavy chinned face of an
early Roman emperor.

  See Vilmós Fraknói, _King Matthias Hunyadi_ (Hung., Budapest, 1890,
  German ed., Freiburg, 1891); Ignácz Acsády, _History of the Hungarian
  Realm_ (Hung. vol. i., Budapest, 1904); József Teleki, _The Age of the
  Hunyadis in Hungary_ (Hung., vols. 3-5, Budapest, 1852-1890); V.
  Fraknói, _Life of János Vitéz_ (Hung. Budapest 1879); Karl Schober,
  _Die Eroberung Niederösterreichs durch Matthias Corvinus_ (Vienna,
  1879); János Huszár, _Matthias's Black Army_ (Hung. Budapest, 1890);
  Antonio Bonfini, _Rerum hungaricarum decades_ (7th ed., Leipzig,
  1771); Aeneas Sylvius, _Opera_ (Frankfort, 1707); _The Correspondence
  of King Matthias_ (Hung. and Lat., Budapest, 1893); V. Fraknói, _The
  Embassies of Cardinal Carvajal to Hungary_ (Hung., Budapest, 1889);
  Marzio Galeotti, _De egregie sapienter et jocose, dictis ac factis
  Matthiae regis (Script. reg. hung. I.)_ (Vienna, 1746). Of the above
  the first is the best general sketch and is rich in notes; the second
  somewhat chauvinistic but excellently written; the third the best work
  for scholars; the seventh, eighth and eleventh are valuable as being
  by contemporaries.     (R. N. B.)




MATTHISSON, FRIEDRICH VON (1761-1831), German poet, was born at
Hohendodeleben near Magdeburg, the son of the village pastor, on the
23rd of January 1761. After studying theology and philology at the
university of Halle, he was appointed in 1781 master at the classical
school Philanthropin in Dessau. This once famous seminary was, however,
then rapidly decaying in public favour, and in 1784 Matthisson was glad
to accept a travelling tutorship. He lived for two years with the Swiss
author Bonstetten at Nyon on the lake of Geneva. In 1794 he was
appointed reader and travelling companion to the princess Louisa of
Anhalt-Dessau. In 1812 he entered the service of the king of
Württemberg, was ennobled, created counsellor of legation, appointed
intendant of the court theatre and chief librarian of the royal library
at Stuttgart. In 1828 he retired and settled at Wörlitz near Dessau,
where he died on the 12th of March 1831. Matthisson enjoyed for a time a
great popularity on account of his poems, _Gedichte_ (1787; 15th ed.,
1851; new ed., 1876), which Schiller extravagantly praised for their
melancholy sweetness and their fine descriptions of scenery. The verse
is melodious and the language musical, but the thought and sentiments
they express are too often artificial and insincere. His _Adelaide_ has
been rendered famous owing to Beethoven's setting of the song. Of his
elegies, _Die Elegie in den Ruinen eines alten Bergschlosses_ is still a
favourite. His reminiscences, _Erinnerungen_ (5 vols., 1810-1816),
contain interesting accounts of his travels.

  Matthisson's _Schriften_ appeared in eight volumes (1825-1829), of
  which the first contains his poems, the remainder his _Erinnerungen_;
  a ninth volume was added in 1833 containing his biography by H.
  Döring. His _Literarischer Nachlass_, with a selection from his
  correspondence, was published in four volumes by F. R. Schoch in 1832.




MATTING, a general term embracing many coarse woven or plaited fibrous
materials used for covering floors or furniture, for hanging as screens,
for wrapping up heavy merchandise and for other miscellaneous purposes.
In the United Kingdom, under the name of "coir" matting, a large amount
of a coarse kind of carpet is made from coco-nut fibre; and the same
material, as well as strips of cane, Manila hemp, various grasses and
rushes, is largely employed in various forms for making door mats. Large
quantities of the coco-nut fibre are woven in heavy looms, then cut up
into various sizes, and finally bound round the edges by a kind of rope
made from the same material. The mats may be of one colour only, or they
may be made of different colours and in different designs. Sometimes the
names of institutions are introduced into the mats. Another type of mat
is made exclusively from the above-mentioned rope by arranging alternate
layers in sinuous and straight paths, and then stitching the parts
together. It is also largely used for the outer covering of ships'
fenders. Perforated and otherwise prepared rubber, as well as wire-woven
material, are also largely utilized for door and floor mats. Matting of
various kinds is very extensively employed throughout India for floor
coverings, the bottoms of bedsteads, fans and fly-flaps, &c.; and a
considerable export trade in such manufactures is carried on. The
materials used are numerous; but the principal substances are straw, the
bulrushes _Typha elephantina_ and _T. angustifolia_, leaves of the date
palm (_Phoenix sylvestris_), of the dwarf palm (_Chamaerops Ritchiana_),
of the Palmyra palm (_Borassus flabelliformis_), of the coco-nut palm
(_Cocos nucifera_) and of the screw pine (_Pandanus odoratissimus_), the
munja or munj grass (_Saccharum Munja_) and allied grasses, and the mat
grasses _Cyperus textilis_ and _C. Pangorei_, from the last of which the
well-known Palghat mats of the Madras Presidency are made. Many of these
Indian grass-mats are admirable examples of elegant design, and the
colours in which they are woven are rich, harmonious and effective in
the highest degree. Several useful household articles are made from the
different kinds of grasses. The grasses are dyed in all shades and
plaited to form attractive designs suitable for the purposes to which
they are to be applied. This class of work obtains in India, Japan and
other Eastern countries. Vast quantities of coarse matting used for
packing furniture, heavy and coarse goods, flax and other plants, &c.,
are made in Russia from the bast or inner bark of the lime tree. This
industry centres in the great forest governments of Viatka,
Nizhniy-Novgorod, Kostroma, Kazan, Perm and Simbirsk.




MATTOCK (O.E. _mattuc_, of uncertain origin), a tool having a double
iron head, of which one end is shaped like an adze, and the other like a
pickaxe. The head has a socket in the centre in which the handle is
inserted transversely to the blades. It is used chiefly for grubbing and
rooting among tree stumps in plantations and copses, where the roots are
too close for the use of a spade, or for loosening hard soil.




MATTO GROSSO, an inland state of Brazil, bounded N. by Amazonas and
Pará, E. by Goyaz, Minas Geraes, São Paulo and Paraná, S. by Paraguay
and S.W. and W. by Bolivia. It ranks next to Amazonas in size, its area,
which is largely unsettled and unexplored, being 532,370 sq. m., and its
population only 92,827 in 1890 and 118,025 in 1900. No satisfactory
estimate of its Indian population can be made. The greater part of the
state belongs to the western extension of the Brazilian plateau, across
which, between the 14th and 16th parallels, runs the watershed which
separates the drainage basins of the Amazon and La Plata. This elevated
region is known as the plateau of Matto Grosso, and its elevations so
far as known rarely exceed 3000 ft. The northern slope of this great
plateau is drained by the Araguaya-Tocantins, Xingú, Tapajos and
Guaporé-Mamoré-Madeira, which flow northward, and, except the first,
empty into the Amazon; the southern slope drains southward through a
multitude of streams flowing into the Paraná and Paraguay. The general
elevation in the south part of the state is much lower, and large areas
bordering the Paraguay are swampy, partially submerged plains which the
sluggish rivers are unable to drain. The lowland elevations in this part
of the state range from 300 to 400 ft. above sea-level, the climate is
hot, humid and unhealthy, and the conditions for permanent settlement
are apparently unfavourable. On the highlands, however, which contain
extensive open _campos_, the climate, though dry and hot, is considered
healthy. The basins of the Paraná and Paraguay are separated by low
mountain ranges extending north from the _sierras_ of Paraguay. In the
north, however, the ranges which separate the river valleys are
apparently the remains of the table-land through which deep valleys have
been eroded. The resources of Matto Grosso are practically undeveloped,
owing to the isolated situation of the state, the costs of
transportation and the small population.

The first industry was that of mining, gold having been discovered in
the river valleys on the southern slopes of the plateau, and diamonds on
the head-waters of the Paraguay, about Diamantino and in two or three
other districts. Gold is found chiefly in placers, and in colonial times
the output was large, but the deposits were long ago exhausted and the
industry is now comparatively unimportant. As to other minerals little
is definitely known. Agriculture exists only for the supply of local
needs, though tobacco of a superior quality is grown. Cattle-raising,
however, has received some attention and is the principal industry of
the landowners. The forest products of the state include fine woods,
rubber, ipecacuanha, sarsaparilla, jaborandi, vanilla and copaiba. There
is little export, however, the only means of communication being down
the Paraguay and Paraná rivers by means of subsidized steamers. The
capital of the state is Cuyabá, and the chief commercial town is Corumbá
at the head of navigation for the larger river boats, and 1986 m. from
the mouth of the La Plata. Communication between these two towns is
maintained by a line of smaller boats, the distance being 517 m.

The first permanent settlements in Matto Grosso seem to have been made
in 1718 and 1719, in the first year at Forquilha and in the second at or
near the site of Cuyabá, where rich placer mines had been found. At this
time all this inland region was considered a part of São Paulo, but in
1748 it was made a separate _capitania_ and was named Matto Grosso
("great woods"). In 1752 its capital was situated on the right bank of
the Guaporé river and was named Villa Bella da Santissima Trindade de
Matto Grosso, but in 1820 the seat of government was removed to Cuyabá
and Villa Bella has fallen into decay. In 1822 Matto Grosso became a
province of the empire and in 1889 a republican state. It was invaded by
the Paraguayans in the war of 1860-65.




MATTOON, a city of Coles county, Illinois, U.S.A., in the east central
part of the state, about 12 m. south-east of Peoria. Pop. (1890), 6833;
(1900), 9622, of whom 430 were foreign-born; (1910 census) 11,456. It is
served by the Illinois Central and Cleveland, Cincinnati, Chicago & St
Louis railways, which have repair shops here, and by inter-urban
electric lines. The city has a public library, a Methodist Episcopal
Hospital, and an Old Folks' Home, the last supported by the Independent
Order of Odd Fellows. Mattoon is an important shipping point for Indian
corn and broom corn, extensively grown in the vicinity, and for fruit
and livestock. Among its manufactures are foundry and machine shop
products, stoves and bricks; in 1905 the factory product was valued at
$1,308,781, an increase of 71.2% over that in 1900. The municipality
owns the waterworks and an electric lighting plant. Mattoon was first
settled about 1855, was named in honour of William Mattoon, an early
landowner, was first chartered as a city in 1857, and was reorganized
under a general state law in 1879.




MATTRESS (O.Fr. _materas_, mod. _matelas_; the origin is the Arab.
_al-materah_, cushion, whence Span. and Port. _almadraque_, Ital.
_materasso_), the padded foundation of a bed, formed of canvas or other
stout material stuffed with wool, hair, flock or straw; in the last case
it is properly known as a "palliasse" (Fr. _paille_, straw; Lat.
_palea_); but this term is often applied to an under-mattress stuffed
with substances other than straw. The padded mattress on which lay the
feather-bed has been replaced by the "wire-mattress," a network of wire
stretched on a light wooden or iron frame, which is either a separate
structure or a component part of the bedstead itself. The
"wire-mattress" has taken the place of the "spring mattress," in which
spiral springs support the stuffing. The term "mattress" is used in
engineering for a mat of brushwood, faggots, &c., corded together and
used as a foundation or as surface in the construction of dams, jetties,
dikes, &c.




MATURIN, CHARLES ROBERT (1782-1824), Irish novelist and dramatist, was
born in Dublin in 1782. His grandfather, Gabriel Jasper Maturin, had
been Swift's successor in the deanery of St Patrick. Charles Maturin was
educated at Trinity College, Dublin, and became curate of Loughrea and
then of St Peter's, Dublin. His first novels, _The Fatal Revenge; or,
the Family of Montorio_ (1807), _The Wild Irish Boy_ (1808), _The
Milesian Chief_ (1812), were issued under the pseudonym of "Dennis
Jasper Murphy." All these were mercilessly ridiculed, but the irregular
power displayed in them attracted the notice of Sir Walter Scott, who
recommended the author to Byron. Through their influence Maturin's
tragedy of _Bertram_ was produced at Drury Lane in 1816, with Kean and
Miss Kelly in the leading parts. A French version by Charles Nodier and
Baron Taylor was produced in Paris at the Théâtre Favart. Two more
tragedies, _Manuel_ (1817) and _Fredolfo_ (1819), were failures, and his
poem _The Universe_ (1821) fell flat. He wrote three more novels,
_Women_ (1818), _Melmoth, the Wanderer_ (1820), and _The Albigenses_
(1824). _Melmoth_, which forms its author's title to remembrance, is the
best of them, and has for hero a kind of "Wandering Jew." Honoré de
Balzac wrote a sequel to it under the title of _Melmoth réconcilié à
l'église_ (1835). Maturin died in Dublin on the 30th of October 1824.




MATVYEEV, ARTAMON SERGYEEVICH ( -1682), Russian statesman and reformer,
was one of the greatest of the precursors of Peter the Great. His
parentage and the date of his birth are uncertain. Apparently his birth
was humble, but when the obscure figure of the young Artamon emerges
into the light of history we find him equipped at all points with the
newest ideas, absolutely free from the worst prejudices of his age, a
ripe scholar, and even an author of some distinction. In 1671 the tsar
Alexius and Artamon were already on intimate terms, and on the
retirement of Orduin-Nashchokin Matvyeev became the tsar's chief
counsellor. It was at his house, full of all the wondrous,
half-forbidden novelties of the west, that Alexius, after the death of
his first consort, Martha, met Matvyeev's favourite pupil, the beautiful
Natalia Naruishkina, whom he married on the 21st of January 1672. At the
end of the year Matvyeev was raised to the rank of _okolnichy_, and on
the 1st of September 1674 attained the still higher dignity of _boyar_.
Matvyeev remained paramount to the end of the reign and introduced
play-acting and all sorts of refining western novelties into Muscovy.
The deplorable physical condition of Alexius's immediate successor,
Theodore III. suggested to Matvyeev the desirability of elevating to
the throne the sturdy little tsarevich Peter, then in his fourth year.
He purchased the allegiance of the _stryeltsi_, or musketeers, and then,
summoning the boyars of the council, earnestly represented to them that
Theodore, scarce able to live, was surely unable to reign, and urged the
substitution of little Peter. But the reactionary boyars, among whom
were the near kinsmen of Theodore, proclaimed him tsar and Matvyeev was
banished to Pustozersk, in northern Russia, where he remained till
Theodore's death (April 27, 1682). Immediately afterwards Peter was
proclaimed tsar by the patriarch, and the first _ukaz_ issued in Peter's
name summoned Matvyeev to return to the capital and act as chief adviser
to the tsaritsa Natalia. He reached Moscow on the 15th of May, prepared
"to lay down his life for the tsar," and at once proceeded to the head
of the Red Staircase to meet and argue with the assembled stryeltsi, who
had been instigated to rebel by the anti-Petrine faction. He had already
succeeded in partially pacifying them, when one of their colonels began
to abuse the still hesitating and suspicious musketeers. Infuriated,
they seized and flung Matvyeev into the square below, where he was
hacked to pieces by their comrades.

  See R. Nisbet Bain, _The First Romanovs_ (London, 1905); M. P.
  Pogodin, _The First Seventeen Years of the Life of Peter the Great_
  (Rus.), (Moscow, 1875); S. M. Solovev, _History of Russia_ (Rus.),
  (vols. 12, 13, (St Petersburg, 1895, &c.); L. Shehepotev, _A. S.
  Matvyeev as an Educational and Political Reformer_ (Rus.), (St
  Petersburg, 1906).     (R. N. B.)




MAUBEUGE, a town of northern France, in the department of Nord, situated
on both banks of the Sambre, here canalized, 23½ m. by rail E. by S. of
Valenciennes, and about 2 m. from the Belgian frontier. Pop. (1906),
town 13,569, commune 21,520. As a fortress Maubeuge has an old enceinte
of bastion trace which serves as the centre of an important entrenched
camp of 18 m. perimeter, constructed for the most part after the war of
1870, but since modernized and augmented. The town has a board of trade
arbitration, a communal college, a commercial and industrial school; and
there are important foundries, forges and blast-furnaces, together with
manufactures of machine-tools, porcelain, &c. It is united by electric
tramway with Hautmont (pop. 12,473), also an important metallurgical
centre.

Maubeuge (_Malbodium_) owes its origin to a double monastery, for men
and women, founded in the 7th century by St Aldegonde relics of whom are
preserved in the church. It subsequently belonged to the territory of
Hainault. It was burnt by Louis XI., by Francis I., and by Henry II.,
and was finally assigned to France by the Treaty of Nijmwegen. It was
fortified at Vauban by the command of Louis XIV., who under Turenne
first saw military service there. Besieged in 1793 by Prince Josias of
Coburg, it was relieved by the victory of Wattignies, which is
commemorated by a monument in the town. It was unsuccessfully besieged
in 1814, but was compelled to capitulate, after a vigorous resistance,
in the Hundred Days.




MAUCH CHUNK, a borough and the county-seat of Carbon county,
Pennsylvania, U.S.A., on the W. bank of the Lehigh river and on the
Lehigh Coal and Navigation Company's Canal, 46 m. by rail W.N.W. of
Easton. Pop. (1800), 4101; (1900), 4029 (571 foreign-born); (1910),
3952. Mauch Chunk is served by the Central of New Jersey railway and, at
East Mauch Chunk, across the river, connected by electric railway, by
the Lehigh Valley railway. The borough lies in the valley of the Lehigh
river, along which runs one of its few streets and in another deeply cut
valley at right angles to the river; through this second valley east and
west runs the main street, on which is an electric railway; parallel to
it on the south is High Street, formerly an Irish settlement; half way
up the steep hill, and on the north at the top of the opposite hill is
the ward of Upper Mauch Chunk, reached by the electric railway. An
incline railway, originally used to transport coal from the mines to the
river and named the "Switch-Back," now carries tourists up the steep
slopes of Mount Pisgah and Mount Jefferson, to Summit Hill, a rich
anthracite coal region, with a famous "burning mine," which has been on
fire since 1832, and then back. An electric railway to the top of
Flagstaff Mountain, built in 1900, was completed in 1901 to Lehighton, 4
m. south-east of Mauch Chunk, where coal is mined and silk and stoves
are manufactured, and which had a population in 1900 of 4629, and in
1910 of 5316. Immediately above Mauch Chunk the river forms a horseshoe;
on the opposite side, connected by a bridge, is the borough of East
Mauch Chunk (pop. 1900, 3458; 1910, 3548); and 2 m. up the river is Glen
Onoko, with fine falls and cascades. The principal buildings in Mauch
Chunk are the county court house, a county gaol, a Young Men's Christian
Association building, and the Dimmick Memorial Library (1890). The
borough was long a famous shipping point for coal. It now has ironworks
and foundries, and in East Mauch Chunk there are silk mills. The name is
Indian and means "Bear Mountain," this English name being used for a
mountain on the east side of the river. The borough was founded by the
Lehigh Coal and Navigation Company in 1818. This company began in 1827
the operation of the "Switch-Back," probably the first railway in the
country to be used for transporting coal. In 1831 the town was opened to
individual enterprise, and in 1850 it was incorporated as a borough.
Mauch Chunk was for many years the home of Asa Packer, the projector and
builder of the Lehigh Valley railroad from Mauch Chunk to Easton.




MAUCHLINE, a town in the division of Kyle, Ayrshire, Scotland. Pop.
(1901), 1767. It lies 8 m. E.S.E. of Kilmarnock and 11 m. E. by N. of
Ayr by the Glasgow and South-Western railway. It is situated on a gentle
slope about 1 m. from the river Ayr, which flows through the south of
the parish of Mauchline. It is noted for its manufacture of snuff-boxes
and knick-knacks in wood, and of curling-stones. There is also some
cabinet-making, besides spinning and weaving, and its horse fairs and
cattle markets have more than local celebrity. The parish church, dating
from 1829, stands in the middle of the village, and on the green a
monument, erected in 1830, marks the spot where five Covenanters were
killed in 1685. Robert Burns lived with his brother Gilbert on the farm
of Mossgiel, about a mile to the north, from 1784 to 1788. Mauchline
kirkyard was the scene of the "Holy Fair"; at "Poosie Nansie's" (Agnes
Gibson's)--still, though much altered, a popular inn--the "Jolly
Beggars" held their high jinks; near the church (in the poet's day an
old, barn-like structure) was the Whiteford Arms inn, where on a pane of
glass Burns wrote the epitaph on John Dove, the landlord; "auld Nanse
Tinnock's" house, with the date of 1744 above the door, nearly faces the
entrance to the churchyard; the Rev. William Auld was minister of
Mauchline, and "Holy Willie," whom the poet scourged in the celebrated
"Prayer," was one of "Daddy Auld's" elders; behind the kirkyard stands
the house of Gavin Hamilton, the lawyer and firm friend of Burns, in
which the poet was married. The braes of Ballochmyle, where he met the
heroine of his song, "The Lass o' Ballochmyle," lie about a mile to the
south-east. Adjoining them is the considerable manufacturing town of
CATRINE (pop. 2340), with cotton factories, bleach fields and brewery,
where Dr Matthew Stewart (1717-1785), the father of Dugald Stewart--had
a mansion, and where there is a big water-wheel said to be inferior in
size only to that of Laxey in the Isle of Man. Barskimming House, 2 m.
south by west of Mauchline, the seat of Lord-President Miller
(1717-1789), was burned down in 1882. Near the confluence of the Fail
and the Ayr was the scene of Burns's parting with Highland Mary.




MAUDE, CYRIL (1862-   ), English actor, was born in London and educated
at Charterhouse. He began his career as an actor in 1883 in America, and
from 1896 to 1905 was co-manager with F. Harrison of the Haymarket
Theatre, London. There he became distinguished for his quietly humorous
acting in many parts. In 1906 he went into management on his own
account, and in 1907 opened his new theatre The Playhouse. In 1888 he
married the actress Winifred Emery (b. 1862), who had made her London
début as a child in 1875, and acted with Irving at the Lyceum between
1881 and 1887. She was a daughter of Samuel Anderson Emery (1817-1881)
and granddaughter of John Emery (1777-1822), both well-known actors in
their day.




MAULE, a coast province of central Chile, bounded N. by Talea, E. by
Linares and Nuble, and S. by Concepción, and lying between the rivers
Maule and Itata, which form its northern and southern boundaries. Pop.
(1895), 119,791; area, 2475 sq. m. Maule is traversed from north to
south by the coast range and its surfaces are much broken. The
Buchupureo river flows westward across the province. The climate is mild
and healthy. Agriculture and stock-raising are the principal
occupations, and hides, cattle, wheat and timber are exported. Transport
facilities are afforded by the Maule and the Itata, which are navigable,
and by a branch of the government railway from Cauquenes to Parral, an
important town of southern Linares. The provincial capital, Cauquenes
(pop., in 1895, 8574; 1902 estimate, 9895), is centrally situated on the
Buchupureo river, on the eastern slopes of the coast cordilleras. The
town and port of Constitución (pop., in 1900, about 7000) on the south
bank of the Maule, one mile above its mouth, was formerly the capital of
the province. The port suffers from a dangerous bar at the mouth of the
river, but is connected with Talca by rail and has a considerable trade.

The Maule river, from which the province takes its name, is of historic
interest because it is said to have marked the southern limits of the
Inca Empire. It rises in the Laguna del Maule, an Andean lake near the
Argentine frontier, 7218 ft. above sea-level, and flows westward about
140 m. to the Pacific, into which it discharges in 35° 18´ S. The upper
part of its drainage basin, to which the _Anuario Hydrografico_ gives an
area of 8000 sq. m., contains the volcanoes of San Pedro (11,800 ft.),
the Descabezado (12,795 ft.), and others of the same group of lower
elevations. The upper course and tributaries of the Maule, principally
in the province of Linares, are largely used for irrigation.




MAULÉON, SAVARI DE (d. 1236), French soldier, was the son of Raoul de
Mauléon, vicomte de Thouars and lord of Mauléon (now Châtillon-sur-Sèvre).
Having espoused the cause of Arthur of Brittany, he was captured at
Mirebeau (1202), and imprisoned in the château of Corfe. But John set him
at liberty in 1204, gained him to his side and named him seneschal of
Poitou (1205). In 1211 Savari de Mauléon assisted Raymond VI. count of
Toulouse, and with him besieged Simon de Montfort in Castelnaudary. Philip
Augustus bought his services in 1212 and gave him command of a fleet which
was destroyed in the Flemish port of Damme. Then Mauléon returned to John,
whom he aided in his struggle with the barons in 1215. He was one of those
whom John designated on his deathbed for a council of regency (1216). Then
he went to Egypt (1219), and was present at the taking of Damietta.
Returning to Poitou he was a second time seneschal for the king of
England. He defended Saintonge against Louis VIII. in 1224, but was
accused of having given La Rochelle up to the king of France, and the
suspicions of the English again threw him back upon the French. Louis
VIII. then turned over to him the defence of La Rochelle and the coast of
Saintonge. In 1227 he took part in the rising of the barons of Poitiers
and Anjou against the young Louis IX. He enjoyed a certain reputation for
his poems in the _langue d'oc_.

  See Chilhaud-Dumaine, "Savari de Mauléon," in _Positions des Thèses
  des élèves de l'École des Chartes_ (1877); _Histoire littéraire de la
  France_, xviii. 671-682.




MAULSTICK, or MAHLSTICK, a stick with a soft leather or padded head,
used by painters to support the hand that holds the brush. The word is
an adaptation of the Dutch _maalstok_, i.e. the painter's stick, from
_malen_, to paint.




MAUNDY THURSDAY (through O.Fr. _mandé_ from Lat. _mandatum_,
commandment, in allusion to Christ's words: "A new commandment give I
unto you," after he had washed the disciples' feet at the Last Supper),
the Thursday before Easter. Maundy Thursday is sometimes known as
_Sheer_ or _Chare_ Thursday, either in allusion, it is thought, to the
"shearing" of heads and beards in preparation for Easter, or more
probably in the word's Middle English sense of "pure," in allusion to
the ablutions of the day. The chief ceremony, as kept from the early
middle ages onwards--the washing of the feet of twelve or more poor men
or beggars--was in the early Church almost unknown. Of Chrysostom and St
Augustine, who both speak of Maundy Thursday as being marked by a
solemn celebration of the Sacrament, the former does not mention the
foot-washing, and the latter merely alludes to it. Perhaps an indication
of it may be discerned as early as the 4th century in a custom, current
in Spain, northern Italy and elsewhere, of washing the feet of the
catechumens towards the end of Lent before their baptism. It was not,
however, universal, and in the 48th canon of the synod of Elvira (A.D.
306) it is expressly prohibited (cf. _Corp. Jur. Can._, c. 104, _caus._
i. _qu._ 1). From the 4th century ceremonial foot-washing became yearly
more common, till it was regarded as a necessary rite, to be performed
by the pope, all Catholic sovereigns, prelates, priests and nobles. In
England the king washed the feet of as many poor men as he was years
old, and then distributed to them meat, money and clothes. At Durham
Cathedral, until the 16th century, every charity-boy had a monk to wash
his feet. At Peterborough Abbey, in 1530, Wolsey made "his maund in Our
Lady's Chapel, having fifty-nine poor men whose feet he washed and
kissed; and after he had wiped them he gave every of the said poor men
twelve pence in money, three ells of good canvas to make them shirts, a
pair of new shoes, a cast of red herrings and three white herrings."
Queen Elizabeth performed the ceremony, the paupers' feet, however,
being first washed by the yeomen of the laundry with warm water and
sweet herbs. James II. was the last English monarch to perform the rite.
William III. delegated the washing to his almoner, and this was usual
until the middle of the 18th century. Since 1754 the foot-washing has
been abandoned, and the ceremony now consists of the presentation of
Maundy money, officially called Maundy Pennies. These were first coined
in the reign of Charles II. They come straight from the Mint, and have
their edges unmilled. The service which formerly took place in the
Chapel Royal, Whitehall, is now held in Westminster Abbey. A procession
is formed in the nave, consisting of the lord high almoner representing
the sovereign, the clergy and the yeomen of the guard, the latter
carrying white and red purses in baskets. The clothes formerly given are
now commuted for in cash. The full ritual is gone through by the Roman
Catholic archbishop of Westminster, and abroad it survives in all
Catholic countries, a notable example being that of the Austrian
emperor. In the Greek Church the rite survives notably at Moscow, St
Petersburg and Constantinople. It is on Maundy Thursday that in the
Church of Rome the sacred oil is blessed, and the chrism prepared
according to an elaborate ritual which is given in the _Pontificale_.




MAUPASSANT, HENRI RENÉ ALBERT GUY DE (1850-1893), French novelist and
poet, was born at the Château of Miromesnil in the department of
Seine-Inférieure on the 5th August 1850. His grandfather, a landed
proprietor of a good Lorraine family, owned an estate at
Neuville-Champ-d'Oisel near Rouen, and bequeathed a moderate fortune to
his son, a Paris stockbroker, who married Mademoiselle Laure Lepoitevin.
Maupassant was educated at Yvetot and at the Rouen lycée. A copy of
verses entitled _Le Dieu créateur_, written during his year of
philosophy, has been preserved and printed. He entered the ministry of
marine, and was promoted by M. Bardoux to the Cabinet de l'Instruction
publique. A pleasant legend says that, in a report by his official
chief, Maupassant is mentioned as not reaching the standard of the
department in the matter of style. He may very well have been an
unsatisfactory clerk, as he divided his time between rowing expeditions
and attending the literary gatherings at the house of Gustave Flaubert,
who was not, as he is often alleged to be, connected with Maupassant by
any blood tie. Flaubert was not his uncle, nor his cousin, nor even his
godfather, but merely an old friend of Madame de Maupassant, whom he had
known from childhood. At the literary meetings Maupassant seldom shared
in the conversation. Upon those who met him--Tourgenieff, Alphonse
Daudet, Catulle Mendès, José-Maria de Heredia and Émile Zola--he left
the impression of a simple young athlete. Even Flaubert, to whom
Maupassant submitted some sketches, was not greatly struck by their
talent, though he encouraged the youth to persevere. Maupassant's first
essay was a dramatic piece twice given at Étretat in 1873 before an
audience which included Tourgenieff, Flaubert and Meilhac. In this
indecorous performance, of which nothing more is heard, Maupassant
played the part of a woman. During the next seven years he served a
severe apprenticeship to Flaubert, who by this time realized his pupil's
exceptional gifts. In 1880 Maupassant published a volume of poems, _Des
Vers_, against which the public prosecutor of Etampes took proceedings
that were finally withdrawn through the influence of the senator
Cordier. From Flaubert, who had himself been prosecuted for his first
book, _Madame Bovary_, there came a letter congratulating the poet on
the similarity between their first literary experiences. _Des Vers_ is
an extremely interesting experiment, which shows Maupassant to us still
hesitating in his choice of a medium; but he recognized that it was not
wholly satisfactory, and that its chief deficiency--the absence of
verbal melody--was fatal. Later in the same year he contributed to the
_Soirées de Médan_, a collection of short stories by MM. Zola, J.-K.
Huysmans, Henry Céard, Léon Hennique and Paul Alexis; and in _Boule de
suif_ the young unknown author revealed himself to his amazed
collaborators and to the public as an admirable writer of prose and a
consummate master of the _conte_. There is perhaps no other instance in
modern literary history of a writer beginning, as a fully equipped
artist, with a genuine masterpiece. This early success was quickly
followed by another. The volume entitled _La Maison Tellier_ (1881)
confirmed the first impression, and vanquished even those who were
repelled by the author's choice of subjects. In _Mademoiselle Fifi_
(1883) he repeated his previous triumphs as a _conteur_, and in this
same year he, for the first time, attempted to write on a larger scale.
Choosing to portray the life of a blameless girl, unfortunate in her
marriage, unfortunate in her son, consistently unfortunate in every
circumstance of existence, he leaves her, ruined and prematurely old,
clinging to the tragic hope, which time, as one feels, will belie, that
she may find happiness in her grandson. This picture of an average woman
undergoing the constant agony of disillusion Maupassant calls _Une Vie_
(1883), and as in modern literature there is no finer example of cruel
observation, so there is no sadder book than this, while the effect of
extreme truthfulness which it conveys justifies its sub-title--_L'Humble
vérité_. Certain passages of _Une Vie_ are of such a character that the
sale of the volume at railway bookstalls was forbidden throughout
France. The matter was brought before the chamber of deputies, with the
result of drawing still more attention to the book, and of advertising
the _Contes de la bécasse_ (1883), a collection of stories as improper
as they are clever. _Au soleil_ (1884), a book of travels which has the
eminent qualities of lucid observation and exact description, was less
read than _Clair de lune_, _Miss Harriet_, _Les Soeurs Rondoli_ and
_Yvette_, all published in 1883-1884 when Maupassant's powers were at
their highest level. Three further collections of short tales, entitled
_Contes et nouvelles_, _Monsieur Parent_, and _Contes du jour et de la
nuit_, issued in 1885, proved that while the author's vision was as
incomparable as ever, his fecundity had not improved his impeccable
form. To 1885 also belongs an elaborate novel, _Bel-ami_, the cynical
history of a particularly detestable, brutal scoundrel who makes his way
in the world by means of his handsome face. Maupassant is here no less
vivid in realizing his literary men, financiers and frivolous women than
in dealing with his favourite peasants, boors and servants, to whom he
returned in _Toine_ (1886) and in _La Petite roque_ (1886). About this
time appeared the first symptoms of the malady which destroyed him; he
wrote less, and though the novel _Mont-Oriol_ (1887) shows him
apparently in undiminished possession of his faculty, _Le Horla_ (1887)
suggests that he was already subject to alarming hallucinations.
Restored to some extent by a sea-voyage, recorded in _Sur l'eau_ (1888),
he went back to short stories in _Le Rosier de Madame Husson_ (1888), a
burst of Rabelaisian humour equal to anything he had ever written. His
novels _Pierre et Jean_ (1888), _Fort comme la mort_ (1889), and _Notre
coeur_ (1890) are penetrating studies touched with a profounder sympathy
than had hitherto distinguished him; and this softening into pity for
the tragedy of life is deepened in some of the tales included in
_Inutile beauté_ (1890). One of these, _Le Champ d'Oliviers_, is an
unsurpassable example of poignant, emotional narrative. With _La Vie
errante_ (1890), a volume of travels, Maupassant's career practically
closed. _Musotte_, a theatrical piece written in collaboration with M.
Jacques Normand, was published in 1891. By this time inherited nervous
maladies, aggravated by excessive physical exercises and by the
imprudent use of drugs, had undermined his constitution. He began to
take an interest in religious problems, and for a while made the
_Imitation_ his handbook; but his misanthropy deepened, and he suffered
from curious delusions as to his wealth and rank. A victim of general
paralysis, of which _La Folie des grandeurs_ was one of the symptoms, he
drank the waters at Aix-les-Bains during the summer of 1891, and retired
to Cannes, where he purposed passing the winter. The singularities of
conduct which had been observed at Aix-les-Bains grew more and more
marked. Maupassant's reason slowly gave way. On the 6th of January 1892
he attempted suicide, and was removed to Paris, where he died in the
most painful circumstances on the 6th of July 1893. He is buried in the
cemetery of Montparnasse. The opening chapters of two projected novels,
_L'Angélus_ and _L'Ame étrangère_, were found among his papers; these,
with _La Paix du ménage_, a comedy in two acts, and two collections of
tales, _Le Père Milon_ (1898) and _Le Colporteur_ (1899), have been
published posthumously. A correspondence, called _Amitié amoureuse_
(1897), and dedicated to his mother, is probably unauthentic. Among the
prefaces which he wrote for the works of others, only one--an
introduction to a French prose version of Mr Swinburne's _Poems and
Ballads_--is likely to interest English readers.

Maupassant began as a follower of Flaubert and of M. Zola, but, whatever
the masters may have called themselves, they both remained essentially
_romantiques_. The pupil is the last of the "naturalists": he even
destroyed naturalism, since he did all that can be done in that
direction. He had no psychology, no theories of art, no moral or strong
social prejudices, no disturbing imagination, no wealth of perplexing
ideas. It is no paradox to say that his marked limitations made him the
incomparable artist that he was. Undisturbed by any external influence,
his marvellous vision enabled him to become a supreme observer, and,
given his literary sense, the rest was simple. He prided himself in
having no invention; he described nothing that he had not seen. The
peasants whom he had known as a boy figure in a score of tales; what he
saw in Government offices is set down in _L'Héritage_; from Algiers he
gathers the material for Maroca; he drinks the waters and builds up
_Mont-Oriol_; he enters journalism, constructs _Bel-ami_, and, for the
sake of precision, makes his brother, Hervé de Maupassant, sit for the
infamous hero's portrait; he sees fashionable society, and, though it
wearied him intensely, he transcribes its life in _Fort comme la mort_
and _Notre coeur_. Fundamentally he finds all men alike. In every grade
he finds the same ferocious, cunning, animal instincts at work: it is
not a gay world, but he knows no other; he is possessed by the dread of
growing old, of ceasing to enjoy; the horror of death haunts him like a
spectre. It is an extremely simple outlook. Maupassant does not prefer
good to bad, one man to another; he never pauses to argue about the
meaning of life, a senseless thing which has the one advantage of
yielding materials for art; his one aim is to discover the hidden aspect
of visible things, to relate what he has observed, to give an objective
rendering of it, and he has seen so intensely and so serenely that he is
the most exact transcriber in literature. And as the substance is, so is
the form: his style is exceedingly simple and exceedingly strong; he
uses no rare or superfluous word, and is content to use the humblest
word if only it conveys the exact picture of the thing seen. In ten
years he produced some thirty volumes. With the exception of _Pierre et
Jean_, his novels, excellent as they are, scarcely represent him at his
best, and of over two hundred _contes_ a proportion must be rejected.
But enough will remain to vindicate his claim to a permanent place in
literature as an unmatched observer and the most perfect master of the
short story.

  See also F. Brunetière, _Le Roman naturaliste_ (1883); T. Lemaître,
  _Les Contemporains_ (vols. i. v. vi.); R. Doumic, _Ecrivains
  d'aujourd'hui_ (1894); an introduction by Henry James to _The Odd
  Number_ ... (1891); a critical preface by the earl of Crewe to _Pierre
  and Jean_ (1902); A. Symons, _Studies in Prose and Verse_ (1904).
  There are many references to Maupassant in the _Journal des Goncourt_,
  and some correspondence with Marie Bashkirtseff was printed with
  _Further Memoirs_ of that lady in 1901.     (J. F. K.)




MAUPEOU, RENÉ NICOLAS CHARLES AUGUSTIN (1714-1792), chancellor of
France, was born on the 25th of February 1714, being the eldest son of
René Charles de Maupeou (1688-1775), who was president of the parlement
of Paris from 1743 to 1757. He married in 1744 a rich heiress, Anne de
Roncherolles, a cousin of Madame d'Épinay. Entering public life, he was
his father's right hand in the conflicts between the parlement and
Christophe de Beaumont, archbishop of Paris, who was supported by the
court. Between 1763 and 1768, dates which cover the revision of the case
of Jean Calas and the trial of the comte de Lally, Maupeou was himself
president of the parlement. In 1768, through the protection of Choiseul,
whose fall two years later was in large measure his work, he became
chancellor in succession to his father, who had held the office for a
few days only. He determined to support the royal authority against the
parlement, which in league with the provincial magistratures was seeking
to arrogate to itself the functions of the states-general. He allied
himself with the duc d'Aiguillon and Madame du Barry, and secured for a
creature of his own, the Abbé Terrai, the office of comptroller-general.
The struggle came over the trial of the case of the duc d'Aiguillon,
ex-governor of Brittany, and of La Chalotais, procureur-général of the
province, who had been imprisoned by the governor for accusations
against his administration. When the parlement showed signs of hostility
against Aiguillon, Maupeou read letters patent from Louis XV. annulling
the proceedings. Louis replied to remonstrances from the parlement by a
_lit de justice_, in which he demanded the surrender of the minutes of
procedure. On the 27th of November 1770 appeared the _Édit de règlement
et de discipline_, which was promulgated by the chancellor, forbidding
the union of the various branches of the parlement and correspondence
with the provincial magistratures. It also made a strike on the part of
the parlement punishable by confiscation of goods, and forbade further
obstruction to the registration of royal decrees after the royal reply
had been given to a first remonstrance. This edict the magistrates
refused to register, and it was registered in a _lit de justice_ held at
Versailles on the 7th of December, whereupon the parlement suspended its
functions. After five summonses to return to their duties, the
magistrates were surprised individually on the night of the 19th of
January 1771 by musketeers, who required them to sign yes or no to a
further request to return. Thirty-eight magistrates gave an affirmative
answer, but on the exile of their former colleagues by _lettres de
cachet_ they retracted, and were also exiled. Maupeou installed the
council of state to administer justice pending the establishment of six
superior courts in the provinces, and of a new parlement in Paris. The
_cour des aides_ was next suppressed.

Voltaire praised this revolution, applauding the suppression of the old
hereditary magistrature, but in general Maupeou's policy was regarded as
the triumph of tyranny. The remonstrances of the princes, of the nobles,
and of the minor courts, were met by exile and suppression, but by the
end of 1771 the new system was established, and the Bar, which had
offered a passive resistance, recommenced to plead. But the death of
Louis XV. in May 1774 ruined the chancellor. The restoration of the
parlements was followed by a renewal of the quarrels between the new
king and the magistrature. Maupeou and Terrai were replaced by
Malesherbes and Turgot. Maupeou lived in retreat until his death at
Thuit on the 29th of July 1792, having lived to see the overthrow of the
_ancien régime_. His work, in so far as it was directed towards the
separation of the judicial and political functions and to the reform of
the abuses attaching to a hereditary magistrature, was subsequently
endorsed by the Revolution; but no justification of his violent methods
or defence of his intriguing and avaricious character is possible. He
aimed at securing absolute power for Louis XV., but his action was in
reality a serious blow to the monarchy.

  The chief authority for the administration of Maupeou is the _compte
  rendu_ in his own justification presented by him to Louis XVI. in
  1789, which included a dossier of his speeches and edicts, and is
  preserved in the Bibliothèque nationale. These documents, in the hands
  of his former secretary, C. F. Lebrun, duc de Plaisance, formed the
  basis of the judicial system of France as established under the
  consulate (cf. C. F. Lebrun, _Opinions, rapports et choix d'écrits
  politiques_, published posthumously in 1829). See further _Maupeouana_
  (6 vols., Paris, 1775), which contains the pamphlets directed against
  him; _Journal hist. de la révolution opérée ... par M. de Maupeou_ (7
  vols., 1775); the official correspondence of Mercy-Argenteau, the
  letters of Mme d'Épinay; and Jules Flammermont, _Le Chancelier Maupeou
  et les parlements_ (1883).




MAUPERTUIS, PIERRE LOUIS MOREAU DE (1698-1759), French mathematician and
astronomer, was born at St Malo on the 17th of July 1698. When twenty
years of age he entered the army, becoming lieutenant in a regiment of
cavalry, and employing his leisure on mathematical studies. After five
years he quitted the army and was admitted in 1723 a member of the
Academy of Sciences. In 1728 he visited London, and was elected a fellow
of the Royal Society. In 1736 he acted as chief of the expedition sent
by Louis XV. into Lapland to measure the length of a degree of the
meridian (see EARTH, FIGURE OF), and on his return home he became a
member of almost all the scientific societies of Europe. In 1740
Maupertuis went to Berlin on the invitation of the king of Prussia, and
took part in the battle of Mollwitz, where he was taken prisoner by the
Austrians. On his release he returned to Berlin, and thence to Paris,
where he was elected director of the Academy of Sciences in 1742, and in
the following year was admitted into the Academy. Returning to Berlin in
1744, at the desire of Frederick II., he was chosen president of the
Royal Academy of Sciences in 1746. Finding his health declining, he
repaired in 1757 to the south of France, but went in 1758 to Basel,
where he died on the 27th of July 1759. Maupertuis was unquestionably a
man of considerable ability as a mathematician, but his restless, gloomy
disposition involved him in constant quarrels, of which his
controversies with König and Voltaire during the latter part of his life
furnish examples.

  The following are his most important works: _Sur la figure de la
  terre_ (Paris, 1738); _Discours sur la parallaxe de la lune_ (Paris,
  1741); _Discours sur la figure des astres_ (Paris, 1742); _Éléments de
  la géographie_ (Paris, 1742); _Lettre sur la comète de 1742_ (Paris,
  1742); _Astronomie nautique_ (Paris, 1745 and 1746); _Vénus physique_
  (Paris, 1745); _Essai de cosmologie_ (Amsterdam, 1750). His _Oeuvres_
  were published in 1752 at Dresden and in 1756 at Lyons.




MAU RANIPUR, a town of British India in Jahnsi district, in the United
Provinces. Pop. (1901), 17,231. It contains a large community of wealthy
merchants and bankers. A special variety of red cotton cloth, known as
_kharua_, is manufactured and exported to all parts of India. Trees line
many of the streets, and handsome temples ornament the town.




MAUREL, ABDIAS (d. 1705), Camisard leader, became a cavalry officer in
the French army and gained distinction in Italy; here he served under
Marshal Catinat, and on this account he himself is sometimes known as
Catinat. In 1702, when the revolt in the Cévennes broke out, he became
one of the Camisard leaders, and in this capacity his name was soon
known and feared. He refused to accept the peace made by Jean Cavalier
in 1704, and after passing a few weeks in Switzerland he returned to
France and became one of the chiefs of those Camisards who were still in
arms. He was deeply concerned in a plot to capture some French towns, a
scheme which, it was hoped, would be helped by England and Holland. But
it failed; Maurel was betrayed, and with three other leaders of the
movement was burned to death at Nîmes on the 22nd of April 1705. He was
a man of great physical strength; but he was very cruel, and boasted he
had killed 200 Roman Catholics with his own hands.




MAUREL, VICTOR (1848-   ), French singer, was born at Marseilles, and
educated in music at the Paris Conservatoire. He made his début in opera
at Paris in 1868, and in London in 1873, and from that time onwards his
admirable acting and vocal method established his reputation as one of
the finest of operatic baritones. He created the leading part in Verdi's
_Otello_, and was equally fine in Wagnerian and Italian opera.




MAURENBRECHER, KARL PETER WILHELM (1838-1892), German historian, was
born at Bonn on the 21st of December, 1838, and studied in Berlin and
Munich under Ranke and Von Sybel, being especially influenced by the
latter historian. After doing some research work at Simancas in Spain,
he became professor of history at the university of Dorpat in 1867; and
was then in turn professor at Königsberg, Bonn and Leipzig. He died at
Leipzig on the 6th of November, 1892.

  Many of Maurenbrecher's works are concerned with the Reformation,
  among them being _England im Reformationszeitalter_ (Düsseldorf,
  1866); _Karl V. und die deutschen Protestanten_ (Düsseldorf, 1865);
  _Studien und Skizzen zur Geschichte der Reformationszeit_ (Leipzig,
  1874); and the incomplete _Geschichte der Katholischen Reformation_
  (Nördlingen, 1880). He also wrote _Don Karlos_ (Berlin, 1876);
  _Gründung des deutschen Reiches 1859-1871_ (Leipzig, 1892, and again
  1902); and _Geschichte der deutschen Königswahlen_ (Leipzig, 1889).
  See G. Wolf, _Wilhelm Maurenbrecher_ (Berlin, 1893).




MAUREPAS, JEAN FRÉDÉRIC PHÉLYPEAUX, COMTE DE (1701-1781), French
statesman, was born on the 9th of July 1701 at Versailles, being the son
of Jérôme de Pontchartrain, secretary of state for the marine and the
royal household. Maurepas succeeded to his father's charge at fourteen,
and began his functions in the royal household at seventeen, while in
1725 he undertook the actual administration of the navy. Although
essentially light and frivolous in character, Maurepas was seriously
interested in scientific matters, and he used the best brains of France
to apply science to questions of navigation and of naval construction.
He was disgraced in 1749, and exiled from Paris for an epigram against
Madame de Pompadour. On the accession of Louis XVI., twenty-five years
later, he became a minister of state and Louis XVI.'s chief adviser. He
gave Turgot the direction of finance, placed Lamoignon-Malesherbes over
the royal household and made Vergennes minister for foreign affairs. At
the outset of his new career he showed his weakness by recalling to
their functions, in deference to popular clamour, the members of the old
parlement ousted by Maupeou, thus reconstituting the most dangerous
enemy of the royal power. This step, and his intervention on behalf of
the American states, helped to pave the way for the French revolution.
Jealous of his personal ascendancy over Louis XVI., he intrigued against
Turgot, whose disgrace in 1776 was followed after six months of disorder
by the appointment of Necker. In 1781 Maurepas deserted Necker as he had
done Turgot, and he died at Versailles on the 21st of November 1781.

  Maurepas is credited with contributions to the collection of facetiae
  known as the _Étrennes de la Saint Jean_ (2nd ed., 1742). Four volumes
  of _Mémoires de Maurepas_, purporting to be collected by his secretary
  and edited by J. L. G. Soulavie in 1792, must be regarded as
  apocryphal. Some of his letters were published in 1896 by the _Soc. de
  l'hist. de Paris_. His _éloge_ in the Academy of Sciences was
  pronounced by Condorcet.




MAURER, GEORG LUDWIG VON (1790-1872), German statesman and historian,
son of a Protestant pastor, was born at Erpolzheim, near Dürkheim, in
the Rhenish Palatinate, on the 2nd of November 1790. Educated at
Heidelberg, he went in 1812 to reside in Paris, where he entered upon a
systematic study of the ancient legal institutions of the Germans.
Returning to Germany in 1814, he received an appointment under the
Bavarian government, and afterwards filled several important official
positions. In 1824 he published at Heidelberg his _Geschichte des
altgermanischen und namentlich altbayrischen öffentlich-mündlichen
Gerichtsverfahrens_, which obtained the first prize of the academy of
Munich, and in 1826 he became professor in the university of Munich. In
1829 he returned to official life, and was soon offered an important
post. In 1832, when Otto (Otho), son of Louis I., king of Bavaria, was
chosen to fill the throne of Greece, a council of regency was nominated
during his minority, and Maurer was appointed a member. He applied
himself energetically to the task of creating institutions adapted to
the requirements of a modern civilized community; but grave difficulties
soon arose and Maurer was recalled in 1834, when he returned to Munich.
This loss was a serious one for Greece. Maurer was the ablest, most
energetic and most liberal-minded member of the council, and it was
through his enlightened efforts that Greece obtained a revised penal
code, regular tribunals and an improved system of civil procedure. Soon
after his recall he published _Das griechische Volk in öffentlicher,
kirchlicher, und privatrechtlicher Beziehung vor und nach dem
Freiheitskampfe bis zum 31 Juli 1834_ (Heidelberg, 1835-1836), a useful
source of information for the history of Greece before Otto ascended the
throne, and also for the labours of the council of regency to the time
of the author's recall. After the fall of the ministry of Karl von Abel
(1788-1859) in 1847, he became chief Bavarian minister and head of the
departments of foreign affairs and of justice, but was overthrown in the
same year. He died at Munich on the 9th of May 1872. His only son,
Conrad von Maurer (1823-1902), was a Scandinavian scholar of some
repute, and like his father was a professor at the university of Munich.

  Maurer's most important contribution to history is a series of books
  on the early institutions of the Germans. These are: _Einleitung zur
  Geschichte der Mark-, Hof-, Dorf-, und Stadtverfassung und der
  öffentlichen Gewalt_ (Munich, 1854); _Geschichte der Markenverfassung
  in Deutschland_ (Erlangen, 1856); _Geschichte der Fronhöfe, der
  Bauernhöfe, und der Hofverfassung in Deutschland_ (Erlangen,
  1862-1863); _Geschichte der Dorfverfassung in Deutschland_ (Erlangen,
  1865-1866); and _Geschichte der Slädteverfassung in Deutschland_
  (Erlangen, 1869-1871). These works are still important authorities for
  the early history of the Germans. Among other works are, _Das Stadt-
  und Landrechtsbuch Ruprechts von Freising, ein Beitrag zur Geschichte
  des Schwabenspiegels_ (Stuttgart, 1839); _Über die Freipflege (plegium
  liberale), und die Entstehung der grossen und kleinen Jury in England_
  (Munich, 1848); and _Über die deutsche Reichsterritorial- und
  Rechtsgeschichte_ (1830).

  Sec K. T. von Heigel, _Denkwürdigkeiten des bayrischen Staatsrats G.
  L. von Maurer_ (Munich, 1903).




MAURETANIA, the ancient name of the north-western angle of the African
continent, and under the Roman Empire also of a large territory eastward
of that angle. The name had different significations at different times;
but before the Roman occupation, Mauretania comprised a considerable
part of the modern Morocco i.e. the northern portion bounded on the east
by Algiers. Towards the south we may suppose it bounded by the Atlas
range, and it seems to have been regarded by geographers as extending
along the coast to the Atlantic as far as the point where that chain
descends to the sea, in about 30 N. lat. (Strabo, p. 825). The
magnificent plateau in which the city of Morocco is situated seems to
have been unknown to ancient geographers, and was certainly never
included in the Roman Empire. On the other hand, the Gaetulians to the
south of the Atlas range, on the date-producing slopes towards the
Sahara, seem to have owned a precarious subjection to the kings of
Mauretania, as afterwards to the Roman government. A large part of the
country is of great natural fertility, and in ancient times produced
large quantities of corn, while the slopes of Atlas were clothed with
forests, which, besides other kinds of timber, produced the celebrated
ornamental wood called _citrum_ (Plin. _Hist. Nat._ 13-96), for tables
of which the Romans gave fabulous prices. (For physical geography, see
MOROCCO.)

  Mauretania, or Maurusia as it was called by Greek writers, signified
  the land of the Mauri, a term still retained in the modern name of
  Moors (q.v.). The origin and ethnical affinities of the race are
  uncertain; but it is probable that all the inhabitants of this
  northern tract of Africa were kindred races belonging to the great
  Berber family, possibly with an intermingled fair-skinned race from
  Europe (see Tissot, _Géographie comparée de la province romaine
  d'Afrique_, i. 400 seq.; also BERBERS). They first appear in history
  at the time of the Jugurthine War (110-106 B.C.), when Mauretania was
  under the government of Bocchus and seems to have been recognized as
  organized state (Sallust, _Jugurtha_, 19). To this Bocchus was given,
  after the war, the western part of Jugurtha's kingdom of Numidia,
  perhaps as far east as Saldae (Bougie). Sixty years later, at the time
  of the dictator Caesar, we find two Mauretanian kingdoms, one to the
  west of the river Mulucha under Bogud, and the other to the east under
  a Bocchus; as to the date or cause of the division we are ignorant.
  Both these kings took Caesar's part in the civil wars, and had their
  territory enlarged by him (Appian, B.C. 4, 54). In 25 B.C., after
  their deaths, Augustus gave the two kingdoms to Juba II. of Numidia
  (see under JUBA), with the river Ampsaga as the eastern frontier
  (Plin. 5. 22; Ptol. 4. 3. 1). Juba and his son Ptolemaeus after him
  reigned till A.D. 40, when the latter was put to death by Caligula,
  and shortly afterwards Claudius incorporated the kingdom into the
  Roman state as two provinces, viz. Mauretania Tingitana to the west
  of the Mulucha and M. Caesariensis to the east of that river, the
  latter taking its name from the city Caesarea (formerly Iol), which
  Juba had thus named and adopted as his capital. Thus the dividing line
  between the two provinces was the same as that which had originally
  separated Mauretania from Numidia (q.v.). These provinces were
  governed until the time of Diocletian by imperial procurators, and
  were occasionally united for military purposes. Under and after
  Diocletian M. Tingitana was attached administratively to the
  _dioicesis_ of Spain, with which it was in all respects closely
  connected; while M. Caesariensis was divided by making its eastern
  part into a separate government, which was called M. Sitifensis from
  the Roman colony Sitifis.

  In the two provinces of Mauretania there were at the time of Pliny a
  number of towns, including seven (possibly eight) Roman colonies in M.
  Tingitana and eleven in M. Caesariensis; others were added later.
  These were mostly military foundations, and served the purpose of
  securing civilization against the inroads of the natives, who were not
  in a condition to be used as material for town-life as in Gaul and
  Spain, but were under the immediate government of the procurators,
  retaining their own clan organization. Of these colonies the most
  important, beginning from the west, were Lixus on the Atlantic, Tingis
  (Tangier), Rusaddir (Melila, Melilla), Cartenna (Tenes), Iol or
  Caesarea (Cherchel), Icosium (Algiers), Saldae (Bougie), Igilgili
  (Jijelli) and Sitifis (Setif). All these were on the coast but the
  last, which was some distance inland. Besides these there were many
  municipia or _oppida civium romanorum_ (Plin. 5. 19 seq.), but, as has
  been made clear by French archaeologists who have explored these
  regions, Roman settlements are less frequent the farther we go west,
  and M. Tingitana has as yet yielded but scanty evidence of Roman
  civilization. On the whole Mauretania was in a flourishing condition
  down to the irruption of the Vandals in A.D. 429; in the _Notitia_
  nearly a hundred and seventy episcopal sees are enumerated here, but
  we must remember that numbers of these were mere villages.

  In 1904 the term Mauretania was revived as an official designation by
  the French government, and applied to the territory north of the lower
  Senegal under French protection (see SENEGAL).

  To the authorities quoted under AFRICA, ROMAN, may be added here
  Göbel, _Die West-küste Afrikas im Alterthum_.     (W. W. F.*)




MAURIAC, a town of central France, capital of an arrondissement in the
department of Cantal, 39 m. N.N.W. of Aurillac by rail. Pop. (1906),
2558. Mauriac, built on the slope of a volcanic hill, has a church of
the 12th century, and the buildings of an old abbey now used as public
offices and dwellings; the town owes its origin to the abbey, founded
during the 6th century. It is the seat of a sub-prefect and has a
tribunal of first instance and a communal college. There are marble
quarries in the vicinity.




MAURICE [or MAURITIUS], ST (d. c. 286), an early Christian martyr, who,
with his companions, is commemorated by the Roman Catholic Church on the
22nd of September. The oldest form of his story is found in the _Passio_
ascribed to Eucherius, bishop of Lyons, c. 450, who relates how the
"Theban" legion commanded by Mauritius was sent to north Italy to
reinforce the army of Maximinian. Maximinian wished to use them in
persecuting the Christians, but as they themselves were of this faith,
they refused, and for this, after having been twice decimated, the
legion was exterminated at Octodurum (Martigny) near Geneva. In late
versions this legend was expanded and varied, the martyrdom was
connected with a refusal to take part in a great sacrifice ordered at
Octodurum and the name of Exsuperius was added to that of Mauritius.
Gregory of Tours (c. 539-593) speaks of a company of the same legion
which suffered at Cologne.

  The _Magdeburg Centuries_, in spite of Mauritius being the patron
  saint of Magdeburg, declared the whole legend fictitious; J. A. du
  Bordien _La Légion thébéenne_ (Amsterdam, 1705); J. J. Hottinger in
  _Helvetische Kirchengeschichte_ (Zürich, 1708); and F. W. Rettberg,
  _Kirchengeschichte Deutschlands_ (Göttingen, 1845-1848) have also
  demonstrated its untrustworthiness, while the Bollandists, De Rivaz
  and Joh. Friedrich uphold it. Apart from the a priori improbability of
  a whole legion being martyred, the difficulties are that in 286
  Christians everywhere throughout the empire were not molested, that at
  no later date have we evidence of the presence of Maximinian in the
  Valais, and that none of the writers nearest to the event (Eusebius,
  Lactantius, Orosius, Sulpicius Severus) know anything of it. It is of
  course quite possible that isolated cases of officers being put to
  death for their faith occurred during Maximinian's reign, and on some
  such cases the legend may have grown up during the century and a half
  between Maximinian and Eucherius. The cult of St Maurice and the
  Theban legion is found in Switzerland (where two places bear the name
  in Valais, besides St Moritz in Grisons), along the Rhine, and in
  north Italy. The foundation of the abbey of St Maurice (Agaunum) in
  the Valais is usually ascribed to Sigismund of Burgundy (515). Relics
  of the saint are preserved here and at Brieg and Turin.




MAURICE (MAURICIUS FLAVIUS TIBERIUS) (c. 539-602), East Roman emperor
from 582 to 602, was of Roman descent, but a native of Arabissus in
Cappadocia. He spent his youth at the court of Justin II., and, having
joined the army, fought with distinction in the Persian War (578-581).
At the age of forty-three he was declared Caesar by the dying emperor
Tiberius II., who bestowed upon him the hand of his daughter
Constantina. Maurice brought the Persian War to a successful close by
the restoration of Chosroes II. to the throne (591). On the northern
frontier he at first bought off the Avars by payments which compelled
him to exercise strict economy in his general administration, but after
595 inflicted several defeats upon them through his general Crispus. By
his strict discipline and his refusal to ransom a captive corps he
provoked to mutiny the army on the Danube. The revolt spread to the
popular factions in Constantinople, and Maurice consented to abdicate.
He withdrew to Chalcedon, but was hunted down and put to death after
witnessing the slaughter of his five sons.

  The work on military art ([Greek: stratêgika]) ascribed to him is a
  contemporary work of unknown authorship (ed. Scheffer, _Arriani
  tactica et Mauricii ars militaris_, Upsala, 1664; see Max Jähns,
  _Gesch. d. Kriegswissensch._, i. 152-156).

  See Theophylactus Simocatta, _Vita Mauricii_ (ed. de Boor, 1887); E.
  Gibbon, _The Decline and Fall of the Roman Empire_ (ed. Bury, London,
  1896, v. 19-21, 57); J. B. Bury, _The Later Roman Empire_ (London,
  1889, ii. 83-94); G. Finlay, _History of Greece_ (ed. 1877, Oxford, i.
  299-306).




MAURICE (1521-1553), elector of Saxony, elder son of Henry, duke of
Saxony, belonging to the Albertine branch of the Wettin family, was born
at Freiberg on the 21st of March 1521. In January 1541 he married Agnes,
daughter of Philip, landgrave of Hesse. In that year he became duke of
Saxony by his father's death, and he continued Henry's work in
forwarding the progress of the Reformation. Duke Henry had decreed that
his lands should be divided between his two sons, but as a partition was
regarded as undesirable the whole of the duchy came to his elder son.
Maurice, however, made generous provision for his brother Augustus, and
the desire to compensate him still further was one of the minor threads
of his subsequent policy. In 1542 he assisted the emperor Charles V.
against the Turks, in 1543 against William, duke of Cleves, and in 1544
against the French; but his ambition soon took a wider range. The
harmonious relations which subsisted between the two branches of the
Wettins were disturbed by the interference of Maurice in Cleves, a
proceeding distasteful to the Saxon elector, John Frederick; and a
dispute over the bishopric of Meissen having widened the breach, war was
only averted by the mediation of Philip of Hesse and Luther. About this
time Maurice seized the idea of securing for himself the electoral
dignity held by John Frederick, and his opportunity came when Charles
was preparing to attack the league of Schmalkalden. Although educated as
a Lutheran, religious questions had never seriously appealed to Maurice.
As a youth he had joined the league of Schmalkalden, but this adhesion,
as well as his subsequent declaration to stand by the confession of
Augsburg, cannot be regarded as the decision of his maturer years. In
June 1546 he took a decided step by making a secret agreement with
Charles at Regensburg. Maurice was promised some rights over the
archbishopric of Magdeburg and the bishopric of Halberstadt; immunity,
in part at least, for his subjects from the Tridentine decrees; and the
question of transferring the electoral dignity was discussed. In return
the duke probably agreed to aid Charles in his proposed attack on the
league as soon as he could gain the consent of the Saxon estates, or at
all events to remain neutral during the impending war. The struggle
began in July 1546, and in October Maurice declared war against John
Frederick. He secured the formal consent of Charles to the transfer of
the electoral dignity and took the field in November. He had gained a
few successes when John Frederick hastened from south Germany to defend
his dominions. Maurice's ally, Albert Alcibiades, prince of Bayreuth,
was taken prisoner at Rochlitz; and the duke, driven from electoral
Saxony, was unable to prevent his own lands from being overrun.
Salvation, however, was at hand. Marching against John Frederick,
Charles V., aided by Maurice, gained a decisive victory at Mühlberg in
April 1547, after which by the capitulation of Wittenberg John Frederick
renounced the electoral dignity in favour of Maurice, who also obtained
a large part of his kinsman's lands. The formal investiture of the new
elector took place at Augsburg in February 1548.

The plans of Maurice soon took a form less agreeable to the emperor. The
continued imprisonment of his father-in-law, Philip of Hesse, whom he
had induced to surrender to Charles and whose freedom he had guaranteed,
was neither his greatest nor his only cause of complaint. The emperor
had refused to complete the humiliation of the family of John Frederick;
he had embarked upon a course of action which boded danger to the
elector's Lutheran subjects, and his increased power was a menace to the
position of Maurice. Assuring Charles of his continued loyalty, the
elector entered into negotiations with the discontented Protestant
princes. An event happened which gave him a base of operations, and
enabled him to mask his schemes against the emperor. In 1550 he had been
entrusted with the execution of the imperial ban against the city of
Magdeburg, and under cover of these operations he was able to collect
troops and to concert measures with his allies. Favourable terms were
granted to Magdeburg, which surrendered and remained in the power of
Maurice, and in January 1552 a treaty was concluded with Henry II. of
France at Chambord. Meanwhile Maurice had refused to recognize the
_Interim_ issued from Augsburg in May 1548 as binding on Saxony; but a
compromise was arranged on the basis of which the Leipzig _Interim_ was
drawn up for his lands. It is uncertain how far Charles was ignorant of
the elector's preparations, but certainly he was unprepared for the
attack made by Maurice and his allies in March 1552. Augsburg was taken,
the pass of Ehrenberg was forced, and in a few days the emperor left
Innsbruck as a fugitive. Ferdinand undertook to make peace, and the
Treaty of Passau, signed in August 1552, was the result. Maurice
obtained a general amnesty and freedom for Philip of Hesse, but was
unable to obtain a perpetual religious peace for the Lutherans. Charles
stubbornly insisted that this question must be referred to the Diet, and
Maurice was obliged to give way. He then fought against the Turks, and
renewed his communications with Henry of France. Returning from Hungary
the elector placed himself at the head of the princes who were seeking
to check the career of his former ally, Albert Alcibiades, whose
depredations were making him a curse to Germany. The rival armies met at
Sievershausen on the 9th of July 1553, where after a fierce encounter
Albert was defeated. The victor, however, was wounded during the fight
and died two days later.

Maurice was a friend to learning, and devoted some of the secularized
church property to the advancement of education. Very different
estimates have been formed of his character. He has been represented as
the saviour of German Protestantism on the one hand, and on the other as
a traitor to his faith and country. In all probability he was neither
the one nor the other, but a man of great ambition who, indifferent to
religious considerations, made good use of the exigencies of the time.
He was generous and enlightened, a good soldier and a clever
diplomatist. He left an only daughter Anna (d. 1577), who became the
second wife of William the Silent, prince of Orange.

  The elector's _Politische Korrespondenz_ has been edited by E.
  Brandenburg (Leipzig, 1900-1904); and a sketch of him is given by
  Roger Ascham in _A Report and Discourse of the Affairs and State of
  Germany_ (London, 1864-1865). See also F. A. von Langenn, _Moritz
  Herzog und Churfürst zu Sachsen_ (Leipzig, 1841); G. Voigt, _Moritz
  von Sachsen_ (Leipzig, 1876); E. Brandenburg, _Moritz von Sachsen_
  (Leipzig, 1898); S. Issleib, _Moritz von Sachsen als protestantischer
  Fürst_ (Hamburg, 1898); J. Witter, _Die Beziehung und der Verkehr des
  Kurfürsten Moritz mit König Ferdinand_ (Jena, 1886); L. von Ranke,
  _Deutsche Geschichte im Zeitalter der Reformation_, Bde. IV. and V.
  (Leipzig, 1882); and W. Maurenbrecher in the _Allgemeine deutsche
  Biographie_, Bd. XXII. (Leipzig, 1885). For bibliography see
  Maurenbrecher; and _The Cambridge Modern History_, vol. ii.
  (Cambridge, 1903).




MAURICE, JOHN FREDERICK DENISON (1805-1872), English theologian, was
born at Normanston, Suffolk, on the 29th of August, 1805. He was the son
of a Unitarian minister, and entered Trinity College, Cambridge, in
1823, though it was then impossible for any but members of the
Established Church to obtain a degree. Together with John Sterling (with
whom he founded the Apostles' Club) he migrated to Trinity Hall, whence
he obtained a first class in civil law in 1827; he then came to London,
and gave himself to literary work, writing a novel, _Eustace Conyers_,
and editing the _London Literary Chronicle_ until 1830, and also for a
short time the _Athenaeum_. At this time he was much perplexed as to his
religious opinions, and he ultimately found relief in a decision to take
a further university course and to seek Anglican orders. Entering Exeter
College, Oxford, he took a second class in classics in 1831. He was
ordained in 1834, and after a short curacy at Bubbenhall in Warwickshire
was appointed chaplain of Guy's Hospital, and became thenceforward a
sensible factor in the intellectual and social life of London. From 1839
to 1841 Maurice was editor of the _Education Magazine_. In 1840 he was
appointed professor of English history and literature in King's College,
and to this post in 1846 was added the chair of divinity. In 1845 he was
Boyle lecturer and Warburton lecturer. These chairs he held till 1853.
In that year he published _Theological Essays_, wherein were stated
opinions which savoured to the principal, Dr R. W. Jelf, and to the
council, of unsound theology in regard to eternal punishment. He had
previously been called on to clear himself from charges of heterodoxy
brought against him in the _Quarterly Review_ (1851), and had been
acquitted by a committee of inquiry. Now again he maintained with great
warmth of conviction that his views were in close accordance with
Scripture and the Anglican standards, but the council, without
specifying any distinct "heresy" and declining to submit the case to the
judgment of competent theologians, ruled otherwise, and he was deprived
of his professorships. He held at the same time the chaplaincy of
Lincoln's Inn, for which he had resigned Guy's (1846-1860), but when he
offered to resign this the benchers refused. Nor was he assailed in the
incumbency of St. Peter's, Vere Street, which he held for nine years
(1860-1869), and where he drew round him a circle of thoughtful people.
During the early years of this period he was engaged in a hot and bitter
controversy with H. L. Mansel (afterwards dean of St Paul's), arising
out of the latter's Bampton lecture upon reason and revelation.

During his residence in London Maurice was specially identified with two
important movements for education. He helped to found Queen's College
for the education of women (1848), and the Working Men's College (1854),
of which he was the first principal. He strongly advocated the abolition
of university tests (1853), and threw himself with great energy into all
that affected the social life of the people. Certain abortive attempts
at co-operation among working men, and the movement known as Christian
Socialism, were the immediate outcome of his teaching. In 1866 Maurice
was appointed professor of moral philosophy at Cambridge, and from 1870
to 1872 was incumbent of St Edward's in that city. He died on the 1st of
April 1872.

He was twice married, first to Anna Barton, a sister of John Sterling's
wife, secondly to a half-sister of his friend Archdeacon Hare. His son
Major-General Sir J. Frederick Maurice (b. 1841), became a distinguished
soldier and one of the most prominent military writers of his time.

Those who knew Maurice best were deeply impressed with the spirituality
of his character. "Whenever he woke in the night," says his wife, "he
was always praying." Charles Kingsley called him "the most beautiful
human soul whom God has ever allowed me to meet with." As regards his
intellectual attainments we may set Julius Hare's verdict "the greatest
mind since Plato" over against Ruskin's "by nature puzzle-headed and
indeed wrong-headed." Such contradictory impressions bespeak a life made
up of contradictory elements. Maurice was a man of peace, yet his life
was spent in a series of conflicts; of deep humility, yet so polemical
that he often seemed biased; of large charity, yet bitter in his attack
upon the religious press of his time; a loyal churchman who detested the
label "Broad," yet poured out criticism upon the leaders of the Church.
With an intense capacity for visualizing the unseen, and a kindly
dignity, he combined a large sense of humour. While most of the "Broad
Churchmen" were influenced by ethical and emotional considerations in
their repudiation of the dogma of everlasting torment, he was swayed by
purely intellectual and theological arguments, and in questions of a
more general liberty he often opposed the proposed Liberal theologians,
though he as often took their side if he saw them hard pressed. He had a
wide metaphysical and philosophical knowledge which he applied to the
history of theology. He was a strenuous advocate of ecclesiastical
control in elementary education, and an opponent of the new school of
higher biblical criticism, though so far an evolutionist as to believe
in growth and development as applied to the history of nations.

  As a preacher, his message was apparently simple; his two great
  convictions were the fatherhood of God, and that all religious systems
  which had any stability lasted because of a portion of truth which had
  to be disentangled from the error differentiating them from the
  doctrines of the Church of England as understood by himself. His love
  to God as his Father was a passionate adoration which filled his whole
  heart. The prophetic, even apocalyptic, note of his preaching was
  particularly impressive. He prophesied in London as Isaiah prophesied
  to the little towns of Palestine and Syria, "often with dark
  foreboding, but seeing through all unrest and convulsion the working
  out of a sure divine purpose." Both at King's College and at Cambridge
  Maurice gathered round him a band of earnest students, to whom he
  directly taught much that was valuable drawn from wide stores of his
  own reading, wide rather than deep, for he never was, strictly
  speaking, a learned man. Still more did he encourage the habit of
  inquiry and research, more valuable than his direct teaching. In his
  Socratic power of convincing his pupils of their ignorance he did more
  than perhaps any other man of his time to awaken in those who came
  under his sway the desire for knowledge and the process of independent
  thought.

  As a social reformer, Maurice was before his time, and gave his eager
  support to schemes for which the world was not ready. From an early
  period of his life in London the condition of the poor pressed upon
  him with consuming force; the enormous magnitude of the social
  questions involved was a burden which he could hardly bear. For many
  years he was the clergyman whom working men of all opinions seemed to
  trust even if their faith in other religious men and all religious
  systems had faded, and he had a marvellous power of attracting the
  zealot and the outcast.

  His works cover nearly 40 volumes, often obscure, often tautological,
  and with no great distinction of style. But their high purpose and
  philosophical outlook give his writings a permanent place in the
  history of the thought of his time. The following are the more
  important works--some of them were rewritten and in a measure recast,
  and the date given is not necessarily that of the first appearance of
  the book, but of its more complete and abiding form: _Eustace Conway,
  or the Brother and Sister_, a novel (1834); _The Kingdom of Christ_
  (1842); _Christmas Day and Other Sermons_ (1843); _The Unity of the
  New Testament_ (1844); _The Epistle to the Hebrews_ (1846); _The
  Religions of the World_ (1847); _Moral and Metaphysical Philosophy_
  (at first an article in the _Encyclopaedia Metropolitana_, 1848); _The
  Church a Family_ (1850); _The Old Testament_ (1851); _Theological
  Essays_ (1853); _The Prophets and Kings of the Old Testament_ (1853);
  _Lectures on Ecclesiastical History_ (1854); _The Doctrine of
  Sacrifice_ (1854); _The Patriarchs and Lawgivers of the Old Testament_
  (1855); _The Epistles of St John_ (1857); _The Commandments as
  Instruments of National Reformation_ (1866); _On the Gospel of St
  Luke_ (1868); _The Conscience: Lectures on Casuistry_ (1868); _The
  Lord's Prayer, a Manual_ (1870). The greater part of these works were
  first delivered as sermons or lectures. Maurice also contributed many
  prefaces and introductions to the works of friends, as to Archdeacon
  Hare's _Charges_, Kingsley's _Saint's Tragedy_, &c.

  See _Life_ by his son (2 vols., London, 1884), and a monograph by C.
  F. G. Masterman (1907) in "Leader of the Church" series; W. E. Collins
  in _Typical English Churchmen_, pp. 327-360 (1902), and T. Hughes in
  _The Friendship of Books_ (1873).




MAURICE OF NASSAU, prince of Orange (1567-1625), the second son of
William the Silent, by Anna, only daughter of the famous Maurice,
elector of Saxony, was born at Dillenburg. At the time of his father's
assassination in 1584 he was being educated at the university of Leiden,
at the expense of the states of Holland and Zeeland. Despite his youth
he was made stadtholder of those two provinces and president of the
council of state. During the period of Leicester's governorship he
remained in the background, engaged in acquiring a thorough knowledge of
the military art, and in 1586 the States of Holland conferred upon him
the title of prince. On the withdrawal of Leicester from the Netherlands
in August 1587, Johan van Oldenbarneveldt, the advocate of Holland,
became the leading statesman of the country, a position which he
retained for upwards of thirty years. He had been a devoted adherent of
William the Silent and he now used his influence to forward the
interests of Maurice. In 1588 he was appointed by the States-General
captain and admiral-general of the Union, in 1590 he was elected
stadtholder of Utrecht and Overysel, and in 1591 of Gelderland. From
this time forward, Oldenbarneveldt at the head of the civil government
and Maurice in command of the armed forces of the republic worked
together in the task of rescuing the United Netherlands from Spanish
domination (for details see HOLLAND). Maurice soon showed himself to be
a general second in skill to none of his contemporaries. He was
especially famed for his consummate knowledge of the science of sieges.
The twelve years' truce on the 9th of April 1609 brought to an end the
cordial relations between Maurice and Oldenbarneveldt. Maurice was
opposed to the truce, but the advocate's policy triumphed and
henceforward there was enmity between them. The theological disputes
between the Remonstrants and contra-Remonstrants found them on different
sides; and the theological quarrel soon became a political one.
Oldenbarneveldt, supported by the states of Holland, came forward as the
champion of provincial sovereignty against that of the states-general;
Maurice threw the weight of his sword on the side of the union. The
struggle was a short one, for the army obeyed the general who had so
often led them to victory. Oldenbarneveldt perished on the scaffold, and
the share which Maurice had in securing the illegal condemnation by a
packed court of judges of the aged patriot must ever remain a stain upon
his memory.

Maurice, who had on the death of his elder brother Philip William, in
February 1618, become prince of Orange, was now supreme in the state,
but during the remainder of his life he sorely missed the wise counsels
of the experienced Oldenbarneveldt. War broke out again in 1621, but
success had ceased to accompany him on his campaigns. His health gave
way, and he died, a prematurely aged man, at the Hague on the 4th of
April 1625. He was buried by his father's side at Delft.

  BIBLIOGRAPHY.--I. Commelin, _Wilhelm en Maurits v. Nassau, pr. v.
  Orangien, haer leven en bedrijf_ (Amsterdam, 1651); G. Groen van
  Prinsterer, _Archives ou correspondance de la maison d'Orange-Nassau_,
  1^e série, 9 vols. (Leiden, 1841-1861); G. Groen van Prinsterer,
  _Maurice et Barneveldt_ (Utrecht, 1875); J. L. Motley, _Life and Death
  of John of Barneveldt_ (2 vols., The Hague, 1894); C. M. Kemp, v.d.
  _Maurits v. Nassau, prins v. Oranje in zijn leven en verdiensten_ (4
  vols., Rotterdam, 1845); M. O. Nutting, _The Days of Prince Maurice_
  (Boston and Chicago, 1894).




MAURISTS, a congregation of French Benedictines called after St Maurus
(d. 565), a disciple of St Benedict and the legendary introducer of the
Benedictine rule and life into Gaul.[1] At the end of the 16th century
the Benedictine monasteries of France had fallen into a state of
disorganization and relaxation. In the abbey of St Vaune near Verdun a
reform was initiated by Dom Didier de la Cour, which spread to other
houses in Lorraine, and in 1604 the reformed congregation of St Vaune
was established, the most distinguished members of which were Ceillier
and Calmet. A number of French houses joined the new congregation; but
as Lorraine was still independent of the French crown, it was considered
desirable to form on the same lines a separate congregation for France.
Thus in 1621 was established the famous French congregation of St Maur.
Most of the Benedictine monasteries of France, except those belonging to
Cluny, gradually joined the new congregation, which eventually embraced
nearly two hundred houses. The chief house was Saint-Germain-des-Prés,
Paris, the residence of the superior-general and centre of the literary
activity of the congregation. The primary idea of the movement was not
the undertaking of literary and historical work, but the return to a
strict monastic régime and the faithful carrying out of Benedictine
life; and throughout the most glorious period of Maurist history the
literary work was not allowed to interfere with the due performance of
the choral office and the other duties of the monastic life. Towards the
end of the 18th century a tendency crept in, in some quarters, to relax
the monastic observances in favour of study; but the constitutions of
1770 show that a strict monastic régime was maintained until the end.
The course of Maurist history and work was checkered by the
ecclesiastical controversies that distracted the French Church during
the 17th and 18th centuries. Some of the members identified themselves
with the Jansenist cause; but the bulk, including nearly all the
greatest names, pursued a middle path, opposing the lax moral theology
condemned in 1679 by Pope Innocent XI., and adhering to those strong
views on grace and predestination associated with the Augustinian and
Thomist schools of Catholic theology; and like all the theological
faculties and schools on French soil, they were bound to teach the four
Gallican articles. It seems that towards the end of the 18th century a
rationalistic and free-thinking spirit invaded some of the houses. The
congregation was suppressed and the monks scattered at the revolution,
the last superior-general with forty of his monks dying on the scaffold
in Paris. The present French congregation of Benedictines initiated by
Dom Guéranger in 1833 is a new creation and has no continuity with the
congregation of St Maur.

The great claim of the Maurists to the gratitude and admiration of
posterity is their historical and critical school, which stands quite
alone in history, and produced an extraordinary number of colossal works
of erudition which still are of permanent value. The foundations of this
school were laid by Dom Tarisse, the first superior-general, who in 1632
issued instructions to the superiors of the monasteries to train the
young monks in the habits of research and of organized work. The
pioneers in production were Ménard and d'Achery.

  The following tables give, divided into groups, the most important
  Maurist works, along with such information as may be useful to
  students. All works are folio when not otherwise noted:--

      I.--THE EDITIONS OF THE FATHERS

    Epistle of Barnabas          Ménard               1645        1 in 4^to
      (editio princeps)
    Lanfranc                     d'Achery             1648        1
    Guibert of Nogent            d'Achery             1651        1
    Robert Pulleyn and Peter
      of Poitiers                Mathou               1655        1
    Bernard                      Mabillon             1667        2
    Anselm                       Gerberon             1675        1
    Cassiodorus                  Garet                1679        1
    Augustine (see Kukula,       Delfau, Blampin,
      _Die Mauriner-Ausgabe        Coustant, Guesnie  1681-1700  11
      des Augustinus_, 1898)
    Ambrose                      du Frische           1686-1690   2
    Acta martyrum sincera        Ruinart              1689        1
    Hilary                       Coustant             1693        1
    Jerome                       Martianay            1693-1706   5
    Athanasius                   Loppin and Mont-
                                   faucon             1698        3
    Gregory of Tours             Ruinart              1699        1
    Gregory the Great            Sainte-Marthe        1705        4
    Hildebert of Tours           Beaugendre           1708        1
    Irenaeus                     Massuet              1710        1
    Chrysostom                   Montfaucon           1718-1738  13
    Cyril of Jerusalem           Touttée and Maran    1720        1
    Epistolae romanorum          Coustant             1721        1
      pontificum[2]
    Basil                        Garnier and Maran    1721-1730   3
    Cyprian                        (Baluze, not a
                                   Maurist) finished
                                   by Maran           1726        1
    Origen                       Ch. de la Rue (1, 2,
                                  3) V. de la Rue (4) 1733-1759   4
    Justin and the Apologists    Maran                1742        1
    Gregory Nazianzen[3]         Maran and Clémencet  1778        1

      II.--BIBLICAL WORKS

    St Jerome's Latin Bible      Martianay            1693        1
    Origen's Hexapla             Montfaucon           1713        2
    Old Latin versions           Sabbathier           1743-1749   3

      III.--GREAT COLLECTIONS OF DOCUMENTS

    Spicilegium                  d'Achery             1655-1677  13 in 4^to
    Veterae analecta             Mabillon             1675-1685   4 in 8^vo
    Musaeum italicum             Mabillon             1687-1689   2 in 4^to
    Collectio nova patrum        Montfaucon           1706        2
      graecorum
    Thesaurus novus              Martène and Durand   1717        5
      anecdotorum
    Veterum scriptorum           Martène and Durand   1724-1733   9
      collectio
    De antiquis                  Martène              1690-1706
      ecclesiaeritibus           (Final form)         1736-1738   4

      IV.--MONASTIC HISTORY

    Acta of the Benedictine      d'Achery, Mabillon
      Saints                       and Ruinart        1668-1701   9
    Benedictine Annals (to       Mabillon (1-4),
      1157)                        Massuet (5),
                                   Martène (6)        1703-1739   6

      V.--ECCLESIASTICAL HISTORY AND ANTIQUITIES OF FRANCE

      A.--_General._

    Gallia Christiana (3 other   Sainte-Marthe
      vols. were published         (1, 2, 3)          1715-1785  13
      1856-1865)
    Monuments de la monarchie    Montfaucon           1729-1733   5
      française
    Histoire littéraire de la    Rivet, Clémencet,
      France (16 other vols.       Clément            1733-1763  12 in 4^to
      were published 1814-1881)
    Recueil des historiens de    Bouquet (1-8), Brial
      la France (4 other vols.    (12-19)             1738-1833  19
      were published 1840-1876)
    Concilia Galliae (the        Labbat               1789        1
      printing of vol. ii. was
      interrupted by the
      Revolution; there were
      to have been 8 vols.)

      B.--HISTORIES OF THE PROVINCES.

    Bretagne                     Lobineau             1707        2
    Paris                        Félibien and
                                   Lobineau           1725        5
    Languedoc                    Vaissette and de Vic 1730-1745   5
    Bourgogne                    Plancher (1-3),      1739-1748   4
                                   Merle (4)          1781
    Bretagne                     Morice               1742-1756   5

      VI.--MISCELLANEOUS WORKS OF TECHNICAL ERUDITION

    De re diplomatica            Mabillon             1681        1
      Ditto Supplement           Mabillon             1704        1
    Nouveau traité de            Toustain and Tassin  1750-1765   6 in 4^to
      diplomatique
    Paleographia graeca          Montfaucon           1708        1
    Bibliotheca coisliniana      Montfaucon           1715        1
    Bibliotheca bibliothecarum   Montfaucon           1739        2
      manuscriptorum nova
    L'Antiquité expliqué         Montfaucon           1719-1724  15
    New ed. of Du Cange's        Dantine and
      glossarium                   Carpentier         1733-1736   6
    Ditto Supplement             Carpentier           1766        4
    Apparatus ad bibliothecam      le Nourry          1703        2
      maximam patrum
    L'Art de vérifier les        Dantine, Durand,
      dates                        Clémencet          1750        1 in 4^to
      Ed. 2                      Clément              1770        1
      Ed. 3                      Clément              1783-1787   3

  The 58 works in the above list comprise 199 great folio volumes and 39
  in 4^to or 8^vo. The full Maurist bibliography contains the names of
  some 220 writers and more than 700 works. The lesser works in large
  measure cover the same fields as those in the list, but the number of
  works of purely religious character, of piety, devotion and
  edification, is very striking. Perhaps the most wonderful phenomenon
  of Maurist work is that what was produced was only a portion of what
  was contemplated and prepared for. The French Revolution cut short
  many gigantic undertakings, the collected materials for which fill
  hundreds of manuscript volumes in the Bibliothèque nationale of Paris
  and other libraries of France. There are at Paris 31 volumes of
  Berthereau's materials for the Historians of the Crusades, not only in
  Latin and Greek, but in the oriental tongues; from them have been
  taken in great measure the _Recueil des historiens des croisades_,
  whereof 15 folio volumes have been published by the Académie des
  Inscriptions. There exist also the preparations for an edition of
  Rufinus and one of Eusebius, and for the continuation of the Papal
  Letters and of the Concilia Galliae. Dom Caffiaux and Dom Villevielle
  left 236 volumes of materials for a _Trésor généalogique_. There are
  Benedictine Antiquities (37 vols.), a Monasticon Gallicanum and a
  Monasticon Benedictinum (54 vols.). Of the Histories of the Provinces
  of France barely half a dozen were printed, but all were in hand, and
  the collections for the others fill 800 volumes of MSS. The materials
  for a geography of Gaul and France in 50 volumes perished in a fire
  during the Revolution.

  When these figures were considered, and when one contemplates the
  vastness of the works in progress during any decade of the century
  1680-1780; and still more, when not only the quantity but the quality
  of the work, and the abiding value of most of it is realized, it will
  be recognized that the output was prodigious and unique in the history
  of letters, as coming from a single society. The qualities that have
  made Maurist work proverbial for sound learning are its fine critical
  tact and its thoroughness.

  The chief source of information on the Maurists and their work is Dom
  Tassin's _Histoire littéraire de la congregation de Saint-Maur_
  (1770); it has been reduced to a bare bibliography and completed by de
  Lama, _Bibliothèque des écrivains de la congr. de S.-M._ (1882). The
  two works of de Broglie, _Mabillon_ (2 vols., 1888) and _Montfaucon_
  (2 vols., 1891), give a charming picture of the inner life of the
  great Maurists of the earlier generation in the midst of their work
  and their friends. Sketches of the lives of a few of the chief
  Maurists will be found in McCarthy's _Principal Writers of the Congr.
  of S. M._ (1868). Useful information about their literary undertakings
  will be found in De Lisle's _Cabinet des MSS. de la Bibl. Nat. Fonds
  St Germain-des-Prés_. General information will be found in the
  standard authorities: Helyot, _Hist. des ordres religieux_ (1718), vi.
  c. 37; Heimbucher, _Orden und Kongregationen_ (1907) i. § 36; Wetzer
  und Welte, Kirchenlexicon (ed. 2) and Herzog-Hauck's
  _Realencyklopädie_ (ed. 3), the latter an interesting appreciation by
  the Protestant historian Otto Zöckler of the spirit and the merits of
  the work of the Maurists.     (E. C. B.)


FOOTNOTES:

  [1] His festival is kept on the 15th of January. He founded the
    monastery of Glanfeuil or St Maur-sur-Loire.

  [2] 14 vols. of materials collected for the continuation are at
    Paris.

  [3] The printing of vol. ii. was impeded by the Revolution.




MAURITIUS, an island and British colony in the Indian Ocean (known
whilst a French possession as the _Île de France_). It lies between 57°
18´ and 57° 49´ E., and 19° 58´ and 20° 32´ S., 550 m. E. of Madagascar,
2300 m. from the Cape of Good Hope, and 9500 m. from England via Suez.
The island is irregularly elliptical--somewhat triangular--in shape, and
is 36 m. long from N.N.E. to S.S.W., and about 23 m. broad. It is 130 m.
in circumference, and its total area is about 710 sq. m. (For map see
MADAGASCAR.) The island is surrounded by coral reefs, so that the ports
are difficult of access.

From its mountainous character Mauritius is a most picturesque island,
and its scenery is very varied and beautiful. It has been admirably
described by Bernardin de St Pierre, who lived in the island towards the
close of the 18th century, in _Paul et Virginie_. The most level
portions of the coast districts are the north and north-east, all the
rest being broken by hills, which vary from 500 to 2700 ft. in height.
The principal mountain masses are the north-western or Pouce range, in
the district of Port Louis; the south-western, in the districts of
Rivière Noire and Savanne; and the south-eastern range, in the Grand
Port district. In the first of these, which consists of one principal
ridge with several lateral spurs, overlooking Port Louis, are the
singular peak of the Pouce (2650 ft.), so called from its supposed
resemblance to the human thumb; and the still loftier Pieter Botte (2685
ft.), a tall obelisk of bare rock, crowned with a globular mass of
stone. The highest summit in the island is in the south-western mass of
hills, the Piton de la Rivière Noire, which is 2711 ft. above the sea.
The south-eastern group of hills consists of the Montagne du Bambou,
with several spurs running down to the sea. In the interior are
extensive fertile plains, some 1200 ft. in height, forming the districts
of Moka, Vacois, and Plaines Wilhelms; and from nearly the centre of the
island an abrupt peak, the Piton du Milieu de l'Île rises to a height of
1932 ft. Other prominent summits are the Trois Mamelles, the Montagne du
Corps de Garde, the Signal Mountain, near Port Louis, and the Morne
Brabant, at the south-west corner of the island.

The rivers are small, and none is navigable beyond a few hundred yards
from the sea. In the dry season little more than brooks, they become
raging torrents in the wet season. The principal stream is the Grande
Rivière, with a course of about 10 m. There is a remarkable and very
deep lake, called Grand Bassin, in the south of the island, it is
probably the extinct crater of an ancient volcano; similar lakes are the
Mare aux Vacois and the Mare aux Joncs, and there are other deep hollows
which have a like origin.

  _Geology._--The island is of volcanic origin, but has ceased to show
  signs of volcanic activity. All the rocks are of basalt and
  greyish-tinted lavas, excepting some beds of upraised coral. Columnar
  basalt is seen in several places. The remains of ancient craters can
  be distinguished, but their outlines have been greatly destroyed by
  denudation. There are many caverns and steep ravines, and from the
  character of the rocks the ascents are rugged and precipitous. The
  island has few minerals, although iron, lead and copper in very small
  quantities have in former times been obtained. The greater part of the
  surface is composed of a volcanic breccia, with here and there
  lava-streams exposed in ravines, and sometimes on the surface. The
  commonest lavas are dolerites. In at least two places sedimentary
  rocks are found at considerable elevations. In the Black River
  Mountains, at a height of about 1200 ft., there is a clay-slate; and
  near Midlands, in the Grand Port group of mountains, a chloritic
  schist occurs about 1700 ft. above the sea, forming the hill of La
  Selle. This schist is much contorted, but seems to have a general dip
  to the south or south-east. Evidence of recent elevation of the island
  is furnished by masses of coral reef and beach coral rock standing at
  heights of 40 ft. above sea-level in the south, 12 ft. in the north
  and 7 ft. on the islands situated on the bank extending to the
  north-east.[1]

  _Climate._--The climate is pleasant during the cool season of the
  year, but oppressively hot in summer (December to April), except in
  the elevated plains of the interior, where the thermometer ranges from
  70° to 80° F., while in Port Louis and on the coast generally it
  ranges from 90° to 96°. The mean temperature for the year at Port
  Louis is 78.6°. There are two seasons, the cool and comparatively dry
  season, from April to November, and the hotter season, during the rest
  of the year. The climate is now less healthy than it was, severe
  epidemics of malarial fever having frequently occurred, so that
  malaria now appears to be endemic among the non-European population.
  The rainfall varies greatly in different parts of the island. Cluny in
  the Grand Port (south-eastern) district has a mean annual rainfall of
  145 in.; Albion on the west coast is the driest station, with a mean
  annual rainfall of 31 in. The mean monthly rainfall for the whole
  island varies from 12 in. in March to 2.6 in. in September and
  October. The Royal Alfred Observatory is situated at Pamplemousses, on
  the north-west or dry side of the island. From January to the middle
  of April, Mauritius, in common with the neighbouring islands and the
  surrounding ocean from 8° to 30° of southern latitude is subject to
  severe cyclones, accompanied by torrents of rain, which often cause
  great destruction to houses and plantations. These hurricanes
  generally last about eight hours, but they appear to be less frequent
  and violent than in former times, owing, it is thought, to the
  destruction of the ancient forests and the consequent drier condition
  of the atmosphere.

  _Fauna and Flora._--Mauritius being an oceanic island of small size,
  its present fauna is very limited in extent. When first seen by
  Europeans it contained no mammals except a large fruit-eating bat
  (_Pteropus vulgaris_), which is plentiful in the woods; but several
  mammals have been introduced, and are now numerous in the uncultivated
  region. Among these are two monkeys of the genera _Macacus_ and
  _Cercopithecus_, a stag (_Cervus hippelaphus_), a small hare, a
  shrew-mouse, and the ubiquitous rat. A lemur and one of the curious
  hedgehog-like _Insectivora_ of Madagascar (_Centetes ecaudatus_) have
  probably both been brought from the larger island. The avifauna
  resembles that of Madagascar; there are species of a peculiar genus of
  caterpillar shrikes (_Campephagidae_), as well as of the genera
  _Pratincola_, _Hypsipetes_, _Phedina_, _Tchitrea_, _Zosterops_,
  _Foudia_, _Collocalia_ and _Coracopsis_, and peculiar forms of doves
  and parakeets. The living reptiles are small and few in number. The
  surrounding seas contain great numbers of fish; the coral reefs abound
  with a great variety of molluscs; and there are numerous land-shells.
  The extinct fauna of Mauritius has considerable interest. In common
  with the other Mascarene islands, it was the home of the dodo (_Didus
  ineptus_); there were also _Aphanapteryx_, a species of rail, and a
  short-winged heron (_Ardea megacephala_), which probably seldom flew.
  The defenceless condition of these birds led to their extinction after
  the island was colonized. Considerable quantities of the bones of the
  dodo and other extinct birds--a rail (_Aphanapteryx_), and a
  short-winged heron--have been discovered in the beds of some of the
  ancient lakes (see DODO). Several species of large fossil tortoises
  have also been discovered; they are quite different from the living
  ones of Aldabra, in the same zoological region.

  Owing to the destruction of the primeval forests for the formation of
  sugar plantations, the indigenous flora is only seen in parts of the
  interior plains, in the river valleys and on the hills; and it is not
  now easy to distinguish between what is native and what has come from
  abroad. The principal timber tree is the ebony (_Diospyros ebeneum_),
  which grows to a considerable size. Besides this there are bois de
  cannelle, olive-tree, benzoin (_Croton Benzoe_), colophane
  (_Colophonia_), and iron-wood, all of which arc useful in carpentry;
  the coco-nut palm, an importation, but a tree which has been so
  extensively planted during the last hundred years that it is extremely
  plentiful; the palmiste (_Palma dactylifera latifolia_), the latanier
  (_Corypha umbraculifera_) and the date-palm. The vacoa or vacois,
  (_Pandanus utilis_) is largely grown, the long tough leaves being
  manufactured into bags for the export of sugar, and the roots being
  also made of use; and in the few remnants of the original forests the
  traveller's tree (_Urania speciosa_), grows abundantly. A species of
  bamboo is very plentiful in the river valleys and in marshy
  situations. A large variety of fruit is produced, including the
  tamarind, mango, banana, pine-apple, guava, shaddock, fig,
  avocado-pear, litchi, custard-apple and the mabolo (_Diospyros
  discolor_), a fruit of exquisite flavour, but very disagreeable odour.
  Many of the roots and vegetables of Europe have been introduced, as
  well as some of those peculiar to the tropics, including maize,
  millet, yams, manioc, dhol, gram, &c. Small quantities of tea, rice
  and sago, have been grown, as well as many of the spices (cloves,
  nutmeg, ginger, pepper and allspice), and also cotton, indigo, betel,
  camphor, turmeric and vanilla. The Royal Botanical Gardens at
  Pamplemousses, which date from the French occupation of the island,
  contain a rich collection of tropical and extra-tropical species.

_Inhabitants._--The inhabitants consist of two great divisions, those of
European blood, chiefly French and British, together with numerous
half-caste people, and those of Asiatic or African blood. The population
of European blood, which calls itself Creole, is greater than that of
any other tropical colony; many of the inhabitants trace their descent
from ancient French families, and the higher and middle classes are
distinguished for their intellectual culture. French is more commonly
spoken than English. The Creole class is, however, diminishing, though
slowly, and the most numerous section of the population is of Indian
blood.

  The introduction of Indian coolies to work the sugar plantations dates
  from the period of the emancipation of the slaves in 1834-1839. At
  that time the negroes who showed great unwillingness to work on their
  late masters' estates, numbered about 66,000. Immigration from India
  began in 1834, and at a census taken in 1846, when the total
  population was 158,462, there were already 56,245 Indians in the
  island. In 1851 the total population had increased to 180,823, while
  in 1861 it was 310,050. This great increase was almost entirely due to
  Indian immigration, the Indian population, 77,996 in 1851, being
  192,634 in 1861. From that year the increase in the Indian population
  has been more gradual but steady, while the non-Indian population has
  decreased. From 102,827 in 1851 it rose to 117,416 in 1861 to sink to
  99,784 in 1871. The figures for the three following census years
  were:--

                 1881.      1891.      1901.

    Indians    248,993    255,920    259,086
    Others     110,881    114,668    111,937
               -------    -------    -------
      Total    359,874    370,588    371,023
               -------    -------    -------

  Including the military and crews of ships in harbour, the total
  population in 1901 was 373,336.[2] This total included 198,958
  Indo-Mauritians, i.e. persons of Indian descent born in Mauritius, and
  62,022 other Indians. There were 3,509 Chinese, while the remaining
  108,847 included persons of European, African or mixed descent,
  Malagasy, Malays and Sinhalese. The Indian female population increased
  from 51,019 in 1861 to 115,986 in 1901. In the same period the
  non-Indian female population but slightly varied, being 56,070 in 1861
  and 55,485 in 1901. The Indo-Mauritians are now dominant in
  commercial, agricultural and domestic callings, and much town and
  agricultural land has been transferred from the Creole planters to
  Indians and Chinese. The tendency to an Indian peasant proprietorship
  is marked. Since 1864 real property to the value of over £1,250,000
  has been acquired by Asiatics. Between 1881 and 1901 the number of
  sugar estates decreased from 171 to 115, those sold being held in
  small parcels by Indians. The average death-rate for the period
  1873-1901 was 32.6 per 1000. The average birth-rate in the Indian
  community is 37 per 1000; in the non-Indian community 34 per 1000.
  Many Mauritian Creoles have emigrated to South Africa. The great
  increase in the population since 1851 has made Mauritius one of the
  most densely peopled regions of the world, having over 520 persons per
  square mile.

  _Chief Towns._--The capital and seat of government, the city of Port
  Louis, is on the north-western side of the island, in 20° 10´ S., 57°
  30´ E. at the head of an excellent harbour, a deep inlet about a mile
  long, available for ships of the deepest draught. This is protected by
  Fort William and Fort George, as well as by the citadel (Fort
  Adelaide), and it has three graving-docks connected with the inner
  harbour, the depths alongside quays and berths being from 12 to 28 ft.
  The trade of the island passes almost entirely through the port.
  Government House is a three-storeyed structure with broad verandas,
  of no particular style of architecture, while the Protestant cathedral
  was formerly a powder magazine, to which a tower and spire have been
  added. The Roman Catholic cathedral is more pretentious in style, but
  is tawdry in its interior. There are, besides the town-hall, Royal
  College, public offices and theatre, large barracks and military
  stores. Port Louis, which is governed by an elective municipal
  council, is surrounded by lofty hills and its unhealthy situation is
  aggravated by the difficulty of effective drainage owing to the small
  amount of tide in the harbour. Though much has been done to make the
  town sanitary, including the provision of a good water-supply, the
  death-rate is generally over 44 per 1000. Consequently all those who
  can make their homes in the cooler uplands of the interior. As a
  result the population of the city decreased from about 70,000 in 1891
  to 53,000 in 1901. The favourite residential town is Curepipe, where
  the climate resembles that of the south of France. It is built on the
  central plateau about 20 m. distant from Port Louis by rail and 1800
  ft. above the sea. Curepipe was incorporated in 1888 and had a
  population (1901) of 13,000. On the railway between Port Louis and
  Curepipe are other residential towns--Beau Bassin, Rose Hill and
  Quatre Bornes. Mahébourg, pop. (1901), 4810, is a town on the shores
  of Grand Port on the south-east side of the island, Souillac a small
  town on the south coast.

  _Industries.--The Sugar Plantations:_ The soil of the island is of
  considerable fertility; it is a ferruginous red clay, but so largely
  mingled with stones of all sizes that no plough can be used, and the
  hoe has to be employed to prepare the ground for cultivation. The
  greater portion of the plains is now a vast sugar plantation. The
  bright green of the sugar fields is a striking feature in a view of
  Mauritius from the sea, and gives a peculiar beauty and freshness to
  the prospect. The soil is suitable for the cultivation of almost all
  kinds of tropical produce, and it is to be regretted that the
  prosperity of the colony depends almost entirely on one article of
  production, for the consequences are serious when there is a failure,
  more or less, of the sugar crop. Guano is extensively imported as a
  manure, and by its use the natural fertility of the soil has been
  increased to a wonderful extent. Since the beginning of the 20th
  century some attention has been paid to the cultivation of tea and
  cotton, with encouraging results. Of the exports, sugar amounts on an
  average to about 95% of the total. The quantity of sugar exported rose
  from 102,000 tons in 1854 to 189,164 tons in 1877. The competition of
  beet-sugar and the effect of bounties granted by various countries
  then began to tell on the production in Mauritius, the average crop
  for the seven years ending 1900-1901 being only 150,449 tons. The
  Brussels Sugar Convention of 1902 led to an increase in production,
  the average annual weight of sugar exported for the three years
  1904-1906 being 182,000 tons. The value of the crop was likewise
  seriously affected by the causes mentioned, and by various diseases
  which attacked the canes. Thus in 1878 the value of the sugar exported
  was £3,408,000; in 1888 it had sunk to £1,911,000, and in 1898 to
  £1,632,000. In 1900 the value was £1,922,000, and in 1905 it had risen
  to £2,172,000. India and the South African colonies between them take
  some two-thirds of the total produce. The remainder is taken chiefly
  by Great Britain, Canada and Hong-Kong. Next to sugar, aloe-fibre is
  the most important export, the average annual export for the five
  years ending 1906 being 1840 tons. In addition, a considerable
  quantity of molasses and smaller quantities of rum, vanilla and
  coco-nut oil are exported. The imports are mainly rice, wheat, cotton
  goods, wine, coal, hardware and haberdashery, and guano. The rice
  comes principally from India and Madagascar; cattle are imported from
  Madagascar, sheep from South Africa and Australia, and frozen meat
  from Australia. The average annual value of the exports for the ten
  years 1896-1905 was £2,153,159; the average annual value of the
  imports for the same period £1,453,089. These figures when compared
  with those in years before the beet and bounty-fed sugar had entered
  into severe competition with cane sugar, show how greatly the island
  had thereby suffered. In 1864 the exports were valued at £2,249,000;
  in 1868 at £2,339,000; in 1877 at £4,201,000 and in 1880 at
  £3,634,000. And in each of the years named the imports exceeded
  £2,000,000 in value. Nearly all the aloe-fibre exported is taken by
  Great Britain, and France, while the molasses goes to India. Among the
  minor exports is that of _bambara_ or sea-slugs, which are sent to
  Hong-Kong and Singapore. This industry is chiefly in Chinese hands.
  The great majority of the imports are from Great Britain or British
  possessions.

  The currency of Mauritius is rupees and cents of a rupee, the Indian
  rupee (= 16d.) being the standard unit. The metric system of weights
  and measures has been in force since 1878.

  _Communications._--There is a regular fortnightly steamship service
  between Marseilles and Port Louis by the Messageries Maritimes, a
  four-weekly service with Southampton via Cape Town by the Union
  Castle, and a four-weekly service with Colombo direct by the British
  India Co.'s boats. There is also frequent communication with
  Madagascar, Réunion and Natal. The average annual tonnage of ships
  entering Port Louis is about 750,000 of which five-sevenths is
  British. Cable communication with Europe, via the Seychelles, Zanzibar
  and Aden, was established in 1893, and the Mauritius section of the
  Cape-Australian cable, via Rodriguez, was completed in 1902.

  Railways connect all the principal places and sugar estates on the
  island, that known as the Midland line, 36 miles long, beginning at
  Port Louis crosses the island to Mahébourg, passing through Curepipe,
  where it is 1822 ft. above the sea. There are in all over 120 miles of
  railway, all owned and worked by the government. The first railway was
  opened in 1864. The roads are well kept and there is an extensive
  system of tramways for bringing produce from the sugar estates to the
  railway lines. Traction engines are also largely used. There is a
  complete telegraphic and telephonic service.

_Government and Revenue._--Mauritius is a crown colony. The governor is
assisted by an executive council of five official and two elected
members, and a legislative council of 27 members, 8 sitting _ex
officio_, 9 being nominated by the governor and 10 elected on a moderate
franchise. Two of the elected members represent St Louis, the 8 rural
districts into which the island is divided electing each one member. At
least one-third of the nominated members must be persons not holding any
public office. The number of registered electors in 1908 was 6186. The
legislative session usually lasts from April to December. Members may
speak either in French or English. The average annual revenue of the
colony for the ten years 1896-1905, was £608,245, the average annual
expenditure during the same period £663,606. Up to 1854 there was a
surplus in hand, but since that time expenditure has on many occasions
exceeded income, and the public debt in 1908 was £1,305,000, mainly
incurred however on reproductive works.

The island has largely retained the old French laws, the _codes civil_,
_de procédure_, _du commerce_, and _d'instruction criminelle_ being
still in force, except so far as altered by colonial ordinances. A
supreme court of civil and criminal justice was established in 1831
under a chief judge and three puisne judges.

  _Religion and Education._--The majority of the European inhabitants
  belong to the Roman Catholic faith. They numbered at the 1901 census
  117,102, and the Protestants 6644. Anglicans, Roman Catholics and the
  Church of Scotland are helped by state grants. At the head of the
  Anglican community is the bishop of Mauritius; the chief Romanist
  dignitary is styled bishop of Port Louis. The Mahommedans number over
  30,000, but the majority of the Indian coolies are Hindus.

  The educational system, as brought into force in 1900, is under a
  director of public instruction assisted by an advisory committee, and
  consists of two branches (1) superior or secondary instruction, (2)
  primary instruction. For primary instruction there are government
  schools and schools maintained by the Roman Catholics, Protestants and
  other faiths, to which the government gives grants in aid. In 1908
  there were 67 government schools with 8400 scholars and 90 grant
  schools with 10,200 scholars, besides Hindu schools receiving no
  grant. The Roman Catholic scholars number 67.72%; the Protestants
  3.80%; Mahommedans 8.37%; and Hindus and others 20.11%. Secondary and
  higher education is given in the Royal College and associated schools
  at Port Louis and Curepipe.

  _Defence._--Mauritius occupies an important strategic position on the
  route between South Africa and India and in relation to Madagascar and
  East Africa, while in Port Louis it possesses one of the finest
  harbours in the Indian Ocean. A permanent garrison of some 3000 men is
  maintained in the island at a cost of about £180,000 per annum. To the
  cost of the troops Mauritius contributes 5½% of its annual
  revenue--about £30,000.

_History._--Mauritius appears to have been unknown to European nations,
if not to all other peoples, until the year 1505, when it was discovered
by Mascarenhas, a Portuguese navigator. It had then no inhabitants, and
there seem to be no traces of a previous occupation by any people. The
island was retained for most of the 16th century by its discoverers, but
they made no settlements in it. In 1598 the Dutch took possession, and
named the island "Mauritius," in honour of their stadtholder, Count
Maurice of Nassau. It had been previously called by the Portuguese "Ilha
do Cerné," from the belief that it was the island so named by Pliny. But
though the Dutch built a fort at Grand Port and introduced a number of
slaves and convicts, they made no permanent settlement in Mauritius,
finally abandoning the island in 1710. From 1715 to 1767 (when the
French government assumed direct control) the island was held by agents
of the French East India Company, by whom its name was again changed to
"Île de France." The Company was fortunate in having several able men as
governors of its colony, especially the celebrated Mahé de Labourdonnais
(q.v.), who made sugar planting the main industry of the
inhabitants.[3] Under his direction roads were made, forts built, and
considerable portions of the forest were cleared, and the present
capital, Port Louis, was founded. Labourdonnais also promoted the
planting of cotton and indigo, and is remembered as the most enlightened
and best of all the French governors. He also put down the maroons or
runaway slaves who had long been the pest of the island. The colony
continued to rise in value during the time it was held by the French
crown, and to one of the intendants,[4] Pierre Poivre, was due the
introduction of the clove, nutmeg and other spices. Another governor was
D'Entrecasteaux, whose name is kept in remembrance by a group of islands
east of New Guinea.

During the long war between France and England, at the commencement of
the 19th century, Mauritius was a continual source of much mischief to
English Indiamen and other merchant vessels; and at length the British
government determined upon an expedition for its capture. This was
effected in 1810; and upon the restoration of peace in 1814 the
possession of the island was confirmed to Britain by the Treaty of
Paris. By the eighth article of capitulation it was agreed that the
inhabitants should retain their own laws, customs, and religion; and
thus the island is still largely French in language, habits, and
predilections; but its name has again been changed to that given by the
Dutch. One of the most distinguished of the British governors was Sir
Robert Farquhar (1810-1823), who did much to abolish the Malagasy slave
trade and to establish friendly relations with the rising power of the
Hova sovereign of Madagascar. Later governors of note were Sir Henry
Barkly (1863-1871), and Sir J. Pope Hennessy (1883-1886 and 1888).

The history of the colony since its acquisition by Great Britain has
been one of social and political evolution. At first all power was
concentrated in the hands of the governor, but in 1832 a legislative
council was constituted on which non-official nominated members served.
In 1884-1885 this council was transformed into a partly elected body. Of
more importance than the constitutional changes were the economic
results which followed the freeing of the slaves (1834-1839)--for the
loss of whose labour the planters received over £2,000,000 compensation.
Coolies were introduced to supply the place of the negroes, immigration
being definitely sanctioned by the government of India in 1842. Though
under government control the system of coolie labour led to many abuses.
A royal commission investigated the matter in 1871 and since that time
the evils which were attendant on the system have been gradually
remedied. One result of the introduction of free labour has been to
reduce the descendants of the slave population to a small and
unimportant class--Mauritius in this respect offering a striking
contrast to the British colonies in the West Indies. The last half of
the 19th century was, however, chiefly notable in Mauritius for the
number of calamities which overtook the island. In 1854 cholera caused
the death of 17,000 persons; in 1867 over 30,000 people died of malarial
fever; in 1892 a hurricane of terrific violence caused immense
destruction of property and serious loss of life; in 1893 a great part
of Port Louis was destroyed by fire. There were in addition several
epidemics of small-pox and plague, and from about 1880 onward the
continual decline in the price of sugar seriously affected the
islanders, especially the Creole population. During 1902-1905 an
outbreak of surra, which caused great mortality among draught animals,
further tried the sugar planters and necessitated government help.
Notwithstanding all these calamities the Mauritians, especially the
Indo-Mauritians, have succeeded in maintaining the position of the
colony as an important sugar-producing country.

  _Dependencies._--Dependent upon Mauritius and forming part of the
  colony are a number of small islands scattered over a large extent of
  the Indian Ocean. Of these the chief is Rodriguez (q.v.), 375 m. east
  of Mauritius. Considerably north-east of Rodriguez lie the Oil Islands
  or Chagos archipelago, of which the chief is Diego Garcia (see
  CHAGOS). The Cargados, Carayos or St Brandon islets, deeps and shoals,
  lie at the south end of the Nazareth Bank about 250 m. N.N.E. of
  Mauritius. Until 1903 the Seychelles, Amirantes, Aldabra and other
  islands lying north of Madagascar were also part of the colony of
  Mauritius. In the year named they were formed into a separate colony
  (see SEYCHELLES). Two islands, Farquhar and Coetivy, though
  geographically within the Seychelles area, remained dependent on
  Mauritius, being owned by residents in that island. In 1908, however,
  Coetivy was transferred to the Seychelles administration. Amsterdam
  and St Paul, uninhabited islands in the South Indian Ocean, included
  in an official list of the dependencies of Mauritius drawn up in 1880,
  were in 1893 annexed by France. The total population of the
  dependencies of Mauritius was estimated in 1905 at 5400.

  AUTHORITIES.--F. Leguat, _Voyages et aventures en deux isles désertes
  des Indes orientales_ (Eng. trans., _A New Voyage to the East Indies_;
  London, 1708); Prudham, "England's Colonial Empire," vol. i., _The
  Mauritius and its Dependencies_ (1846); C. P. Lucas, _A Historical
  Geography of the British Colonies_, vol. i. (Oxford, 1888); Ch. Grant,
  _History of Mauritius, or the Isle of France and Neighbouring Islands_
  (1801); J. Milbert, _Voyage pittoresque à l'Île-de-France, &c._, 4
  vols. (1812); Aug. Billiard, _Voyage aux colonies orientales_ (1822);
  P. Beaton, _Creoles and Coolies, or Five Years in Mauritius_ (1859);
  Paul Chasteau, _Histoire et description de l'île Maurice_ (1860); F.
  P. Flemyng, _Mauritius, or the Isle of France_ (1862); Ch. J. Boyle,
  _Far Away, or Sketches of Scenery and Society in Mauritius_ (1867); L.
  Simonin, _Les Pays lointains, notes de voyage (Maurice, &c.)_ (1867);
  N. Pike, _Sub-Tropical Rambles in the Land of the Aphanapteryx_
  (1873); A. R. Wallace. "The Mascarene Islands," in ch. xi. vol. i. of
  _The Geographical Distribution of Animals_ (1876); K. Möbius, F.
  Richter and E. von Martens, _Beiträge zur Meeresfauna der Insel
  Mauritius und der Seychellen_ (Berlin, 1880); G. Clark, _A Brief
  Notice of the Fauna of Mauritius_ (1881); A. d'Épinay, _Renseignements
  pour servir à l'histoire de l'Île de France jusqu'à 1810_ (Mauritius,
  1890); N. Decotter, _Geography of Mauritius and its Dependencies_
  (Mauritius, 1892); H. de Haga Haig, "The Physical Features and Geology
  of Mauritius" in vol. li., _Q. J. Geol. Soc._ (1895); the Annual
  Reports on Mauritius issued by the Colonial Office, London; _The
  Mauritius Almanack_ published yearly at Port Louis. A map of the
  island in six sheets on the scale of one inch to a mile was issued by
  the War Office in 1905.     (J. Si.*)


FOOTNOTES:

  [1] See _Geog. Journ._ (June 1895), p. 597.

  [2] The total population of the colony (including dependencies) on
    the 1st of January 1907 was estimated at 383,206.

  [3] Labourdonnais is credited by several writers with the
    introduction of the sugar cane into the island. Leguat, however,
    mentions it as being cultivated during the Dutch occupation.

  [4] The régime introduced in 1767 divided the administration between
    a governor, primarily charged with military matters, and an
    intendant.




MAURY, JEAN SIFFREIN (1746-1817), French cardinal and archbishop of
Paris, the son of a poor cobbler, was born on the 26th of June 1746 at
Valréas in the Comtat-Venaissin, the district in France which belonged
to the pope. His acuteness was observed by the priests of the seminary
at Avignon, where he was educated and took orders. He tried his fortune
by writing _éloges_ of famous persons, then a favourite practice; and in
1771 his _éloge_ on Fénelon was pronounced next best to Laharpe's by the
Academy. The real foundation of his fortunes was the success of a
panegyric on St Louis delivered before the Academy in 1772, which caused
him to be recommended for an abbacy. In 1777 he published under the
title of _Discours choisis_ his panegyrics on Saint Louis, Saint
Augustine and Fénelon, his remarks on Bossuet and his _Essai sur
l'éloquence de la chaire_, a volume which contains much good criticism,
and remains a French classic. The book was often reprinted as _Principes
de l'éloquence_. He became a favourite preacher in Paris, and was Lent
preacher at court in 1781, when King Louis XVI. said of his sermon: "If
the abbé had only said a few words on religion he would have discussed
every possible subject." In 1781 he obtained the rich priory of Lyons,
near Péronne, and in 1785 he was elected to the Academy, as successor of
Lefranc de Pompignan. His morals were as loose as those of his great
rival Mirabeau, but he was famed in Paris for his wit and gaiety. In
1789 he was elected a member of the states-general by the clergy of the
bailliage of Péronne, and from the first proved to be the most able and
persevering defender of the _ancien régime_, although he had drawn up
the greater part of the _cahier_ of the clergy of Péronne, which
contained a considerable programme of reform. It is said that he
attempted to emigrate both in July and in October 1789; but after that
time he held firmly to his place, when almost universally deserted by
his friends. In the Constituent Assembly he took an active part in every
important debate, combating with especial vigour the alienation of the
property of the clergy. His life was often in danger, but his ready wit
always saved it, and it was said that one _bon mot_ would preserve him
for a month. When he did emigrate in 1792 he found himself regarded as
a martyr to the church and the king, and was at once named archbishop
_in partibus_, and extra nuncio to the diet at Frankfort, and in 1794
cardinal. He was finally made bishop of Montefiascone, and settled down
in that little Italian town--but not for long, for in 1798 the French
drove him from his retreat, and he sought refuge in Venice and St
Petersburg. Next year he returned to Rome as ambassador of the exiled
Louis XVIII. at the papal court. In 1804 he began to prepare his return
to France by a well-turned letter to Napoleon, congratulating him on
restoring religion to France once more. In 1806 he did return; in 1807
he was again received into the Academy; and in 1810, on the refusal of
Cardinal Fesch, was made archbishop of Paris. He was presently ordered
by the pope to surrender his functions as archbishop of Paris. This he
refused to do. On the restoration of the Bourbons he was summarily
expelled from the Academy and from the archiepiscopal palace. He retired
to Rome, where he was imprisoned in the castle of St Angelo for six
months for his disobedience to the papal orders, and died in 1817, a
year or two after his release, of disease contracted in prison and of
chagrin. As a critic he was a very able writer, and Sainte-Beuve gives
him the credit of discovering Father Jacques Bridayne, and of giving
Bossuet his rightful place as a preacher above Massillon; as a
politician, his wit and eloquence make him a worthy rival of Mirabeau.
He sacrificed too much to personal ambition, yet it would have been a
graceful act if Louis XVIII. had remembered the courageous supporter of
Louis XVI., and the pope the one intrepid defender of the Church in the
states-general.

  The _Oeuvres choisies du Cardinal Maury_ (5 vols., 1827) contain what
  is worth preserving. Mgr Ricard has published Maury's _Correspondance
  diplomatique_ (2 vols., Lille, 1891). For his life and character see
  _Vie du Cardinal Maury_, by Louis Siffrein Maury, his nephew (1828);
  J. J. F. Poujoulat, _Cardinal Maury, sa vie et ses oeuvres_ (1855);
  Sainte-Beuve, _Causeries du lundi_ (vol. iv.); Mgr Ricard, _L'Abbé
  Maury_ (1746-1791), _L'Abbé Maury avant 1789, L'Abbé Maury et
  Mirabeau_ (1887); G. Bonet-Maury, _Le Cardinal Maury d'après ses
  mémoires et sa correspondance inédits_ (Paris, 1892); A. Aulard, _Les
  Orateurs de la constituante_ (Paris, 1882). Of the many libels written
  against him during the Revolution the most noteworthy are the _Petit
  carême de l'abbé Maury_, with a supplement called the _Seconde année_
  (1790), and the _Vie privée de l'abbé Maury_ (1790), claimed by J. R.
  Hébert, but attributed by some writers to Restif de la Bretonne. For
  further bibliographical details see J. M. Quérard, _La France
  littéraire_, vol. v. (1833).




MAURY, LOUIS FERDINAND ALFRED (1817-1892), French scholar, was born at
Meaux on the 23rd of March 1817. In 1836, having completed his
education, he entered the Bibliothèque Nationale, and afterwards the
Bibliothèque de l'Institut (1844), where he devoted himself to the study
of archaeology, ancient and modern languages, medicine and law. Gifted
with a great capacity for work, a remarkable memory and an unbiassed and
critical mind, he produced without great effort a number of learned
pamphlets and books on the most varied subjects. He rendered great
service to the Académie des Inscriptions et Belles Lettres, of which he
had been elected a member in 1857. Napoleon III. employed him in
research work connected with the _Histoire de César_, and he was
rewarded, proportionately to his active, if modest, part in this work,
with the positions of librarian of the Tuileries (1860), professor at
the College of France (1862) and director-general of the Archives
(1868). It was not, however, to the imperial favour that he owed these
high positions. He used his influence for the advancement of science and
higher education, and with Victor Duruy was one of the founders of the
École des Hautes Études. He died at Paris four years after his
retirement from the last post, on the 11th of February 1892.

  BIBLIOGRAPHY.--His works are numerous: _Les Fées au moyen âge_ and
  _Histoire des légendes pieuses au moyen âge_; two books filled with
  ingenious ideas, which were published in 1843, and reprinted after the
  death of the author, with numerous additions under the title
  _Croyances et légendes du moyen âge_ (1896); _Histoire des grandes
  forêts de la Gaule et de l'ancienne France_ (1850, a 3rd ed. revised
  appeared in 1867 under the title _Les Forêts de la Gaule et de
  l'ancienne France); La Terre et l'homme_, a general historical sketch
  of geology, geography and ethnology, being the introduction to the
  _Histoire universelle_, by Victor Duruy (1854); _Histoire des
  religions de la_ _Grèce antique_, (3 vols., 1857-1859); _La Magie et
  l'astrologie dans l'antiquité et dans le moyen âge_ (1863); _Histoire
  de l'ancienne académie des sciences_ (1864); _Histoire de l'Académie
  des Inscriptions et Belles Lettres_ (1865); a learned paper on the
  reports of French archaeology, written on the occasion of the
  universal exhibition (1867); a number of articles in the _Encyclopédie
  moderne_ (1846-1851), in Michaud's _Biographie universelle_ (1858 and
  seq.), in the _Journal des savants_ in the _Revue des deux mondes_
  (1873, 1877, 1879-1880, &c.). A detailed bibliography of his works has
  been placed by Auguste Longnon at the beginning of the volume _Les
  Croyances et légendes du moyen âge_.




MAURY, MATTHEW FONTAINE (1806-1873), American naval officer and
hydrographer, was born near Fredericksburg in Spottsylvania county,
Virginia, on the 24th of January 1806. He was educated at Harpeth
academy, and in 1825 entered the navy as midshipman, circumnavigating
the globe in the "Vincennes," during a cruise of four years (1826-1830).
In 1831 he was appointed master of the sloop "Falmouth" on the Pacific
station, and subsequently served in other vessels before returning home
in 1834, when he married his cousin, Ann Herndon. In 1835-1836 he was
actively engaged in producing for publication a treatise on navigation,
a remarkable achievement at so early a stage in his career; he was at
this time made lieutenant, and gazetted astronomer to a South Sea
exploring expedition, but resigned this position and was appointed to
the survey of southern harbours. In 1839 he met with an accident which
resulted in permanent lameness, and unfitted him for active service. In
the same year, however, he began to write a series of articles on naval
reform and other subjects, under the title of _Scraps from the
Lucky-Bag_, which attracted much attention; and in 1841 he was placed in
charge of the Dépôt of Charts and Instruments, out of which grew the
United States Naval Observatory and the Hydrographie Office. He laboured
assiduously to obtain observations as to the winds and currents by
distributing to captains of vessels specially prepared log-books; and in
the course of nine years he had collected a sufficient number of logs to
make two hundred manuscript volumes, each with about two thousand five
hundred days' observations. One result was to show the necessity for
combined action on the part of maritime nations in regard to ocean
meteorology. This led to an international conference at Brussels in
1853, which produced the greatest benefit to navigation as well as
indirectly to meteorology. Maury attempted to organize co-operative
meteorological work on land, but the government did not at this time
take any steps in this direction. His oceanographical work, however,
received recognition in all parts of the civilized world, and in 1855 it
was proposed in the senate to remunerate him, but in the same year the
Naval Retiring Board, erected under an act to promote the efficiency of
the navy, placed him on the retired list. This action aroused wide
opposition, and in 1858 he was reinstated with the rank of commander as
from 1855. In 1853 Maury had published his _Letters on the Amazon and
Atlantic Slopes of South America_, and the most widely popular of his
works, the _Physical Geography of the Sea_, was published in London in
1855, and in New York in 1856; it was translated into several European
languages. On the outbreak of the American Civil War in 1861, Maury
threw in his lot with the South, and became head of coast, harbour and
river defences. He invented an electric torpedo for harbour defence, and
in 1862 was ordered to England to purchase torpedo material, &c. Here he
took active part in organizing a petition for peace to the American
people, which was unsuccessful. Afterwards he became imperial
commissioner of emigration to the emperor Maximilian of Mexico, and
attempted to form a Virginian colony in that country. Incidentally he
introduced there the cultivation of cinchona. The scheme of colonization
was abandoned by the emperor (1866), and Maury, who had lost nearly his
all during the war, settled for a while in England, where he was
presented with a testimonial raised by public subscription, and among
other honours received the degree of LL.D. of Cambridge University
(1868). In the same year, a general amnesty admitting of his return to
America, he accepted the professorship of meteorology in the Virginia
Military Institute, and settled at Lexington, Virginia, where he died on
the 1st of February 1873.

  Among works published by Maury, in addition to those mentioned, are
  the papers contributed by him to the _Astronomical Observations_ of
  the United States Observatory, _Letter concerning Lanes for Steamers
  crossing the Atlantic_ (1855); _Physical Geography_ (1864) and _Manual
  of Geography_ (1871). In 1859 he began the publication of a series of
  _Nautical Monographs_.

  See Diana Fontaine Maury Corbin (his daughter), _Life of Matthew
  Fontaine Maury_ (London, 1888).




MAUSOLEUM, the term given to a monument erected to receive the remains
of a deceased person, which may sometimes take the form of a sepulchral
chapel. The term _cenotaph_ ([Greek: kenos], empty, [Greek: taphos],
tomb) is employed for a similar monument where the body is not buried in
the structure. The term "mausoleum" originated with the magnificent
monument erected by Queen Artemisia in 353 B.C. in memory of her husband
King Mausolus, of which the remains were brought to England in 1859 by
Sir Charles Newton and placed in the British Museum. The tombs of
Augustus and of Hadrian in Rome are perhaps the largest monuments of the
kind ever erected.




MAUSOLUS (more correctly MAUSSOLLUS), satrap and practically ruler of
Caria (377-353 B.C.). The part he took in the revolt against Artaxerxes
Mnemon, his conquest of a great part of Lycia, Ionia and of several of
the Greek islands, his co-operation with the Rhodians and their allies
in the war against Athens, and the removal of his capital from Mylasa,
the ancient seat of the Carian kings, to Halicarnassus are the leading
facts of his history. He is best known from the tomb erected for him by
his widow Artemisia. The architects Satyrus and Pythis, and the
sculptors Scopas, Leochares, Bryaxis and Timotheus, finished the work
after her death. (See HALICARNASSUS.) An inscription discovered at
Mylasa (Böckh, _Inscr. gr._ ii. 2691 _c._) details the punishment of
certain conspirators who had made an attempt upon his life at a festival
in a temple at Labranda in 353.

  See Diod. Sic. xv. 90, 3, xvi. 7, 4, 36, 2; Demosthenes, _De Rhodiorum
  libertate_; J. B. Bury, _Hist. of Greece_ (1902), ii. 271; W. Judeich,
  _Kleinasiatische Studien_ (Marburg, 1892), pp. 226-256, and
  authorities under HALICARNASSUS.




MAUVE, ANTON (1838-1888), Dutch landscape painter, was born at Zaandam,
the son of a Baptist minister. Much against the wish of his parents he
took up the study of art and entered the studio of Van Os, whose dry
academic manner had, however, but little attraction for him. He
benefited far more by his intimacy with his friends Jozef Israels and W.
Maris. Encouraged by their example he abandoned his early tight and
highly finished manner for a freer, looser method of painting, and the
brilliant palette of his youthful work for a tender lyric harmony which
is generally restricted to delicate greys, greens, and light blue. He
excelled in rendering the soft hazy atmosphere that lingers over the
green meadows of Holland, and devoted himself almost exclusively to
depicting the peaceful rural life of the fields and country lanes of
Holland--especially of the districts near Oosterbeck and Wolfhezen, the
sand dunes of the coast at Scheveningen, and the country near Laren,
where he spent the last years of his life. A little sad and melancholy,
his pastoral scenes are nevertheless conceived in a peaceful soothing
lyrical mood, which is in marked contrast to the epic power and almost
tragic intensity of J. F. Millet. There are fourteen of Mauve's pictures
at the Mesdag Museum at the Hague, and two ("Milking Time" and "A
Fishing Boat putting to Sea") at the Ryks Museum in Amsterdam. The
Glasgow Corporation Gallery owns his painting of "A Flock of Sheep." The
finest and most representative private collection of pictures by Mauve
was made by Mr J. C. J. Drucker, London.




MAVROCORDATO, MAVROCORDAT or MAVROGORDATO, the name of a family of
Phanariot Greeks, distinguished in the history of Turkey, Rumania and
modern Greece. The family was founded by a merchant of Chios, whose son
Alexander Mavrocordato (c. 1636-1709), a doctor of philosophy and
medicine of Bologna, became dragoman to the sultan in 1673, and was much
employed in negotiations with Austria. It was he who drew up the treaty
of Karlowitz (1699). He became a secretary of state, and was created a
count of the Holy Roman Empire. His authority, with that of Hussein
Kupruli and Rami Pasha, was supreme at the court of Mustapha II., and he
did much to ameliorate the condition of the Christians in Turkey. He
was disgraced in 1703, but was recalled to court by Sultan Ahmed III. He
left some historical, grammatical, &c. treatises of little value.

His son NICHOLAS MAVROCORDATO (1670-1730) was grand dragoman to the
Divan (1697), and in 1708 was appointed hospodar (prince) of Moldavia.
Deposed, owing to the sultan's suspicions, in favour of Demetrius
Cantacuzene, he was restored in 1711, and soon afterwards became
hospodar of Walachia. In 1716 he was deposed by the Austrians, but was
restored after the peace of Passarowitz. He was the first Greek set to
rule the Danubian principalities, and was responsible for establishing
the system which for a hundred years was to make the name of Greek
hateful to the Rumanians. He introduced Greek manners, the Greek
language and Greek costume, and set up a splendid court on the Byzantine
model. For the rest he was a man of enlightenment, founded libraries and
was himself the author of a curious work entitled [Greek: Peri
kathêkontôn] (Bucharest, 1719). He was succeeded as grand dragoman
(1709) by his son John (Ioannes), who was for a short while hospodar of
Moldavia, and died in 1720.

Nicholas Mavrocordato was succeeded as prince of Walachia in 1730 by his
son Constantine. He was deprived in the same year, but again ruled the
principality from 1735 to 1741 and from 1744 to 1748; he was prince of
Moldavia from 1741 to 1744 and from 1748 to 1749. His rule was
distinguished by numerous tentative reforms in the fiscal and
administrative systems. He was wounded and taken prisoner in the affair
of Galati during the Russo-Turkish War, on the 5th of November 1769, and
died in captivity.

PRINCE ALEXANDER MAVROCORDATO (1791-1865), Greek statesman, a descendant
of the hospodars, was born at Constantinople on the 11th of February
1791. In 1812 he went to the court of his uncle Ioannes Caradja,
hospodar of Walachia, with whom he passed into exile in Russia and Italy
(1817). He was a member of the Hetairia Philike and was among the
Phanariot Greeks who hastened to the Morea on the outbreak of the War of
Independence in 1821. He was active in endeavouring to establish a
regular government, and in January 1822 presided over the first Greek
national assembly at Epidaurus. He commanded the advance of the Greeks
into western Hellas the same year, and suffered a defeat at Peta on the
16th of July, but retrieved this disaster somewhat by his successful
resistance to the first siege of Missolonghi (Nov. 1822 to Jan. 1823).
His English sympathies brought him, in the subsequent strife of
factions, into opposition to the "Russian" party headed by Demetrius
Ypsilanti and Kolokotrones; and though he held the portfolio of foreign
affairs for a short while under the presidency of Petrobey (Petros
Mavromichales), he was compelled to withdraw from affairs until February
1825, when he again became a secretary of state. The landing of Ibrahim
Pasha followed, and Mavrocordato again joined the army, only escaping
capture in the disaster at Sphagia (Spakteria), on the 9th of May 1815,
by swimming to Navarino. After the fall of Missolonghi (April 22, 1826)
he went into retirement, until President Capo d'Istria made him a member
of the committee for the administration of war material, a position he
resigned in 1828. After Capo d'Istria's murder (Oct. 9, 1831) and the
resignation of his brother and successor, Agostino Capo d'Istria (April
13, 1832), Mavrocordato became minister of finance. He was
vice-president of the National Assembly at Argos (July, 1832), and was
appointed by King Otto minister of finance, and in 1833 premier. From
1834 onwards he was Greek envoy at Munich, Berlin, London and--after a
short interlude as premier in Greece in 1841--Constantinople. In 1843,
after the revolution of September, he returned to Athens as minister
without portfolio in the Metaxas cabinet, and from April to August 1844
was head of the government formed after the fall of the "Russian" party.
Going into opposition, he distinguished himself by his violent attacks
on the Kolettis government. In 1854-1855 he was again head of the
government for a few months. He died in Aegina on the 18th of August
1865.

  See E. Legrand, _Genealogie des Mavrocordato_ (Paris, 1886).




MAWKMAI (Burmese _Maukmè_), one of the largest states in the eastern
division of the southern Shan States of Burma. It lies approximately
between 19° 30´ and 20° 30´ N. and 97° 30´ and 98° 15´ E., and has an
area of 2,787 sq. m. The central portion of the state consists of a wide
plain well watered and under rice cultivation. The rest is chiefly hills
in ranges running north and south. There is a good deal of teak in the
state, but it has been ruinously worked. The sawbwa now works as
contractor for government, which takes one-third of the net profits.
Rice is the chief crop, but much tobacco of good quality is grown in the
Langkö district on the Têng river. There is also a great deal of
cattle-breeding. The population in 1901 was 29,454, over two-thirds of
whom were Shans and the remainder Taungthu, Burmese, Yangsek and Red
Karens. The capital, MAWKMAI, stands in a fine rice plain in 20° 9´ N.
and 97° 25´ E. It had about 150 houses when it first submitted in 1887,
but was burnt out by the Red Karens in the following year. It has since
recovered. There are very fine orange groves a few miles south of the
town at Kantu-awn, called Kadugate by the Burmese.




MAXENTIUS, MARCUS AURELIUS VALERIUS, Roman emperor from A.D. 306 to 312,
was the son of Maximianus Herculius, and the son-in-law of Galerius.
Owing to his vices and incapacity he was left out of account in the
division of the empire which took place in 305. A variety of causes,
however, had produced strong dissatisfaction at Rome with many of the
arrangements established by Diocletian, and on the 28th of October 306,
the public discontent found expression in the massacre of those
magistrates who remained loyal to Flavius Valerius Severus and in the
election of Maxentius to the imperial dignity. With the help of his
father, Maxentius was enabled to put Severus to death and to repel the
invasion of Galerius; his next steps were first to banish Maximianus,
and then, after achieving a military success in Africa against the
rebellious governor, L. Domitius Alexander, to declare war against
Constantine as having brought about the death of his father Maximianus.
His intention of carrying the war into Gaul was anticipated by
Constantine, who marched into Italy. Maxentius was defeated at Saxa
Rubra near Rome and drowned in the Tiber while attempting to make his
way across the Milvian bridge into Rome. He was a man of brutal and
worthless character; but although Gibbon's statement that he was "just,
humane and even partial towards the afflicted Christians" may be
exaggerated, it is probable that he never exhibited any special
hostility towards them.

  See De Broglie, _L'Église et l'empire Romain au quatrième siècle_
  (1856-1866), and on the attitude of the Romans towards Christianity
  generally, app. 8 in vol. ii. of J. B. Bury's edition of Gibbon
  (Zosimus ii. 9-18; Zonaras xii. 33, xiii. 1; Aurelius Victor, _Epit._
  40; Eutropius, x. 2).




MAXIM, SIR HIRAM STEVENS (1840-   ), Anglo-American engineer and
inventor, was born at Sangerville, Maine, U.S.A., on the 5th of February
1840. After serving an apprenticeship with a coachbuilder, he entered
the machine works of his uncle, Levi Stevens, at Fitchburg,
Massachusetts, in 1864, and four years later he became a draughtsman in
the Novelty Iron Works and Shipbuilding Company in New York City. About
this period he produced several inventions connected with illumination
by gas; and from 1877 he was one of the numerous inventors who were
trying to solve the problem of making an efficient and durable
incandescent electric lamp, in this connexion introducing the
widely-used process of treating the carbon filaments by heating them in
an atmosphere of hydrocarbon vapour. In 1880 he came to Europe, and soon
began to devote himself to the construction of a machine-gun which
should be automatically loaded and fired by the energy of the recoil
(see MACHINE-GUN). In order to realize the full usefulness of the
weapon, which was first exhibited in an underground range at Hatton
Garden, London, in 1884, he felt the necessity of employing a smokeless
powder, and accordingly he devised maximite, a mixture of
trinitrocellulose, nitroglycerine and castor oil, which was patented in
1889. He also undertook to make a flying machine, and after numerous
preliminary experiments constructed an apparatus which was tried at
Bexley Heath, Kent, in 1894. (See FLIGHT.) Having been naturalized as a
British subject, he was knighted in 1901. His younger brother, Hudson
Maxim (b. 1853), took out numerous patents in connexion with explosives.




MAXIMA AND MINIMA, in mathematics. By the _maximum_ or _minimum_ value
of an expression or quantity is meant primarily the "greatest" or
"least" value that it can receive. In general, however, there are points
at which its value ceases to increase and begins to decrease; its value
at such a point is called a maximum. So there are points at which its
value ceases to decrease and begins to increase; such a value is called
a minimum. There may be several maxima or minima, and a minimum is not
necessarily less than a maximum. For instance, the expression (x² + x +
2)/(x - 1) can take all values from -[oo] to -1 and from +7 to +[oo],
but has, so long as x is real, no value between -1 and +7. Here -1 is a
maximum value, and +7 is a minimum value of the expression, though it
can be made greater or less than any assignable quantity.

The first general method of investigating maxima and minima seems to
have been published in A.D. 1629 by Pierre Fermat. Particular cases had
been discussed. Thus Euclid in book III. of the _Elements_ finds the
greatest and least straight lines that can be drawn from a point to the
circumference of a circle, and in book VI. (in a proposition generally
omitted from editions of his works) finds the parallelogram of greatest
area with a given perimeter. Apollonius investigated the greatest and
least distances of a point from the perimeter of a conic section, and
discovered them to be the normals, and that their feet were the
intersections of the conic with a rectangular hyperbola. Some remarkable
theorems on maximum areas are attributed to Zenodorus, and preserved by
Pappus and Theon of Alexandria. The most noteworthy of them are the
following:--

  1. Of polygons of n sides with a given perimeter the regular polygon
  encloses the greatest area.

  2. Of two regular polygons of the same perimeter, that with the
  greater number of sides encloses the greater area.

  3. The circle encloses a greater area than any polygon of the same
  perimeter.

  4. The sum of the areas of two isosceles triangles on given bases, the
  sum of whose perimeters is given, is greatest when the triangles are
  similar.

  5. Of segments of a circle of given perimeter, the semicircle encloses
  the greatest area.

  6. The sphere is the surface of given area which encloses the greatest
  volume.

Serenus of Antissa investigated the somewhat trifling problem of finding
the triangle of greatest area whose sides are formed by the
intersections with the base and curved surface of a right circular cone
of a plane drawn through its vertex.

The next problem on maxima and minima of which there appears to be any
record occurs in a letter from Regiomontanus to Roder (July 4, 1471),
and is a particular numerical example of the problem of finding the
point on a given straight line at which two given points subtend a
maximum angle. N. Tartaglia in his _General trattato de numeri et
mesuri_ (c. 1556) gives, without proof, a rule for dividing a number
into two parts such that the continued product of the numbers and their
difference is a maximum.

Fermat investigated maxima and minima by means of the principle that in
the neighbourhood of a maximum or minimum the differences of the values
of a function are insensible, a method virtually the same as that of the
differential calculus, and of great use in dealing with geometrical
maxima and minima. His method was developed by Huygens, Leibnitz, Newton
and others, and in particular by John Hudde, who investigated maxima and
minima of functions of more than one independent variable, and made some
attempt to discriminate between maxima and minima, a question first
definitely settled, so far as one variable is concerned, by Colin
Maclaurin in his _Treatise on Fluxions_ (1742). The method of the
differential calculus was perfected by Euler and Lagrange.

John Bernoulli's famous problem of the "brachistochrone," or curve of
quickest descent from one point to another under the action of gravity,
proposed in 1696, gave rise to a new kind of maximum and minimum problem
in which we have to find a curve and not points on a given curve. From
these problems arose the "Calculus of Variations." (See VARIATIONS,
CALCULUS OF.)

The only general methods of attacking problems on maxima and minima are
those of the differential calculus or, in geometrical problems, what is
practically Fermat's method. Some problems may be solved by algebra;
thus if y = f(x) ÷ [phi](x), where f(x) and [phi](x) are polynomials in
x, the limits to the values of y[phi] may be found from the
consideration that the equation y[phi](x) - f(x) = 0 must have real
roots. This is a useful method in the case in which [phi](x) and f(x)
are quadratics, but scarcely ever in any other case. The problem of
finding the maximum product of n positive quantities whose sum is given
may also be found, algebraically, thus. If a and b are any two real
unequal quantities whatever {½(a + b)}² > ab, so that we can increase
the product leaving the sum unaltered by replacing any two terms by half
their sum, and so long as any two of the quantities are unequal we can
increase the product. Now, the quantities being all positive, the
product cannot be increased without limit and must somewhere attain a
maximum, and no other form of the product than that in which they are
all equal can be the maximum, so that the product is a maximum when they
are all equal. Its minimum value is obviously zero. If the restriction
that all the quantities shall be positive is removed, the product can be
made equal to any quantity, positive or negative. So other theorems of
algebra, which are stated as theorems on inequalities, may be regarded
as algebraic solutions of problems on maxima and minima.

For purely geometrical questions the only general method available is
practically that employed by Fermat. If a quantity depends on the
position of some point P on a curve, and if its value is equal at two
neighbouring points P and P´, then at some position between P and P´ it
attains a maximum or minimum, and this position may be found by making P
and P´ approach each other indefinitely. Take for instance the problem
of Regiomontanus "to find a point on a given straight line which
subtends a maximum angle at two given points A and B." Let P and P´ be
two near points on the given straight line such that the angles APB and
AP´B are equal. Then ABPP´ lie on a circle. By making P and P´ approach
each other we see that for a maximum or minimum value of the angle APB,
P is a point in which a circle drawn through AB touches the given
straight line. There are two such points, and unless the given straight
line is at right angles to AB the two angles obtained are not the same.
It is easily seen that both angles are maxima, one for points on the
given straight line on one side of its intersection with AB, the other
for points on the other side. For further examples of this method
together with most other geometrical problems on maxima and minima of
any interest or importance the reader may consult such a book as J. W.
Russell's _A Sequel lo Elementary Geometry_ (Oxford, 1907).

  The method of the differential calculus is theoretically very simple.
  Let u be a function of several variables x1, x2, x3 ... x_n, supposed
  for the present independent; if u is a maximum or minimum for the set
  of values x1, x2, x3, ... x_n, and u becomes u + [delta]u, when x1,
  x2, x3 ... x_n receive small increments [delta]x1, [delta]x2, ...
  [delta]x_n; then [delta]u must have the same sign for all possible
  values of [delta]x1, [delta]2 ... [delta]x_n.

  Now
                                            _                                                                   _
                __ [delta]u                |   __ [delta]²u       __      [delta]³u                              |
    [delta]u = \   --------- [delta]x1 + ½ |  \   ---------- + 2 \   ------------------- [delta]x1 [delta]x2 ... | + ...
               /__ [delta]x1               |_ /__ [delta]x1²     /__ [delta]x1 [delta]x2                        _|

  The sign of this expression in general is that of
  [Sigma]([delta]u/[delta]x1)[delta]x1, which cannot be one-signed when
  x1, x2, ... x_n can take all possible values, for a set of increments
  [delta]x1, [delta]x2 ... [delta]x_n, will give an opposite sign to the
  set -[delta]x1, -[delta]x2, ... -[delta]x_n. Hence
  [Sigma]([delta]u/[delta]x1)[delta]x1 must vanish for all sets of
  increments [delta]x1, ... [delta]x_n, and since these are independent,
  we must have [delta]u/[delta]x1 = 0, [delta]u/[delta]x2 = 0, ...
  [delta]u/[delta]x_n = 0. A value of u given by a set of solutions of
  these equations is called a "critical value" of u. The value of
  [delta]u now becomes
       _                                                                               _
      |   __ [delta]²u                 __      [delta]²u                                |
    ½ |  \   --------- [delta]x1² + 2 \   ------------------- [delta]x1 [delta]x2 + ... |;
      |_ /__ [delta]x1²               /__ [delta]x1 [delta]x2                          _|

  for u to be a maximum or minimum this must have always the same sign.
  For the case of a single variable x, corresponding to a value of x
  given by the equation du/dx = 0, u is a maximum or minimum as d²u/dx²
  is negative or positive. If d²u/dx² vanishes, then there is no maximum
  or minimum unless d²u/dx² vanishes, and there is a maximum or minimum
  according as d^4u/dx^4 is negative or positive. Generally, if the
  first differential coefficient which does not vanish is even, there is
  a maximum or minimum according as this is negative or positive. If it
  is odd, there is no maximum or minimum.

  In the case of several variables, the quadratic

     __ [delta]²u                  __      [delta]²u
    \   ---------- [delta]x1² + 2 \   ------------------- + ...
    /__ [delta]x1²                /__ [delta]x1 [delta]x2

  must be one-signed. The condition for this is that the series of
  discriminants

    a11 , | a11  a12 | , | a11  a12  a13 | , ...
          | a21  a22 |   | a21  a22  a23 |
                         | a31  a32  a33 |

  where a_pq denotes [delta]²u/[delta]a_p[delta]a_q should be all
  positive, if the quadratic is always positive, and alternately
  negative and positive, if the quadratic is always negative. If the
  first condition is satisfied the critical value is a minimum, if the
  second it is a maximum. For the case of two variables the conditions
  are

     [delta]²u   [delta]²u     /       [delta]²      \²
    ---------- · ---------- > (  -------------------  )
    [delta]x1²   [delta]x2²    \ [delta]x1 [delta]x2 /

  for a maximum or minimum at all and [delta]²u/[delta]x1² and
  [delta]²u/[delta]x2² both negative for a maximum, and both positive
  for a minimum. It is important to notice that by the quadratic being
  one-signed is meant that it cannot be made to vanish except when
  [delta]x1, [delta]x2, ... [delta]x_n all vanish. If, in the case of
  two variables,

    [delta]²u    [delta]²u     /      [delta]²u      \²
    ---------- · ---------- = (  -------------------  )
    [delta]x1²   [delta]x2²    \ [delta]x1 [delta]x2 /

  then the quadratic is one-signed unless it vanishes, but the value of
  u is not necessarily a maximum or minimum, and the terms of the third
  and possibly fourth order must be taken account of.

  Take for instance the function u = x² - xy² + y². Here the values x =
  0, y = 0 satisfy the equations [delta]u/[delta]x = 0,
  [delta]u/[delta]y = 0, so that zero is a critical value of u, but it
  is neither a maximum nor a minimum although the terms of the second
  order are ([delta]x)², and are never negative. Here [delta]u =
  [delta]x² - [delta]x[delta]y² + [delta]y², and by putting [delta]x = 0
  or an infinitesimal of the same order as [delta]y², we can make the
  sign of [delta]u depend on that of [delta]y², and so be positive or
  negative as we please. On the other hand, if we take the function u =
  x² - xy² + y^4, x = 0, y = 0 make zero a critical value of u, and here
  [delta]u = [delta]x² - [delta]x[delta]y² + [delta]y^4, which is always
  positive, because we can write it as the sum of two squares, viz.
  ([delta]x - ½[delta]y²)² + ¾[delta]y^4; so that in this case zero is a
  minimum value of u.

  A critical value usually gives a maximum or minimum in the case of a
  function of one variable, and often in the case of several independent
  variables, but all maxima and minima, particularly absolutely greatest
  and least values, are not necessarily critical values. If, for
  example, x is restricted to lie between the values a and b and
  [phi]´(x) = 0 has no roots in this interval, it follows that [phi]´(x)
  is one-signed as x increases from a to b, so that [phi](x) is
  increasing or diminishing all the time, and the greatest and least
  values of [phi](x) are [phi](a) and [phi](b), though neither of them
  is a critical value. Consider the following example: A person in a
  boat a miles from the nearest point of the beach wishes to reach as
  quickly as possible a point b miles from that point along the shore.
  The ratio of his rate of walking to his rate of rowing is cosec
  [alpha]. Where should he land?

  Here let AB be the direction of the beach, A the nearest point to the
  boat O, and B the point he wishes to reach. Clearly he must land, if
  at all, between A and B. Suppose he lands at P. Let the angle AOP be
  [theta], so that OP = a sec[theta], and PB = b - a tan [theta]. If his
  rate of rowing is V miles an hour his time will be a sec [theta]/V +
  (b - a tan [theta]) sin [alpha]/V hours. Call this T. Then to the
  first power of [delta][theta], [delta]T = (a/V) sec²[theta] (sin
  [theta] - sin [alpha])[delta][theta], so that if AOB > [alpha],
  [delta]T and [delta][theta] have opposite signs from [theta] = 0 to
  [theta] = [alpha], and the same signs from [theta] = [alpha] to
  [theta] = AOB. So that when AOB is > [alpha], T decreases from [theta]
  = 0 to [theta] = [alpha], and then increases, so that he should land
  at a point distant a tan [alpha] from A, unless a tan [alpha] > b.
  When this is the case, [delta]T and [delta][theta] have opposite signs
  throughout the whole range of [theta], so that T decreases as [theta]
  increases, and he should row direct to B. In the first case the
  minimum value of T is also a critical value; in the second case it is
  not.

  The greatest and least values of the bending moments of loaded rods
  are often at the extremities of the divisions of the rods and not at
  points given by critical values.

  In the case of a function of several variables, X1, x2, ... x_n, not
  independent but connected by m functional relations u1 = 0, u2 = 0,
  ..., u_m = 0, we might proceed to eliminate m of the variables; but
  Lagrange's "Method of undetermined Multipliers" is more elegant and
  generally more useful.

  We have [delta]u1 = 0, [delta]u2 = 0, ..., [delta]u_m = 0. Consider
  instead of [delta]u, what is the same thing, viz., [delta]u +
  [lambda]1[delta]u1 + [lambda]2[delta]u2 + ... + [lambda]_m[delta]u_m,
  where [lambda]1, [lambda]2, ... [lambda]_m, are arbitrary multipliers.
  The terms of the first order in this expression are

     __ [delta]u                        __  [delta]u1                               __ [delta]u_m
    \   --------- [delta]x1 + [lambda]1 \   --------- [delta]x1 + ... + [lambda]_m \   ---------- [delta]x1.
    /__ [delta]x1                       /__ [delta]x1                              /__ [delta]x1

  We can choose [lambda]1, ... [lambda]_m, to make the coefficients of
  [delta]x1, [delta]x2, ... [delta]x_m, vanish, and the remaining
  [delta]x_(m+1) to [delta]x_n may be regarded as independent, so that,
  when u has a critical value, their coefficients must also vanish. So
  that we put

     [delta]u              [delta]u1                     [delta]u_m
    ---------- + [lambda]1 ---------- + ... + [lambda]_m ---------- = 0
    [delta]x_r             [delta]x_r                    [delta]x_r

  for all values of r. These equations with the equations u1 = 0, ...,
  u_m = 0 are exactly enough to determine [lambda]1, ..., [lambda]_m, x1
  x2, ..., x_n, so that we find critical values of u, and examine the
  terms of the second order to decide whether we obtain a maximum or
  minimum.

  To take a very simple illustration; consider the problem of
  determining the maximum and minimum radii vectors of the ellipsoid
  x²/a² + y²/b² + z²/c² = 1, where a² > b² > c². Here we require the
  maximum and minimum values of x² + y² + z² where x²/a² + y²/b² + z²/c²
  = 1.

  We have

                            /    [lambda]\                 /    [lambda]\                 /    [lambda]\
    [delta]u = 2x [delta]x ( 1 + -------- ) + 2y [delta]y ( 1 + -------- ) + 2z [delta]z ( 1 + -------- )
                            \       a²   /                 \       b²   /                 \       c²   /

                 /    [lambda]\               /    [lambda]\               /    [lambda]\
    + [delta]x² ( 1 + -------- ) + [delta]y² ( 1 + -------- ) + [delta]z² ( 1 + -------- ).
                 \       a²   /               \       b²   /               \       c²   /

  To make the terms of the first order disappear, we have the three
  equations:--

  x(1 + [lambda]/a²) = 0, y(1 + [lambda]/b²) = 0, z(1 + [lambda]/c²) =
  0.

  These have three sets of solutions consistent with the conditions
  x²/a² + y²/b² + z²/c² = 1, a² > b² > c², viz.:--

    (1) y = 0, z = 0, [lambda] = -a²; (2) z = 0, x = 0, [lambda] = -b²;

    (3) x = 0, y = 0, [lambda] = -c².

  In the case of (1) [delta]u = [delta]y² (1 - a²/b²) + [delta]z² (1 -
  a²/c²), which is always negative, so that u = a² gives a maximum.

  In the case of (3) [delta]u = [delta]x² (1 - c²/a²) + [delta]y² (1 -
  c²/b²), which is always positive, so that u = c² gives a minimum.

  In the case of (2) [delta]u = [delta]x²(1 - b²/a²) - [delta]z²(b²/c² -
  1), which can be made either positive or negative, or even zero if we
  move in the planes x²(1 - b²/a²) = z²(b²/c² - 1), which are well known
  to be the central planes of circular section. So that u = b², though a
  critical value, is neither a maximum nor minimum, and the central
  planes of circular section divide the ellipsoid into four portions in
  two of which a² > r² > b², and in the other two b² > r² > c².
       (A. E. J.)




MAXIMIANUS, a Latin elegiac poet who flourished during the 6th century
A.D. He was an Etruscan by birth, and spent his youth at Rome, where he
enjoyed a great reputation as an orator. At an advanced age he was sent
on an important mission to the East, perhaps by Theodoric, if he is the
Maximianus to whom that monarch addressed a letter preserved in
Cassiodorus (_Variarum_, i. 21). The six elegies extant under his name,
written in old age, in which he laments the loss of his youth, contain
descriptions of various amours. They show the author's familiarity with
the best writers of the Augustan age.

  Editions by J. C. Wernsdorf, _Poetae latini minores_, vi.; E. Bährens,
  _Poetae latini minores_, v.; M. Petschenig (1890), in C. F.
  Ascherson's _Berliner Studien_, xi.; R. Webster (Princeton, 1901; see
  _Classical Review_, Oct. 1901), with introduction and commentary; see
  also Robinson Ellis in _American Journal of Philology_, v. (1884) and
  Teuffel-Schwabe, _Hist. of Roman Literature_ (Eng. trans.), § 490.
  There is an English version (as from Cornelius Gallus), by Hovenden
  Walker (1689), under the title of _The Impotent Lover_.




MAXIMIANUS, MARCUS AURELIUS VALERIUS, surnamed Herculius, Roman emperor
from A.D. 286 to 305, was born of humble parents at Sirmium in Pannonia.
He achieved distinction during long service in the army, and having been
made Caesar by Diocletian in 285, received the title of Augustus in the
following year (April 1, 286). In 287 he suppressed the rising of the
peasants (Bagaudae) in Gaul, but in 289, after a three years' struggle,
his colleague and he were compelled to acquiesce in the assumption by
his lieutenant Carausius (who had crossed over to Britain) of the title
of Augustus. After 293 Maximianus left the care of the Rhine frontier to
Constantius Chlorus, who had been designated Caesar in that year, but in
297 his arms achieved a rapid and decisive victory over the barbarians
of Mauretania, and in 302 he shared at Rome the triumph of Diocletian,
the last pageant of the kind ever witnessed by that city. On the 1st of
May 305, the day of Diocletian's abdication, he also, but without his
colleague's sincerity, divested himself of the imperial dignity at
Mediolanum (Milan), which had been his capital, and retired to a villa
in Lucania; in the following year, however, he was induced by his son
Maxentius to reassume the purple. In 307 he brought the emperor Flavius
Valerius Severus a captive to Rome, and also compelled Galerius to
retreat, but in 308 he was himself driven by Maxentius from Italy into
Illyricum, whence again he was compelled to seek refuge at Arelate
(Arles), the court of his son-in-law, Constantine. Here a false report
was received, or invented, of the death of Constantine, at that time
absent on the Rhine. Maximianus at once grasped at the succession, but
was soon driven to Massilia (Marseilles), where, having been delivered
up to his pursuers, he strangled himself.

  See Zosimus ii. 7-11; Zonaras xii. 31-33; Eutropius ix. 20, x. 2, 3;
  Aurelius Victor p. 39. For the emperor Galerius Valerius Maximianus
  see GALERIUS.




MAXIMILIAN I. (1573-1651), called "the Great," elector and duke of
Bavaria, eldest son of William V. of Bavaria, was born at Munich on the
17th of April 1573. He was educated by the Jesuits at the university of
Ingolstadt, and began to take part in the government in 1591. He married
in 1595 his cousin, Elizabeth, daughter of Charles II., duke of
Lorraine, and became duke of Bavaria upon his father's abdication in
1597. He refrained from any interference in German politics until 1607,
when he was entrusted with the duty of executing the imperial ban
against the free city of Donauwörth, a Protestant stronghold. In
December 1607 his troops occupied the city, and vigorous steps were
taken to restore the supremacy of the older faith. Some Protestant
princes, alarmed at this action, formed a union to defend their
interests, which was answered in 1609 by the establishment of a league,
in the formation of which Maximilian took an important part. Under his
leadership an army was set on foot, but his policy was strictly
defensive and he refused to allow the league to become a tool in the
hands of the house of Habsburg. Dissensions among his colleagues led the
duke to resign his office in 1616, but the approach of trouble brought
about his return to the league about two years later.

Having refused to become a candidate for the imperial throne in 1619,
Maximilian was faced with the complications arising from the outbreak of
war in Bohemia. After some delay he made a treaty with the emperor
Ferdinand II. in October 1619, and in return for large concessions
placed the forces of the league at the emperor's service. Anxious to
curtail the area of the struggle, he made a treaty of neutrality with
the Protestant Union, and occupied Upper Austria as security for the
expenses of the campaign. On the 8th of November 1620 his troops under
Count Tilly defeated the forces of Frederick, king of Bohemia and count
palatine of the Rhine, at the White Hill near Prague. In spite of the
arrangement with the union Tilly then devastated the Rhenish Palatinate,
and in February 1623 Maximilian was formally invested with the electoral
dignity and the attendant office of imperial steward, which had been
enjoyed since 1356 by the counts palatine of the Rhine. After receiving
the Upper Palatinate and restoring Upper Austria to Ferdinand,
Maximilian became leader of the party which sought to bring about
Wallenstein's dismissal from the imperial service. At the diet of
Regensburg in 1630 Ferdinand was compelled to assent to this demand, but
the sequel was disastrous both for Bavaria and its ruler. Early in 1632
the Swedes marched into the duchy and occupied Munich, and Maximilian
could only obtain the assistance of the imperialists by placing himself
under the orders of Wallenstein, now restored to the command of the
emperor's forces. The ravages of the Swedes and their French allies
induced the elector to enter into negotiations for peace with Gustavus
Adolphus and Cardinal Richelieu. He also proposed to disarm the
Protestants by modifying the Restitution edict of 1629; but these
efforts were abortive. In March 1647 he concluded an armistice with
France and Sweden at Ulm, but the entreaties of the emperor Ferdinand
III. led him to disregard his undertaking. Bavaria was again ravaged,
and the elector's forces defeated in May 1648 at Zusmarshausen. But the
peace of Westphalia soon put an end to the struggle. By this treaty it
was agreed that Maximilian should retain the electoral dignity, which
was made hereditary in his family; and the Upper Palatinate was
incorporated with Bavaria. The elector died at Ingolstadt on the 27th of
September 1651. By his second wife, Maria Anne, daughter of the emperor
Ferdinand II., he left two sons, Ferdinand Maria, who succeeded him, and
Maximilian Philip. In 1839 a statue was erected to his memory at Munich
by Louis I., king of Bavaria. Weak in health and feeble in frame,
Maximilian had high ambitions both for himself and his duchy, and was
tenacious and resourceful in prosecuting his designs. As the ablest
prince of his age he sought to prevent Germany from becoming the
battleground of Europe, and although a rigid adherent of the Catholic
faith, was not always subservient to the priest.

  See P. P. Wolf, _Geschichte Kurfürst Maximilians I. und seiner Zeit_
  (Munich, 1807-1809); C. M. Freiherr von Aretin, _Geschichte des
  bayerschen Herzogs und Kurfürsten Maximilian des Ersten_ (Passau,
  1842); M. Lossen, _Die Reichstadt Donauwörth und Herzog Maximilian_
  (Munich, 1866); F. Stieve, _Kurfürst Maximilian I. von Bayern_
  (Munich, 1882); F. A. W. Schreiber, _Maximilian I. der Katholische
  Kurfürst von Bayern, und der dreissigjährige Krieg_ (Munich, 1868); M.
  Högl, _Die Bekehrung der Oberpfalz durch Kurfürst Maximilian I._
  (Regensburg, 1903).




MAXIMILIAN I. (MAXIMILIAN JOSEPH) (1756-1825), king of Bavaria, was the
son of the count palatine Frederick of Zweibrücken-Birkenfeld, and was
born on the 27th of May 1756. He was carefully educated under the
supervision of his uncle, Duke Christian IV. of Zweibrücken, took
service in 1777 as a colonel in the French army, and rose rapidly to the
rank of major-general. From 1782 to 1789 he was stationed at Strassburg,
but at the outbreak of the revolution he exchanged the French for the
Austrian service, taking part in the opening campaigns of the
revolutionary wars. On the 1st of April 1795 he succeeded his brother,
Charles II., as duke of Zweibrücken, and on the 16th of February 1799
became elector of Bavaria on the extinction of the Sulzbach line with
the death of the elector Charles Theodore.

The sympathy with France and with French ideas of enlightenment which
characterized his reign was at once manifested. In the newly organized
ministry Count Max Josef von Montgelas (q.v.), who, after falling into
disfavour with Charles Theodore, had acted for a time as Maximilian
Joseph's private secretary, was the most potent influence, an influence
wholly "enlightened" and French. Agriculture and commerce were fostered,
the laws were ameliorated, a new criminal code drawn up, taxes and
imposts equalized without regard to traditional privileges, while a
number of religious houses were suppressed and their revenues used for
educational and other useful purposes. In foreign politics Maximilian
Joseph's attitude was from the German point of view less commendable.
With the growing sentiment of German nationality he had from first to
last no sympathy, and his attitude throughout was dictated by wholly
dynastic, or at least Bavarian, considerations. Until 1813 he was the
most faithful of Napoleon's German allies, the relation being cemented
by the marriage of his daughter to Eugène Beauharnais. His reward came
with the treaty of Pressburg (Dec. 26, 1805), by the terms of which he
was to receive the royal title and important territorial acquisitions in
Swabia and Franconia to round off his kingdom. The style of king he
actually assumed on the 1st of January 1806.

The new king of Bavaria was the most important of the princes belonging
to the Confederation of the Rhine, and remained Napoleon's ally until
the eve of the battle of Leipzig, when by the convention of Ried (Oct.
8, 1813) he made the guarantee of the integrity of his kingdom the price
of his joining the Allies. By the first treaty of Paris (June 3, 1814),
however, he ceded Tirol to Austria in exchange for the former duchy of
Würzburg. At the congress of Vienna, too, which he attended in person,
Maximilian had to make further concessions to Austria, ceding the
quarters of the Inn and Hausruck in return for a part of the old
Palatinate. The king fought hard to maintain the contiguity of the
Bavarian territories as guaranteed at Ried; but the most he could obtain
was an assurance from Metternich in the matter of the Baden succession,
in which he was also doomed to be disappointed (see BADEN: _History_,
iii. 506).

At Vienna and afterwards Maximilian sturdily opposed any reconstitution
of Germany which should endanger the independence of Bavaria, and it
was his insistence on the principle of full sovereignty being left to
the German reigning princes that largely contributed to the loose and
weak organization of the new German Confederation. The Federal Act of
the Vienna congress was proclaimed in Bavaria, not as a law but as an
international treaty. It was partly to secure popular support in his
resistance to any interference of the federal diet in the internal
affairs of Bavaria, partly to give unity to his somewhat heterogeneous
territories, that Maximilian on the 26th of May 1818 granted a liberal
constitution to his people. Montgelas, who had opposed this concession,
had fallen in the previous year, and Maximilian had also reversed his
ecclesiastical policy, signing on the 24th of October 1817 a concordat
with Rome by which the powers of the clergy, largely curtailed under
Montgelas's administration, were restored. The new parliament proved so
intractable that in 1819 Maximilian was driven to appeal to the powers
against his own creation; but his Bavarian "particularism" and his
genuine popular sympathies prevented him from allowing the Carlsbad
decrees to be strictly enforced within his dominions. The suspects
arrested by order of the Mainz Commission he was accustomed to examine
himself, with the result that in many cases the whole proceedings were
quashed, and in not a few the accused dismissed with a present of money.
Maximilian died on the 13th of October 1825 and was succeeded by his son
Louis I.

In private life Maximilian was kindly and simple. He loved to play the
part of _Landesvater_, walking about the streets of his capital _en
bourgeois_ and entering into conversation with all ranks of his
subjects, by whom he was regarded with great affection. He was twice
married: (1) in 1785 to Princess Wilhelmine Auguste of Hesse-Darmstadt,
(2) in 1797 to Princess Caroline Friederike of Baden.

  See G. Freiherr von Lerchenfeld, _Gesch. Bayerns unter König
  Maximilian Joseph I._ (Berlin, 1854); J. M. Söltl, _Max Joseph, König
  von Bayern_ (Stuttgart, 1837); L. von Kobell, _Unter den vier ersten
  Königen Bayerns. Nach Briefen und eigenen Erinnerungen_ (Munich,
  1894).




MAXIMILIAN II. (1811-1864), king of Bavaria, son of king Louis I. and of
his consort Theresa of Saxe-Hildburghausen, was born on the 28th of
November 1811. After studying at Göttingen and Berlin and travelling in
Germany, Italy and Greece, he was introduced by his father into the
council of state (1836). From the first he showed a studious
disposition, declaring on one occasion that had he not been born in a
royal cradle his choice would have been to become a professor. As crown
prince, in the château of Hohenschwangau near Füssen, which he had
rebuilt with excellent taste, he gathered about him an intimate society
of artists and men of learning, and devoted his time to scientific and
historical study. When the abdication of Louis I. (March 28, 1848)
called him suddenly to the throne, his choice of ministers promised a
liberal régime. The progress of the revolution, however, gave him pause.
He strenuously opposed the unionist plans of the Frankfort parliament,
refused to recognize the imperial constitution devised by it, and
assisted Austria in restoring the federal diet and in carrying out the
federal execution in Hesse and Holstein. Although, however, from 1850
onwards his government tended in the direction of absolutism, he refused
to become the tool of the clerical reaction, and even incurred the
bitter criticism of the Ultramontanes by inviting a number of celebrated
men of learning and science (e.g. Liebig and Sybel) to Munich,
regardless of their religious views. Finally, in 1859, he dismissed the
reactionary ministry of von der Pfordten, and met the wishes of his
people for a moderate constitutional government. In his German policy he
was guided by the desire to maintain the union of the princes, and hoped
to attain this as against the perilous rivalry of Austria and Prussia by
the creation of a league of the "middle" and small states--the so-called
Trias. In 1863, however, seeing what he thought to be a better way, he
supported the project of reform proposed by Austria at the Fürstentag of
Frankfort. The failure of this proposal, and the attitude of Austria
towards the Confederation and in the Schleswig-Holstein question,
undeceived him; but before he could deal with the new situation created
by the outbreak of the war with Denmark he died suddenly at Munich, on
the 10th of March 1864.

Maximilian was a man of amiable qualities and of intellectual
attainments far above the average, but as a king he was hampered by
constant ill-health, which compelled him to be often abroad, and when at
home to live much in the country. By his wife, Maria Hedwig, daughter of
Prince William of Prussia, whom he married in 1842, he had two sons,
Louis II., king of Bavaria, and Otto, king of Bavaria, both of whom lost
their reason.

  See J. M. Söltl, _Max der Zweite, König von Bayern_ (Munich, 1865);
  biography by G. K. Heigel in _Allgem. Deutsche Biographie_, vol. xxi.
  (Leipzig, 1885). Maximilian's correspondence with Schlegel was
  published at Stuttgart in 1890.




MAXIMILIAN I. (1459-1519), Roman emperor, son of the emperor Frederick
III. and Leonora, daughter of Edward, king of Portugal, was born at
Vienna Neustadt on the 22nd of March 1459. On the 18th of August 1477,
by his marriage at Ghent to Mary, who had just inherited Burgundy and
the Netherlands from her father Charles the Bold, duke of Burgundy, he
effected a union of great importance in the history of the house of
Habsburg. He at once undertook the defence of his wife's dominions from
an attack by Louis XI., king of France, and defeated the French forces
at Guinegatte, the modern Enguinegatte, on the 7th of August 1479. But
Maximilian was regarded with suspicion by the states of Netherlands, and
after suppressing a rising in Gelderland his position was further
weakened by the death of his wife on the 27th of March 1482. He claimed
to be recognized as guardian of his young son Philip and as regent of
the Netherlands, but some of the states refused to agree to his demands
and disorder was general. Maximilian was compelled to assent to the
treaty of Arras in 1482 between the states of the Netherlands and Louis
XI. This treaty provided that Maximilian's daughter Margaret should
marry Charles, the dauphin of France, and have for her dowry Artois and
Franche-Comté, two of the provinces in dispute, while the claim of Louis
on the duchy of Burgundy was tacitly admitted. Maximilian did not,
however, abandon the struggle in the Netherlands. Having crushed a
rebellion at Utrecht, he compelled the burghers of Ghent to restore
Philip to him in 1485, and returning to Germany was chosen king of the
Romans, or German king, at Frankfort on the 16th of February 1486, and
crowned at Aix-la-Chapelle on the 9th of the following April. Again in
the Netherlands, he made a treaty with Francis II., duke of Brittany,
whose independence was threatened by the French regent, Anne of Beaujeu,
and the struggle with France was soon renewed. This war was very
unpopular with the trading cities of the Netherlands, and early in 1488
Maximilian, having entered Bruges, was detained there as a prisoner for
nearly three months, and only set at liberty on the approach of his
father with a large force. On his release he had promised he would
maintain the treaty of Arras and withdraw from the Netherlands; but he
delayed his departure for nearly a year and took part in a punitive
campaign against his captors and their allies. On his return to Germany
he made peace with France at Frankfort in July 1489, and in October
several of the states of the Netherlands recognized him as their ruler
and as guardian of his son. In March 1490 the county of Tirol was added
to his possessions through the abdication of his kinsman, Count
Sigismund, and this district soon became his favourite residence.

Meanwhile the king had formed an alliance with Henry VII. king of
England, and Ferdinand II., king of Aragon, to defend the possessions of
the duchess Anne, daughter and successor of Francis, duke of Brittany.
Early in 1490 he took a further step and was betrothed to the duchess,
and later in the same year the marriage was celebrated by proxy; but
Brittany was still occupied by French troops, and Maximilian was unable
to go to the assistance of his bride. The sequel was startling. In
December 1491 Anne was married to Charles VIII., king of France, and
Maximilian's daughter Margaret, who had resided in France since her
betrothal, was sent back to her father. The inaction of Maximilian at
this time is explained by the condition of affairs in Hungary, where
the death of king Matthias Corvinus had brought about a struggle for
this throne. The Roman king, who was an unsuccessful candidate, took up
arms, drove the Hungarians from Austria, and regained Vienna, which had
been in the possession of Matthias since 1485; but he was compelled by
want of money to retreat, and on the 7th of November 1491 signed the
treaty of Pressburg with Ladislaus, king of Bohemia, who had obtained
the Hungarian throne. By this treaty it was agreed that Maximilian
should succeed to the crown in case Ladislaus left no legitimate male
issue. Having defeated the invading Turks at Villach in 1492, the king
was eager to take revenge upon the king of France; but the states of the
Netherlands would afford him no assistance. The German diet was
indifferent, and in May 1493 he agreed to the peace of Senlis and
regained Artois and Franche-Comté.

In August 1493 the death of the emperor left Maximilian sole ruler of
Germany and head of the house of Habsburg; and on the 16th of March 1494
he married at Innsbruck Bianca Maria Sforza, daughter of Galeazzo
Sforza, duke of Milan (d. 1476). At this time Bianca's uncle, Ludovico
Sforza, was invested with the duchy of Milan in return for the
substantial dowry which his niece brought to the king. Maximilian
harboured the idea of driving the Turks from Europe; but his appeal to
all Christian sovereigns was ineffectual. In 1494 he was again in the
Netherlands, where he led an expedition against the rebels of
Gelderland, assisted Perkin Warbeck to make a descent upon England, and
formally handed over the government of the Low Countries to Philip. His
attention was next turned to Italy, and, alarmed at the progress of
Charles VIII. in the peninsula, he signed the league of Venice in March
1495, and about the same time arranged a marriage between his son Philip
and Joanna, daughter of Ferdinand and Isabella, king and queen of
Castile and Aragon. The need for help to prosecute the war in Italy
caused the king to call the diet to Worms in March 1495, when he urged
the necessity of checking the progress of Charles. As during his
father's lifetime Maximilian had favoured the reforming party among the
princes, proposals for the better government of the empire were brought
forward at Worms as a necessary preliminary to financial and military
support. Some reforms were adopted, the public peace was proclaimed
without any limitation of time and a general tax was levied. The three
succeeding years were mainly occupied with quarrels with the diet, with
two invasions of France, and a war in Gelderland against Charles, count
of Egmont, who claimed that duchy, and was supported by French troops.
The reforms of 1495 were rendered abortive by the refusal of Maximilian
to attend the diets or to take any part in the working of the new
constitution, and in 1497 he strengthened his own authority by
establishing an Aulic Council (_Reichshofrath_), which he declared was
competent to deal with all business of the empire, and about the same
time set up a court to centralize the financial administration of
Germany.

In February 1499 the king became involved in a war with the Swiss, who
had refused to pay the imperial taxes or to furnish a contribution for
the Italian expedition. Aided by France they defeated the German troops,
and the peace of Basel in September 1499 recognized them as virtually
independent of the empire. About this time Maximilian's ally, Ludovico
of Milan, was taken prisoner by Louis XII., king of France, and
Maximilian was again compelled to ask the diet for help. An elaborate
scheme for raising an army was agreed to, and in return a council of
regency (_Reichsregiment_) was established, which amounted, in the words
of a Venetian envoy, to a deposition of the king. The relations were now
very strained between the reforming princes and Maximilian, who, unable
to raise an army, refused to attend the meetings of the council at
Nuremberg, while both parties treated for peace with France. The
hostility of the king rendered the council impotent. He was successful
in winning the support of many of the younger princes, and in
establishing a new court of justice, the members of which were named by
himself. The negotiations with France ended in the treaty of Blois,
signed in September 1504, when Maximilian's grandson Charles was
betrothed to Claude, daughter of Louis XII., and Louis, invested with
the duchy of Milan, agreed to aid the king of the Romans to secure the
imperial crown. A succession difficulty in Bavaria-Landshut was only
decided after Maximilian had taken up arms and narrowly escaped with his
life at Regensburg. In the settlement of this question, made in 1505, he
secured a considerable increase of territory, and when the king met the
diet at Cologne in 1505 he was at the height of his power. His enemies
at home were crushed, and their leader, Berthold, elector of Mainz, was
dead; while the outlook abroad was more favourable than it had been
since his accession.

It is at this period that Ranke believes Maximilian to have entertained
the idea of a universal monarchy; but whatever hopes he may have had
were shattered by the death of his son Philip and the rupture of the
treaty of Blois. The diet of Cologne discussed the question of reform in
a halting fashion, but afforded the king supplies for an expedition into
Hungary, to aid his ally Ladislaus, and to uphold his own influence in
the East. Having established his daughter Margaret as regent for Charles
in the Netherlands, Maximilian met the diet at Constance in 1507, when
the imperial chamber (_Reichskammergericht_) was revised and took a more
permanent form, and help was granted for an expedition to Italy. The
king set out for Rome to secure his coronation, but Venice refused to
let him pass through her territories; and at Trant, on the 4th of
February 1508, he took the important step of assuming the title of Roman
Emperor Elect, to which he soon received the assent of pope Julius II.
He attacked the Venetians, but finding the war unpopular with the
trading cities of southern Germany, made a truce with the republic for
three years. The treaty of Blois had contained a secret article
providing for an attack on Venice, and this ripened into the league of
Cambray, which was joined by the emperor in December 1509. He soon took
the field, but after his failure to capture Padua the league broke up;
and his sole ally, the French king, joined him in calling a general
council at Pisa to discuss the question of Church reform. A breach with
pope Julius followed, and at this time Maximilian appears to have
entertained, perhaps quite seriously, the idea of seating himself in the
chair of St Peter. After a period of vacillation he deserted Louis and
joined the Holy League, which had been formed to expel the French from
Italy; but unable to raise troops, he served with the English forces as
a volunteer and shared in the victory gained over the French at the
battle of the Spurs near Thérouanne on the 16th of August 1513. In 1500
the diet had divided Germany into six circles, for the maintenance of
peace, to which the emperor at the diet of Cologne in 1512 added four
others. Having made an alliance with Christian II., king of Denmark, and
interfered to protect the Teutonic Order against Sigismund I., king of
Poland, Maximilian was again in Italy early in 1516 fighting the French
who had overrun Milan. His want of success compelled him on the 4th of
December 1516 to sign the treaty of Brussels, which left Milan in the
hands of the French king, while Verona was soon afterwards transferred
to Venice. He attempted in vain to secure the election of his grandson
Charles as king of the Romans, and in spite of increasing infirmity was
eager to lead the imperial troops against the Turks. At the diet of
Augsburg in 1518 the emperor heard warnings of the Reformation in the
shape of complaints against papal exactions, and a repetition of the
complaints preferred at the diet of Mainz in 1517 about the
administration of Germany. Leaving the diet, he travelled to Wels in
Upper Austria, where he died on the 12th of January 1519. He was buried
in the church of St George in Vienna Neustadt, and a superb monument,
which may still be seen, was raised to his memory at Innsbruck.

  Maximilian had many excellent personal qualities. He was not handsome,
  but of a robust and well-proportioned frame. Simple in his habits,
  conciliatory in his bearing, and catholic in his tastes, he enjoyed
  great popularity and rarely made a personal enemy. He was a skilled
  knight and a daring huntsman, and although not a great general, was
  intrepid on the field of battle. His mental interests were extensive.
  He knew something of six languages, and could discuss art, music,
  literature or theology. He reorganized the university of Vienna and
  encouraged the development of the universities of Ingolstadt and
  Freiburg. He was the friend and patron of scholars, caused manuscripts
  to be copied and medieval poems to be collected. He was the author of
  military reforms, which included the establishment of standing troops,
  called _Landsknechte_, the improvement of artillery by making cannon
  portable, and some changes in the equipment of the cavalry. He was
  continually devising plans for the better government of Austria, and
  although they ended in failure, he established the unity of the
  Austrian dominions. Maximilian has been called the second founder of
  the house of Habsburg, and certainly by bringing about marriages
  between Charles and Joanna and between his grandson Ferdinand and
  Anna, daughter of Ladislaus, king of Hungary and Bohemia, he paved the
  way for the vast empire of Charles V. and for the influence of the
  Habsburgs in eastern Europe. But he had many qualities less desirable.
  He was reckless and unstable, resorting often to lying and deceit, and
  never pausing to count the cost of an enterprise or troubling to adapt
  means to ends. For absurd and impracticable schemes in Italy and
  elsewhere he neglected Germany, and sought to involve its princes in
  wars undertaken solely for private aggrandizement or personal
  jealousy. Ignoring his responsibilities as ruler of Germany, he only
  considered the question of its government when in need of money and
  support from the princes. As the "last of the knights" he could not
  see that the old order of society was passing away and a new order
  arising, while he was fascinated by the glitter of the medieval empire
  and spent the better part of his life in vague schemes for its
  revival. As "a gifted amateur in politics" he increased the disorder
  of Germany and Italy and exposed himself and the empire to the jeers
  of Europe.

  Maximilian was also a writer of books, and his writings display his
  inordinate vanity. His _Geheimes Jagdbuch_, containing about 2500
  words, is a treatise purporting to teach his grandsons the art of
  hunting. He inspired the production of _The Dangers and Adventures of
  the Famous Hero and Knight Sir Teuerdank_, an allegorical poem
  describing his adventures on his journey to marry Mary of Burgundy.
  The emperor's share in the work is not clear, but it seems certain
  that the general scheme and many of the incidents are due to him. It
  was first published at Nuremberg by Melchior Pfintzing in 1517, and
  was adorned with woodcuts by Hans Leonhard Schäufelein. The
  _Weisskunig_ was long regarded as the work of the emperor's secretary,
  Marx Treitzsaurwein, but it is now believed that the greater part of
  the book at least is the work of the emperor himself. It is an
  unfinished autobiography containing an account of the achievements of
  Maximilian, who is called "the young white king." It was first
  published at Vienna in 1775. He also is responsible for _Freydal_, an
  allegorical account of the tournaments in which he took part during
  his wooing of Mary of Burgundy; _Ehrenpforten_, _Triumphwagen_ and
  _Der weisen könige Stammbaum_, books concerning his own history and
  that of the house of Habsburg, and works on various subjects, as _Das
  Stahlbuch_, _Die Baumeisterei_ and _Die Gärtnerei_. These works are
  all profusely illustrated, some by Albrecht Dürer, and in the
  preparation of the woodcuts Maximilian himself took the liveliest
  interest. A facsimile of the original editions of Maximilian's
  autobiographical and semi-autobiographical works has been published in
  nine volumes in the _Jahrbücher der kunsthistorischen Sammlungen des
  Kaiserhauses_ (Vienna, 1880-1888). For this edition S. Laschitzer
  wrote an introduction to _Sir Teuerdank_, Q. von Leitner to _Freydal_,
  and N. A. von Schultz to _Der Weisskunig_. The Holbein society issued
  a facsimile of _Sir Teuerdank_ (London, 1884) and _Triumphwagen_
  (London, 1883).

  See _Correspondance de l'empereur Maximilien I. et de Marguerite
  d'Autriche, 1507-1519_, edited by A. G. le Glay (Paris, 1839);
  _Maximilians I. vertraulicher Briefwechsel mit Sigmund Prüschenk_,
  edited by V. von Kraus (Innsbruck, 1875); J. Chmel, _Urkunden, Briefe
  und Aktenstücke zur Geschichte Maximilians I. und seiner Zeit_.
  (Stuttgart, 1845) and _Aktenstücke und Briefe zur Geschichte des
  Hauses Habsburg im Zeitalter Maximilians I._ (Vienna, 1854-1858); K.
  Klüpfel, _Kaiser Maximilian I._ (Berlin, 1864); H. Ulmann, _Kaiser
  Maximilian I._ (Stuttgart, 1884); L. P. Gachard, _Lettres inédites de
  Maximilien I. sur les affaires des Pays Bas_ (Brussels, 1851-1852); L.
  von Ranke, _Geschichte der romanischen und germanischen Völker,
  1494-1514_ (Leipzig, 1874); R. W. S. Watson, _Maximilian I._ (London,
  1902); A. Jäger, _Über Kaiser Maximilians I. Verhältnis zum Papstthum_
  (Vienna, 1854); H. Ulmann, _Kaiser Maximilians I. Absichten auf das
  Papstthum_ (Stuttgart, 1888), and A. Schulte, _Kaiser Maximilian I.
  als Kandidat für den päpstlichen Stuhl_ (Leipzig, 1906).
       (A. W. H.*)




MAXIMILIAN II. (1527-1576), Roman emperor, was the eldest son of the
emperor Ferdinand I. by his wife Anne, daughter of Ladislaus, king of
Hungary and Bohemia, and was born in Vienna on the 31st of July 1527.
Educated principally in Spain, he gained some experience of warfare
during the campaign of Charles V. against France in 1544, and also
during the war of the league of Schmalkalden, and soon began to take
part in imperial business. Having in September 1548 married his cousin
Maria, daughter of Charles V., he acted as the emperor's representative
in Spain from 1548 to 1550, returning to Germany in December 1550 in
order to take part in the discussion over the imperial succession.
Charles V. wished his son Philip (afterwards king of Spain) to succeed
him as emperor, but his brother Ferdinand, who had already been
designated as the next occupant of the imperial throne, and Maximilian
objected to this proposal. At length a compromise was reached. Philip
was to succeed Ferdinand, but during the former's reign Maximilian, as
king of the Romans, was to govern Germany. This arrangement was not
carried out, and is only important because the insistence of the emperor
seriously disturbed the harmonious relations which had hitherto existed
between the two branches of the Habsburg family; and the estrangement
went so far that an illness which befell Maximilian in 1552 was
attributed to poison given to him in the interests of his cousin and
brother-in-law, Philip of Spain. About this time he took up his
residence in Vienna, and was engaged mainly in the government of the
Austrian dominions and in defending them against the Turks. The
religious views of the king of Bohemia, as Maximilian had been called
since his recognition as the future ruler of that country in 1549, had
always been somewhat uncertain, and he had probably learned something of
Lutheranism in his youth; but his amicable relations with several
Protestant princes, which began about the time of the discussion over
the succession, were probably due more to political than to religious
considerations. However, in Vienna he became very intimate with
Sebastian Pfauser (1520-1569), a court preacher with strong leanings
towards Lutheranism, and his religious attitude caused some uneasiness
to his father. Fears were freely expressed that he would definitely
leave the Catholic Church, and when Ferdinand became emperor in 1558 he
was prepared to assure Pope Paul IV. that his son should not succeed him
if he took this step. Eventually Maximilian remained nominally an
adherent of the older faith, although his views were tinged with
Lutheranism until the end of his life. After several refusals he
consented in 1560 to the banishment of Pfauser, and began again to
attend the services of the Catholic Church. This uneasiness having been
dispelled, in November 1562 Maximilian was chosen king of the Romans, or
German king, at Frankfort, where he was crowned a few days later, after
assuring the Catholic electors of his fidelity to their faith, and
promising the Protestant electors that he would publicly accept the
confession of Augsburg when he became emperor. He also took the usual
oath to protect the Church, and his election was afterwards confirmed by
the papacy. In September 1563 he was crowned king of Hungary, and on his
father's death, in July 1564, succeeded to the empire and to the
kingdoms of Hungary and Bohemia.

The new emperor had already shown that he believed in the necessity for
a thorough reform of the Church. He was unable, however, to obtain the
consent of Pope Pius IV. to the marriage of the clergy, and in 1568 the
concession of communion in both kinds to the laity was withdrawn. On his
part Maximilian granted religious liberty to the Lutheran nobles and
knights in Austria, and refused to allow the publication of the decrees
of the council of Trent. Amid general expectations on the part of the
Protestants he met his first Diet at Augsburg in March 1566. He refused
to accede to the demands of the Lutheran princes; on the other hand,
although the increase of sectarianism was discussed, no decisive steps
were taken to suppress it, and the only result of the meeting was a
grant of assistance for the Turkish War, which had just been renewed.
Collecting a large and splendid army Maximilian marched to defend his
territories; but no decisive engagement had taken place when a truce was
made in 1568, and the emperor continued to pay tribute to the sultan for
Hungary. Meanwhile the relations between Maximilian and Philip of Spain
had improved; and the emperor's increasingly cautious and moderate
attitude in religious matters was doubtless due to the fact that the
death of Philip's son, Don Carlos, had opened the way for the succession
of Maximilian, or of one of his sons, to the Spanish throne. Evidence
of this friendly feeling was given in 1570, when the emperor's daughter,
Anne, became the fourth wife of Philip; but Maximilian was unable to
moderate the harsh proceedings of the Spanish king against the revolting
inhabitants of the Netherlands. In 1570 the emperor met the diet at
Spires and asked for aid to place his eastern borders in a state of
defence, and also for power to repress the disorder caused by troops in
the service of foreign powers passing through Germany. He proposed that
his consent should be necessary before any soldiers for foreign service
were recruited in the empire; but the estates were unwilling to
strengthen the imperial authority, the Protestant princes regarded the
suggestion as an attempt to prevent them from assisting their
coreligionists in France and the Netherlands, and nothing was done in
this direction, although some assistance was voted for the defence of
Austria. The religious demands of the Protestants were still
unsatisfied, while the policy of toleration had failed to give peace to
Austria. Maximilian's power was very limited; it was inability rather
than unwillingness that prevented him from yielding to the entreaties of
Pope Pius V. to join in an attack on the Turks both before and after the
victory of Lepanto in 1571; and he remained inert while the authority of
the empire in north-eastern Europe was threatened. His last important
act was to make a bid for the throne of Poland, either for himself or
for his son Ernest. In December 1575 he was elected by a powerful
faction, but the diet which met at Regensburg was loath to assist; and
on the 12th of October 1576 the emperor died, refusing on his deathbed
to receive the last sacraments of the Church.

By his wife Maria he had a family of nine sons and six daughters. He was
succeeded by his eldest surviving son, Rudolph, who had been chosen king
of the Romans in October 1575. Another of his sons, Matthias, also
became emperor; three others, Ernest, Albert and Maximilian, took some
part in the government of the Habsburg territories or of the
Netherlands, and a daughter, Elizabeth, married Charles IX. king of
France.

  The religious attitude of Maximilian has given rise to much
  discussion, and on this subject the writings of W. Maurenbrecher, W.
  Goetz and E. Reimann in the _Historische Zeitschrift_, Bände VII.,
  XV., XXXII. and LXXVII. (Munich, 1870 fol.) should be consulted, and
  also O. H. Hopfen, _Maximilian II. und der Kompromisskatholizismus_
  (Munich, 1895); C. Haupt, _Melanchthons und seiner Lehrer Einfluss auf
  Maximilian II._ (Wittenberg, 1897); F. Walter, _Die Wahl Maximilians
  II._ (Heidelberg, 1892); W. Goetz, _Maximilians II. Wahl zum römischen
  Könige_ (Würzburg, 1891), and T. J. Scherg, _Über die religiöse
  Entwickelung Kaiser Maximilians II. bis zu seiner Wahl zum römischen
  Könige_ (Würzburg, 1903). For a more general account of his life and
  work see _Briefe und Akten zur Geschichte Maximilians II._, edited by
  W. E. Schwarz (Paderborn, 1889-1891); M. Koch, _Quellen zur Geschichte
  des Kaisers Maximilian II. in Archiven gesammelt_ (Leipzig,
  1857-1861); R. Holtzmann, _Kaiser Maximilian II. bis zu seiner
  Thronbesteigung_ (Berlin, 1903); E. Wertheimer, _Zur Geschichte der
  Türkenkriege Maximilians II._ (Vienna, 1875); L. von Ranke, _Über die
  Zeiten Ferdinands I. und Maximilians II._ in Band VII. of his
  _Sämmtliche Werke_ (Leipzig, 1874), and J. Janssen, _Geschichte des
  deutschen Volkes seit dem Ausgang des Mittelalters,_ Bände IV. to
  VIII. (Freiburg, 1885-1894), English translation by M. A. Mitchell and
  A. M. Christie (London, 1896 fol.).




MAXIMILIAN (1832-1867), emperor of Mexico, second son of the archduke
Francis Charles of Austria, was born in the palace of Schönbrunn, on the
6th of July 1832. He was a particularly clever boy, showed considerable
taste for the arts, and early displayed an interest in science,
especially botany. He was trained for the navy, and threw himself into
this career with so much zeal that he quickly rose to high command, and
was mainly instrumental in creating the naval port of Trieste and the
fleet with which Tegethoff won his victories in the Italian War. He had
some reputation as a Liberal, and this led, in February 1857, to his
appointment as viceroy of the Lombardo-Venetian kingdom; in the same
year he married the Princess Charlotte, daughter of Leopold I., king of
the Belgians. On the outbreak of the war of 1859 he retired into private
life, chiefly at Trieste, near which he built the beautiful chateau of
Miramar. In this same year he was first approached by Mexican exiles
with the proposal to become the candidate for the throne of Mexico. He
did not at first accept, but sought to satisfy his restless desire for
adventure by a botanical expedition to the tropical forests of Brazil.
In 1863, however, under pressure from Napoleon III., and after General
Forey's capture of the city of Mexico and the plebiscite which confirmed
his proclamation of the empire, he consented to accept the crown. This
decision was contrary to the advice of his brother, the emperor Francis
Joseph, and involved the loss of all his rights in Austria. Maximilian
landed at Vera Cruz on the 28th of May 1864; but from the very outset he
found himself involved in difficulties of the most serious kind, which
in 1866 made apparent to almost every one outside of Mexico the
necessity for his abdicating. Though urged to this course by Napoleon
himself, whose withdrawal from Mexico was the final blow to his cause,
Maximilian refused to desert his followers. Withdrawing, in February
1867, to Querétaro, he there sustained a siege for several weeks, but on
the 15th of May resolved to attempt an escape through the enemy's lines.
He was, however, arrested before he could carry out this resolution, and
after trial by court-martial was condemned to death. The sentence was
carried out on the 19th of June 1867. His remains were conveyed to
Vienna, where they were buried in the imperial vault early in the
following year. (See MEXICO.)

  Maximilian's papers were published at Leipzig in 1867, in seven
  volumes, under the title _Aus meinem Leben, Reiseskizzen, Aphorismen,
  Gedichte._ See Pierre de la Gorce, _Hist. du Second Empire_, IV., liv.
  xxv. ii. (Paris, 1904); article by von Hoffinger in _Allgemeine
  Deutsche Biographie_, xxi. 70, where authorities are cited.




MAXIMINUS, GAIUS JULIUS VERUS, Roman emperor from A.D. 235 to 238, was
born in a village on the confines of Thrace. He was of barbarian
parentage and was brought up as a shepherd. His immense stature and
enormous feats of strength attracted the attention of the emperor
Septimius Severus. He entered the army, and under Caracalla rose to the
rank of centurion. He carefully absented himself from court during the
reign of Heliogabalus, but under his successor Alexander Severus, was
appointed supreme commander of the Roman armies. After the murder of
Alexander in Gaul, hastened, it is said, by his instigation, Maximinus
was proclaimed emperor by the soldiers on the 19th of March 235. The
three years of his reign, which were spent wholly in the camp, were
marked by great cruelty and oppression; the widespread discontent thus
produced culminated in a revolt in Africa and the assumption of the
purple by Gordian (q.v.). Maximinus, who was in Pannonia at the time,
marched against Rome, and passing over the Julian Alps descended on
Aquileia; while detained before that city he and his son were murdered
in their tent by a body of praetorians. Their heads were cut off and
despatched to Rome, where they were burnt on the Campus Martius by the
exultant crowd.

  Capitolinus, _Maximini duo_; Herodian vi. 8, vii., viii. 1-5; Zosimus
  i. 13-15.




MAXIMINUS [MAXIMIN], GALERIUS VALERIUS, Roman emperor from A.D. 308 to
314, was originally an Illyrian shepherd named Daia. He rose to high
distinction after he had joined the army, and in 305 he was raised by
his uncle, Galerius, to the rank of Caesar, with the government of Syria
and Egypt. In 308, after the elevation of Licinius, he insisted on
receiving the title of Augustus; on the death of Galerius, in 311, he
succeeded to the supreme command of the provinces of Asia, and when
Licinius and Constantine began to make common cause with one another
Maximinus entered into a secret alliance with Maxentius. He came to an
open rupture with Licinius in 313, sustained a crushing defeat in the
neighbourhood of Heraclea Pontica on the 30th of April, and fled, first
to Nicomedia and afterwards to Tarsus, where he died in August
following. His death was variously ascribed "to despair, to poison, and
to the divine justice." Maximinus has a bad name in Christian annals, as
having renewed persecution after the publication of the toleration edict
of Galerius, but it is probable that he has been judged too harshly.

  See MAXENTIUS; Zosimus ii. 8; Aurelius Victor, _Epit_. 40.




MAXIMS, LEGAL. A maxim is an established principle or proposition. The
Latin term _maxima_ is not to be found in Roman law with any meaning
exactly analogous to that of a legal maxim in the modern sense of the
word, but the treatises of many of the Roman jurists on _Regulae
definitiones_, and _Sententiae juris_ are, in some measure, collections
of maxims (see an article on "Latin Maxims in English Law" in _Law Mag.
and Rev._ xx. 285); Fortescue (_De laudibus_, c. 8) and Du Cange treat
_maxima_ and _regula_ as identical. The attitude of early English
commentators towards the maxims of the law was one of unmingled
adulation. In _Doctor and Student_ (p. 26) they are described as "of the
same strength and effect in the law as statutes be." Coke (Co. _Litt._
11 A) says that a maxim is so called "Quia maxima est ejus dignitas et
certissima auctoritas, atque quod maxime omnibus probetur." "Not only,"
observes Bacon in the Preface to his _Collection of Maxims_, "will the
use of maxims be in deciding doubt and helping soundness of judgment,
but, further, in gracing argument, in correcting unprofitable subtlety,
and reducing the same to a more sound and substantial sense of law, in
reclaiming vulgar errors, and, generally, in the amendment in some
measure of the very nature and complexion of the whole law." A similar
note was sounded in Scotland; and it has been well observed that "a
glance at the pages of Morrison's _Dictionary_ or at other early reports
will show how frequently in the older Scots law questions respecting the
rights, remedies and liabilities of individuals were determined by an
immediate reference to legal maxims" (J. M. Irving, _Encyclo. Scots
Law_, s.v. "Maxims"). In later times less value has been attached to the
maxims of the law, as the development of civilization and the increasing
complexity of business relations have shown the necessity of qualifying
the propositions which they enunciate (see Stephen, _Hist. Crim. Law_,
ii. 94 _n: Yarmouth_ v. _France_, 1887, 19 Q.B.D., per Lord Esher, at p.
653, and American authorities collected in Bouvier's _Law Dict._ s.v.
"Maxim"). But both historically and practically they must always possess
interest and value.

  A brief reference need only be made here, with examples by way of
  illustration, to the field which the maxims of the law cover.

  Commencing with rules founded on public policy, we may note the famous
  principle--_Salus populi suprema lex_ (xii. Tables: Bacon, _Maxims_,
  reg. 12)--"the public welfare is the highest law." It is on this maxim
  that the coercive action of the State towards individual liberty in a
  hundred matters is based. To the same category belong the
  maxims--_Summa ratio est quae pro religione facit_ (Co. _Litt._ 341
  a)--"the best rule is that which advances religion"--a maxim which
  finds its application when the enforcement of foreign laws or
  judgments supposed to violate our own laws or the principles of
  natural justice is in question; and _Dies dominicus non est
  juridicus_, which exempts Sunday from the lawful days for juridical
  acts. Among the maxims relating to the crown, the most important are
  _Rex non potest peccare_ (2 Rolle R. 304)--"The King can do no
  wrong"--which enshrines the principle of ministerial responsibility,
  and _Nullum tempus occurrit regi_ (2 Co. Inst. 273)--"lapse of time
  does not bar the crown," a maxim qualified by various enactments in
  modern times. Passing to the judicial office and the administration of
  justice, we may refer to the rules--_Audi alteram partem_--a
  proposition too familiar to need either translation or comment; _Nemo
  debet esse judex in propriâ suâ causâ_ (12 Co. _Rep._ 114)--"no man
  ought to be judge in his own cause"--a maxim which French law, and the
  legal systems based upon or allied to it, have embodied in an
  elaborate network of rules for judicial challenge; and the maxim which
  defines the relative functions of judge and jury, _Ad quaestionem
  facti non respondent judices, ad quaestionem legis non respondent
  juratores_ (8 Co. _Rep._ 155). The maxim _Boni judicis est ampliare
  jurisdictionem_ (Ch. Prec. 329) is certainly erroneous as it stands,
  as a judge has no right to "extend his jurisdiction." If _justitiam_
  is substituted for _jurisdictionem_, as Lord Mansfield said it should
  be (1 Burr. 304), the maxim is near the truth. A group of maxims
  supposed to embody certain fundamental principles of legal right and
  obligations may next be referred to: (a) _Ubi jus ibi remedium_ (see
  Co. _Litt._ 197 b)--a maxim to which the evolution of the flexible
  "action on the case," by which wrongs unknown to the "original writs"
  were dealt with, was historically due, but which must be taken with
  the gloss _Damnum absque injuria_--"there are forms of actual damage
  which do not constitute legal injury" for which the law supplies no
  remedy; (b) _Actus Dei nemini facit injuriam_ (2 Blackstone, 122)--and
  its allied maxim, _Lex non cogit ad impossibilia_ (Co. _Litt._ 231
  b)--on which the whole doctrine of _vis major_ (_force majeure_) and
  impossible conditions in the law of contract has been built up. In
  this category may also be classed _Volenti non fit injuria_ (Wingate,
  _Maxims_), out of which sprang the theory--now profoundly modified by
  statute--of "common employment" in the law of employers' liability;
  see _Smith_ v. _Baker_, 1891, A.C. 325. Other maxims deal with rights
  of property--_Qui prior est tempore, potior est jure_ (Co. _Litt._ 14
  a), which consecrates the position of the _beati possidentes_ alike in
  municipal and in international law; _Sic utere tuo ut alienum non
  laedas_ (9 Co. _Rep._ 59), which has played its part in the
  determination of the rights of adjacent owners; and _Domus sua cuique
  est tutissimum refugium_ (5 Co. _Rep._ 92)--"a man's house is his
  castle," a doctrine which has imposed limitations on the rights of
  execution creditors (see EXECUTION). In the laws of family relations
  there are the maxims _Consensus non concubitus facit matrimonium_ (Co.
  _Litt._ 33 a)--the canon law of Europe prior to the council of Trent,
  and still law in Scotland, though modified by legislation in England;
  and _Pater is est quem nuptiae demonstrant_ (see Co. _Litt._ 7 b), on
  which, in most civilized countries, the presumption of legitimacy
  depends. In the interpretation of written instruments, the maxim
  _Noscitur a sociis_ (3 _Term Reports_, 87), which proclaims the
  importance of the context, still applies. So do the rules _Expressio
  unius est exclusio alterius_ (Co. _Litt._ 210 a), and _Contemporanea
  expositio est optima et fortissima in lege_ (2 Co. _Inst._ 11), which
  lets in evidence of contemporaneous user as an aid to the
  interpretation of statutes or documents; see _Van Diemen's Land Co._
  v. _Table Cape Marine Board_, 1906, A.C. 92, 98. We may conclude this
  sketch with a miscellaneous summary: _Caveat emptor_ (Hob. 99)--"let
  the purchaser beware"; _Qui facit per alium facile per se_, which
  affirms the principal's liability for the acts of his agent;
  _Ignorantia juris neminem excusat_, on which rests the ordinary
  citizen's obligation to know the law; and _Vigilantibus non
  dormientibus jura subveniunt_ (2 Co. _Inst._ 690), one of the maxims
  in accordance with which courts of equity administer relief. Among
  other "maxims of equity" come the rules that "he that seeks equity
  must do equity," i.e. must act fairly, and that "equity looks upon
  that as done which ought to be done"--a principle from which the
  "conversion" into money of land directed to be sold, and of money
  directed to be invested in the purchase of land, is derived.

  The principal collections of legal maxims are: _English Law_: Bacon,
  _Collection of Some Principal Rules and Maxims of the Common Law_
  (1630); Noy, _Treatise of the principal Grounds and Maxims of the Law
  of England_ (1641, 8th ed., 1824); Wingate, _Maxims of Reason_ (1728);
  Francis, _Grounds and Rudiments of Law and Equity_ (2nd ed. 1751);
  Lofft (annexed to his Reports, 1776); Broom, _Legal Maxims_ (7th ed.
  London, 1900). _Scots Law_: Lord Trayner, _Latin Maxims and Phrases_
  (2nd ed., 1876); Stair, _Institutions of the Law of Scotland_, with
  Index by More (Edinburgh, 1832). _American Treatises_: A. I. Morgan,
  _English Version of Legal Maxims_ (Cincinnati, 1878); S. S. Peloubet,
  _Legal Maxims in Law and Equity_ (New York, 1880).     (A. W. R.)




MAXIMUS, the name of four Roman emperors.

I. M. CLODIUS PUPIENUS MAXIMUS, joint emperor with D. Caelius Calvinus
Balbinus during a few months of the year A.D. 238. Pupienus was a
distinguished soldier, who had been proconsul of Bithynia, Achaea, and
Gallia Narbonensis. At the advanced age of seventy-four, he was chosen by
the senate with Balbinus to resist the barbarian Maximinus. Their complete
equality is shown by the fact that each assumed the titles of pontifex
maximus and princeps senatus. It was arranged that Pupienus should take
the field against Maximinus, while Balbinus remained at Rome to maintain
order, a task in which he signally failed. A revolt of the praetorians was
not repressed till much blood had been shed and a considerable part of the
city reduced to ashes. On his march, Pupienus, having received the news
that Maximinus had been assassinated by his own troops, returned in
triumph to Rome. Shortly afterwards, when both emperors were on the point
of leaving the city on an expedition--Pupienus against the Persians and
Balbinus against the Goths--the praetorians, who had always resented the
appointment of the senatorial emperors and cherished the memory of the
soldier-emperor Maximinus, seized the opportunity of revenge. When most of
the people were at the Capitoline games, they forced their way into the
palace, dragged Balbinus and Pupienus through the streets, and put them to
death.

  See Capitolinus, _Life of Maximus and Balbinus_; Herodian vii. 10,
  viii. 6; Zonaras xii. 16; Orosius vii. 19; Eutropius ix. 2; Zosimus i.
  14; Aurelius Victor, _Caesares_, 26, _epit._ 26; H. Schiller,
  _Geschichte der römischen Kaiserzeit_, i. 2; Gibbon, _Decline and
  Fall_, ch. 7 and (for the chronology) appendix 12 (Bury's edition).

II. MAGNUS MAXIMUS, a native of Spain, who had accompanied Theodosius on
several expeditions and from 368 held high military rank in Britain. The
disaffected troops having proclaimed Maximus emperor, he crossed over
to Gaul, attacked Gratian (q.v.), and drove him from Paris to Lyons,
where he was murdered by a partisan of Maximus. Theodosius being unable
to avenge the death of his colleague, an agreement was made (384 or 385)
by which Maximus was recognized as Augustus and sole emperor in Gaul,
Spain and Britain, while Valentinian II. was to remain unmolested in
Italy and Illyricum, Theodosius retaining his sovereignty in the East.
In 387 Maximus crossed the Alps, Valentinian was speedily put to flight,
while the invader established himself in Milan and for the time became
master of Italy. Theodosius now took vigorous measures. Advancing with a
powerful army, he twice defeated the troops of Maximus--at Siscia on the
Save, and at Poetovio on the Danube. He then hurried on to Aquileia,
where Maximus had shut himself up, and had him beheaded. Under the name
of Maxen Wledig, Maximus appears in the list of Welsh royal heroes (see
R. Williams, _Biog. Dict. of Eminent Welshmen_, 1852; "The Dream of
Maxen Wledig," in the _Mabinogion_).

  Full account with classical references in H. Richter, _Das
  weströmische Reich, besonders unter den Kaisern Gratian, Valentinian
  II. und Maximus_ (1865); see also H. Schiller, _Geschichte der
  römischen Kaiserzeit_, ii. (1887); Gibbon, _Decline and Fall_, ch. 27;
  Tillemont, _Hist. des empereurs_, v.

III. MAXIMUS TYRANNUS, made emperor in Spain by the Roman general,
Gerontius, who had rebelled against the usurper Constantine in 408.
After the defeat of Gerontius at Arelate (Arles) and his death in 411
Maximus renounced the imperial title and was permitted by Constantine to
retire into private life. About 418 he rebelled again, but, failing in
his attempt, was seized, carried into Italy, and put to death at Ravenna
in 422.

  See Orosius vii. 42; Zosimus vi. 5; Sozomen ix. 3; E. A. Freeman, "The
  Tyrants of Britain, Gaul and Spain, A.D. 406-411," in _English
  Historical Review_, i. (1886).

IV. PETRONIUS MAXIMUS, a member of the higher Roman nobility, had held
several court and public offices, including those of _praefectus Romae_
(420) and _Italiae_ (439-441 and 445), and consul (433, 443). He was one
of the intimate associates of Valentinian III., whom he assisted in the
palace intrigues which led to the death of Aëtius in 454; but an outrage
committed on the wife of Maximus by the emperor turned his friendship
into hatred. Maximus was proclaimed emperor immediately after
Valentinian's murder (March 16, 455), but after reigning less than three
months, he was murdered by some Burgundian mercenaries as he was fleeing
before the troops of Genseric, who, invited by Eudoxia, the widow of
Valentinian, had landed at the mouth of the Tiber (May or June 455).

  See Procopius, _Vand._ i. 4; Sidonius Apollinaris, _Panegyr. Aviti_,
  ep. ii. 13; the various _Chronicles_; Gibbon, _Decline and Fall_, chs.
  35, 36; Tillemont, _Hist. des empereurs_, vi.




MAXIMUS, ST (c. 580-662), abbot of Chrysopolis, known as "the Confessor"
from his orthodox zeal in the Monothelite (q.v.) controversy, or as "the
monk," was born of noble parentage at Constantinople about the year 580.
Educated with great care, he early became distinguished by his talents
and acquirements, and some time after the accession of the emperor
Heraclius in 610 was made his private secretary. In 630 he abandoned the
secular life and entered the monastery of Chrysopolis (Scutari),
actuated, it was believed, less by any longing for the life of a recluse
than by the dissatisfaction he felt with the Monothelite leanings of his
master. The date of his promotion to the abbacy is uncertain. In 633 he
was one of the party of Sophronius of Jerusalem (the chief original
opponent of the Monothelites) at the council of Alexandria; and in 645
he was again in Africa, when he held in presence of the governor and a
number of bishops the disputation with Pyrrhus, the deposed and banished
patriarch of Constantinople, which resulted in the (temporary)
conversion of his interlocutor to the Dyothelite view. In the following
year several African synods, held under the influence of Maximus,
declared for orthodoxy. In 649, after the accession of Martin I., he
went to Rome, and did much to fan the zeal of the new pope, who in
October of that year held the (first) Lateran synod, by which not only
the Monothelite doctrine but also the moderating _ecthesis_ of Heraclius
and _typus_ of Constans II. were anathematized. About 653 Maximus, for
the part he had taken against the latter document especially, was
apprehended (together with the pope) by order of Constans and carried a
prisoner to Constantinople. In 655, after repeated examinations, in
which he maintained his theological opinions with memorable constancy,
he was banished to Byzia in Thrace, and afterwards to Perberis. In 662
he was again brought to Constantinople and was condemned by a synod to
be scourged, to have his tongue cut out by the root, and to have his
right hand chopped off. After this sentence had been carried out he was
again banished to Lazica, where he died on the 13th of August 662. He is
venerated as a saint both in the Greek and in the Latin Churches.
Maximus was not only a leader in the Monothelite struggle but a mystic
who zealously followed and advocated the system of Pseudo-Dionysius,
while adding to it an ethical element in the conception of the freedom
of the will. His works had considerable influence in shaping the system
of John Scotus Erigena.

  The most important of the works of Maximus will be found in Migne,
  _Patrologia graeca_, xc. xci., together with an anonymous life; an
  exhaustive list in Wagenmann's article in vol. xii. (1903) of
  Hauck-Herzog's _Realencyklopädie_ where the following classification
  is adopted: (a) exegetical, (b) scholia on the Fathers, (c) dogmatic
  and controversial, (d) ethical and ascetic, (e) miscellaneous. The
  details of the disputation with Pyrrhus and of the martyrdom are given
  very fully and clearly in Hefele's _Conciliengeschichte_, iii. For
  further literature see H. Gelzer in C. Krumbacher's _Geschichte der
  byzantinischen Litteratur_ (1897).




MAXIMUS OF SMYRNA, a Greek philosopher of the Neo-platonist school, who
lived towards the end of the 4th century A.D. He was perhaps the most
important of the followers of Iamblichus. He is said to have been of a
rich and noble family, and exercised great influence over the emperor
Julian, who was commended to him by Aedesius. He pandered to the
emperor's love of magic and theurgy, and by judicious administration of
the omens won a high position at court. His overbearing manner made him
numerous enemies, and, after being imprisoned on the death of Julian, he
was put to death by Valens. He is a representative of the least
attractive side of Neoplatonism. Attaching no value to logical proof and
argument, he enlarged on the wonders and mysteries of nature, and
maintained his position by the working of miracles. In logic he is
reported to have agreed with Eusebius, Iamblichus and Porphyry in
asserting the validity of the second and third figures of the syllogism.




MAXIMUS OF TYRE (CASSIUS MAXIMUS TYRIUS), a Greek rhetorician and
philosopher who flourished in the time of the Antonines and Commodus
(2nd century A.D.). After the manner of the sophists of his age, he
travelled extensively, delivering lectures on the way. His writings
contain many allusions to the history of Greece, while there is little
reference to Rome; hence it is inferred that he lived longer in Greece,
perhaps as a professor at Athens. Although nominally a Platonist, he is
really an Eclectic and one of the precursors of Neoplatonism. There are
still extant by him forty-one essays or discourses ([Greek: dialexeis])
on theological, ethical, and other philosophical commonplaces. With him
God is the supreme being, one and indivisible though called by many
names, accessible to reason alone; but as animals form the intermediate
stage between plants and human beings, so there exist intermediaries
between God and man, viz. daemons, who dwell on the confines of heaven
and earth. The soul in many ways bears a great resemblance to the
divinity; it is partly mortal, partly immortal, and, when freed from the
fetters of the body, becomes a daemon. Life is the sleep of the soul,
from which it awakes at death. The style of Maximus is superior to that
of the ordinary sophistical rhetorician, but scholars differ widely as
to the merits of the essays themselves.

Maximus of Tyre must be distinguished from the Stoic Maximus, tutor of
Marcus Aurelius.

  Editions by J. Davies, revised with valuable notes by J. Markland
  (1740); J. J. Reiske (1774); F. Dübner (1840, with Theophrastus, &c.,
  in the Didot series). Monographs by R. Rohdich (Beuthen, 1879); H.
  Hobein, _De Maximo Tyrio quaestiones philol._ (Jena, 1895). There is
  an English translation (1804) by Thomas Taylor, the Platonist.




MAX MÜLLER, FRIEDRICH (1823-1900), Anglo-German orientalist and
comparative philologist, was born at Dessau on the 6th of December 1823,
being the son of Wilhelm Müller (1794-1827), the German poet, celebrated
for his phil-Hellenic lyrics, who was ducal librarian at Dessau. The
elder Müller had endeared himself to the most intellectual circles in
Germany by his amiable character and his genuine poetic gift; his songs
had been utilized by musical composers, notably Schubert; and it was his
son's good fortune to meet in his youth with a succession of eminent
friends, who, already interested in him for his father's sake, and
charmed by the qualities which they discovered in the young man himself,
powerfully aided him by advice and patronage. Mendelssohn, who was his
godfather, dissuaded him from indulging his natural bent to the study of
music; Professor Brockhaus of the University of Leipzig, where Max
Müller matriculated in 1841, induced him to take up Sanskrit; Bopp, at
the University of Berlin (1844), made the Sanskrit student a scientific
comparative philologist; Schelling at the same university, inspired him
with a love for metaphysical speculation, though failing to attract him
to his own philosophy; Burnouf, at Paris in the following year, by
teaching him Zend, started him on the track of inquiry into the science
of comparative religion, and impelled him to edit the _Rig Veda_; and
when, in 1846, Max Müller came to England upon this errand, Bunsen, in
conjunction with Professor H. H. Wilson, prevailed upon the East India
Company to undertake the expense of publication. Up to this time Max
Müller had lived the life of a poor student, supporting himself partly
by copying manuscripts, but Bunsen's introductions to Queen Victoria and
the prince consort, and to Oxford University, laid the foundation for
him of fame and fortune. In 1848 the printing of his _Rig Veda_ at the
University Press obliged him to settle in Oxford, a step which decided
his future career. He arrived at a favourable conjuncture: the
Tractarian strife, which had so long thrust learning into the
background, was just over, and Oxford was becoming accessible to modern
ideas. The young German excited curiosity and interest, and it was soon
discovered that, although a genuine scholar, he was no mere bookworm.
Part of his social success was due to his readiness to exert his musical
talents at private parties. Max Müller was speedily subjugated by the
_genius loci_. He was appointed deputy Taylorian professor of modern
languages in 1850, and the German government failed to tempt him back to
Strassburg. In the following year he was made M.A. and honorary fellow
of Christ Church, and in 1858 he was elected a fellow of All Souls. In
1854 the Crimean War gave him the opportunity of utilizing his oriental
learning in vocabularies and schemes of transliteration. In 1857 he
successfully essayed another kind of literature in his beautiful story
_Deutsche Liebe_, written both in German and English. He had by this
time become an extensive contributor to English periodical literature,
and had written several of the essays subsequently collected as _Chips
from a German Workshop_. The most important of them was the fascinating
essay on "Comparative Mythology" in the _Oxford Essays_ for 1856. His
valuable _History of Ancient Sanskrit Literature_, so far as it
illustrates the primitive religion of the Brahmans (and hence the Vedic
period only), was published in 1850.

Though Max Müller's reputation was that of a comparative philologist and
orientalist, his professional duties at Oxford were long confined to
lecturing on modern languages, or at least their medieval forms. In 1860
the death of Horace Hayman Wilson, professor of Sanskrit, seemed to open
a more congenial sphere to him. His claims to the succession seemed
incontestable, for his opponent, Monier Williams, though well qualified
as a Sanskritist, lacked Max Müller's brilliant versatility, and
although educated at Oxford, had held no University office. But Max
Müller was a Liberal, and the friend of Liberals in university matters,
in politics, and in theology, and this consideration united with his
foreign birth to bring the country clergy in such hosts to the poll that
the voice of resident Oxford was overborne, and Monier Williams was
elected by a large majority. It was the one great disappointment of Max
Müller's life, and made a lasting impression upon him. It was,
nevertheless, serviceable to his influence and reputation by permitting
him to enter upon a wider field of subjects than would have been
possible otherwise. Directly, Sanskrit philology received little more
from him, except in connexion with his later undertaking of _The Sacred
Books of the East_; but indirectly he exalted it more than any
predecessor by proclaiming its commanding position in the history of the
human intellect by his _Science of Language_, two courses of lectures
delivered at the Royal Institution in 1861 and 1863. Max Müller ought
not to be described as "the introducer of comparative philology into
England." Prichard had proved the Aryan affinities of the Celtic
languages by the methods of comparative philology so long before as
1831; Winning's _Manual of Comparative Philology_ had been published in
1838; the discoveries of Bopp and Pott and Pictet had been recognized in
brilliant articles in the _Quarterly Review_, and had guided the
researches of Rawlinson. But Max Müller undoubtedly did far more to
popularize the subject than had been done, or could have been done, by
any predecessor. He was on less sure ground in another department of the
study of language--the problem of its origin. He wrote upon it as a
disciple of Kant, whose _Critique of Pure Reason_ he translated. His
essays on mythology are among the most delightful of his writings, but
their value is somewhat impaired by a too uncompromising adherence to
the seductive generalization of the solar myth.

Max Müller's studies in mythology led him to another field of activity
in which his influence was more durable and extensive, that of the
comparative science of religions. Here, so far as Great Britain is
concerned, he does deserve the fame of an originator, and his
_Introduction to the Science of Religion_ (1873: the same year in which
he lectured on the subject, at Dean Stanley's invitation, in Westminster
Abbey, this being the only occasion on which a layman had given an
address there) marks an epoch. It was followed by other works of
importance, especially the four volumes of Gifford lectures, delivered
between 1888 and 1892; but the most tangible result of the impulse he
had given was the publication under his editorship, from 1875 onwards,
of _The Sacred Books of the East_, in fifty-one volumes, including
indexes, all but three of which appeared under his superintendence
during his lifetime. These comprise translations by the most competent
scholars of all the really important non-Christian scriptures of
Oriental nations, which can now be appreciated without a knowledge of
the original languages. Max Müller also wrote on Indian philosophy in
his latter years, and his exertions to stimulate search for Oriental
manuscripts and inscriptions were rewarded with important discoveries of
early Buddhist scriptures, in their Indian form, made in Japan. He was
on particularly friendly terms with native Japanese scholars, and after
his death his library was purchased by the university of Tôkyô.

In 1868 Max Müller had been indemnified for his disappointment over the
Sanskrit professorship by the establishment of a chair of Comparative
Philology to be filled by him. He retired, however, from the actual
duties of the post in 1875, when entering upon the editorship of _The
Sacred Books of the East_. The most remarkable external events of his
latter years were his delivery of lectures at the restored university of
Strassburg in 1872, when he devoted his honorarium to the endowment of a
Sanskrit lectureship, and his presidency over the International Congress
of Orientalists in 1892. But his days, if uneventful, were busy. He
participated in every movement at Oxford of which he could approve, and
was intimate with nearly all its men of light and leading; he was a
curator of the Bodleian Library, and a delegate of the University Press.
He was acquainted with most of the crowned heads

of Europe, and was an especial favourite with the English royal family.
His hospitality was ample, especially to visitors from India, where he
was far better known than any other European Orientalist. His
distinctions, conferred by foreign governments and learned societies,
were innumerable, and, having been naturalized shortly after his arrival
in England, he received the high honour of being made a privy
councillor. In 1898 and 1899 he published autobiographical reminiscences
under the title of _Auld Lang Syne_. He was writing a more detailed
autobiography when overtaken by death on the 28th of October 1900. Max
Müller married in 1859 Georgiana Adelaide Grenfell, sister of the wives
of Charles Kingsley and J. A. Froude. One of his daughters, Mrs
Conybeare, distinguished herself by a translation of Scherer's _History
of German Literature_.

Though undoubtedly a great scholar, Max Müller did not so much represent
scholarship pure and simple as her hybrid types--the scholar-author and
the scholar-courtier. In the former capacity, though manifesting little
of the originality of genius, he rendered vast service by popularizing
high truths among high minds. In his public and social character he
represented Oriental studies with a brilliancy, and conferred upon them
a distinction, which they had not previously enjoyed in Great Britain.
There were drawbacks in both respects: the author was too prone to build
upon insecure foundations, and the man of the world incurred censure for
failings which may perhaps be best indicated by the remark that he
seemed too much of a diplomatist. But the sum of foibles seems
insignificant in comparison with the life of intense labour dedicated to
the service of culture and humanity.

  Max Müller's _Collected Works_ were published in 1903.     (R. G.)




MAXWELL, the name of a Scottish family, members of which have held the
titles of earl of Morton, earl of Nithsdale, Lord Maxwell, and Lord
Herries. The name is taken probably from Maccuswell, or Maxwell, near
Kelso, whither the family migrated from England about 1100. Sir Herbert
Maxwell won great fame by defending his castle of Carlaverock against
Edward I. in 1300; another Sir Herbert was made a lord of the Scottish
parliament before 1445; and his great-grandson John, 3rd Lord Maxwell,
was killed at Flodden in 1513. John's son Robert, the 4th lord (d.
1546), was a member of the royal council under James V.; he was also an
extraordinary lord of session, high admiral, and warden of the west
marches, and was taken prisoner by the English at the rout of Solway
Moss in 1542. Robert's grandson John, 7th Lord Maxwell (1553-1593), was
the second son of Robert, the 5th lord (d. 1552), and his wife Beatrix,
daughter of James Douglas, 3rd earl of Morton. After the execution of
the regent Morton, the 4th earl, in 1581 this earldom was bestowed upon
Maxwell, but in 1586 the attainder of the late earl was reversed and he
was deprived of his new title. He had helped in 1585 to drive the royal
favourite James Stewart, earl of Arran, from power, and he made active
preparations to assist the invading Spaniards in 1588. His son John, the
8th lord (c. 1586-1613), was at feud with the Johnstones, who had killed
his father in a skirmish, and with the Douglases over the earldom of
Morton, which he regarded as his inheritance. After a life of
exceptional and continuous lawlessness he escaped from Scotland and in
his absence was sentenced to death; having returned to his native
country he was seized and was beheaded in Edinburgh. In 1618 John's
brother and heir Robert (d. 1646) was restored to the lordship of
Maxwell, and in 1620 was created earl of Nithsdale, surrendering at this
time his claim to the earldom of Morton. He and his son Robert,
afterwards the 2nd earl, fought under Montrose for Charles I. during the
Civil War. Robert died without sons in October 1667, when a cousin John
Maxwell, 7th Lord Herries (d. 1677), became third earl.

William, 5th earl of Nithsdale (1676-1744), a grandson of the third
earl, was like his ancestor a Roman Catholic and was attached to the
cause of the exiled house of Stuart. In 1715 he joined the Jacobite
insurgents, being taken prisoner at the battle of Preston and sentenced
to death. He escaped, however, from the Tower of London through the
courage and devotion of his wife Winifred (d. 1749), daughter of William
Herbert, 1st marquess of Powis. He was attainted in 1716 and his titles
became extinct, but his estates passed to his son William (d. 1776),
whose descendant, William Constable-Maxwell, regained the title of Lord
Herries in 1858. The countess of Nithsdale wrote an account of her
husband's escape, which is published in vol. i. of the _Transactions of
the Society of Antiquaries of Scotland_.

  A few words may be added about other prominent members of the Maxwell
  family. John Maxwell (c. 1590-1647), archbishop of Tuam, was a
  Scottish ecclesiastic who took a leading part in helping Archbishop
  Laud in his futile attempt to restore the liturgy in Scotland. He was
  bishop of Ross from 1633 until 1638, when he was deposed by the
  General Assembly; then crossing over to Ireland he was bishop of
  Killala and Achonry from 1640 to 1645, and archbishop of Tuam from
  1645 until his death. James Maxwell of Kirkconnell (c. 1708-1762), the
  Jacobite, wrote the _Narrative of Charles Prince of Wales's Expedition
  to Scotland in 1745_, which was printed for the Maitland Club in 1841.
  Robert Maxwell (1695-1765) was the author of _Select Transactions of
  the Society of Improvers_ and was a great benefactor to Scottish
  agriculture. Sir Murray Maxwell (1775-1831), a naval officer, gained
  much fame by his conduct when his ship the "Alceste" was wrecked in
  Gaspar Strait in 1817. William Hamilton Maxwell (1792-1850), the Irish
  novelist, wrote, in addition to several novels, a _Life of the Duke of
  Wellington_ (1839-1841 and again 1883), and a _History of the Irish
  Rebellion in 1798_ (1845 and 1891). Sir Herbert Maxwell, 7th bart. (b.
  1845), member of parliament for Wigtownshire from 1880 to 1906, and
  president of the Society of Antiquaries of Scotland, became well known
  as a writer, his works including _Life and Times of the Right Hon. W.
  H. Smith_ (1893); _Life of the Duke of Wellington_ (1899); _The House
  of Douglas_ (1902); _Robert the Bruce_ (1897) and _A Duke of Britain_
  (1895).




MAXWELL, JAMES CLERK (1831-1879), British physicist, was the last
representative of a younger branch of the well-known Scottish family of
Clerk of Penicuik, and was born at Edinburgh on the 13th of November
1831. He was educated at the Edinburgh Academy (1840-1847) and the
university of Edinburgh (1847-1850). Entering at Cambridge in 1850, he
spent a term or two at Peterhouse, but afterwards migrated to Trinity.
In 1854 he took his degree as second wrangler, and was declared equal
with the senior wrangler of his year (E. J. Routh, q.v.) in the higher
ordeal of the Smith's prize examination. He held the chair of Natural
Philosophy in Marischal College, Aberdeen, from 1856 till the fusion of
the two colleges there in 1860. For eight years subsequently he held the
chair of Physics and Astronomy in King's College, London, but resigned
in 1868 and retired to his estate of Glenlair in Kirkcudbrightshire. He
was summoned from his seclusion in 1871 to become the first holder of
the newly founded professorship of Experimental Physics in Cambridge;
and it was under his direction that the plans of the Cavendish
Laboratory were prepared. He superintended every step of the progress of
the building and of the purchase of the very valuable collection of
apparatus with which it was equipped at the expense of its munificent
founder the seventh duke of Devonshire (chancellor of the university,
and one of its most distinguished alumni). He died at Cambridge on the
5th of November 1879.

For more than half of his brief life he held a prominent position in the
very foremost rank of natural philosophers. His contributions to
scientific societies began in his fifteenth year, when Professor J. D.
Forbes communicated to the Royal Society of Edinburgh a short paper of
his on a mechanical method of tracing Cartesian ovals. In his eighteenth
year, while still a student in Edinburgh, he contributed two valuable
papers to the _Transactions_ of the same society--one of which, "On the
Equilibrium of Elastic Solids," is remarkable, not only on account of
its intrinsic power and the youth of its author, but also because in it
he laid the foundation of one of the most singular discoveries of his
later life, the temporary double refraction produced in viscous liquids
by shearing stress. Immediately after taking his degree, he read to the
Cambridge Philosophical Society a very novel memoir, "On the
Transformation of Surfaces by Bending." This is one of the few purely
mathematical papers he published, and it exhibited at once to experts
the full genius of its author. About the same time appeared his
elaborate memoir, "On Faraday's Lines of Force," in which he gave the
first indication of some of those extraordinary electrical
investigations which culminated in the greatest work of his life. He
obtained in 1859 the Adams prize in Cambridge for a very original and
powerful essay, "On the Stability of Saturn's Rings." From 1855 to 1872
he published at intervals a series of valuable investigations connected
with the "Perception of Colour" and "Colour-Blindness," for the earlier
of which he received the Rumford medal from the Royal Society in 1860.
The instruments which he devised for these investigations were simple
and convenient, but could not have been thought of for the purpose
except by a man whose knowledge was co-extensive with his ingenuity. One
of his greatest investigations bore on the "Kinetic Theory of Gases."
Originating with D. Bernoulli, this theory was advanced by the
successive labours of John Herapath, J. P. Joule, and particularly R.
Clausius, to such an extent as to put its general accuracy beyond a
doubt; but it received enormous developments from Maxwell, who in this
field appeared as an experimenter (on the laws of gaseous friction) as
well as a mathematician. He wrote an admirable textbook of the _Theory
of Heat_ (1871), and a very excellent elementary treatise on _Matter and
Motion_ (1876).

But the great work of his life was devoted to electricity. He began by
reading, with the most profound admiration and attention, the whole of
Faraday's extraordinary self-revelations, and proceeded to translate the
ideas of that master into the succinct and expressive notation of the
mathematicians. A considerable part of this translation was accomplished
during his career as an undergraduate in Cambridge. The writer had the
opportunity of perusing the MS. of "On Faraday's Lines of Force," in a
form little different from the final one, a year before Maxwell took his
degree. His great object, as it was also the great object of Faraday,
was to overturn the idea of action at a distance. The splendid
researches of S. D. Poisson and K. F. Gauss had shown how to reduce all
the phenomena of statical electricity to mere attractions and repulsions
exerted at a distance by particles of an imponderable on one another.
Lord Kelvin (Sir W. Thomson) had, in 1846, shown that a totally
different assumption, based upon other analogies, led (by its own
special mathematical methods) to precisely the same results. He treated
the resultant electric force at any point as analogous to the _flux of
heat_ from sources distributed in the same manner as the supposed
electric particles. This paper of Thomson's, whose ideas Maxwell
afterwards developed in an extraordinary manner, seems to have given the
first hint that there are at least two perfectly distinct methods of
arriving at the known formulae of statical electricity. The step to
magnetic phenomena was comparatively simple; but it was otherwise as
regards electro-magnetic phenomena, where current electricity is
essentially involved. An exceedingly ingenious, but highly artificial,
theory had been devised by W. E. Weber, which was found capable of
explaining all the phenomena investigated by Ampère as well as the
induction currents of Faraday. But this was based upon the assumption of
a distance-action between electric particles, the intensity of which
depended on their relative motion as well as on their position. This
was, of course, even more repugnant to Maxwell's mind than the statical
distance-action developed by Poisson. The first paper of Maxwell's in
which an attempt at an admissible physical theory of electromagnetism
was made was communicated to the Royal Society in 1867. But the theory,
in a fully developed form, first appeared in 1873 in his great treatise
on _Electricity and Magnetism_. This work was one of the most splendid
monuments ever raised by the genius of a single individual. Availing
himself of the admirable generalized co-ordinate system of Lagrange,
Maxwell showed how to reduce all electric and magnetic phenomena to
stresses and motions of a material medium, and, as one preliminary, but
excessively severe, test of the truth of his theory, he pointed out that
(if the electro-magnetic medium be that which is required for the
explanation of the phenomena of light) the velocity of light in vacuo
should be numerically the same as the ratio of the electro-magnetic and
electrostatic units. In fact, the means of the best determinations of
each of these quantities separately agree with one another more closely
than do the various values of either.

One of Maxwell's last great contributions to science was the editing
(with copious original notes) of the _Electrical Researches of the Hon.
Henry Cavendish_, from which it appeared that Cavendish, already famous
by many other researches (such as the mean density of the earth, the
composition of water, &c.), must be looked on as, in his day, a man of
Maxwell's own stamp as a theorist and an experimenter of the very first
rank.

In private life Clerk Maxwell was one of the most lovable of men, a
sincere and unostentatious Christian. Though perfectly free from any
trace of envy or ill-will, he yet showed on fit occasion his contempt
for that pseudo-science which seeks for the applause of the ignorant by
professing to reduce the whole system of the universe to a fortuitous
sequence of uncaused events.

  His collected works, including the series of articles on the
  properties of matter, such as "Atom," "Attraction," "Capillary
  Action," "Diffusion," "Ether," &c., which he contributed to the 9th
  edition of this encyclopaedia, were issued in two volumes by the
  Cambridge University Press in 1890; and an extended biography, by his
  former schoolfellow and lifelong friend Professor Lewis Campbell, was
  published in 1882.     (P. G. T.)




MAXWELLTOWN, a burgh of barony and police burgh of Kirkcudbrightshire,
Scotland. Pop. (1901), 5796. It lies on the Nith, opposite to Dumfries,
with which it is connected by three bridges, being united with it for
parliamentary purposes. It has a station on the Glasgow & South-Western
line from Dumfries to Kirkcudbright. Its public buildings include a
court-house, the prison for the south-west of Scotland, and an
observatory and museum, housed in a disused windmill. The chief
manufactures are woollens and hosiery, besides dyeworks and sawmills. It
was a hamlet known as Bridgend up till 1810, in which year it was
erected into a burgh of barony under its present name. To the north-west
lies the parish of Terregles, said to be a corruption of Tir-eglwys
(_terra ecclesia_, that is, "Kirk land"). The parish contains the
beautiful ruin of Lincluden Abbey (see DUMFRIES), and Terregles House,
once the seat of William Maxwell, last earl of Nithsdale. In the parish
of Lochrutton, a few miles south-west of Maxwelltown, there is a good
example of a stone circle, the "Seven Grey Sisters," and an old
peel-tower in the Mains of Hills.




MAY, PHIL (1864-1903), English caricaturist, was born at Wortley, near
Leeds, on the 22nd of April 1864, the son of an engineer. His father
died when the child was nine years old, and at twelve he had begun to
earn his living. Before he was fifteen he had acted as time-keeper at a
foundry, had tried to become a jockey, and had been on the stage at
Scarborough and Leeds. When he was about seventeen he went to London
with a sovereign in his pocket. He suffered extreme want, sleeping out
in the parks and streets, until he obtained employment as designer to a
theatrical costumier. He also drew posters and cartoons, and for about
two years worked for the _St Stephen's Review_, until he was advised to
go to Australia for his health. During the three years he spent there he
was attached to the _Sydney Bulletin_, for which many of his best
drawings were made. On his return to Europe he went to Paris by way of
Rome, where he worked hard for some time before he appeared in 1892 in
London to resume his interrupted connexion with the _St Stephen's
Review_. His studies of the London "guttersnipe" and the coster-girl
rapidly made him famous. His overflowing sense of fun, his genuine
sympathy with his subjects, and his kindly wit were on a par with his
artistic ability. It was often said that the extraordinary economy of
line which was a characteristic feature of his drawings had been forced
upon him by the deficiencies of the printing machines of the _Sydney
Bulletin_. It was in fact the result of a laborious process which
involved a number of preliminary sketches, and of a carefully considered
system of elimination. His later work included some excellent political
portraits. He became a regular member of the staff of _Punch_ in 1896,
and in his later years his services were retained exclusively for
_Punch_ and the _Graphic_. He died on the 5th of August 1903.

  There was an exhibition of his drawings at the Fine Arts Society in
  1895, and another at the Leicester Galleries in 1903. A selection of
  his drawings contributed to the periodical press and from _Phil May's
  Annual_ and _Phil May's Sketch Books_, with a portrait and biography
  of the artist, entitled _The Phil May Folio_, appeared in 1903.




MAY, THOMAS (1595-1650), English poet and historian, son of Sir Thomas
May of Mayfield, Sussex, was born in 1595. He entered Sidney Sussex
College, Cambridge, in 1609, and took his B.A. degree three years later.
His father having lost his fortune and sold the family estate, Thomas
May, who was hampered by an impediment in his speech, made literature
his profession. In 1620 he produced _The Heir_, an ingeniously
constructed comedy, and, probably about the same time, _The Old Couple_,
which was not printed until 1658. His other dramatic works are classical
tragedies on the subjects of Antigone, Cleopatra, and Agrippina. F. G.
Fleay has suggested that the more famous anonymous tragedy of _Nero_
(printed 1624, reprints in A. H. Bullen's _Old English Plays_ and the
_Mermaid Series_) should also be assigned to May. But his most important
work in the department of pure literature was his translation (1627)
into heroic couplets of the _Pharsalia_ of Lucan. Its success led May to
write a continuation of Lucan's narrative down to the death of Caesar.
Charles I. became his patron, and commanded him to write metrical
histories of Henry II. and Edward III., which were completed in 1635.
When the earl of Pembroke, then lord chamberlain, broke his staff across
May's shoulders at a masque, the king took him under his protection as
"my poet," and Pembroke made him an apology accompanied with a gift of
£50. These marks of the royal favour seem to have led May to expect the
posts of poet-laureate and city chronologer when they fell vacant on the
death of Ben Jonson in 1637, but he was disappointed, and he forsook the
court and attached himself to the party of the Parliament. In 1646 he is
styled one of the "secretaries" of the Parliament, and in 1647 he
published his best known work, _The History of the Long Parliament_. In
this official apology for the moderate or Presbyterian party, he
professes to give an impartial statement of facts, unaccompanied by any
expression of party or personal opinion. If he refrained from actual
invective, he accomplished his purpose, according to Guizot, by
"omission, palliation and dissimulation." Accusations of this kind were
foreseen by May, who says in his preface that if he gives more
information about the Parliament men than their opponents it is that he
was more conversant with them and their affairs. In 1650 he followed
this with another work written with a more definite bias, a _Breviary of
the History of the Parliament of England_, in Latin and English, in
which he defended the position of the Independents. He stopped short of
the catastrophe of the king's execution, and it seems likely that his
subservience to Cromwell was not quite voluntary. In February 1650 he
was brought to London from Weymouth under a strong guard for having
spread false reports of the Parliament and of Cromwell. He died on the
13th of November in the same year, and was buried in Westminster Abbey,
but after the Restoration his remains were exhumed and buried in a pit
in the yard of St Margaret's, Westminster. May's change of side made him
many bitter enemies, and he is the object of scathing condemnation from
many of his contemporaries.

  There is a long notice of May in the _Biographia Britannica_. See also
  W. J. Courthope, _Hist. of Eng. Poetry_, vol. 3; and Guizot, _Études
  biographiques sur la révolution d'Angleterre_ (pp. 403-426, ed. 1851).




MAY, or MEY(E), WILLIAM (d. 1560), English divine, was the brother of
John May, bishop of Carlisle. He was educated at Cambridge, where he was
a fellow of Trinity Hall, and in 1537, president of Queen's College. May
heartily supported the Reformation, signed the Ten Articles in 1536, and
helped in the production of _The Institution of a Christian Man_. He had
close connexion with the diocese of Ely, being successively chancellor,
vicar-general and prebendary. In 1545 he was made a prebendary of St
Paul's, and in the following year dean. His favourable report on the
Cambridge colleges saved them from dissolution. He was dispossessed
during the reign of Mary, but restored to the deanery on Elizabeth's
accession. He died on the day of his election to the archbishopric of
York.




MAY, the fifth month of our modern year, the third of the old Roman
calendar. The origin of the name is disputed; the derivation from Maia,
the mother of Mercury, to whom the Romans were accustomed to sacrifice
on the first day of this month, is usually accepted. The ancient Romans
used on May Day to go in procession to the grotto of Egeria. From the
28th of April to the 2nd of May was kept the festival in honour of
Flora, goddess of flowers. By the Romans the month was regarded as
unlucky for marriages, owing to the celebration on the 9th, 11th and
13th of the Lemuria, the festival of the unhappy dead. This superstition
has survived to the present day.

In medieval and Tudor England, May Day was a great public holiday. All
classes of the people, young and old alike, were up with the dawn, and
went "a-Maying" in the woods. Branches of trees and flowers were borne
back in triumph to the towns and villages, the centre of the procession
being occupied by those who shouldered the maypole, glorious with
ribbons and wreaths. The maypole was usually of birch, and set up for
the day only; but in London and the larger towns the poles were of
durable wood and permanently erected. They were special eyesores to the
Puritans. John Stubbes in his _Anatomy of Abuses_ (1583) speaks of them
as those "stinckyng idols," about which the people "leape and daunce, as
the heathen did." Maypoles were forbidden by the parliament in 1644, but
came once more into favour at the Restoration, the last to be erected in
London being that set up in 1661. This pole, which was of cedar, 134 ft.
high, was set up by twelve British sailors under the personal
supervision of James II., then duke of York and lord high admiral, in
the Strand on or about the site of the present church of St
Mary's-in-the-Strand. Taken down in 1717, it was conveyed to Wanstead
Park in Essex, where it was fixed by Sir Isaac Newton as part of the
support of a large telescope, presented to the Royal Society by a French
astronomer.

  For an account of the May Day survivals in rural England see P. H.
  Ditchfield, _Old English Customs extant at Present Times_ (1897).




MAY, ISLE OF, an island belonging to Fifeshire, Scotland, at the
entrance to the Firth of Forth, 5 m. S.E. of Crail and Anstruther. It
has a N.W. to S.E. trend, is more than 1 m. long, and measures at its
widest about 1/3 m. St Adrian, who had settled here, was martyred by the
Danes about the middle of the 9th century. The ruins of the small chapel
dedicated to him, which was a favourite place of pilgrimage, still
exist. The place where the pilgrims--of whom James IV. was often
one--landed is yet known as Pilgrims' Haven, and traces may yet be seen
of the various wells of St Andrew, St John, Our Lady, and the Pilgrims,
though their waters have become brackish. In 1499 Sir Andrew Wood of
Largo, with the "Yellow Carvel" and "Mayflower," captured the English
seaman Stephen Bull, and three ships, after a fierce fight which took
place between the island and the Bass Rock. In 1636 a coal beacon was
lighted on the May and maintained by Alexander Cunningham of Barns. The
oil light substituted for it in 1816 was replaced in 1888 by an electric
light.




MAYA, an important tribe and stock of American Indians, the dominant
race of Yucatan and other states of Mexico and part of Central America
at the time of the Spanish conquest. They were then divided into many
nations, chief among them being the Maya proper, the Huastecs, the
Tzental, the Pokom, the Mame and the Cakchiquel and Quiché. They were
spread over Yucatan, Vera Cruz, Tabasco, Campeche, and Chiapas in
Mexico, and over the greater part of Guatemala and Salvador. In
civilization the Mayan peoples rivalled the Aztecs. Their traditions
give as their place of origin the extreme north; thence a migration took
place, perhaps at the beginning of the Christian era. They appear to
have reached Yucatan as early as the 5th century. From the evidence of
the Quiché chronicles, which are said to date back to about A.D. 700,
Guatemala was shortly afterwards overrun. Physically the Mayans are a
dark-skinned, round-headed, short and sturdy type. Although they were
already decadent when the Spaniards arrived they made a fierce
resistance. They still form the bulk of the inhabitants of Yucatan. For
their culture, ruined cities, &c. see CENTRAL AMERICA and MEXICO.




MAYAGUEZ, the third largest city of Porto Rico, a seaport, and the seat
of government of the department of Mayaguez, on the west coast, at the
mouth of Rio Yaguez, about 72 m. W. by S. of San Juan. Pop. of the city
(1899), 15,187, including 1381 negroes and 4711 of mixed races; (1910),
16,591; of the municipal district, 35,700 (1899), of whom 2687 were
negroes and 9933 were of mixed races. Mayaguez is connected by the
American railroad of Porto Rico with San Juan and Ponce, and it is
served regularly by steamboats from San Juan, Ponce and New York,
although its harbour is not accessible to vessels drawing more than 16
ft. of water. It is situated at the foot of Las Mesas mountains and
commands picturesque views. The climate is healthy and good water is
obtained from the mountain region. From the shipping district along the
water-front a thoroughfare leads to the main portion of the city, about
1 m. distant. There are four public squares, in one of which is a statue
of Columbus. Prominent among the public buildings are the City Hall
(containing a public library), San Antonio Hospital, Roman Catholic
churches, a Presbyterian church, the court-house and a theatre. The
United States has an agricultural experiment station here, and the
Insular Reform School is 1 m. south of the city. Coffee, sugar-cane and
tropical fruits are grown in the surrounding country; and the business
of the city consists chiefly in their export and the import of flour.
Among the manufactures are sugar, tobacco and chocolate. Mayaguez was
founded about the middle of the 18th century on the site of a hamlet
which was first settled about 1680. It was incorporated as a town in
1836, and became a city in 1873. In 1841 it was nearly all destroyed by
fire.




MAYAVARAM, a town of British India, in the Tanjore district of Madras,
on the Cauvery river; junction on the South Indian railway, 174 m. S.W.
of Madras. Pop. (1901), 24,276. It possesses a speciality of fine cotton
and silk cloth, known as Kornad from the suburb in which the weavers
live. During October and November the town is the scene of a great
pilgrimage to the holy waters of the Cauvery.




MAYBOLE, a burgh of barony and police burgh of Ayrshire, Scotland. Pop.
(1901), 5892. It is situated 9 m. S. of Ayr and 50¼ m. S.W. of Glasgow
by the Glasgow & South-Western railway. It is an ancient place, having
received a charter from Duncan II. in 1193. In 1516 it was made a burgh
of regality, but for generations it remained under the subjection of the
Kennedys, afterwards earls of Cassillis and marquesses of Ailsa, the
most powerful family in Ayrshire. Of old Maybole was the capital of the
district of Carrick, and for long its characteristic feature was the
family mansions of the barons of Carrick. The castle of the earls of
Cassillis still remains. The public buildings include the town-hall, the
Ashgrove and the Lumsden fresh-air fortnightly homes, and the Maybole
combination poorhouse. The leading manufactures are of boots and shoes
and agricultural implements. Two miles to the south-west are the ruins
of Crossraguel (Cross of St Regulus) Abbey, founded about 1240.
KIRKOSWALD, where Burns spent his seventeenth year, learning
land-surveying, lies a little farther west. In the parish churchyard lie
"Tam o' Shanter" (Douglas Graham) and "Souter Johnnie" (John Davidson).
Four miles to the west of Maybole on the coast is Culzean Castle, the
chief seat of the marquess of Ailsa, dating from 1777; it stands on a
basaltic cliff, beneath which are the Coves of Culzean, once the retreat
of outlaws and a resort of the fairies. Farther south are the ruins of
Turnberry Castle, where Robert Bruce is said to have been born. A few
miles to the north of Culzean are the ruins of Dunure Castle, an ancient
stronghold of the Kennedys.




MAYEN, a town of Germany, in the Prussian Rhine province, on the
northern declivity of the Eifel range, 16 m. W. from Coblenz, on the
railway Andernach-Gerolstein. Pop. (1905), 13,435. It is still partly
surrounded by medieval walls, and the ruins of a castle rise above the
town. There are some small industries, embracing textile manufactures,
oil mills and tanneries, and a trade in wine, while near the town are
extensive quarries of basalt. Having been a Roman settlement, Mayen
became a town in 1291. In 1689 it was destroyed by the French.




MAYENNE, CHARLES OF LORRAINE, DUKE OF (1554-1611), second son of Francis
of Lorraine, second duke of Guise, was born on the 26th of March 1554.
He was absent from France at the time of the massacre of Saint
Bartholomew, but took part in the siege of La Rochelle in the following
year, when he was created duke and peer of France. He went with Henry of
Valois, duke of Anjou (afterwards Henry III.), on his election as king
of Poland, but soon returned to France to become the energetic supporter
and lieutenant of his brother, the 3rd duke of Guise. In 1577 he gained
conspicuous successes over the Huguenot forces in Poitou. As governor of
Burgundy he raised his province in the cause of the League in 1585. The
assassination of his brothers at Blois on the 23rd and 24th of December
1588 left him at the head of the Catholic party. The Venetian
ambassador, Mocenigo, states that Mayenne had warned Henry III. that
there was a plot afoot to seize his person and to send him by force to
Paris. At the time of the murder he was at Lyons, where he received a
letter from the king saying that he had acted on his warning, and
ordering him to retire to his government. Mayenne professed obedience,
but immediately made preparations for marching on Paris. After a vain
attempt to recover the persons of those of his relatives who had been
arrested at Blois he proceeded to recruit troops in his government of
Burgundy and in Champagne. Paris was devoted to the house of Guise and
had been roused to fury by the news of the murder. When Mayenne entered
the city in February 1589 he found it dominated by representatives of
the sixteen quarters of Paris, all fanatics of the League. He formed a
council general to direct the affairs of the city and to maintain
relations with the other towns faithful to the League. To this council
each quarter sent four representatives, and Mayenne added
representatives of the various trades and professions of Paris in order
to counterbalance this revolutionary element. He constituted himself
"lieutenant-general of the state and crown of France," taking his oath
before the parlement of Paris. In April he advanced on Tours. Henry III.
in his extremity sought an alliance with Henry of Navarre, and the
allied forces drove the leaguers back, and had laid siege to Paris, when
the murder of Henry III. by a Dominican fanatic changed the face of
affairs and gave new strength to the Catholic party.

Mayenne was urged to claim the crown for himself, but he was faithful to
the official programme of the League and proclaimed Charles, cardinal of
Bourbon, at that time a prisoner in the hands of Henry IV., as Charles
X. Henry IV. retired to Dieppe, followed by Mayenne, who joined his
forces with those of his cousin Charles, duke of Aumale, and Charles de
Cossé, comte de Brissac, and engaged the royal forces in a succession of
fights in the neighbourhood of Arques (September 1589). He was defeated
and out-marched by Henry IV., who moved on Paris, but retreated before
Mayenne's forces. In 1590 Mayenne received additions to his army from
the Spanish Netherlands, and took the field again, only to suffer
complete defeat at Ivry (March 14, 1590). He then escaped to Mantes, and
in September collected a fresh army at Meaux, and with the assistance of
Alexander Farnese, prince of Parma, sent by Philip II., raised the siege
of Paris, which was about to surrender to Henry IV. Mayenne feared with
reason the designs of Philip II., and his difficulties were increased by
the death of Charles X., the "king of the league." The extreme section
of the party, represented by the Sixteen, urged him to proceed to the
election of a Catholic king and to accept the help and the claims of
their Spanish allies. But Mayenne, who had not the popular gifts of his
brother, the duke of Guise, had no sympathy with the demagogues, and
himself inclined to the moderate side of his party, which began to urge
reconciliation with Henry IV. He maintained the ancient forms of the
constitution against the revolutionary policy of the Sixteen, who during
his absence from Paris took the law into their own hands and in November
1591 executed one of the leaders of the more moderate party, Barnabé
Brisson, president of the parlement. He returned to Paris and executed
four of the chief malcontents. The power of the Sixteen diminished from
that time, but with it the strength of the League.[1]

Mayenne entered into negotiations with Henry IV. while he was still
appearing to consider with Philip II. the succession to the French crown
of the Infanta Elizabeth, granddaughter, through her mother Elizabeth of
Valois, of Henry II. He demanded that Henry IV. should accomplish his
conversion to Catholicism before he was recognized by the leaguers. He
also desired the continuation to himself of the high offices which had
accumulated in his family and the reservation of their provinces to his
relatives among the leaguers. In 1593 he summoned the States General to
Paris and placed before them the claims of the Infanta, but they
protested against foreign intervention. Mayenne signed a truce at La
Villette on the 31st of July 1593. The internal dissensions of the
league continued to increase, and the principal chiefs submitted.
Mayenne finally made his peace only in October 1595. Henry IV. allowed
him the possession of Chalon-sur-Saône, of Seurre and Soissons for three
years, made him governor of the Isle of France and paid a large
indemnity. Mayenne died at Soissons on the 3rd of October 1611.

  A _Histoire de la vie et de la mort du duc de Mayenne_ appeared at
  Lyons in 1618. See also J. B. H. Capefigue, _Hist. de la Réforme, de
  la ligue et du règne de Henri IV._ (8 vols., 1834-1835) and the
  literature dealing with the house of Guise (q.v.).


FOOTNOTE:

  [1] The estates of the League in 1593 were the occasion of the famous
    _Satire Ménippée_, circulated in MS. in that year, but only printed
    at Tours in 1594. It was the work of a circle of men of letters who
    belonged to the _politiques_ or party of the centre and ridiculed the
    League. The authors were Pierre Le Roy, Jean Passerat, Florent
    Chrestien, Nicolas Rapin and Pierre Pithou. It opened with "La vertu
    du catholicon," in which a Spanish quack (the cardinal of Plaisance)
    vaunts the virtues of his drug "catholicon composé," manufactured in
    the Escurial, while a Lorrainer rival (the cardinal of Pellevé) tries
    to sell a rival cure. A mock account of the estates, with harangues
    delivered by Mayenne and the other chiefs of the League, followed.
    Mayenne's discourse is said to have been written by the jurist
    Pithou.




MAYENNE, a department of north-western France, three-fourths of which
formerly belonged to Lower Maine and the remainder to Anjou, bounded on
the N. by Manche and Orne, E. by Sarthe, S. by Maine-et-Loire and W. by
Ille-et-Vilaine. Area, 2012 sq. m. Pop. (1906), 305,457. Its ancient
geological formations connect it with Brittany. The surface is agreeably
undulating; forests are numerous, and the beauty of the cultivated
portions is enhanced by the hedgerows and lines of trees by which the
farms are divided. The highest point of the department, and indeed of
the whole north-west of France, is the Mont des Avaloirs (1368 ft.).
Hydrographically Mayenne belongs to the basins of the Loire, the Vilaine
and the Sélune, the first mentioned draining by far the larger part of
the entire area. The principal stream is the Mayenne, which passes
successively from north to south through Mayenne, Laval and
Château-Gontier; by means of weirs and sluices it is navigable below
Mayenne, but traffic is inconsiderable. The chief affluents are the
Jouanne on the left, and on the right the Colmont, the Ernée and the
Oudon. A small area in the east of the department drains by the Erve
into the Sarthe; the Vilaine rises in the west, and in the north-west
two small rivers flow into the Sélune. The climate of Mayenne is
generally healthy except in the neighbourhood of the numerous marshes.
The temperature is lower and the moisture of the atmosphere greater than
in the neighbouring departments; the rainfall (about 32 in. annually) is
above the average for France.

  Agriculture and stock-raising are prosperous. A large number of horned
  cattle are reared, and in no other French department are so many
  horses found within the same area; the breed, that of Craon, is famed
  for its strength. Craon has also given its name to the most prized
  breed of pigs in western France. Mayenne produces excellent butter and
  poultry and a large quantity of honey. The cultivation of the vine is
  very limited, and the most common beverage is cider. Wheat, oats,
  barley and buckwheat, in the order named, are the most important
  crops, and a large quantity of flax and hemp is produced. Game is
  abundant. The timber grown is chiefly beech, oak, birch, elm and
  chestnut. The department produces antimony, auriferous quartz and
  coal. Marble, slate and other stone are quarried. There are several
  chalybeate springs. The industries include flour-milling, brick and
  tile making, brewing, cotton and wool spinning, and the production of
  various textile fabrics (especially ticking) for which Laval and
  Château-Gontier are the centres, agricultural implement making, wood
  and marble sawing, tanning and dyeing. The exports include
  agricultural produce, live-stock, stone and textiles; the chief
  imports are coal, brandy, wine, furniture and clothing. The department
  is served by the Western railway. It forms part of the
  circumscriptions of the IV. army corps, the académie (educational
  division) of Rennes, and the court of appeal of Angers. It comprises
  three arrondissements (Laval, Château-Gontier and Mayenne), with 27
  cantons and 276 communes. Laval, the capital, is the seat of a
  bishopric of the province of Tours. The other principal towns are
  Château-Gontier and Mayenne, which are treated under separate
  headings. The following places are also of interest: Evron, which has
  a church of the 12th and 13th centuries; Jublains, with a Roman fort
  and other Roman remains; Lassay, with a fine château of the 14th and
  16th centuries; and Ste Suzanne, which has remains of medieval
  ramparts and a fortress with a keep of the Romanesque period.




MAYENNE, a town of north-western France, capital of an arrondissement in
the department of Mayenne, 19 m. N.N.E. of Laval by rail. Pop., town
7003, commune 10,020. Mayenne is an old feudal town, irregularly built
on hills on both sides of the river Mayenne. Of the old castle
overlooking the river several towers remain, one of which has retained
its conical roof; the vaulted chambers and chapel are ornamented in the
style of the 13th century; the building is now used as a prison. The
church of Notre-Dame, beside which there is a statue of Joan of Arc,
dates partly from the 12th century; the choir was rebuilt in the 19th
century. In the Place de Cheverus is a statue, by David of Angers, to
Cardinal Jean de Cheverus (1768-1836), who was born in Mayenne. Mayenne
has a subprefecture, tribunals of first instance and of commerce, a
chamber of arts and manufactures, and a board of trade-arbitration.
There is a school of agriculture in the vicinity. The chief industry of
the place is the manufacture of tickings, linen, handkerchiefs and
calicoes.

Mayenne had its origin in the castle built here by Juhel, baron of
Mayenne, the son of Geoffrey of Maine, in the beginning of the 11th
century. It was taken by the English in 1424, and several times suffered
capture by the opposing parties in the wars of religion and the Vendée.
At the beginning of the 16th century the territory passed to the family
of Guise, and in 1573 was made a duchy in favour of Charles of Mayenne,
leader of the League.




MAYER, JOHANN TOBIAS (1723-1762), German astronomer, was born at
Marbach, in Würtemberg, on the 17th of February 1723, and brought up at
Esslingen in poor circumstances. A self-taught mathematician, he had
already published two original geometrical works when, in 1746, he
entered J. B. Homann's cartographic establishment at Nuremberg. Here he
introduced many improvements in map-making, and gained a scientific
reputation which led (in 1751) to his election to the chair of economy
and mathematics in the university of Göttingen. In 1754 he became
superintendent of the observatory, where he laboured with great zeal and
success until his death, on the 20th of February 1762. His first
important astronomical work was a careful investigation of the libration
of the moon (_Kosmographische Nachrichten_, Nuremberg, 1750), and his
chart of the full moon (published in 1775) was unsurpassed for half a
century. But his fame rests chiefly on his lunar tables, communicated in
1752, with new solar tables, to the Royal Society of Göttingen, and
published in their _Transactions_ (vol. ii.). In 1755 he submitted to
the English government an amended body of MS. tables, which James
Bradley compared with the Greenwich observations, and found to be
sufficiently accurate to determine the moon's place to 75´´, and
consequently the longitude at sea to about half a degree. An improved
set was afterwards published in London (1770), as also the theory
(_Theoria lunae juxta systema Newtonianum_, 1767) upon which the tables
are based. His widow, by whom they were sent to England, received in
consideration from the British government a grant of £3000. Appended to
the London edition of the solar and lunar tables are two short
tracts--the one on determining longitude by lunar distances, together
with a description of the repeating circle (invented by Mayer in 1752),
the other on a formula for atmospheric refraction, which applies a
remarkably accurate correction for temperature.

Mayer left behind him a considerable quantity of manuscript, part of
which was collected by G. C. Lichtenberg and published in one volume
(_Opera inedita_, Göttingen, 1775). It contains an easy and accurate
method for calculating eclipses; an essay on colour, in which three
primary colours are recognized; a catalogue of 998 zodiacal stars; and a
memoir, the earliest of any real value, on the proper motion of eighty
stars, originally communicated to the Göttingen Royal Society in 1760.
The manuscript residue includes papers on atmospheric refraction (dated
1755), on the motion of Mars as affected by the perturbations of Jupiter
and the Earth (1756), and on terrestrial magnetism (1760 and 1762). In
these last Mayer sought to explain the magnetic action of the earth by a
modification of Euler's hypothesis, and made the first really definite
attempt to establish a mathematical theory of magnetic action (C.
Hansteen, _Magnetismus der Erde_, i. 283). E. Klinkerfuss published in
1881 photo-lithographic reproductions of Mayer's local charts and
general map of the moon; and his star-catalogue was re-edited by F.
Baily in 1830 (_Memoirs Roy. Astr. Soc._ iv. 391) and by G. F. J. A.
Auvers in 1894.

  AUTHORITIES.--A. G. Kästner, _Elogium Tobiae Mayeri_ (Göttingen,
  1762); _Connaissance des temps, 1767_, p. 187 (J. Lalande);
  _Monatliche Correspondenz_ viii. 257, ix. 45, 415, 487, xi. 462;
  _Allg. Geographische Ephemeriden_ iii. 116, 1799 (portrait); _Berliner
  Astr. Jahrbuch_, Suppl. Bd. iii. 209, 1797 (A. G. Kästner); J. B. J.
  Delambre, _Hist. de l'Astr. au XVIII^e siècle_, p. 429; R. Grant,
  _Hist. of Phys. Astr._ pp. 46, 488, 555; A. Berry, _Short Hist. of
  Astr._ p. 282; J. S. Pütter, _Geschichte von der Universität zu
  Göttingen_, i. 68; J. Gehler, _Physik. Wörterbuch neu bearbeitet_, vi.
  746, 1039; Allg. _Deutsche Biographie_ (S. Günther).     (A. M. C.)




MAYER, JULIUS ROBERT (1814-1878), German physicist, was born at
Heilbronn on the 25th of November 1814, studied medicine at Tübingen,
Munich and Paris, and after a journey to Java in 1840 as surgeon of a
Dutch vessel obtained a medical post in his native town. He claims
recognition as an independent a priori propounder of the "First Law of
Thermodynamics," but more especially as having early and ably applied
that law to the explanation of many remarkable phenomena, both cosmical
and terrestrial. His first little paper on the subject, "_Bemerkungen
über die Kräfte der unbelebten Natur_," appeared in 1842 in Liebig's
_Annalen_, five years after the republication, in the same journal, of
an extract from K. F. Mohr's paper on the nature of heat, and three
years later he published _Die organische Bewegung in ihren Zusammenhange
mit dem Stoffwechsel_.

  It has been repeatedly claimed for Mayer that he calculated the value
  of the dynamical equivalent of heat, indirectly, no doubt, but in a
  manner altogether free from error, and with a result according almost
  exactly with that obtained by J. P. Joule after years of patient
  labour in direct experimenting. This claim on Mayer's behalf was first
  shown to be baseless by W. Thomson (Lord Kelvin) and P. G. Tait in an
  article on "Energy," published in _Good Words_ in 1862, which gave
  rise to a long but lively discussion. A calm and judicial annihilation
  of the claim is to be found in a brief article by Sir G. G. Stokes,
  _Proc. Roy. Soc._, 1871, p. 54. See also Maxwell's _Theory of Heat_,
  chap. xiii. Mayer entirely ignored the grand fundamental principle
  laid down by Sadi Carnot--that nothing can be concluded as to the
  relation between heat and work from an experiment in which the working
  substance is left at the end of an operation in a different physical
  state from that in which it was at the commencement. Mayer has also
  been styled the discoverer of the fact that heat consists in (the
  energy of) motion, a matter settled at the very end of the 18th
  century by Count Rumford and Sir H. Davy; but in the teeth of this
  statement we have Mayer's own words, "We might much rather assume the
  contrary--that in order to become heat motion must cease to be
  motion."

  Mayer's real merit consists in the fact that, having for himself made
  out, on inadequate and even questionable grounds, the conservation of
  energy, and having obtained (though by inaccurate reasoning) a
  numerical result correct so far as his data permitted, he applied the
  principle with great power and insight to the explanation of numerous
  physical phenomena. His papers, which were republished in a single
  volume with the title _Die Mechanik der Wärme_ (3rd ed., 1893), are of
  unequal merit. But some, especially those on _Celestial Dynamics_ and
  _Organic Motion_, are admirable examples of what really valuable work
  may be effected by a man of high intellectual powers, in spite of
  imperfect information and defective logic.

  Different, and it would appear exaggerated, estimates of Mayer are
  given in John Tyndall's papers in the _Phil. Mag._, 1863-1864 (whose
  avowed object was "to raise a noble and a suffering man to the
  position which his labours entitled him to occupy"), and in E.
  Dühring's _Robert Mayer, der Galilei des neunzehnten Jahrhunderts_,
  Chemnitz, 1880. Some of the simpler facts of the case are summarized
  by Tait in the _Phil. Mag._, 1864, ii. 289.




MAYFLOWER, the vessel which carried from Southampton, England, to
Plymouth, Massachusetts, the Pilgrims who established the first
permanent colony in New England. It was of about 180 tons burden, and in
company with the "Speedwell" sailed from Southampton on the 5th of
August 1620, the two having on board 120 Pilgrims. After two trials the
"Speedwell" was pronounced unseaworthy, and the "Mayflower" sailed alone
from Plymouth, England, on the 6th of September with the 100 (or 102)
passengers, some 41 of whom on the 11th of November (O.S.) signed the
famous "Mayflower Compact" in Provincetown Harbor, and a small party of
whom, including William Bradford, sent to choose a place for settlement,
landed at what is now Plymouth, Massachusetts, on the 11th of December
(21st N.S.), an event which is celebrated, as Forefathers' Day, on the
22nd of December. A "General Society of Mayflower Descendants" was
organized in 1894 by lineal descendants of passengers of the "Mayflower"
to "preserve their memory, their records, their history, and all facts
relating to them, their ancestors and their posterity." Every lineal
descendant, over eighteen years of age, of any passenger of the
"Mayflower" is eligible to membership. Branch societies have since been
organized in several of the states and in the District of Columbia, and
a triennial congress is held in Plymouth.

  See Azel Ames, _The May-Flower and Her Log_ (Boston, 1901); Blanche
  McManus, _The Voyage of the Mayflower_ (New York, 1897); _The General
  Society of Mayflower: Meetings, Officers and Members, arranged in
  State Societies, Ancestors and their Descendants_ (New York, 1901).
  Also the articles PLYMOUTH, MASS.; MASSACHUSETTS, §_History_; PILGRIM;
  and PROVINCETOWN, MASS.




MAY-FLY. The Mayflies belong to the Ephemeridae, a remarkable family of
winged insects, included by Linnaeus in his order Neuroptera, which
derive their scientific name from [Greek: ephêmeros], in allusion to
their very short lives. In some species it is possible that they have
scarcely more than one day's existence, but others are far longer lived,
though the extreme limit is probably rarely more than a week. The family
has very sharply defined characters, which separate its members at once
from all other neuropterous (or pseudo-neuropterous) groups.

These insects are universally aquatic in their preparatory states. The
eggs are dropped into the water by the female in large masses,
resembling, in some species, bunches of grapes in miniature. Probably
several months elapse before the young larvae are excluded. The
sub-aquatic condition lasts a considerable time: in _Cloeon_, a genus of
small and delicate species, Sir J. Lubbock (Lord Avebury) proved it to
extend over more than six months; but in larger and more robust genera
(e.g. _Palingenia_) there appears reason to believe that the greater
part of three years is occupied in preparatory conditions.

  The larva is elongate and campodeiform. The head is rather large, and
  is furnished at first with five simple eyes of nearly equal size; but
  as it increases in size the homologues of the facetted eyes of the
  imago become larger, whereas those equivalent to the ocelli remain
  small. The antennae are long and thread-like, composed at first of few
  joints, but the number of these latter apparently increases at each
  moult. The mouth parts are well developed, consisting of an upper lip,
  powerful mandibles, maxillae with three-jointed palpi, and a deeply
  quadrifid labium or lower lip with three-jointed labial palpi.
  Distinct and conspicuous maxillulae are associated with the tongue or
  hypopharynx. There are three distinct and large thoracic segments,
  whereof the prothorax is narrower than the others; the legs are much
  shorter and stouter than in the winged insect, with monomerous tarsi
  terminated by a single claw. The abdomen consists of ten segments, the
  tenth furnished with long and slender multi-articulate tails, which
  appear to be only two in number at first, but an intermediate one
  gradually develops itself (though this latter is often lost in the
  winged insect). Respiration is effected by means of external gills
  placed along both sides of the dorsum of the abdomen and hinder
  segments of the thorax. These vary in form: in some species they are
  entire plates, in others they are cut up into numerous divisions, in
  all cases traversed by numerous tracheal ramifications. According to
  the researches of Lubbock and of E. Joly, the very young larvae have
  no breathing organs, and respiration is effected through the skin.
  Lubbock traced at least twenty moults in _Cloeon_; at about the tenth
  rudiments of the wing-cases began to appear. These gradually become
  larger, and when so the creature may be said to have entered its
  "nymph" stage; but there is no condition analogous to the pupa-stage
  of insects with complete metamorphoses.

  There may be said to be three or four different modes of life in these
  larvae: some are fossorial, and form tubes in the mud or clay in which
  they live; others are found on or beneath stones; while others again
  swim and crawl freely among water plants. It is probable that some are
  carnivorous, either attacking other larvae or subsisting on more
  minute forms of animal life; but others perhaps feed more exclusively
  on vegetable matters of a low type, such as diatoms.

  The most aberrant type of larva is that of the genus _Prosopistoma_,
  which was originally described as an entomostracous crustacean on
  account of the presence of a large carapace overlapping the greater
  part of the body. The dorsal skeletal elements of the thorax and of
  the anterior six abdominal segments unite with the wing-cases to form
  a large respiratory chamber, containing five pairs of tracheal gills,
  with lateral slits for the inflow and a posterior orifice for the
  outflow of water. Species of this genus occur in Europe, Africa and
  Madagascar.

When the aquatic insect has reached its full growth it emerges from the
water or seeks its surface; the thorax splits down the back and the
winged form appears. But this is not yet perfect, although it has all
the form of a perfect insect and is capable of flight; it is what is
variously termed a "pseud-imago," "sub-imago" or "pro-imago." Contrary
to the habits of all other insects, there yet remains a pellicle that
has to be shed, covering every part of the body. This final moult is
effected soon after the insect's appearance in the winged form; the
creature seeks a temporary resting-place, the pellicle splits down the
back, and the now perfect insect comes forth, often differing very
greatly in colours and markings from the condition in which it was only
a few moments before. If the observer takes up a suitable position near
water, his coat is often seen to be covered with the cast sub-imaginal
skins of these insects, which had chosen him as a convenient object upon
which to undergo their final change. In some few genera of very low type
it appears probable that, at any rate in the female, this final change
is never effected and that the creature dies a sub-imago.

  The winged insect differs considerably in form from its sub-aquatic
  condition. The head is smaller, often occupied almost entirely above
  in the male by the very large eyes, which in some species are
  curiously double in that sex, one portion being pillared, and forming
  what is termed a "turban," the mouth parts are aborted, for the
  creature is now incapable of taking nutriment either solid or fluid;
  the antennae are mere short bristles, consisting of two rather large
  basal joints and a multi-articulate thread. The prothorax is much
  narrowed, whereas the other segments (especially the mesothorax) are
  greatly enlarged; the legs long and slender, the anterior pair often
  very much longer in the male than in the female; the tarsi four- or
  five-jointed; but in some genera (e.g. _Oligoneuria_ and allies) the
  legs are aborted, and the creatures are driven helplessly about by the
  wind. The wings are carried erect: the anterior pair large, with
  numerous longitudinal nervures, and usually abundant transverse
  reticulation; the posterior pair very much smaller, often lanceolate,
  and frequently wanting absolutely. The abdomen consists of ten
  segments; at the end are either two or three long multi-articulate
  tails; in the male the ninth joint bears forcipated appendages; in the
  female the oviducts terminate at the junction of the seventh and
  eighth ventral segments. The independent opening of the genital ducts
  and the absence of an ectodermal vagina and ejaculatory duct are
  remarkable archaic features of these insects, as has been pointed out
  by J. A. Palmén. The sexual act takes place in the air, and is of very
  short duration, but is apparently repeated several times, at any rate
  in some cases.

_Ephemeridae_ are found all over the world, even up to high northern
latitudes. F. J. Pictet, A. E. Eaton and others have given us valuable
works or monographs on the family; but the subject still remains little
understood, partly owing to the great difficulty of preserving such
delicate insects; and it appears probable they can only be
satisfactorily investigated as moist preparations. The number of
described species is less than 200, spread over many genera.

From the earliest times attention has been drawn to the enormous
abundance of species of the family in certain localities. Johann Anton
Scopoli, writing in the 18th century, speaks of them as so abundant in
one place in Carniola that in June twenty cartloads were carried away
for manure! _Polymitarcys virgo_, which, though not found in England,
occurs in many parts of Europe (and is common at Paris), emerges from
the water soon after sunset, and continues for several hours in such
myriads as to resemble snow showers, putting out lights, and causing
inconvenience to man, and annoyance to horses by entering their
nostrils. In other parts of the world they have been recorded in
multitudes that obscured passers-by on the other side of the street. And
similar records might be multiplied almost to any extent. In Britain,
although they are often very abundant, we have scarcely anything
analogous.

Fish, as is well known, devour them greedily, and enjoy a veritable
feast during the short period in which any particular species appears.
By anglers the common English species of _Ephemera_ (_vulgata_ and
_danica_, but more especially the latter, which is more abundant) is
known as the "may-fly," but the terms "green drake" and "bastard drake"
are applied to conditions of the same species. Useful information on
this point will be found in Ronalds's _Fly-Fisher's Entomology_, edited
by Westwood.

Ephemeridae belong to a very ancient type of insects, and fossil
imprints of allied forms occur even in the Devonian and Carboniferous
formations.

There is much to be said in favour of the view entertained by some
entomologists that the structural and developmental characteristics of
may-flies are sufficiently peculiar to warrant the formation for them of
a special order of insects, for which the names Agnatha, Plectoptera and
Ephemeroptera have been proposed. (See HEXAPODA, NEUROPTERA.)

  BIBLIOGRAPHY.--Of especial value to students of these insects are A.
  E. Eaton's monograph (_Trans. Linn. Soc._ (2) iii. 1883-1885) and A.
  Vayssière's "Recherches sur l'organisation des larves" (_Ann. Sci.
  Nat. Zool._ (6) xiii. 1882 (7) ix. 1890). J. A. Palmén's memoirs _Zur
  Morphologie des Tracheensystems_ (Leipzig, 1877) and _Über paarige
  Ausführungsgänge der Geschlechtsorgane bei Insekten_ (Helsingfors,
  1884), contain important observations on may-flies. See also L. C.
  Miall, _Nat. Hist. Aquatic Insects_ (London, 1895); J. G. Needham and
  others (New York State Museum, Bull. 86, 1905).     (R. M'L.; G. H. C.)




MAYHEM (for derivation see MAIMING), an old Anglo-French term of the law
signifying an assault whereby the injured person is deprived of a member
proper for his defence in fight, e.g. an arm, a leg, a fore tooth, &c.
The loss of an ear, jaw tooth, &c., was not mayhem. The most ancient
punishment in English law was retaliative--_membrum pro membro_, but
ultimately at common law fine and imprisonment. Various statutes were
passed aimed at the offence of maiming and disfiguring, which is now
dealt with by section 18 of the Offences against the Person Act 1861.
Mayhem may also be the ground of a civil action, which had this
peculiarity that the court on sight of the wound might increase the
damages awarded by the jury.




MAYHEW, HENRY (1812-1887), English author and journalist, son of a
London solicitor, was born in 1812. He was sent to Westminster school,
but ran away to sea. He sailed to India, and on his return studied law
for a short time under his father. He began his journalistic career by
founding, with Gilbert à Beckett, in 1831, a weekly paper, _Figaro in
London_. This was followed in 1832 by a short-lived paper called _The
Thief_; and he produced one or two successful farces. His brothers
Horace (1816-1872) and Augustus Septimus (1826-1875) were also
journalists, and with them Henry occasionally collaborated, notably with
the younger in _The Greatest Plague of Life_ (1847) and in _Acting
Charades_ (1850). In 1841 Henry Mayhew was one of the leading spirits
in the foundation of _Punch_, of which he was for the first two years
joint-editor with Mark Lemon. He afterwards wrote on all kinds of
subjects, and published a number of volumes of no permanent
reputation--humorous stories, travel and practical handbooks. He is
credited with being the first to "write up" the poverty side of London
life from a philanthropic point of view; with the collaboration of John
Binny and others he published _London Labour and London Poor_ (1851;
completed 1864) and other works on social and economic questions. He
died in London, on the 25th of July 1887. Horace Mayhew was for some
years sub-editor of _Punch_, and was the author of several humorous
publications and plays. The books of Horace and Augustus Mayhew owe
their survival chiefly to Cruikshank's illustrations.




MAYHEW, JONATHAN (1720-1766), American clergyman, was born at Martha's
Vineyard on the 8th of October 1720, being fifth in descent from Thomas
Mayhew (1592-1682), an early settler and the grantee (1641) of Martha's
Vineyard. Thomas Mayhew (c. 1616-1657), the younger, his son John (d.
1689) and John's son, Experience (1673-1758), were active missionaries
among the Indians of Martha's Vineyard and the vicinity. Jonathan, the
son of Experience, graduated at Harvard in 1744. So liberal were his
theological views that when he was to be ordained minister of the West
Church in Boston in 1747 only two ministers attended the first council
called for the ordination, and it was necessary to summon a second
council. Mayhew's preaching made his church practically the first
"Unitarian" Congregational church in New England, though it was never
officially Unitarian. In 1763 he published _Observations on the Charter
and Conduct of the Society for Propagating the Gospel in Foreign Parts_,
an attack on the policy of the society in sending missionaries to New
England contrary to its original purpose of "Maintaining Ministers of
the Gospel" in places "wholly destitute and unprovided with means for
the maintenance of ministers and for the public worship of God;" the
_Observations_ marked him as a leader among those in New England who
feared, as Mayhew said (1762), "that there is a scheme forming for
sending a bishop into this part of the country, and that our
Governor,[1] a true churchman, is deeply in the plot." To an American
reply to the _Observations_, entitled _A Candid Examination_ (1763),
Mayhew wrote a _Defense_; and after the publication of an _Answer_,
anonymously published in London in 1764 and written by Thomas Seeker,
archbishop of Canterbury, he wrote a _Second Defense_. He bitterly
opposed the Stamp Act, and urged the necessity of colonial union (or
"communion") to secure colonial liberties. He died on the 9th of July
1766. Mayhew was Dudleian lecturer at Harvard in 1765, and in 1749 had
received the degree of D.D. from the University of Aberdeen.

  See Alden Bradford, _Memoir of the Life and Writings of Rev. Jonathan
  Mayhew_ (Boston, 1838), and "An Early Pulpit Champion of Colonial
  Rights," chapter vi., in vol. i. of M. C. Tyler's _Literary History of
  the American Revolution_ (2 vols., New York, 1897).


FOOTNOTE:

  [1] Francis Bernard, whose project for a college at Northampton
    seemed to Mayhew and others a move to strengthen Anglicanism.




MAYHEW, THOMAS, English 18th century cabinet-maker. Mayhew was the less
distinguished partner of William Ince (q.v.). The chief source of
information as to his work is supplied by his own drawings in the volume
of designs, _The universal system of household furniture_, which he
published in collaboration with his partner. The name of the firm
appears to have been Mayhew and Ince, but on the title page of this book
the names are reversed, perhaps as an indication that Ince was the more
extensive contributor. In the main Mayhew's designs are heavy and
clumsy, and often downright extravagant, but he had a certain lightness
of accomplishment in his applications of the bizarre Chinese style. Of
original talent he possessed little, yet it is certain that much of his
Chinese work has been attributed to Chippendale. It is indeed often only
by reference to books of design that the respective work of the English
cabinet-makers of the second half of the 18th century can be correctly
attributed.




MAYMYO, a hill sanatorium in India, in the Mandalay district of Upper
Burma, 3500 ft. above the sea, with a station on the Mandalay-Lashio
railway 422 m. from Rangoon. Pop. (1901), 6223. It consists of an
undulating plateau, surrounded by hills, which are covered with thin oak
forest and bracken. Though not entirely free from malaria, it has been
chosen for the summer residence of the lieutenant-governor; and it is
also the permanent headquarters of the lieutenant-general commanding the
Burma division, and of other officials.




MAYNARD, FRANÇOIS DE (1582-1646), French poet, was born at Toulouse in
1582. His father was _conseiller_ in the parlement of the town, and
François was also trained for the law, becoming eventually president of
Aurillac. He became secretary to Margaret of Valois, wife of Henry IV.,
for whom his early poems are written. He was a disciple of Malherbe, who
said that in the workmanship of his lines he excelled Racan, but lacked
his rival's energy. In 1634 he accompanied the Cardinal de Noailles to
Rome and spent about two years in Italy. On his return to France he made
many unsuccessful efforts to obtain the favour of Richelieu, but was
obliged to retire to Toulouse. He never ceased to lament his exile from
Paris and his inability to be present at the meetings of the Academy, of
which he was one of the earliest members. The best of his poems is in
imitation of Horace, "Alcippe, reviens dans nos bois." He died at
Toulouse on the 23rd of December 1646.

  His works consist of odes, epigrams, songs and letters, and were
  published in 1646 by Marin le Roy de Gomberville.




MAYNE, JASPER (1604-1672), English author, was baptized at Hatherleigh,
Devonshire, on the 23rd of November 1604. He was educated at Westminster
School and at Christ Church, Oxford, where he had a distinguished
career. He was presented to two college livings in Oxfordshire, and was
made D.D. in 1646. During the Commonwealth he was dispossessed, and
became chaplain to the duke of Devonshire. At the Restoration he was
made canon of Christ Church, archdeacon of Chichester and chaplain in
ordinary to the king. He wrote a farcical domestic comedy, _The City
Match_ (1639), which is reprinted in vol. xiii. of Hazlitt's edition of
Dodsley's _Old Plays_, and a fantastic tragi-comedy entitled _The
Amorous War_ (printed 1648). After receiving ecclesiastical preferment
he gave up poetry as unbefitting his profession. His other works
comprise some occasional gems, a translation of Lucian's _Dialogues_
(printed 1664) and a number of sermons. He died on the 6th of December
1672 at Oxford.




MAYNOOTH, a small town of county Kildare, Ireland, on the Midland Great
Western railway and the Royal Canal, 15 m. W. by N. of Dublin. Pop.
(1901), 948. The Royal Catholic College of Maynooth, founded by an Act
of the Irish parliament in 1795, is the chief seminary for the education
of the Roman Catholic clergy of Ireland. The building is a fine Gothic
structure by A. W. Pugin, erected by a parliamentary grant obtained in
1846. The chapel, with fine oak choir-stalls, mosaic pavements, marble
altars and stained glass, and with adjoining cloisters, was dedicated in
1890. The average number of students is about 500--the number specified
under the act of 1845--and the full course of instruction is eight
years. Near the college stand the ruins of Maynooth Castle, probably
built in 1176, but subsequently extended, and formerly the residence of
the Fitzgerald family. It was besieged in the reigns of Henry VIII. and
Edward VI., and during the Cromwellian Wars, when it was demolished. The
beautiful mansion of Carton is about a mile from the town.




MAYO, RICHARD SOUTHWELL BOURKE, 6TH EARL OF (1822-1872), British
statesman, son of Robert Bourke, the 5th earl (1797-1867), was born in
Dublin on the 21st of February, 1822, and was educated at Trinity
College, Dublin. After travelling in Russia he entered parliament, and
sat successively for Kildare, Coleraine and Cockermouth. He was chief
secretary for Ireland in three administrations, in 1852, 1858 and 1866,
and was appointed viceroy of India in January 1869. He consolidated the
frontiers of India and met Shere Ali, amir of Afghanistan, in durbar at
Umballa in March 1869. His reorganization of the finances of the country
put India on a paying basis; and he did much to promote irrigation,
railways, forests and other useful public works. Visiting the convict
settlement at Port Blair in the Andaman Islands, for the purpose of
inspection, the viceroy was assassinated by a convict on the 8th of
February 1872. His successor was his son, Dermot Robert Wyndham Bourke
(b. 1851) who became 7th earl of Mayo.

  See Sir W. W. Hunter, _Life of the Earl of Mayo_, (1876), and _The
  Earl of Mayo_ in the Rulers of India Series (1891).




MAYO, a western county of Ireland, in the province of Connaught, bounded
N. and W. by the Atlantic Ocean, N.E. by Sligo, E. by Roscommon, S.E.
and S. by Galway. The area is 1,380,390 acres, or about 2157 sq. m., the
county being the largest in Ireland after Cork and Galway. About
two-thirds of the boundary of Mayo is formed by sea, and the coast is
very much indented, and abounds in picturesque scenery. The principal
inlets are Killary Harbour between Mayo and Galway; Clew Bay, in which
are the harbours of Westport and Newport; Blacksod Bay and Broad Haven,
which form the peninsula of the Mullet; and Killala Bay between Mayo and
Sligo. The islands are very numerous, the principal being Inishturk,
near Killary Harbour; Clare Island, at the mouth of Clew Bay, where
there are many islets, all formed of drift; and Achill, the largest
island off Ireland. The coast scenery is not surpassed by that of
Donegal northward and Connemara southward, and there are several small
coast-towns, among which may be named Killala on the north coast,
Belmullet on the isthmus between Blacksod Bay and Broad Haven, Newport
and Westport on Clew Bay, with the watering-place of Mallaranny. The
majestic cliffs of the north coast, however, which reach an extreme
height in Benwee Head (892 ft.), are difficult of access and rarely
visited. In the eastern half of the county the surface is comparatively
level, with occasional hills; the western half is mountainous. Mweelrea
(2688 ft.) is included in a mountain range lying between Killary Harbour
and Lough Mask. The next highest summits are Nephin (2646 ft.), to the
west of Lough Conn, and Croagh Patrick (2510 ft.), to the south of Clew
Bay. The river Moy flows northwards, forming part of the boundary of the
county with Sligo, and falls into Killala Bay. The courses of the other
streams are short, and except when swollen by rains their volume is
small. The principal lakes are Lough Mask and Lough Corrib, on the
borders of the county with Galway, and Loughs Conn in the east,
Carrowmore in the north-west, Beltra in the west, and Carra adjoining
Lough Mask. These loughs and the smaller loughs, with the streams
generally, afford admirable sport with salmon, sea-trout and brown
trout, and Ballina is a favourite centre.

  _Geology._--The wild and barren west of this county, including the
  great hills on Achill Island, is formed of "Dalradian" rocks, schists
  and quartzites, highly folded and metamorphosed, with intrusions of
  granite near Belmullet. At Blacksod Bay the granite has been quarried
  as an ornamental stone. Nephin Beg, Nephin and Croagh Patrick are
  typical quartzite summits, the last named belonging possibly to a
  Silurian horizon but rising from a metamorphosed area on the south
  side of Clew Bay. The schists and gneisses of the Ox Mountain axis
  also enter the county north of Castlebar. The Muilrea and Ben Gorm
  range, bounding the fine fjord of Killary Harbour, is formed of
  terraced Silurian rocks, from Bala to Ludlow age. These beds, with
  intercalated lavas, form the mountainous west shore of Lough Mask, the
  east, like that of Lough Corrib, being formed of low Carboniferous
  Limestone ground. Silurian rocks, with Old Red Sandstone over them,
  come out at the west end of the Curlew range at Ballaghaderreen. Clew
  Bay, with its islets capped by glacial drift, is a submerged part of a
  synclinal of Carboniferous strata, and Old Red Sandstone comes out on
  the north side of this, from near Achill to Lough Conn. The country
  from Lough Conn northward to the sea is a lowland of Carboniferous
  Limestone, with L. Carboniferous Sandstone against the Dalradian on
  the west.

  _Industries._--There are some very fertile regions in the level
  portions of the county, but in the mountainous districts the soil is
  poor, the holdings are subdivided beyond the possibility of affording
  proper sustenance to their occupiers, and, except where fishing is
  combined with agricultural operations, the circumstances of the
  peasantry are among the most wretched of any district of Ireland. The
  proportion of tillage to pasturage is roughly as 1 to 3½. Oats and
  potatoes are the principal crops. Cattle, sheep, pigs and poultry are
  reared. Coarse linen and woollen cloths are manufactured to a small
  extent. At Foxford woollen-mills are established at a nunnery, in
  connexion with a scheme of technical instruction. Keel, Belmullet and
  Ballycastle are the headquarters of sea and coast fishing districts,
  and Ballina of a salmon-fishing district, and these fisheries are of
  some value to the poor inhabitants. A branch of the Midland Great
  Western railway enters the county from Athlone, in the south-east, and
  runs north to Ballina and Killala on the coast, branches diverging
  from Claremorris to Ballinrobe, and from Manulla to Westport and
  Achill on the west coast. The Limerick and Sligo line of the Great
  Southern and Western passes from south to north-east by way of
  Claremorris.

_Population and Administration._--The population was 218,698 in 1891,
and 199,166 in 1901. The decrease of population and the number of
emigrants are slightly below the average of the Irish counties. Of the
total population about 97% are rural, and about the same percentage are
Roman Catholics. The chief towns are Ballina (pop. 4505), Westport
(3892) and Castlebar (3585), the county town. Ballaghaderreen,
Claremorris (Clare), Crossmolina and Swineford are lesser market towns;
and Newport and Westport are small seaports on Clew Bay. The county
includes nine baronies. Assizes are held at Castlebar, and quarter
sessions at Ballina, Ballinrobe, Belmullet, Castlebar, Claremorris,
Swineford and Westport. In the Irish parliament two members were
returned for the county, and two for the borough of Castlebar, but at
the union Castlebar was disfranchised. The division since 1885 is into
north, south, east and west parliamentary divisions, each returning one
member. The county is in the Protestant diocese of Tuam and the Roman
Catholic dioceses of Taum, Achonry, Galway and Kilmacduagh, and Killala.

_History and Antiquities._--Erris in Mayo was the scene of the landing
of the chief colony of the Firbolgs, and the battle which is said to
have resulted in the overthrow and almost annihilation of this tribe
took place also in this county, at Moytura near Cong. At the close of
the 12th century what is now the county of Mayo was granted, with other
lands, by king John to William, brother of Hubert de Burgh. After the
murder of William de Burgh, 3rd earl of Ulster (1333), the Bourkes (de
Burghs) of the collateral male line, rejecting the claim of William's
heiress (the wife of Lionel, son of King Edward III.) to the succession,
succeeded in holding the bulk of the De Burgh possessions, what is now
Mayo falling to the branch known by the name of "MacWilliam Oughter,"
who maintained their virtual independence till the time of Elizabeth.
Sir Henry Sydney, during his first viceroyalty, after making efforts to
improve communications between Dublin and Connaught in 1566, arranged
for the shiring of that province, and Mayo was made shire ground, taking
its name from the monastery of Maio or Mageo, which was the seat of a
bishop. Even after this period the MacWilliams continued to exercise
very great authority, which was regularized in 1603, when "the
MacWilliam Oughter," Theobald Bourke, surrendered his lands and received
them back, to hold them by English tenure, with the title of Viscount
Mayo (see BURGH, DE). Large confiscations of the estates in the county
were made in 1586, and on the termination of the wars of 1641; and in
1666 the restoration of his estates to the 4th Viscount Mayo involved
another confiscation, at the expense of Cromwell's settlers. Killala was
the scene of the landing of a French squadron in connexion with the
rebellion of 1798. In 1879 the village of Knock in the south-east
acquired notoriety from a story that the Virgin Mary had appeared in the
church, which became the resort of many pilgrims.

There are round towers at Killala, Turlough, Meelick and Balla, and an
imperfect one at Aughagower. Killala was formerly a bishopric. The
monasteries were numerous, and many of them of considerable importance:
the principal being those at Mayo, Ballyhaunis, Cong, Ballinrobe,
Ballintober, Burrishoole, Cross or Holycross in the peninsula of Mullet,
Moyne, Roserk or Rosserick and Templemore or Strade. Of the old castles
the most notable are Carrigahooly near Newport, said to have been built
by the celebrated Grace O'Malley, and Deel Castle near Ballina, at one
time the residence of the earls of Arran.

  See Hubert Thomas Knox, _History of the County of Mayo_ (1908).




MAYOR, JOHN EYTON BICKERSTETH (1825-   ), English classical scholar, was
born at Baddegama, Ceylon, on the 28th of January 1825, and educated in
England at Shrewsbury School and St John's College, Cambridge. From 1863
to 1867 he was librarian of the university, and in 1872 succeeded H. A.
J. Munro in the professorship of Latin. His best-known work, an edition
of thirteen satires of Juvenal, is marked by an extraordinary wealth of
illustrative quotations. His _Bibliographical Clue to Latin Literature_
(1873), based on E. Hübner's _Grundriss zu Vorlesungen über die römische
Litteraturgeschichte_ is a valuable aid to the student, and his edition
of Cicero's _Second Philippic_ is widely used. He also edited the
English works of J. Fisher, bishop of Rochester, i. (1876); Thomas
Baker's _History of St John's College, Cambridge_ (1869); Richard of
Cirencester's _Speculum historiale de gestis regum Angliae 447-1066_
(1863-1869); Roger Ascham's _Schoolmaster_ (new ed., 1883); the _Latin
Heptateuch_ (1889); and the _Journal of Philology_.

His brother, JOSEPH BICKERSTETH MAYOR (1828-   ), classical scholar and
theologian, was educated at Rugby and St John's College, Cambridge, and
from 1870 to 1879 was professor of classics at King's College, London.
His most important classical works are an edition of Cicero's _De natura
deorum_ (3 vols., 1880-1885) and _Guide to the Choice of Classical
Books_ (3rd ed., 1885, with supplement, 1896). He also devoted attention
to theological literature and edited the epistles of St James (2nd ed.,
1892), St Jude and St Peter (1907), and the _Miscellanies_ of Clement of
Alexandria (with F. J. A. Hort, 1902). From 1887 to 1893 he was editor
of the _Classical Review_. His _Chapters on English Metre_ (1886)
reached a second edition in 1901.




MAYOR (Lat. _major_, greater), in modern times the title of a municipal
officer who discharges judicial and administrative functions. The French
form of the word is _maire_. In Germany the corresponding title is
_Bürgermeister_, in Italy _sindico_, and in Spain _alcalde_. "Mayor" had
originally a much wider significance. Among the nations which arose on
the ruins of the Roman empire of the West, and which made use of the
Latin spoken by their "Roman" subjects as their official and legal
language, _major_ and the Low Latin feminine _majorissa_ were found to
be very convenient terms to describe important officials of both sexes
who had the superintendence of others. Any female servant or slave in
the household of a barbarian, whose business it was to overlook other
female servants or slaves, would be quite naturally called a
_majorissa_. So the male officer who governed the king's household would
be the _major domus_. In the households of the Frankish kings of the
Merovingian line, the _major domus_, who was also variously known as the
_gubernator_, _rector_, _moderator_ or _praefectus palatii_, was so
great an officer that he ended by evicting his master. He was the "mayor
of the palace" (q.v.). The fact that his office became hereditary in the
family of Pippin of Heristal made the fortune of the Carolingian line.
But besides the _major domus_ (the major-domo), there were other
officers who were _majores_, the _major cubiculi_, mayor of the
bedchamber, and _major equorum_, mayor of the horse. In fact a word
which could be applied so easily and with accuracy in so many
circumstances was certain to be widely used by itself, or in its
derivatives. The post-Augustine _majorinus_, "one of the larger kind,"
was the origin of the medieval Spanish _merinus_, who in Castillian is
the _merino_, and sometimes the _merino mayor_, or chief merino. He was
a judicial and administrative officer of the king's. The _gregum
merinus_ was the superintendent of the flocks of the corporation of
sheep-owners called the _mesta_. From him the sheep, and then the wool,
have come to be known as _merinos_--a word identical in origin with the
municipal title of mayor. The latter came directly from the heads of
gilds, and other associations of freemen, who had their banner and
formed a group on the populations of the towns, the _majores baneriae_
or _vexilli_.

In England the major is the modern representative of the lord's bailiff
or reeve (see BOROUGH). We find the chief magistrate of London bearing
the title of portreeve for considerably more than a century after the
Conquest. This official was elected by popular choice, a privilege
secured from king John. By the beginning of the 11th century the title
of portreeve[1] gave way to that of mayor as the designation of the
chief officer of London,[2] and the adoption of the title by other
boroughs followed at various intervals.

  A mayor is now in England and America the official head of a municipal
  government. In the United Kingdom the Municipal Corporations Act,
  1882, s. 15, regulates the election of mayors. He is to be a fit
  person elected annually on the 9th of November by the council of the
  borough from among the aldermen or councillors or persons qualified to
  be such. His term of office is one year, but he is eligible for
  re-election. He may appoint a deputy to act during illness or absence,
  and such deputy must be either an alderman or councillor. A mayor who
  is absent from the borough for more than two months becomes
  disqualified and vacates his office. A mayor is _ex officio_ during
  his year of office and the next year a justice of the peace for the
  borough. He receives such remuneration as the council thinks
  reasonable. The office of mayor in an English borough does not entail
  any important administrative duties. It is generally regarded as an
  honour conferred for past services. The mayor is expected to devote
  much of his time to ornamental functions and to preside over meetings
  which have for their object the advancement of the public welfare. His
  administrative duties are merely to act as returning officer at
  municipal elections, and as chairman of the meetings of the council.

  The position and power of an English mayor contrast very strongly with
  those of the similar official in the United States. The latter is
  elected directly by the voters within the city, usually for several
  years; and he has extensive administrative powers.

  The English method of selecting a mayor by the council is followed for
  the corresponding functionaries in France (except Paris), the more
  important cities of Italy, and in Germany, where, however, the central
  government must confirm the choice of the council. Direct appointment
  by the central government exists in Belgium, Holland, Denmark, Norway,
  Sweden and the smaller towns of Italy and Spain. As a rule, too, the
  term of office is longer in other countries than in the United
  Kingdom. In France election is for four years, in Holland for six, in
  Belgium for an indefinite period, and in Germany usually for twelve
  years, but in some cases for life. In Germany the post may be said to
  be a professional one, the burgomaster being the head of the city
  magistracy, and requiring, in order to be eligible, a training in
  administration. German burgomasters are most frequently elected by
  promotion from another city. In France the _maire_, and a number of
  experienced members termed "adjuncts," who assist him as an executive
  committee, are elected directly by the municipal council from among
  their own number. Most of the administrative work is left in the hands
  of the _maire_ and his adjuncts, the full council meeting
  comparatively seldom. The _maire_ and the adjuncts receive no salary.

  Further information will be found in the sections on local government
  in the articles on the various countries; see also A. Shaw, _Municipal
  Government in Continental Europe_; J. A. Fairlie, _Municipal
  Administration_; S. and B. Webb, _English Local Government_; Redlich
  and Hirst, _Local Government in England_; A. L. Lowell, _The
  Government of England_.


FOOTNOTES:

  [1] If a place was of mercantile importance it was called a port
    (from _porta_, the city gate), and the reeve or bailiff, a
    "portreeve."

  [2] The mayors of certain cities in the United Kingdom (London, York,
    Dublin) have acquired by prescription the prefix of "lord." In the
    case of London it seems to date from 1540. It has also been conferred
    during the closing years of the 19th century by letters patent on
    other cities--Birmingham, Liverpool, Manchester, Bristol, Sheffield,
    Leeds, Cardiff, Bradford, Newcastle-on-Tyne, Belfast, Cork. In 1910
    it was granted to Norwich. Lord mayors are entitled to be addressed
    as "right honourable."




MAYOR OF THE PALACE.--The office of mayor of the palace was an
institution peculiar to the Franks of the Merovingian period. A
landowner who did not manage his own estate placed it in the hands of a
steward (_major_), who superintended the working of the estate and
collected its revenues. If he had several estates, he appointed a chief
steward, who managed the whole of the estates and was called the _major
domus_. Each great personage had a _major domus_--the queen had hers,
the king his; and since the royal house was called the palace, this
officer took the name of "mayor of the palace." The mayor of the palace,
however, did not remain restricted to domestic functions; he had the
discipline of the palace and tried persons who resided there. Soon his
functions expanded. If the king were a minor, the mayor of the palace
supervised his education in the capacity of guardian (_nutricius_), and
often also occupied himself with affairs of state. When the king came of
age, the mayor exerted himself to keep this power, and succeeded. In the
7th century he became the head of the administration and a veritable
prime minister. He took part in the nomination of the counts and dukes;
in the king's absence he presided over the royal tribunal; and he often
commanded the armies. When the custom of commendation developed, the
king charged the mayor of the palace to protect those who had commended
themselves to him and to intervene at law on their behalf. The mayor of
the palace thus found himself at the head of the _commendati_, just as
he was at the head of the functionaries.

It is difficult to trace the names of some of the mayors of the palace,
the post being of almost no significance in the time of Gregory of
Tours. When the office increased in importance the mayors of the palace
did not, as has been thought, pursue an identical policy. Some--for
instance, Otto, the mayor of the palace of Austrasia towards 640--were
devoted to the Crown. On the other hand, mayors like Flaochat (in
Burgundy) and Erkinoald (in Neustria) stirred up the great nobles, who
claimed the right to take part in their nomination, against the king.
Others again, sought to exercise the power in their own name both
against the king and against the great nobles--such as Ebroïn (in
Neustria), and, later, the Carolingians Pippin II., Charles Martel, and
Pippin III., who, after making use of the great nobles, kept the
authority for themselves. In 751 Pippin III., fortified by his
consultation with Pope Zacharias, could quite naturally exchange the
title of mayor for that of king; and when he became king, he suppressed
the title of mayor of the palace. It must be observed that from 639
there were generally separate mayors of Neustria, Austrasia and
Burgundy, even when Austrasia and Burgundy formed a single kingdom; the
mayor was a sign of the independence of the region. Each mayor, however,
sought to supplant the others; the Pippins and Charles Martel succeeded,
and their victory was at the same time the victory of Austrasia over
Neustria and Burgundy.

  See G. H. Pertz, _Geschichte der merowingischen Hausmeier_ (Hanover,
  1819); H. Bonnell, _De dignitate majoris domus_ (Berlin, 1858); E.
  Hermann, _Das Hausmeieramt, ein echt germanisches Amt_, vol. ix. of
  _Untersuchungen zur deutschen Staats- und Rechtsgeschichte_, ed. by O.
  Gierke (Breslau, 1878, seq.); G. Waitz, _Deutsche
  Verfassungsgeschichte_, 3rd ed., revised by K. Zeumer; and Fustel de
  Coulanges, _Histoire des institutions politiques de l'ancienne France:
  La monarchie franque_ (Paris, 1888).     (C. Pf.)




MAYORUNA, a tribe of South American Indians of Panoan stock. Their
country is between the Ucayali and Javari rivers, north-eastern Peru.
They are a fine race, roaming the forests and living by hunting. They
cut their hair in a line across the forehead and let it hang down their
backs. Many have fair skins and beards, a peculiarity sometimes
explained by their alleged descent from Ursua's soldiers, but this
theory is improbable. They are famous for the potency of their blow-gun
poison.




MAYO-SMITH, RICHMOND (1854-1901), American economist, was born in Troy,
Ohio, on the 9th of February 1854. Educated at Amherst, and at Berlin
and Heidelberg, he became assistant professor of economics at Columbia
University in 1877. He was an adjunct professor from 1878 to 1883, when
he was appointed professor of political economy and social science, a
post which he held until his death on the 11th of November 1901. He
devoted himself especially to the study of statistics, and was
recognized as one of the foremost authorities on the subject. His works
include _Emigration and Immigration_ (1890); _Sociology and Statistics_
(1895), and _Statistics and Economics_ (1899).




MAYOTTE, one of the Comoro Islands, in the Mozambique Channel between
Madagascar and the African mainland. It has belonged to France since
1843 (see COMORO ISLANDS).




MAYOW, JOHN (1643-1679), English chemist and physiologist, was born in
London in May 1643. At the age of fifteen he went up to Wadham College,
Oxford, of which he became a scholar a year later, and in 1660 he was
elected to a fellowship at All Souls. He graduated in law (bachelor,
1665, doctor, 1670), but made medicine his profession, and "became noted
for his practice therein, especially in the summer time, in the city of
Bath." In 1678, on the proposal of R. Hooke, he was chosen a fellow of
the Royal Society. The following year, after a marriage which was "not
altogether to his content," he died in London in September 1679. He
published at Oxford in 1668 two tracts, on respiration and rickets, and
in 1674 these were reprinted, the former in an enlarged and corrected
form, with three others "De sal-nitro et spiritu nitro-aereo," "De
respiratione foetus in utero et ovo," and "De motu musculari et
spiritibus animalibus" as _Tractatus quinque medico-physici_. The
contents of this work, which was several times republished and
translated into Dutch, German and French, show him to have been an
investigator much in advance of his time.

  Accepting as proved by Boyle's experiments that air is necessary for
  combustion, he showed that fire is supported not by the air as a whole
  but by a "more active and subtle part of it." This part he called
  _spiritus igneo-aereus_, or sometimes _nitro-aereus_; for he
  identified it with one of the constituents of the acid portion of
  nitre which he regarded as formed by the union of fixed alkali with a
  _spiritus acidus_. In combustion the _particulae nitro-aereae_--either
  pre-existent in the thing consumed or supplied by the air--combined
  with the material burnt; as he inferred from his observation that
  antimony, strongly heated with a burning glass, undergoes an increase
  of weight which can be attributed to nothing else but these particles.
  In respiration he argued that the same particles are consumed, because
  he found that when a small animal and a lighted candle were placed in
  a closed vessel full of air the candle first went out and soon
  afterwards the animal died, but if there was no candle present it
  lived twice as long. He concluded that this constituent of the air is
  absolutely necessary for life, and supposed that the lungs separate it
  from the atmosphere and pass it into the blood. It is also necessary,
  he inferred, for all muscular movements, and he thought there was
  reason to believe that the sudden contraction of muscle is produced by
  its combination with other combustible (salino-sulphureous) particles
  in the body; hence the heart, being a muscle, ceases to beat when
  respiration is stopped. Animal heat also is due to the union of
  nitro-aerial particles, breathed in from the air, with the combustible
  particles in the blood, and is further formed by the combination of
  these two sets of particles in muscle during violent exertion. In
  effect, therefore, Mayow--who also gives a remarkably correct
  anatomical description of the mechanism of respiration--preceded
  Priestley and Lavoisier by a century in recognizing the existence of
  oxygen, under the guise of his _spiritus nitro-aereus_, as a separate
  entity distinct from the general mass of the air; he perceived the
  part it plays in combustion and in increasing the weight of the calces
  of metals as compared with metals themselves; and, rejecting the
  common notions of his time that the use of breathing is to cool the
  heart, or assist the passage of the blood from the right to the left
  side of the heart, or merely to agitate it, he saw in inspiration a
  mechanism for introducing oxygen into the body, where it is consumed
  for the production of heat and muscular activity, and even vaguely
  conceived of expiration as an excretory process.




MAYSVILLE, a city and the county-seat of Mason county, Kentucky, U.S.A.,
on the Ohio river, 60 m. by rail S.E. of Cincinnati. Pop. (1890) 5358;
(1900) 6423 (1155 negroes); (1910) 6141. It is served by the Louisville
& Nashville, and the Chesapeake & Ohio railways, and by steamboats on
the Ohio river. Among its principal buildings are the Mason county
public library (1878), the Federal building and Masonic and Odd Fellows'
temples. The city lies between the river and a range of hills; at the
back of the hills is a fine farming country, of which tobacco of
excellent quality is a leading product. There is a large plant of the
American Tobacco Company at Maysville, and among the city's manufactures
are pulleys, ploughs, whisky, flour, lumber, furniture, carriages,
cigars, foundry and machine-shop products, bricks and cotton goods. The
city is a distributing point for coal and other products brought to it
by Ohio river boats. Formerly it was one of the principal hemp markets
of the country. The place early became a landing point for immigrants to
Kentucky, and in 1784 a double log cabin and a blockhouse were erected
here. It was then called Limestone, from the creek which flows into the
Ohio here, but several years later the present name was adopted in
honour of John May, who with Simon Kenton laid out the town in 1787, and
who in 1790 was killed by the Indians. Maysville was incorporated as a
town in 1787, was chartered as a city in 1833, and became the
county-seat in 1848.

  In 1830, when the question of "internal improvements" by the National
  government was an important political issue, Congress passed a bill
  directing the government to aid in building a turnpike road from
  Maysville to Lexington. President Andrew Jackson vetoed the bill on
  the ground that the proposed improvement was a local rather than a
  national one; but one-half the capital was then furnished privately,
  the other half was furnished through several state appropriations, and
  the road was completed in 1835 and marked the beginning of a system of
  turnpike roads built with state aid.




MAZAGAN (_El Jadida_), a port on the Atlantic coast of Morocco in 33°
16´ N. 8° 26´ W. Pop. (1908), about 12,000, of whom a fourth are Jews
and some 400 Europeans. It is the port for Marrákesh, from which it is
110 m. nearly due north, and also for the fertile province of Dukálla.
Mazagan presents from the sea a very un-Moorish appearance; it has
massive Portuguese walls of hewn stone. The exports, which include
beans, almonds, maize, chick-peas, wool, hides, wax, eggs, &c., were
valued at £360,000 in 1900, £364,000 in 1904, and £248,000 in 1906. The
imports (cotton goods, sugar, tea, rice, &c.) were valued at £280,000 in
1900, £286,000 in 1904, and £320,000 in 1906. About 46% of the trade is
with Great Britain and 34% with France. Mazagan was built in 1506 by the
Portuguese, who abandoned it to the Moors in 1769 and established a
colony, New Mazagan, on the shores of Para in Brazil.

  See A. H. Dyé, "Les ports du Maroc" in _Bull. Soc. Geog. Comm. Paris_,
  xxx. 325-332 (1908), and British consular reports.




MAZAMET, an industrial town of south-western France in the department of
Tarn, 41 m. S.S.E. of Albi by rail. Pop. (1906), town, 11,370; commune,
14,386. Mazamet is situated on the northern slope of the Montagnes
Noires and on the Arnette, a small sub-tributary of the Agout. Numerous
establishments are employed in wool-spinning and in the manufacture of
"swan-skins" and flannels, and clothing for troops, and hosiery, and
there are important tanneries and leather-dressing, glove and dye works.
Extensive commerce is carried on in wool and raw hides from Argentina,
Australia and Cape Colony.




MAZANDARAN, a province of northern Persia, lying between the Caspian Sea
and the Elburz range, and bounded E. and W. by the provinces of
Astarabad and Gilan respectively, 220 m. in length and 60 m. in (mean)
breadth, with an area of about 10,000 sq. m. and a population estimated
at from 150,000 to 200,000. Mazandaran comprises two distinct natural
regions presenting the sharpest contrasts in their relief, climate and
products. In the north the Caspian is encircled by the level and swampy
lowlands, varying in breadth from 10 to 30 m., partly under impenetrable
jungle, partly under rice, cotton, sugar and other crops. This section
is fringed northwards by the sandy beach of the Caspian, here almost
destitute of natural harbours, and rises somewhat abruptly inland to the
second section, comprising the northern slopes and spurs of the Elburz,
which approach at some points within 1 or 2 m. of the sea, and are
almost everywhere covered with dense forest. The lowlands, rising but a
few feet above the Caspian, and subject to frequent floodings, are
extremely malarious, while the highlands, culminating with the
magnificent Demavend (19,400 ft.), enjoy a tolerably healthy climate.
But the climate, generally hot and moist in summer, is everywhere
capricious and liable to sudden changes of temperature, whence the
prevalence of rheumatism, dropsy and especially ophthalmia, noticed by
all travellers. Snow falls heavily in the uplands, where it often lies
for weeks on the ground. The direction of the long sandbanks at the
river mouths, which project with remarkable uniformity from west to
east, shows that the prevailing winds blow from the west and north-west.
The rivers themselves, of which there are as many as fifty, are little
more than mountain torrents, all rising on the northern slopes of
Elburz, flowing mostly in independent channels to the Caspian, and
subject to sudden freshets and inundations along their lower course. The
chief are the Sardab-rud, Chalus, Herhaz (Lar in its upper course),
Babul, Tejen and Nika, and all are well stocked with trout, salmon
(_azad-mahi_), perch (_safid-mahi_), carp (_kupur_), bream (_subulu_),
sturgeon (_sag-mahi_) and other fish, which with rice form the staple
food of the inhabitants; the sturgeon supplies the caviare for the
Russian market. Near their mouths the rivers, running counter to the
prevailing winds and waves of the Caspian, form long sand-hills 20 to 30
ft. high and about 200 yds. broad, behind which are developed the
so-called _múrd-áb_, or "dead waters," stagnant pools and swamps
characteristic of this coast, and a main cause of its unhealthiness.

The chief products are rice, cotton, sugar, a little silk, and fruits in
great variety, including several kinds of the orange, lemon and citron.
Some of the slopes are covered with extensive thickets of the
pomegranate, and the wild vine climbs to a great height round the trunks
of the forest trees. These woodlands are haunted by the tiger, panther,
bear, wolf and wild boar in considerable numbers. Of the domestic
animals, all remarkable for their small size, the chief are the black,
humped cattle somewhat resembling the Indian variety, and sheep and
goats.

  Kinneir, Fraser and other observers speak unfavourably of the
  Mazandarani people, whom they describe as very ignorant and bigoted,
  arrogant, rudely inquisitive and almost insolent towards strangers.
  The peasantry, however, are far from dull, and betray much shrewdness
  where their interests are concerned. In the healthy districts they are
  stout and well made, and are considered a warlike race, furnishing
  some cavalry (800 men) and eight battalions of infantry (5600 men) to
  government. They speak a marked Persian dialect, but a Turki idiom
  closely akin to the Turkoman is still current amongst the tribes,
  although they have mostly already passed from the nomad to the settled
  state. Of these tribes the most numerous are the Modaunlu, Khojehvand
  and Abdul Maleki, originally of Lek or Kurd stock, besides branches of
  the royal Afshar and Kajar tribes of Turki descent. All these are
  exempt from taxes in consideration of their military service.

  The export trade is chiefly with Russia from Meshed-i-Sar, the
  principal port of the province, to Baku, where European goods are
  taken in exchange for the white and coloured calicoes, caviare, rice,
  fruits and raw cotton of Mazandaran. Great quantities of rice are also
  exported to the interior of Persia, principally to Teheran and Kazvin.
  Owing to the almost impenetrable character of the country there are
  scarcely any roads accessible to wheeled carriages, and the great
  causeway of Shah Abbas along the coast has in many places even
  disappeared under the jungle. Two routes, however, lead to Teheran,
  one by Firuz Kuh, 180 m. long, the other by Larijan, 144 m. long, both
  in tolerably good repair. Except where crossed by these routes the
  Elburz forms an almost impassable barrier to the south.

  The administration is in the hands of a governor, who appoints the
  sub-governors of the nine districts of Amol, Barfarush, Meshed-i-Sar,
  Sari, Ashref, Farah-abad, Tunakabun, Kelarrustak and Kujur into which
  the province is divided. There is fair security for life and property;
  and, although otherwise indifferently administered, the country is
  quite free from marauders; but local disturbances have latterly been
  frequent in the two last-named districts. The revenue is about
  £30,000, of which little goes to the state treasury, most being
  required for the governors, troops and pensions. The capital is Sari,
  the other chief towns being Barfarush, Meshed-i-Sar, Ashref and
  Farah-abad.     (A. H.-S.)




MAZARIN, JULES (1602-1661), French cardinal and statesman, elder son of
a Sicilian, Pietro Mazarini, the intendant of the household of Philip
Colonna, and of his wife Ortensia Buffalini, a connexion of the
Colonnas, was born at Piscina in the Abruzzi on the 14th of July 1602.
He was educated by the Jesuits at Rome till his seventeenth year, when
he accompanied Jerome Colonna as chamberlain to the university of Alcala
in Spain. There he distinguished himself more by his love of gambling
and his gallant adventures than by study, but made himself a thorough
master, not only of the Spanish language and character, but also of that
romantic fashion of Spanish love-making which was to help him greatly in
after life, when he became the servant of a Spanish queen. On his return
to Rome, about 1622, he took his degree as Doctor _utriusque juris_, and
then became captain of infantry in the regiment of Colonna, which took
part in the war in the Valtelline. During this war he gave proofs of
much diplomatic ability, and Pope Urban VIII. entrusted him, in 1629,
with the difficult task of putting an end to the war of the Mantuan
succession. His success marked him out for further distinction. He was
presented to two canonries in the churches of St John Lateran and Sta
Maria Maggiore, although he had only taken the minor orders, and had
never been consecrated priest; he negotiated the treaty of Turin between
France and Savoy in 1632, became vice-legate at Avignon in 1634, and
nuncio at the court of France from 1634 to 1636. But he began to wish
for a wider sphere than papal negotiations, and, seeing that he had no
chance of becoming a cardinal except by the aid of some great power, he
accepted Richelieu's offer of entering the service of the king of
France, and in 1639 became a naturalized Frenchman.

In 1640 Richelieu sent him to Savoy, where the regency of Christine, the
duchess of Savoy, and sister of Louis XIII., was disputed by her
brothers-in-law, the princes Maurice and Thomas of Savoy, and he
succeeded not only in firmly establishing Christine but in winning over
the princes to France. This great service was rewarded by his promotion
to the rank of cardinal on the presentation of the king of France in
December 1641. On the 4th of December 1642 Cardinal Richelieu died, and
on the very next day the king sent a circular letter to all officials
ordering them to send in their reports to Cardinal Mazarin, as they had
formerly done to Cardinal Richelieu. Mazarin was thus acknowledged
supreme minister, but he still had a difficult part to play. The king
evidently could not live long, and to preserve power he must make
himself necessary to the queen, who would then be regent, and do this
without arousing the suspicions of the king or the distrust of the
queen. His measures were ably taken, and when the king died, on the 14th
of May 1643, to everyone's surprise her husband's minister remained the
queen's. The king had by a royal edict cumbered the queen-regent with a
council and other restrictions, and it was necessary to get the
parlement of Paris to overrule the edict and make the queen absolute
regent, which was done with the greatest complaisance. Now that the
queen was all-powerful, it was expected she would at once dismiss
Mazarin and summon her own friends to power. One of them, Potier, bishop
of Beauvais, already gave himself airs as prime minister, but Mazarin
had had the address to touch both the queen's heart by his Spanish
gallantry and her desire for her son's glory by his skilful policy
abroad, and he found himself able easily to overthrow the clique of
Importants, as they were called. That skilful policy was shown in every
arena on which the great Thirty Years' War was being fought out. Mazarin
had inherited the policy of France during the Thirty Years' War from
Richelieu. He had inherited his desire for the humiliation of the house
of Austria in both its branches, his desire to push the French frontier
to the Rhine and maintain a counterpoise of German states against
Austria, his alliances with the Netherlands and with Sweden, and his
four theatres of war--on the Rhine, in Flanders, in Italy and in
Catalonia.

During the last five years of the great war it was Mazarin alone who
directed the French diplomacy of the period. He it was who made the
peace of Brömsebro between the Danes and the Swedes, and turned the
latter once again against the empire; he it was who sent Lionne to make
the peace of Castro, and combine the princes of North Italy against the
Spaniards, and who made the peace of Ulm between France and Bavaria,
thus detaching the emperor's best ally. He made one fatal mistake--he
dreamt of the French frontier being the Rhine and the Scheldt, and that
a Spanish princess might bring the Spanish Netherlands as dowry to Louis
XIV. This roused the jealousy of the United Provinces, and they made a
separate peace with Spain in January 1648; but the valour of the French
generals made the skill of the Spanish diplomatists of no avail, for
Turenne's victory at Zusmarshausen, and Condé's at Lens, caused the
peace of Westphalia to be definitely signed in October 1648. This
celebrated treaty belongs rather to the history of Germany than to a
life of Mazarin; but two questions have been often asked, whether
Mazarin did not delay the peace as long as possible in order to more
completely ruin Germany, and whether Richelieu would have made a similar
peace. To the first question Mazarin's letters, published by M. Chéruel,
prove a complete negative, for in them appears the zeal of Mazarin for
the peace. On the second point, Richelieu's letters in many places
indicate that his treatment of the great question of frontier would have
been more thorough, but then he would not have been hampered in France
itself.

At home Mazarin's policy lacked the strength of Richelieu's. The Frondes
were largely due to his own fault. The arrest of Broussel threw the
people on the side of the parlement. His avarice and unscrupulous
plundering of the revenues of the realm, the enormous fortune which he
thus amassed, his supple ways, his nepotism, and the general lack of
public interest in the great foreign policy of Richelieu, made Mazarin
the especial object of hatred both by bourgeois and nobles. The
irritation of the latter was greatly Mazarin's own fault; he had tried
consistently to play off the king's brother Gaston of Orleans against
Condé, and their respective followers against each other, and had also,
as his _carnets_ prove, jealously kept any courtier from getting into
the good graces of the queen-regent except by his means, so that it was
not unnatural that the nobility should hate him, while the queen found
herself surrounded by his creatures alone. Events followed each other
quickly; the day of the barricades was followed by the peace of Ruel,
the peace of Ruel by the arrest of the princes, by the battle of Rethel,
and Mazarin's exile to Brühl before the union of the two Frondes. It was
while in exile at Brühl that Mazarin saw the mistake he had made in
isolating himself and the queen, and that his policy of balancing every
party in the state against each other had made every party distrust him.
So by his counsel the queen, while nominally in league with De Retz and
the parliamentary Fronde, laboured to form a purely royal party, wearied
by civil dissensions, who should act for her and her son's interest
alone, under the leadership of Mathieu Molé, the famous premier
president of the parlement of Paris. The new party grew in strength, and
in January 1652, after exactly a year's absence, Mazarin returned to the
court. Turenne had now become the royal general, and out-manoeuvred
Condé, while the royal party at last grew to such strength in Paris that
Condé had to leave the capital and France. In order to promote a
reconciliation with the parlement of Paris Mazarin had again retired
from court, this time to Sedan, in August 1652, but he returned finally
in February 1653. Long had been the trial, and greatly had Mazarin been
to blame in allowing the Frondes to come into existence, but he had
retrieved his position by founding that great royal party which steadily
grew until Louis XIV. could fairly have said "L'État, c'est moi." As the
war had progressed, Mazarin had steadily followed Richelieu's policy of
weakening the nobles on their country estates. Whenever he had an
opportunity he destroyed a feudal castle, and by destroying the towers
which commanded nearly every town in France, he freed such towns as
Bourges, for instance, from their long practical subjection to the
neighbouring great lord.

The Fronde over, Mazarin had to build up afresh the power of France at
home and abroad. It is to his shame that he did so little at home.
Beyond destroying the brick-and-mortar remains of feudalism, he did
nothing for the people. But abroad his policy was everywhere successful,
and opened the way for the policy of Louis XIV. He at first, by means of
an alliance with Cromwell, recovered the north-western cities of France,
though at the price of yielding Dunkirk to the Protector. On the Baltic,
France guaranteed the Treaty of Oliva between her old allies Sweden,
Poland and Brandenburg, which preserved her influence in that quarter.
In Germany he, through Hugues de Lionne, formed the league of the Rhine,
by which the states along the Rhine bound themselves under the headship
of France to be on their guard against the house of Austria. By such
measures Spain was induced to sue for peace, which was finally signed in
the Isle of Pheasants on the Bidassoa, and is known as the Treaty of the
Pyrenees. By it Spain recovered Franche Comté, but ceded to France
Roussillon, and much of French Flanders; and, what was of greater
ultimate importance to Europe, Louis XIV. was to marry a Spanish
princess, who was to renounce her claims to the Spanish succession if
her dowry was paid, which Mazarin knew could not happen at present from
the emptiness of the Spanish exchequer. He returned to Paris in
declining health, and did not long survive the unhealthy sojourn on the
Bidassoa; after some political instruction to his young master he passed
away at Vincennes on the 9th of March 1661, leaving a fortune estimated
at from 18 to 40 million livres behind him, and his nieces married into
the greatest families of France and Italy.

  The man who could have had such success, who could have made the
  Treaties of Westphalia and the Pyrenees, who could have weathered the
  storm of the Fronde, and left France at peace with itself and with
  Europe to Louis XIV., must have been a great man; and historians,
  relying too much on the brilliant memoirs of his adversaries, like De
  Retz, are apt to rank him too low. That he had many a petty fault
  there can be no doubt; that he was avaricious and double-dealing was
  also undoubted; and his _carnets_ show to what unworthy means he had
  recourse to maintain his influence over the queen. What that influence
  was will be always debated, but both his _carnets_ and the Brühl
  letters show that a real personal affection, amounting to passion on
  the queen's part, existed. Whether they were ever married may be
  doubted; but that hypothesis is made more possible by M. Chéruel's
  having been able to prove from Mazarin's letters that the cardinal
  himself had never taken more than the minor orders, which could always
  be thrown off. With regard to France he played a more patriotic part
  than Condé or Turenne, for he never treated with the Spaniards, and
  his letters show that in the midst of his difficulties he followed
  with intense eagerness every movement on the frontiers. It is that
  immense mass of letters that prove the real greatness of the
  statesman, and disprove De Retz's portrait, which is carefully
  arranged to show off his enemy against the might of Richelieu. To
  concede that the master was the greater man and the greater statesman
  does not imply that Mazarin was but a foil to his predecessor. It is
  true that we find none of those deep plans for the internal prosperity
  of France which shine through Richelieu's policy. Mazarin was not a
  Frenchman, but a citizen of the world, and always paid most attention
  to foreign affairs; in his letters all that could teach a diplomatist
  is to be found, broad general views of policy, minute details
  carefully elaborated, keen insight into men's characters, cunning
  directions when to dissimulate or when to be frank. Italian though he
  was by birth, education and nature, France owed him a great debt for
  his skilful management during the early years of Louis XIV., and the
  king owed him yet more, for he had not only transmitted to him a
  nation at peace, but had educated for him his great servants Le
  Tellier, Lionne and Colbert. Literary men owed him also much; not only
  did he throw his famous library open to them, but he pensioned all
  their leaders, including Descartes, Vincent Voiture (1598-1648), Jean
  Louis Guez de Balzac (1597-1654) and Pierre Corneille. The last-named
  applied, with an adroit allusion to his birthplace, in the dedication
  of his _Pompée_, the line of Virgil:--

    "Tu regere imperio populos, Romane, memento."     (H. M. S.)

  AUTHORITIES.--All the earlier works on Mazarin, and early accounts of
  his administration, of which the best were Bazin's _Histoire de France
  sous Louis XIII. et sous le Cardinal Mazarin_, 4 vols. (1846), and
  Saint-Aulaire's _Histoire de la Fronde_, have been superseded by P. A.
  Chéruel's admirable _Histoire de France pendant la minorité de Louis
  XIV._, 4 vols. (1879-1880), which covers from 1643-1651, and its
  sequel _Histoire de France sous le ministère de Cardinal Mazarin_, 2
  vols. (1881-1882), which is the first account of the period written by
  one able to sift the statements of De Retz and the memoir writers, and
  rest upon such documents as Mazarin's letters and _carnets_. Mazarin's
  _Lettres_, which must be carefully studied by any student of the
  history of France, have appeared in the _Collection des documents
  inédits_, 9 vols. For his _carnets_ reference must be made to V.
  Cousin's articles in the _Journal des Savants_, and Chéruel in _Revue
  historique_ (1877), see also Chéruel's _Histoire de France pendant la
  minorité_, &c., app. to vol. iii.; for his early life to Cousin's
  _Jeunesse de Mazarin_ (1865) and for the careers of his nieces to
  Renée's _Les Nièces de Mazarin_ (1856). For the Mazarinades or squibs
  written against him in Paris during the Fronde, see C. Moreau's
  _Bibliographie des mazarinades_ (1850), containing an account of 4082
  Mazarinades. See also A. Hassall, _Mazarin_ (1903).




MAZAR-I-SHARIF, a town of Afghanistan, the capital of the province of
Afghan Turkestan. Owing to the importance of the military cantonment of
Takhtapul, and its religious sanctity, it has long ago supplanted the
more ancient capital of Balkh. It is situated in a malarious, almost
desert plain, 9 m. E. of Balkh, and 30 m. S. of the Pata Kesar ferry on
the Oxus river. In this neighbourhood is concentrated most of the Afghan
army north of the Hindu Kush mountains, the fortified cantonment of
Dehdadi having been completed by Sirdar Ghulam Ali Khan and incorporated
with Mazar. Mazar-i-Sharif also contains a celebrated mosque, from which
the town takes its name. It is a huge ornate building with minarets and
a lofty cupola faced with shining blue tiles. It was built by Sultan Ali
Mirza about A.D. 1420, and is held in great veneration by all
Mussulmans, and especially by Shiites, because it is supposed to be the
tomb of Ali, the son-in-law of Mahomet.




MAZARRÓN, a town of eastern Spain, in the province of Murcia, 19 m. W.
of Cartagena. Pop. (1900), 23,284. There are soap and flour mills and
metallurgic factories in the town, and iron, copper and lead mines in
the neighbouring Sierra de Almenara. A railway 5 m. long unites Mazarron
to its port on the Mediterranean, where there is a suburb with 2500
inhabitants (mostly engaged in fisheries and coasting trade), containing
barracks, a custom-house, and important leadworks. Outside of the suburb
there are saltpans, most of the proceeds of which are exported to
Galicia.




MAZATLÁN, a city and port of the state of Sinaloa, Mexico, 120 m.
(direct) W.S.W. of the city of Durango, in lat. 23° 12´ N., long 106°
24´ W. Pop. (1895), 15,852; (1900), 17,852. It is the Pacific coast
terminus of the International railway which crosses northern Mexico from
Ciudad Porfirio Diaz, and a port of call for the principal steamship
lines on this coast. The harbour is spacious, but the entrance is
obstructed by a bar. The city is built on a small peninsula. Its public
buildings include a fine town-hall, chamber of commerce, a custom-house
and two hospitals, besides which there is a nautical school and a
meteorological station, one of the first established in Mexico. The
harbour is provided with a sea-wall at Olas Altas. A government wireless
telegraph service is maintained between Mazatlán and La Paz, Lower
California. Among the manufactures are saw-mills, foundries, cotton
factories and ropeworks, and the exports are chiefly hides, ixtle, dried
and salted fish, gold, silver and copper (bars and ores), fruit, rubber,
tortoise-shell, and gums and resins.




MAZE, a network of winding paths, a labyrinth (q.v.). The word means
properly a state of confusion or wonder, and is probably of Scandinavian
origin; cf. Norw. _mas_, exhausting labour, also chatter, _masa_, to be
busy, also to worry, annoy; Swed. _masa_, to lounge, move slowly and
lazily, to dream, muse. Skeat (_Etym._ Dict.) takes the original sense
to be probably "to be lost in thought," "to dream," and connects with
the root _ma-man_-, to think, cf. "mind," "man," &c. The word "maze"
represents the addition of an intensive suffix.




MAZEPA-KOLEDINSKY, IVAN STEPANOVICH (1644?-1709), hetman of the
Cossacks, belonging to a noble Orthodox family, was born possibly at
Mazeptsina, either in 1629 or 1644, the latter being the more probable
date. He was educated at the court of the Polish king, John Casimir, and
completed his studies abroad. An intrigue with a Polish married lady
forced him to fly into the Ukraine. There is a trustworthy tradition
that the infuriated husband tied the naked youth to the back of a wild
horse and sent him forth into the steppe. He was rescued and cared for
by the Dnieperian Cossacks, and speedily became one of their ablest
leaders. In 1687, during a visit to Moscow, he won the favour of the
then all-powerful Vasily Golitsuin, from whom he virtually purchased the
hetmanship of the Cossacks (July 25). He took a very active part in the
Azov campaigns of Peter the Great and won the entire confidence of the
young tsar by his zeal and energy. He was also very serviceable to Peter
at the beginning of the Great Northern War, especially in 1705 and 1706,
when he took part in the Volhynian campaign and helped to construct the
fortress of Pechersk. The power and influence of Mazepa were fully
recognized by Peter the Great. No other Cossack hetman had ever been
treated with such deference at Moscow. He ranked with the highest
dignitaries in the state; he sat at the tsar's own table. He had been
made one of the first cavaliers of the newly established order of St
Andrew, and Augustus of Poland had bestowed upon him, at Peter's earnest
solicitation, the universally coveted order of the White Eagle. Mazepa
had no temptations to be anything but loyal, and loyal he would
doubtless have remained had not Charles XII. crossed the Russian
frontier. Then it was that Mazepa, who had had doubts of the issue of
the struggle all along, made up his mind that Charles, not Peter, was
going to win, and that it was high time he looked after his own
interests. Besides, he had his personal grievances against the tsar. He
did not like the new ways because they interfered with his old ones. He
was very jealous of the favourite (Menshikov), whom he suspected of a
design to supplant him. But he proceeded very cautiously. Indeed, he
would have preferred to remain neutral, but he was not strong enough to
stand alone. The crisis came when Peter ordered him to co-operate
actively with the Russian forces in the Ukraine. At this very time he
was in communication with Charles's first minister, Count Piper, and had
agreed to harbour the Swedes in the Ukraine and close it against the
Russians (Oct. 1708). The last doubt disappeared when Menshikov was sent
to supervise Mazepa. At the approach of his rival the old hetman
hastened to the Swedish outposts at Horki, in Severia. Mazepa's treason
took Peter completely by surprise. He instantly commanded Menshikov to
get a new hetman elected and raze Baturin, Mazepa's chief stronghold in
the Ukraine, to the ground. When Charles, a week later, passed Baturin
by, all that remained of the Cossack capital was a heap of smouldering
mills and ruined houses. The total destruction of Baturin, almost in
sight of the Swedes, overawed the bulk of the Cossacks into obedience,
and Mazepa's ancient prestige was ruined in a day when the metropolitan
of Kiev solemnly excommunicated him from the high altar, and his effigy,
after being dragged with contumely through the mud at Kiev, was publicly
burnt by the common hangman. Henceforth Mazepa, perforce, attached
himself to Charles. What part he took at the battle of Poltava is not
quite clear. After the catastrophe he accompanied Charles to Turkey with
some 1500 horsemen (the miserable remnant of his 80,000 warriors). The
sultan refused to surrender him to the tsar, though Peter offered
300,000 ducats for his head. He died at Bender on the 22nd of August
1709.

  See N. I. Kostomarov, _Mazepa and the Mazepanites_ (Russ.) (St
  Petersburg), 1885; R. Nisbet Bain, _The First Romanovs_ (London,
  1905); S. M. Solovev, _History of Russia_ (Russ.), vol. xv. (St
  Petersburg, 1895).     (R. N. B.)




MAZER, the name of a special type of drinking vessel, properly made of
maple-wood, and so-called from the spotted or "birds-eye" marking on the
wood (Ger. _Maser_, spot, marking, especially on wood; cf. "measles").
These drinking vessels are shallow bowls without handles, with a broad
flat foot and a knob or boss in the centre of the inside, known
technically as the "print." They were made from the 13th to the 16th
centuries, and were the most prized of the various wooden cups in use,
and so were ornamented with a rim of precious metal, generally of silver
or silver gilt; the foot and the "print" being also of metal. The depth
of the mazers seems to have decreased in course of time, those of the
16th century that survive being much shallower than the earlier
examples. There are examples with wooden covers with a metal handle,
such as the Flemish and German mazers in the Franks Bequest in the
British Museum. On the metal rim is usually an inscription, religious or
bacchanalian, and the "print" was also often decorated. The later mazers
sometimes had metal straps between the rim and the foot.

  A very fine mazer with silver gilt ornamentation 3 in. deep and 9½ in.
  in diameter was sold in the Braikenridge collection in 1908 for £2300.
  It bears the London hall-mark of 1534. This example is illustrated in
  the article PLATE: see also DRINKING VESSELS.




MAZURKA (Polish for a woman of the province of Mazovia), a lively dance,
originating in Poland, somewhat resembling the polka.It is danced in
couples, the music being in 3/8 or ¾ time.




MAZZARA DEL VALLO, a town of Sicily, in the province of Trapani, on the
south-west coast of the island, 32 m. by rail S. of Trapani. Pop.
(1901), 20,130. It is the seat of a bishop; the cathedral, founded in
1093, was rebuilt in the 17th century. The castle, at the south-eastern
angle of the town walls, was erected in 1073. The mouth of the river,
which bears the same name, serves as a port for small ships only.
Mazzara was in origin a colony of Selinus: it was destroyed in 409, but
it is mentioned again as a Carthaginian fortress in the First Punic War
and as a post station on the Roman coast road, though whether it had
municipal rights is doubtful.[1] A few inscriptions of the imperial
period exist, but no other remains of importance. On the west bank of
the river are grottoes cut in the rock, of uncertain date: and there are
quarries in the neighbourhood resembling those of Syracuse, but on a
smaller scale.

  See A. Castiglione, _Sulle cose antiche della città di Mazzara_
  (Alcamo, 1878).


FOOTNOTE:

  [1] Th. Mommsen in _Corpus inscr. lat._ (Berlin, 1883), x. 739.




MAZZINI, GIUSEPPE (1805-1872), Italian patriot, was born on the 22nd of
June 1805 at Genoa, where his father, Giacomo Mazzini, was a physician
in good practice, and a professor in the university. His mother is
described as having been a woman of great personal beauty, as well as of
active intellect and strong affections. During infancy and childhood his
health was extremely delicate, and it appears that he was nearly six
years of age before he was quite able to walk; but he had already begun
to devour books of all kinds and to show other signs of great
intellectual precocity. He studied Latin with his first tutor, an old
priest, but no one directed his extensive course of reading. He became a
student at the university of Genoa at an unusually early age, and
intended to follow his father's profession, but being unable to conquer
his horror of practical anatomy, he decided to graduate in law (1826).
His exceptional abilities, together with his remarkable generosity,
kindness and loftiness of character, endeared him to his fellow
students. As to his inner life during this period, we have only one
brief but significant sentence; "for a short time," he says, "my mind
was somewhat tainted by the doctrines of the foreign materialistic
school; but the study of history and the intuitions of conscience--the
only tests of truth--soon led me back to the spiritualism of our Italian
fathers."

The natural bent of his genius was towards literature, and, in the
course of the four years of his nominal connexion with the legal
profession, he wrote a considerable number of essays and reviews, some
of which have been wholly or partially reproduced in the critical and
literary volumes of his _Life and Writings_. His first essay,
characteristically enough on "Dante's Love of Country," was sent to the
editor of the _Antologia fiorentina_ in 1826, but did not appear until
some years afterwards in the _Subalpino_. He was an ardent supporter of
romanticism as against what he called "literary servitude under the name
of classicism"; and in this interest all his critiques (as, for example,
that of Giannoni's "Exile" in the _Indicatore Livornese_, 1829) were
penned. But in the meantime the "republican instincts" which he tells us
he had inherited from his mother had been developing, and his sense of
the evils under which Italy was groaning had been intensified; and at
the same time he became possessed with the idea that Italians, and he
himself in particular, "_could_ and therefore _ought_ to struggle for
liberty of country." Therefore, he at once put aside his dearest
ambition, that of producing a complete history of religion, developing
his scheme of a new theology uniting the spiritual with the practical
life, and devoted himself to political thought. His literary articles
accordingly became more and more suggestive of advanced liberalism in
politics, and led to the suppression by government of the _Indicatore
Genovese_ and the _Indicatore Livornese_ successively. Having joined the
Carbonari, he soon rose to one of the higher grades in their hierarchy,
and was entrusted with a special secret mission into Tuscany; but, as
his acquaintance grew, his dissatisfaction with the organization of the
society increased, and he was already meditating the formation of a new
association stripped of foolish mysterious and theatrical formulae,
which instead of merely combating existing authorities should have a
definite and purely patriotic aim, when shortly after the French
revolution of 1830 he was betrayed, while initiating a new member, to
the Piedmontese authorities. He was imprisoned in the fortress of Savona
on the western Riviera for about six months, when, a conviction having
been found impracticable through deficiency of evidence, he was
released, but upon conditions involving so many restrictions of his
liberty that he preferred the alternative of leaving the country. He
withdrew accordingly into France, living chiefly in Marseilles.

While in his lonely cell at Savona, in presence of "those symbols of the
infinite, the sky and the sea," with a greenfinch for his sole
companion, and having access to no books but "a Tacitus, a Byron, and a
Bible," he had finally become aware of the great mission or "apostolate"
(as he himself called it) of his life; and soon after his release his
prison meditations took shape in the programme of the organization which
was destined soon to become so famous throughout Europe, that of _La
Giovine Italia_, or Young Italy. Its publicly avowed aims were to be the
liberation of Italy both from foreign and domestic tyranny, and its
unification under a republican form of government; the means to be used
were education, and, where advisable, insurrection by guerrilla bands;
the motto was to be "God and the people," and the banner was to bear on
one side the words "Unity" and "Independence" and on the other
"Liberty," "Equality," and "Humanity," to describe respectively the
national and the international aims. In April 1831 Charles Albert, "the
ex-Carbonaro conspirator of 1821," succeeded Charles Felix on the
Sardinian throne, and towards the close of that year Mazzini, making
himself, as he afterwards confessed, "the interpreter of a hope which he
did not share," wrote the new king a letter, published at Marseilles,
urging him to take the lead in the impending struggle for Italian
independence. Clandestinely reprinted, and rapidly circulated all over
Italy, its bold and outspoken words produced a great sensation, but so
deep was the offence it gave to the Sardinian government that orders
were issued for the immediate arrest and imprisonment of the author
should he attempt to cross the frontier. Towards the end of the same
year appeared the important Young Italy "Manifesto," the substance of
which is given in the first volume of the _Life and Writings_ of
Mazzini; and this was followed soon afterwards by the society's
_Journal_, which, smuggled across the Italian frontier, had great
success in the objects for which it was written, numerous
"congregations" being formed at Genoa, Leghorn, and elsewhere.
Representations were consequently made by the Sardinian to the French
government, which issued in an order for Mazzini's withdrawal from
Marseilles (Aug. 1832); he lingered for a few months in concealment, but
ultimately found it necessary to retire into Switzerland.

From this point it is somewhat difficult to follow the career of the
mysterious and terrible conspirator who for twenty years out of the next
thirty led a life of voluntary imprisonment (as he himself tells us)
"within the four walls of a room," and "kept no record of dates, made no
biographical notes, and preserved no copies of letters." In 1833,
however, he is known to have been concerned in an abortive revolutionary
movement which took place in the Sardinian army; several executions took
place, and he himself was laid under sentence of death. Before the close
of the same year a similar movement in Genoa had been planned, but
failed through the youth and inexperience of the leaders. At Geneva,
also in 1833, Mazzini set on foot _L'Europe Centrale_, a journal of
which one of the main objects was the emancipation of Savoy; but he did
not confine himself to a merely literary agitation for this end. Chiefly
through his agency a considerable body of German, Polish and Italian
exiles was organized, and an armed invasion of the duchy planned. The
frontier was actually crossed on the 1st of February 1834, but the
attack ignominiously broke down without a shot having been fired.
Mazzini, who personally accompanied the expedition, is no doubt correct
in attributing the failure to dissensions with the Carbonari leaders in
Paris, and to want of a cordial understanding between himself and the
Savoyard Ramorino, who had been chosen as military leader.

In April 1834 the "Young Europe" association "of men believing in a
future of liberty, equality and fraternity for all mankind, and desirous
of consecrating their thoughts and actions to the realization of that
future" was formed also under the influence of Mazzini's enthusiasm; it
was followed soon afterwards by a "Young Switzerland" society, having
for its leading idea the formation of an Alpine confederation, to
include Switzerland, Tyrol, Savoy and the rest of the Alpine chain as
well. But _La Jeune Suisse_ newspaper was compelled to stop within a
year, and in other respects the affairs of the struggling patriot became
embarrassed. He was permitted to remain at Grenchen in Solothurn for a
while, but at last the Swiss diet, yielding to strong and persistent
pressure from abroad, exiled him about the end of 1836. In January 1837
he arrived in London, where for many months he had to carry on a hard
fight with poverty and the sense of spiritual loneliness, so touchingly
described by himself in the first volume of the _Life and Writings_.
Ultimately, as he gained command of the English language, he began to
earn a livelihood by writing review articles, some of which have since
been reprinted, and are of a high order of literary merit; they include
papers on "Italian Literature since 1830" and "Paolo Sarpi" in the
_Westminster Review_, articles on "Lamennais," "George Sand," "Byron and
Goethe" in the _Monthly Chronicle_, and on "Lamartine," "Carlyle," and
"The Minor Works of Dante" in the _British and Foreign Review_. In 1839
he entered into relations with the revolutionary committees sitting in
Malta and Paris, and in 1840 he originated a working men's association,
and the weekly journal entitled _Apostolato Popolare_, in which the
admirable popular treatise "On the Duties of Man" was commenced. Among
the patriotic and philanthropic labours undertaken by Mazzini during
this period of retirement in London may be mentioned a free evening
school conducted by himself and a few others for some years, at which
several hundreds of Italian children received at least the rudiments of
secular and religious education. He also exposed and combated the
infamous traffic carried on in southern Italy, where scoundrels bought
small boys from poverty-stricken parents and carried them off to England
and elsewhere to grind organs and suffer martyrdom at the hands of cruel
taskmasters.

The most memorable episode in his life during the same period was
perhaps that which arose out of the conduct of Sir James Graham, the
home secretary, in systematically, for some months, opening Mazzini's
letters as they passed through the British post office, and
communicating their contents to the Neapolitan government--a proceeding
which was believed at the time to have led to the arrest and execution
of the brothers Bandiera, Austrian subjects, who had been planning an
expedition against Naples, although the recent publication of Sir James
Graham's life seems to exonerate him from the charge. The prolonged
discussions in parliament, and the report of the committee appointed to
inquire into the matter, did not, however, lead to any practical result,
unless indeed the incidental vindication of Mazzini's character, which
had been recklessly assailed in the course of debate. In this connexion
Thomas Carlyle wrote to _The Times_: "I have had the honour to know Mr
Mazzini for a series of years, and, whatever I may think of his
practical insight and skill in worldly affairs, I can with great freedom
testify that he, if I have ever seen one such, is a man of genius and
virtue, one of those rare men, numerable unfortunately but as units in
this world, who are worthy to be called martyr souls; who in silence,
piously in their daily life, practise what is meant by that."

Mazzini did not share the enthusiastic hopes everywhere raised in the
ranks of the Liberal party throughout Europe by the first acts of Pius
IX., in 1846, but at the same time he availed himself, towards the end
of 1847, of the opportunity to publish a letter addressed to the new
pope, indicating the nature of the religious and national mission which
the Liberals expected him to undertake. The leaders of the revolutionary
outbreaks in Milan and Messina in the beginning of 1848 had long been in
secret correspondence with Mazzini; and their action, along with the
revolution in Paris, brought him early in the same year to Italy, where
he took a great and active interest in the events which dragged Charles
Albert into an unprofitable war with Austria; he actually for a short
time bore arms under Garibaldi immediately before the reoccupation of
Milan, but ultimately, after vain attempts to maintain the insurrection
in the mountain districts, found it necessary to retire to Lugano. In
the beginning of the following year he was nominated a member of the
short-lived provisional government of Tuscany formed after the flight of
the grand-duke, and almost simultaneously, when Rome had, in consequence
of the withdrawal of Pius IX., been proclaimed a republic, he was
declared a member of the constituent assembly there. A month afterwards,
the battle of Novara having again decided against Charles Albert in the
brief struggle with Austria, into which he had once more been drawn,
Mazzini was appointed a member of the Roman triumvirate, with supreme
executive power (March 23, 1849). The opportunity he now had for showing
the administrative and political ability which he was believed to
possess was more apparent than real, for the approach of the professedly
friendly French troops soon led to hostilities, and resulted in a siege
which terminated, towards the end of June, with the assembly's
resolution to discontinue the defence, and Mazzini's indignant
resignation. That he succeeded, however, for so long a time, and in
circumstances so adverse, in maintaining a high degree of order within
the turbulent city is a fact that speaks for itself. His diplomacy,
backed as it was by no adequate physical force, naturally showed at the
time to very great disadvantage, but his official correspondence and
proclamations can still be read with admiration and intellectual
pleasure, as well as his eloquent vindication of the revolution in his
published "Letter to MM. de Tocqueville and de Falloux." The surrender
of the city on the 30th of June was followed by Mazzini's not too
precipitate flight by way of Marseilles into Switzerland, whence he once
more found his way to London. Here in 1850 he became president of the
National Italian Committee, and at the same time entered into close
relations with Ledru-Rollin and Kossuth. He had a firm belief in the
value of revolutionary attempts, however hopeless they might seem; he
had a hand in the abortive rising at Mantua in 1852, and again, in
February 1853, a considerable share in the ill-planned insurrection at
Milan on the 6th of February 1853, the failure of which greatly weakened
his influence; once more, in 1854, he had gone far with preparations for
renewed action when his plans were completely disconcerted by the
withdrawal of professed supporters, and by the action of the French and
English governments in sending ships of war to Naples.

The year 1857 found him yet once more in Italy, where, for complicity in
short-lived émeutes which took place at Genoa, Leghorn and Naples, he
was again laid under sentence of death. Undiscouraged in the pursuit of
the one great aim of his life by any such incidents as these, he
returned to London, where he edited his new journal _Pensiero ed
Azione_, in which the constant burden of his message to the overcautious
practical politicians of Italy was: "I am but a voice crying _Action_;
but the state of Italy cries for it also. So do the best men and people
of her cities. Do you wish to destroy my influence? _Act_." The same
tone was at a somewhat later date assumed in the letter he wrote to
Victor Emmanuel, urging him to put himself at the head of the movement
for Italian unity, and promising republican support. As regards the
events of 1859-1860, however, it may be questioned whether, through his
characteristic inability to distinguish between the ideally perfect and
the practically possible, he did not actually hinder more than he helped
the course of events by which the realization of so much of the great
dream of his life was at last brought about. If Mazzini was the prophet
of Italian unity, and Garibaldi its knight errant, to Cavour alone
belongs the honour of having been the statesman by whom it was finally
accomplished. After the irresistible pressure of the popular movement
had led to the establishment not of an Italian republic but of an
Italian kingdom, Mazzini could honestly enough write, "I too have
striven to realize unity under a monarchical flag," but candour
compelled him to add, "The Italian people are led astray by a delusion
at the present day, a delusion which has induced them to substitute
material for moral unity and their own reorganization. Not so I. I bow
my head sorrowfully to the sovereignty of the national will; but
monarchy will never number me amongst its servants or followers." In
1865, by way of protest against the still uncancelled sentence of death
under which he lay, Mazzini was elected by Messina as delegate to the
Italian parliament, but, feeling himself unable to take the oath of
allegiance to the monarchy, he never took his seat. In the following
year, when a general amnesty was granted after the cession of Venice to
Italy, the sentence of death was at last removed, but he declined to
accept such an "offer of oblivion and pardon for having loved Italy
above all earthly things." In May 1869 he was again expelled from
Switzerland at the instance of the Italian government for having
conspired with Garibaldi; after a few months spent in England he set out
(1870) for Sicily, but was promptly arrested at sea and carried to
Gaeta, where he was imprisoned for two months. Events soon made it
evident that there was little danger to fear from the contemplated
rising, and the occasion of the birth of a prince was seized for
restoring him to liberty. The remainder of his life, spent partly in
London and partly at Lugano, presents no noteworthy incidents. For some
time his health had been far from satisfactory, but the immediate cause
of his death was an attack of pleurisy with which he was seized at Pisa,
and which terminated fatally on the 10th of March 1872. The Italian
parliament by a unanimous vote expressed the national sorrow with which
the tidings of his death had been received, the president pronouncing an
eloquent eulogy on the departed patriot as a model of disinterestedness
and self-denial, and one who had dedicated his whole life ungrudgingly
to the cause of his country's freedom. A public funeral took place at
Pisa on the 14th of March, and the remains were afterwards conveyed to
Genoa.     (J. S. Bl.)

  The published writings of Mazzini, mostly occasional, are very
  voluminous. An edition was begun by himself and continued by A. Saffi,
  _Scritti editi e inediti di Giuseppe Mazzini_, in 18 vols. (Milan and
  Rome, 1861-1891); many of the most important are found in the
  partially autobiographical _Life and Writings of Joseph Mazzini_
  (1864-1870) and the two most systematic--_Thoughts upon Democracy in
  Europe_, a remarkable series of criticisms on Benthamism, St
  Simonianism, Fourierism, and other economic and socialistic schools of
  the day, and the treatise _On the Duties of Man_, an admirable primer
  of ethics, dedicated to the Italian working class--will be found in
  _Joseph Mazzini: a Memoir_, by Mrs E. A. Venturi (London, 1875).
  Mazzini's "first great sacrifice," he tells us, was "the renunciation
  of the career of literature for the more direct path of political
  action," and as late as 1861 we find him still recurring to the
  long-cherished hope of being able to leave the stormy arena of
  politics and consecrate the last years of his life to the dream of his
  youth. He had specially contemplated three considerable literary
  undertakings--a volume of _Thoughts on Religion_, a popular _History
  of Italy_, to enable the working classes to apprehend what he
  conceived to be the "mission" of Italy in God's providential ordering
  of the world, and a comprehensive collection of translations of
  ancient and modern classics into Italian. None of these was actually
  achieved. No one, however, can read even the briefest and most
  occasional writing of Mazzini without gaining some impression of the
  simple grandeur of the man, the lofty elevation of his moral tone, his
  unwavering faith in the living God, who is ever revealing Himself in
  the progressive development of humanity. His last public utterance is
  to be found in a highly characteristic article on Renan's _Réforme
  Morale et Intellectuelle_, finished on the 3rd of March 1872, and
  published in the _Fortnightly Review_ for February 1874. Of the 40,000
  letters of Mazzini only a small part have been published. In 1887 two
  hundred unpublished letters were printed at Turin (_Duecento lettere
  inedite di Giuseppe Mazzini_), in 1895 the _Lettres intimes_ were
  published in Paris, and in 1905 Francesco Rosso published _Lettre
  inedite di Giuseppe Mazzini_ (Turin, 1905). A popular edition of
  Mazzini's writings has been undertaken by order of the Italian
  government.

  For Mazzini's biography see Jessie White Mario, _Della vita di
  Giuseppe Mazzini_ (Milan, 1886), a useful if somewhat too enthusiastic
  work; Bolton King, _Mazzini_ (London, 1903); Count von Schack, _Joseph
  Mazzini und die italienische Einheit_ (Stuttgart, 1891). A. Luzio's
  _Giuseppe Mazzini_ (Milan, 1905) contains a great deal of valuable
  information, bibliographical and other, and Dora Melegari in _La
  giovine Italia e Giuseppe Mazzini_ (Milan, 1906) publishes the
  correspondence between Mazzini and Luigi A. Melegari during the early
  days of "Young Italy." For the literary side of Mazzini's life see
  Peretti, _Gli scritti letterarii di Giuseppe Mazzini_ (Turin, 1904).
       (L. V.*)




MAZZONI, GIACOMO (1548-1598), Italian philosopher, was born at Cesena
and died at Ferrara. A member of a noble family and highly educated, he
was one of the most eminent savants of the period. He occupied chairs in
the universities of Pisa and Rome, was one of the founders of the Della
Crusca Academy, and had the distinction, it is said, of thrice
vanquishing the Admirable Crichton in dialectic. His chief work in
philosophy was an attempt to reconcile Plato and Aristotle, and in this
spirit he published in 1597 a treatise _In universam Platonis et
Aristotelis philosophiam praecludia_. He wrote also _De triplici hominum
vita_, wherein he outlined a theory of the infinite perfection and
development of nature. Apart from philosophy, he was prominent in
literature as the champion of Dante, and produced two works in the
poet's defence: _Discorso composto in difesa della comedia di Dante_
(1572), and _Della difesa della comedia di Dante_ (1587, reprinted
1688). He was an authority on ancient languages and philology, and gave
a great impetus to the scientific study of the Italian language.




MAZZONI, GUIDO (1859-   ), Italian poet, was born at Florence, and
educated at Pisa and Bologna. In 1887 he became professor of Italian at
Padua, and in 1894 at Florence. He was much influenced by Carducci, and
became prominent both as a prolific and well-read critic and as a poet
of individual distinction. His chief volumes of verse are _Versi_
(1880), _Nuove poesie_ (1886), _Poesie_ (1891), _Voci della vita_
(1893).




MEAD, LARKIN GOLDSMITH (1835-   ), American sculptor, was born at
Chesterfield, New Hampshire, on the 3rd of January 1835. He was a pupil
(1853-1855) of Henry Kirke Brown. During the early part of the Civil
War he was at the front for six months, with the army of the Potomac, as
an artist for _Harper's Weekly_; and in 1862-1865 he was in Italy, being
for part of the time attached to the United States consulate at Venice,
while William D. Howells, his brother-in-law, was consul. He returned to
America in 1865, but subsequently went back to Italy and lived at
Florence. His first important work was a statue of Ethan Allen, now at
the State House, Montpelier, Vermont. His principal works are: the
monument to President Lincoln, Springfield, Illinois; "Ethan Allen"
(1876), National Hall of Statuary, Capitol, Washington; an heroic marble
statue, "The Father of Waters," New Orleans; and "Triumph of Ceres,"
made for the Columbian Exposition, Chicago.

His brother, WILLIAM RUTHERFORD MEAD (1846-   ), graduated at Amherst
College in 1867, and studied architecture in New York under Russell
Sturgis, and also abroad. In 1879 he and J. F. McKim, with whom he had
been in partnership for two years as architects, were joined by Stanford
White, and formed the well-known firm of McKim, Mead & White.




MEAD, RICHARD (1673-1754), English physician, eleventh child of Matthew
Mead (1630-1699), Independent divine, was born on the 11th of August
1673 at Stepney, London. He studied at Utrecht for three years under J.
G. Graevius; having decided to follow the medical profession, he then
went to Leiden and attended the lectures of Paul Hermann and Archibald
Pitcairne. In 1695 he graduated in philosophy and physic at Padua, and
in 1696 he returned to London, entering at once on a successful
practice. His _Mechanical Account of Poisons_ appeared in 1702, and in
1703 he was admitted to the Royal Society, to whose _Transactions_ he
contributed in that year a paper on the parasitic nature of scabies. In
the same year he was elected physician to St Thomas's Hospital, and
appointed to read anatomical lectures at the Surgeons' Hall. On the
death of John Radcliffe in 1714 Mead became the recognized head of his
profession; he attended Queen Anne on her deathbed, and in 1727 was
appointed physician to George II., having previously served him in that
capacity when he was prince of Wales. He died in London on the 16th of
February 1754.

  Besides the _Mechanical Account of Poisons_ (2nd ed., 1708), Mead
  published a treatise _De imperio solis et lunae in corpora humana et
  morbis inde oriundis_ (1704), _A Short Discourse concerning
  Pestilential Contagion, and the Method to be used to prevent it_
  (1720), _De variolis et morbillis dissertatio_ (1747), _Medica sacra,
  sive de morbis insignioribus qui in bibliis memorantur commentarius_
  (1748), _On the Scurvy_ (1749), and _Monita et praecepta medica_
  (1751). A _Life_ of Mead by Dr Matthew Maty appeared in 1755.




MEAD. (1) A word now only used more or less poetically for the commoner
form "meadow," properly land laid down for grass and cut for hay, but
often extended in meaning to include pasture-land. "Meadow" represents
the oblique case, _maédwe_, of O. Eng. _maéd_, which comes from the root
seen in "mow"; the word, therefore, means "mowed land." Cognate words
appear in other Teutonic languages, a familiar instance being Ger.
_matt_, seen in place-names such as Zermatt, Andermatt, &c. (See Grass.)
(2) The name of a drink made by the fermentation of honey mixed with
water. Alcoholic drinks made from honey were common in ancient times,
and during the middle ages throughout Europe. The Greeks and Romans knew
of such under the names of [Greek: hodromeli] and _hydromel_; _mulsum_
was a form of mead with the addition of wine. The word is common to
Teutonic languages (cf. Du. _mede_, Ger. _Met_ or _Meth_), and is
cognate with Gr. [Greek: methu], wine, and Sansk. _mádhu_, sweet drink.
"Metheglin," another word for mead, properly a medicated or spiced form
of the drink, is an adaptation of the Welsh _meddyglyn_, which is
derived from _meddyg_, healing (Lat. _medicus_) and _llyn_, liquor. It
therefore means "spiced or medicated drink," and is not etymologically
connected with "mead."




MEADE, GEORGE GORDON (1815-1872), American soldier, was born of American
parentage at Cadiz, Spain, on the 31st of December 1815. On graduation
at the United States Military Academy in 1835, he served in Florida with
the 3rd Artillery against the Seminoles. Resigning from the army in
1836, he became a civil engineer and constructor of railways, and was
engaged under the war department in survey work. In 1842 he was
appointed a second lieutenant in the corps of the topographical
engineers. In the war with Mexico he was on the staffs successively of
Generals Taylor, J. Worth and Robert Patterson, and was brevetted for
gallant conduct at Monterey. Until the Civil War he was engaged in
various engineering works, mainly in connexion with lighthouses, and
later as a captain of topographical engineers in the survey of the
northern lakes. In 1861 he was appointed brigadier-general of
volunteers, and had command of the 2nd brigade of the Pennsylvania
Reserves in the Army of the Potomac under General M'Call. He served in
the Seven Days, receiving a severe wound at the action of Frazier's
Farm. He was absent from his command until the second battle of Bull
Run, after which he obtained the command of his division. He
distinguished himself greatly at the battles of South Mountain and
Antietam. At Fredericksburg he and his division won great distinction by
their attack on the position held by Jackson's corps, and Meade was
promoted major-general of volunteers, to date from the 29th of November.
Soon afterwards he was placed in command of the V. corps. At
Chancellorsville he displayed great intrepidity and energy, and on the
eve of the battle of Gettysburg was appointed to succeed Hooker. The
choice was unexpected, but Meade justified it by his conduct of the
operations, and in the famous three days' battle he inflicted a complete
defeat on General Lee's army. His reward was the commission of
brigadier-general in the regular army. In the autumn of 1863 a war of
manoeuvre was fought between the two commanders, on the whole favourably
to the Union arms. Grant, commanding all the armies of the United
States, joined the Army of the Potomac in the spring of 1864, and
remained with it until the end of the war; but he continued Meade in his
command, and successfully urged his appointment as major-general in the
regular army (Aug. 18, 1864), eulogizing him as the commander who had
successfully met and defeated the best general and the strongest army on
the Confederate side. After the war Meade commanded successively the
military division of the Atlantic, the department of the east, the third
military district (Georgia and Alabama) and the department of the south.
He died at Philadelphia on the 6th of November, 1872. The degree of
LL.D. was conferred upon him by Harvard University, and his scientific
attainments were recognized by the American Philosophical Society and
the Philadelphia Academy of Natural Sciences. There are statues of
General Meade in Philadelphia and at Gettysburg.

  See I. R. Pennypacker, _General Meade_ ("Great Commanders" series, New
  York, 1901).




MEADE, WILLIAM (1789-1862), American Protestant Episcopal bishop, the
son of Richard Kidder Meade (1746-1805), one of General Washington's
aides during the War of Independence, was born on the 11th of November
1789, near Millwood, in that part of Frederick county which is now
Clarke county, Virginia. He graduated as valedictorian in 1808 at the
college of New Jersey (Princeton); studied theology under the Rev.
Walter Addison of Maryland, and in Princeton; was ordained deacon in
1811 and priest in 1814; and preached both in the Stone Chapel,
Millwood, and in Christ Church, Alexandria, for some time. He became
assistant bishop of Virginia in 1829; was pastor of Christ Church,
Norfolk, in 1834-1836; in 1841 became bishop of Virginia; and in
1842-1862 was president of the Protestant Episcopal Theological Seminary
in Virginia, near Alexandria, delivering an annual course of lectures on
pastoral theology. In 1819 he had acted as the agent of the American
Colonization Society to purchase slaves, illegally brought into Georgia,
which had become the property of that state and were sold publicly at
Milledgeville. He had been prominent in the work of the Education
Society, which was organized in 1818 to advance funds to needy students
for the ministry of the American Episcopal Church, and in the
establishment of the Theological Seminary near Alexandria, as he was
afterwards in the work of the American Tract Society, and the Bible
Society. He was a founder and president of the Evangelical Knowledge
Society (1847), which, opposing what it considered the heterodoxy of
many of the books published by the Sunday School Union, attempted to
displace them by issuing works of a more evangelical type. A low
Churchman, he strongly opposed Tractarianism. He was active in the case
against Bishop Henry Ustick Onderdonk (1789-1858) of Pennsylvania, who
because of intemperance was forced to resign and was suspended from the
ministry in 1844; in that against Bishop Benjamin Tredwell Onderdonk
(1791-1861) of New York, who in 1845 was suspended from the ministry on
the charge of intoxication and improper conduct; and in that against
Bishop G. W. Doane of New Jersey. He fought against the threatening
secession of Virginia, but acquiesced in the decision of the state and
became presiding bishop of the Southern Church. He died in Richmond,
Virginia, on the 14th of March 1862.

  Among his publications, besides many sermons, were _A Brief Review of
  the Episcopal Church in Virginia_ (1845); _Wilberforce, Cranmer,
  Jewett and the Prayer Book on the Incarnation_ (1850); _Reasons for
  Loving the Episcopal Church_ (1852); and _Old Churches, Ministers and
  Families of Virginia_ (1857); a storehouse of material on the
  ecclesiastical history of the state.

  See the _Life_ by John Johns (Baltimore, 1867).




MEADVILLE, a city and the county-seat of Crawford county, Pennsylvania,
U.S.A., on French Creek, 36 m. S. of Erie. Pop. (1900), 10,291, of whom
912 were foreign-born and 173 were negroes; (1910 census) 12,780. It is
served by the Erie, and the Bessemer & Lake Erie railways. Meadville has
three public parks, two general hospitals and a public library, and is
the seat of the Pennsylvania College of Music, of a commercial college,
of the Meadville Theological School (1844, Unitarian), and of Allegheny
College (co-educational), which was opened in 1815, came under the
general patronage of the Methodist Episcopal Church in 1833, and in 1909
had 322 students (200 men and 122 women). Meadville is the commercial
centre of a good agricultural region, which also abounds in oil and
natural gas. The Erie Railroad has extensive shops here, which in 1905
employed 46.7% of the total number of wage-earners, and there are
various manufactures. The factory product in 1905 was valued at
$2,074,600, being 24.4% more than that of 1900. Meadville, the oldest
settlement in N.W. Pennsylvania, was founded as a fortified post by
David Mead in 1793, laid out as a town in 1795, incorporated as a
borough in 1823 and chartered as a city in 1866.




MEAGHER, THOMAS FRANCIS (1823-1867), Irish nationalist and American
soldier, was born in Waterford, Ireland, on the 3rd of August 1823. He
graduated at Stonyhurst College, Lancashire, in 1843, and in 1844 began
the study of law at Dublin. He became a member of the Young Ireland
Party in 1845, and in 1847 was one of the founders of the Irish
Confederation. In March 1848 he made a speech before the Confederation
which led to his arrest for sedition, but at his trial the jury failed
to agree and he was discharged. In the following July the Confederation
created a "war directory" of five, of which Meagher was a member, and he
and William Smith O'Brien travelled through Ireland for the purpose of
starting a revolution. The attempt proved abortive; Meagher was arrested
in August, and in October was tried for high treason before a special
commission at Clonmel. He was found guilty and was condemned to death,
but his sentence was commuted to life imprisonment in Van Diemen's Land,
whither he was transported in the summer of 1849. Early in 1852 he
escaped, and in May reached New York City. He made a tour of the cities
of the United States as a popular lecturer, and then studied law and was
admitted to the New York bar in 1855. He made two unsuccessful ventures
in journalism, and in 1857 went to Central America, where he acquired
material for another series of lectures. In 1861 he was captain of a
company (which he had raised) in the 69th regiment of New York
volunteers and fought at the first battle of Bull Run; he then organized
an Irish brigade, of whose first regiment he was colonel until the 3rd
of February 1862, when he was appointed to the command of this
organization with the rank of brigadier-general. He took part in the
siege of Yorktown, the battle of Fair Oaks, the seven days' battle
before Richmond, and the battles of Antietam, Fredericksburg, where he
was wounded, and Chancellorsville, where his brigade was reduced in
numbers to less than a regiment, and General Meagher resigned his
commission. On the 23rd of December 1863 his resignation was cancelled,
and he was assigned to the command of the military district of Etowah,
with headquarters at Chattanooga. At the close of the war he was
appointed by President Johnson secretary of Montana Territory, and
there, in the absence of the territorial governor, he acted as governor
from September 1866 until his death from accidental drowning in the
Missouri River near Fort Benton, Montana, on the 1st of July 1867. He
published _Speeches on the Legislative Independence of Ireland_ (1852).

  W. F. Lyons, in _Brigadier-General Thomas Francis Meagher_ (New York,
  1870), gives a eulogistic account of his career.




MEAL. (1) (A word common to Teutonic languages, cf. Ger. _Mehl_, Du.
meel; the ultimate source is the root seen in various Teutonic words
meaning "to grind," and in Eng. "mill," Lat. _mola_, _molere_, Gr.
[Greek: mylê]), a powder made from the edible part of any grain or
pulse, with the exception of wheat, which is known as "flour." In
America the word is specifically applied to the meal produced from
Indian corn or maize, as in Scotland and Ireland to that produced from
oats, while in South Africa the ears of the Indian corn itself are
called "mealies." (2) Properly, eating and drinking at regular stated
times of the day, as breakfast, dinner, &c., hence taking of food at any
time and also the food provided. The word was in O.E. _mael_, which also
had the meanings (now lost) of time, mark, measure, &c., which still
appear in many forms of the word in Teutonic languages; thus Ger. _mal_,
time, mark, cf. _Denkmal_, monument, _Mahl_, meal, repast, or Du.
_maal_, Swed. _mal_, also with both meanings. The ultimate source is the
pre-Teutonic root _me-_ _ma-_, to measure, and the word thus stood for a
marked-out point of time.




MEALIE, the South African name for Indian corn or maize. The word as
spelled represents the pronunciation of the Cape Dutch _milje_, an
adaptation of _milho_ (_da India_), the millet of India, the Portuguese
name for millet, used in South Africa for maize.




MEAN, an homonymous word, the chief uses of which may be divided thus.
(1) A verb with two principal applications, to intend, purpose or
design, and to signify. This word is in O.E. _maenan_, and cognate forms
appear in other Teutonic languages, cf. Du. _meenen_, Ger. _meinen_. The
ultimate origin is usually taken to be the root _men-_, to think, the
root of "mind." (2) An adjective and substantive meaning "that which is
in the middle." This is derived through the O. Fr. _men_, _meien_ or
_moien_, modern _moyen_, from the late Lat. adjective _medianus_, from
_medius_, middle. The law French form _mesne_ is still preserved in
certain legal phrases (see MESNE). The adjective "mean" is chiefly used
in the sense of "average," as in mean temperature, mean birth or death
rate, &c.

"Mean" as a substantive has the following principal applications; it is
used of that quality, course of action, condition, state, &c., which is
equally distant from two extremes, as in such phrases as the "golden (or
happy) mean." For the philosophic application see ARISTOTLE and ETHICS.

In mathematics, the term "mean," in its most general sense, is given to
some function of two or more quantities which (1) becomes equal to each
of the quantities when they themselves are made equal, and (2) is
unaffected in value when the quantities suffer any transpositions. The
three commonest means are the arithmetical, geometrical, and harmonic;
of less importance are the contraharmonical, arithmetico-geometrical,
and quadratic.

From the sense of that which stands between two things, "mean," or the
plural "means," often with a singular construction, takes the further
significance of agency, instrument, &c., of which that produces some
result, hence resources capable of producing a result, particularly the
pecuniary or other resources by which a person is enabled to live, and
so used either of employment or of property, wealth, &c. There are many
adverbial phrases, such as "by all means," "by no means," &c., which are
extensions of "means" in the sense of agency.

The word "mean" (like the French _moyen_) had also the sense of
middling, moderate, and this considerably influenced the uses of "mean"
(3). This, which is now chiefly used in the sense of inferior, low,
ignoble, or of avaricious, penurious, "stingy," meant originally that
which is common to more persons or things than one. The word in O. E. is
_gemaéne_, and is represented in the modern Ger. _gemein_, common. It is
cognate with Lat. _communis_, from which "common" is derived. The
descent in meaning from that which is shared alike by several to that
which is inferior, vulgar or low, is paralleled by the uses of "common."

In astronomy the "mean sun" is a fictitious sun which moves uniformly in
the celestial equator and has its right ascension always equal to the
sun's mean longitude. The time recorded by the mean sun is termed
mean-solar or clock time; it is regular as distinct from the non-uniform
solar or sun-dial time. The "mean moon" is a fictitious moon which moves
around the earth with a uniform velocity and in the same time as the
real moon. The "mean longitude" of a planet is the longitude of the
"mean" planet, i.e. a fictitious planet performing uniform revolutions
in the same time as the real planet.

  The arithmetical mean of n quantities is the sum of the quantities
  divided by their number n. The geometrical mean of n quantities is the
  nth root of their product. The harmonic mean of n quantities is the
  arithmetical mean of their reciprocals. The significance of the word
  "mean," i.e., middle, is seen by considering 3 instead of n
  quantities; these will be denoted by a, b, c. The arithmetic mean b,
  is seen to be such that the terms a, b, c are in arithmetical
  progression, i.e. b = ½(a + c); the geometrical mean b places a, b, c
  in geometrical progression, i.e. in the proportion a : b :: b : c or
  b² = ac; and the harmonic mean places the quantities in harmonic
  proportion, i.e. a : c :: a - b : b - c, or b = 2ac/(a + c). The
  contraharmonical mean is the quantity b given by the proportion a : c
  :: b - c : a - b, i.e. b = (a² + c²)/(a + c). The
  arithmetico-geometrical mean of two quantities is obtained by first
  forming the geometrical and arithmetical means, then forming the means
  of these means, and repeating the process until the numbers become
  equal. They were invented by Gauss to facilitate the computation of
  elliptic integrals. The quadratic mean of n quantities is the square
  root of the arithmetical mean of their squares.




MEASLES, (_Morbilli_, _Rubeola_; the M. E. word is _maseles_, properly a
diminutive of a word meaning "spot," O.H.G. _masa_, cf. "mazer"; the
equivalent is Ger. _Masern_; Fr. _Rougeole_), an acute infectious
disease occurring mostly in children. It is mentioned in the writings of
Rhazes and others of the Arabian physicians in the 10th century. For
long, however, it was held to be a variety of small-pox. After the
non-identity of these two diseases had been established, measles and
scarlet-fever continued to be confounded with each other; and in the
account given by Thomas Sydenham of epidemics of measles in London in
1670 and 1674 it is evident that even that accurate observer had not as
yet clearly perceived their pathological distinction, although it would
seem to have been made a century earlier by Giovanni Filippo Ingrassias
(1510-1580), a physician of Palermo. The specific micro-organism
responsible for measles has not been definitely isolated.

Its progress is marked by several stages more or less sharply defined.
After the reception of the contagion into the system, there follows a
period of incubation or latency during which scarcely any disturbance of
the health is perceptible. This period generally lasts for from ten to
fourteen days, when it is followed by the invasion of the symptoms
specially characteristic of measles. These consist in the somewhat
sudden onset of acute catarrh of the mucous membranes. At this stage
minute white spots in the buccal mucous membrane frequently occur; when
they do, they are diagnostic of the disease. Sneezing, accompanied with
a watery discharge, sometimes bleeding, from the nose, redness and
watering of the eyes, cough of a short, frequent, and noisy character,
with little or no expectoration, hoarseness of the voice, and
occasionally sickness and diarrhoea, are the chief local phenomena of
this stage. With these there is well-marked febrile disturbance, the
temperature being elevated (102°-104° F.), and the pulse rapid, while
headache, thirst, and restlessness are usually present. In some
instances, these initial symptoms are slight, and the child is allowed
to associate with others at a time when, as will be afterwards seen,
the contagion of the disease is most active. In rare cases, especially
in young children, convulsions usher in, or occur in the course of, this
stage of invasion, which lasts as a rule for four or five days, the
febrile symptoms, however, showing some tendency to undergo abatement
after the second day. On the fourth or fifth day after the invasion,
sometimes later, rarely earlier, the characteristic eruption appears on
the skin, being first noticed on the brow, cheeks, chin, also behind the
ears, and on the neck. It consists of small spots of a dusky red or
crimson colour, just like flea-bites, slightly elevated above the
surface, at first isolated, but tending to become grouped into patches
of irregular, occasionally crescentic, outline, with portions of skin
free from the eruption intervening. The face acquires a swollen and
bloated appearance, which, taken with the catarrh of the nostrils and
eyes, is almost characteristic, and renders the diagnosis at this stage
a matter of no difficulty. The eruption spreads downwards over the body
and limbs, which are soon thickly studded with the red spots or patches.
Sometimes these become confluent over a considerable surface. The rash
continues to come out for two or three days, and then begins to fade in
the order in which it first showed itself, namely from above downwards.
By the end of about a week after its first appearance scarcely any trace
of the eruption remains beyond a faint staining of the skin. Usually
during convalescence slight peeling of the epidermis takes place, but
much less distinctly than is the case in scarlet fever. At the
commencement of the eruptive stage the fever, catarrh, and other
constitutional disturbance, which were present from the beginning,
become aggravated, the temperature often rising to 105° or more, and
there is headache, thirst, furred tongue, and soreness of the throat,
upon which red patches similar to those on the surface of the body may
be observed. These symptoms usually decline as soon as the rash has
attained its maximum, and often there occurs a sudden and extensive fall
of temperature, indicating that the crisis of the disease has been
reached. In favourable cases convalescence proceeds rapidly, the patient
feeling perfectly well even before the rash has faded from the skin.

Measles may, however, occur in a very malignant form, in which the
symptoms throughout are of urgent character, the rash but feebly
developed, and of dark purple hue, while there is great prostration,
accompanied with intense catarrh of the respiratory or gastro-intestinal
mucous membrane. Such cases are rare, occurring mostly in circumstances
of bad hygiene, both as regards the individual and his surroundings. On
the other hand, cases of measles are often of so mild a form throughout
that the patient can scarcely be persuaded to submit to treatment.

Measles as a disease derives its chief importance from the risk, by no
means slight, of certain complications which are apt to arise during its
course, more especially inflammatory affections of the respiratory
organs. These are most liable to occur in the colder seasons of the year
and in very young and delicate children. It has been already stated that
irritation of the respiratory passages is one of the symptoms
characteristic of measles, but that this subsides with the decline of
the eruption. Not unfrequently, however, these symptoms, instead of
abating, become aggravated, and bronchitis of the capillary form (see
BRONCHITIS), or pneumonia, generally of the diffuse or lobular variety
(see PNEUMONIA), supervene. By far the greater proportion of the
mortality in measles is due to its complications, of which those just
mentioned are the most common, but which also include inflammatory
affections of the larynx, with attacks resembling croup, and also
diarrhoea assuming a dysenteric character. Or there may remain as direct
results of the disease chronic ophthalmia, or discharge from the ears
with deafness, and occasionally a form of gangrene affecting the tissues
of the mouth or cheeks and other parts of the body, leading to
disfigurement and gravely endangering life.

Apart from those immediate risks there appears to be a tendency in many
cases for the disease to leave behind a weakened and vulnerable
condition of the general health, which may render children, previously
robust, delicate and liable to chest complaints, and is in not a few
instances the precursor of some of those tubercular affections to which
the period of childhood and youth is liable. These various effects or
sequelae of measles indicate that although in itself a comparatively
mild ailment, it should not be regarded with indifference. Indeed it is
doubtful whether any other disease of early life demands more careful
watching as to its influence on the health. Happily many of those
attending evils may by proper management be averted.

Measles is a disease of the earlier years of childhood. Like other
infectious maladies, it is admittedly rare, though not unknown, in
nurslings or infants under six months old. It is comparatively seldom
met with in adults, but this is due to the fact that most persons have
undergone an attack in early life. Where this has not been the case, the
old suffer equally with the young. All races of men appear liable to
this disease, provided that which constitutes the essential factor in
its origin and spread exists, namely, contagion. Some countries enjoy
long immunity from outbreaks of measles, but it has frequently been
found in such cases that when the contagion has once been introduced the
disease extends with great rapidity and virulence. This was shown by the
epidemic in the Faroe Islands in 1846, where, within six months after
the arrival of a single case of measles, more than three-fourths of the
entire population were attacked and many perished; and the similarly
produced and still more destructive outbreak in Fiji in 1875, in which
it was estimated that about one-fourth of the inhabitants died from the
disease in about three months. In both these cases the great mortality
was due to the complications of the malady, specially induced by
overcrowding, insanitary surroundings, the absence of proper nourishment
and nursing for the sick, and the utter prostration and terror of the
people, and to the disease being specially malignant, occurring on what
might be termed virgin soil.[1] It may be regarded as an invariable rule
that the first epidemic of any disease in a community is specially
virulent, each successive attack conferring a certain immunity.

In many lands, such as the United Kingdom, measles is rarely absent,
especially from large centres of population, where sporadic cases are
found at all seasons. Every now and then epidemics arise from the
extension of the disease among those members of a community who have not
been in some measure protected by a previous attack. There are few
diseases so contagious as measles, and its rapid spread in epidemic
outbreaks is no doubt due to the well-ascertained fact that contagion is
most potent in the earlier stages, even before its real nature has been
evinced by the characteristic appearances on the skin. Hence the
difficulty of timely isolation, and the readiness with which the disease
is spread in schools and families. The contagion is present in the skin
and the various secretions. While the contagion is generally direct, it
can also be conveyed by the particles from the nose and mouth which,
after being expelled, become dry and are conveyed as dust on clothes,
toys, &c. Fortunately the germs of measles do not retain their virulence
long under such conditions, comparing favourably with those of some
other diseases.

_Treatment._--The treatment embraces the preventive measures to be
adopted by the isolation of the sick at as early a period as possible.
Epidemics have often, especially in limited localities, been curtailed
by such a precaution. In families with little house accommodation this
measure is frequently, for the reason given regarding the communicable
period of the disease, ineffectual; nevertheless where practicable it
ought to be tried. The unaffected children should be kept from school
for a time (probably about three weeks from the outbreak in the family
would suffice if no other case occur in the interval), and all clothing
in contact with the patient or nurses should be disinfected. In
extensive epidemics it is often desirable to close the schools for a
time. As regards special treatment, in an ordinary case of measles
little is required beyond what is necessary in febrile conditions
generally. Confinement to bed in a somewhat darkened room, into which,
however, air is freely admitted; light, nourishing, liquid diet (soups,
milk, &c.), water almost _ad lib._ to drink, and mild diaphoretic
remedies such as the acetate of ammonia or ipecacuanha, are all that is
necessary in the febrile stage. When the fever is very severe, sponging
the body generally or the chest and arms affords relief. The serious
chest complications of measles are to be dealt with by those measures
applicable for the relief of the particular symptoms (see BRONCHITIS;
PNEUMONIA). The preparations of ammonia are of special efficacy. During
convalescence the patient must be guarded from exposure to cold, and for
a time after recovery the state of the health ought to be watched with a
view of averting the evils, both local and constitutional, which too
often follow this disease.

  "German measles" (_Rötheln_, or _Epidemic Roseola_) is a term applied
  to a contagious eruptive disorder having certain points of resemblance
  to measles, and also to scarlet fever, but exhibiting its distinct
  individuality in the fact that it protects from neither of these
  diseases. It occurs most commonly in children, but frequently in
  adults also, and is occasionally seen in extensive epidemics. Beyond
  confinement to the house in the eruptive stage, which, from the slight
  symptoms experienced, is often difficult of accomplishment, no special
  treatment is called for. There is little doubt that the disease is
  often mistaken for true measles, and many of the alleged second
  attacks of the latter malady are probably cases of rötheln. The chief
  points of difference are the following: (1) The absence of distinct
  premonitory symptoms, the stage of invasion, which in measles is
  usually of four days' duration, and accompanied with well-marked fever
  and catarrh, being in rötheln either wholly absent or exceedingly
  slight, enduring only for one day. (2) The eruption of rötheln, which,
  although as regards its locality and manner of progress similar to
  measles, differs somewhat in its appearance, the spots being of
  smaller size, paler colour, and with less tendency to grouping in
  crescentic patches. The rash attains its maximum in about one day, and
  quickly disappears. There is not the same increase of temperature in
  this stage as in measles. (3) The presence of white spots on the
  buccal mucous membrane, in the case of measles. (4) The milder
  character of the symptoms of rötheln throughout its whole course, and
  the absence of complications and of liability to subsequent impairment
  of health such as have been seen to appertain to measles.


FOOTNOTE:

  [1] _Transactions of the Epidemiological Society_ (London, 1877).




MEAT, a word originally applied to food in general, and so still used in
such phrases as "meat and drink"; but now, except as an archaism,
generally used of the flesh of certain domestic animals, slaughtered for
human food by butchers, "butcher's meat," as opposed to "game," that of
wild animals, "fish" or "poultry." Cognate forms of the O. Eng. _mete_
are found in certain Teutonic languages, e.g. Swed. _mat_, Dan. _mad_
and O. H. Ger. _Maz_. The ultimate origin has been disputed; the _New
English Dictionary_ considers probable a connexion with the root _med-_,
"to be fat," seen in Sansk. _meda_, Lat. _madere_, "to be wet," and Eng.
"mast," the fruit of the beech as food for pigs.

  See DIETETICS; FOOD PRESERVATION; PUBLIC HEALTH; AGRICULTURE; and the
  sections dealing with agricultural statistics under the names of the
  various countries.




MEATH (pronounced with _th_ soft, as in _the_), a county of Ireland in
the province of Leinster, bounded E. by the Irish Sea, S.E. by Dublin,
S. by Kildare and King's County, W. by Westmeath, N.W. by Cavan and
Monaghan, and N.E. by Louth. Area 579,320 acres, or about 905 sq. m. In
some districts the surface is varied by hills and swells, which to the
west reach a considerable elevation, although the general features of a
fine champain country are never lost. The coast, low and shelving,
extends about 10 m., but there is no harbour of importance. Laytown is a
small seaside resort, 5 m. S.E. of Drogheda. The Boyne enters the county
at its south-western extremity, and flowing north-east to Drogheda
divides it into two almost equal parts. At Navan it receives the
Blackwater, which flows south-west from Cavan. Both these rivers are
noted for their trout, and salmon are taken in the Boyne. The Boyne is
navigable for barges as far as Navan whence a canal is carried to Trim.
The Royal Canal passes along the southern boundary of the county from
Dublin.

  In the north is a broken country of Silurian rocks with much igneous
  material, partly contemporaneous, partly intrusive, near Slane.
  Carboniferous Limestone stretches from the Boyne valley to the Dublin
  border, giving rise to a flat plain especially suitable for grazing.
  Outliers of higher Carboniferous strata occur on the surface; but the
  Coal Measures have all been removed by denudation.

  The climate is genial and favourable for all kinds of crops, there
  being less rain than even in the neighbouring counties. Except a small
  portion occupied by the Bog of Allen, the county is verdant and
  fertile. The soil is principally a rich deep loam resting on limestone
  gravel, but varies from a strong clayey loam to a light sandy gravel.
  The proportion of tillage to pasturage is roughly as 1 to 3½. Oats,
  potatoes and turnips are the principal crops, but all decrease. The
  numbers of cattle, sheep and poultry, however, are increasing or well
  maintained. Agriculture is almost the sole industry, but coarse linen
  is woven by hand-looms, and there are a few woollen manufactories. The
  main line of the Midland Great Western railway skirts the southern
  boundary, with a branch line north from Clonsilla to Navan and
  Kingscourt (county Cavan). From Kilmessan on this line a branch serves
  Trim and Athboy. From Drogheda (county Louth) a branch of the Great
  Northern railway crosses the county from east to West by Navan and
  Kells to Oldcastle.

  The population (76,111 in 1891; 67,497 in 1901) suffers a large
  decrease, considerably above the average of Irish counties, and
  emigration is heavy. Nearly 93% are Roman Catholics. The chief towns
  are Navan (pop. 3839), Kells (2428) and Trim (1513), the county town.
  Lesser market towns are Oldcastle and Athboy, an ancient town which
  received a charter from Henry IV. The county includes eighteen
  baronies. Assizes are held at Trim, and quarter sessions at Kells,
  Navan and Trim. The county is in the Protestant dioceses of Armagh,
  Kilmore and Meath, and in the Roman Catholic dioceses of Armagh and
  Meath. Before the Union in 1800 it sent fourteen members to
  parliament, but now only two members are returned, for the north and
  south divisions of the county respectively.

_History and Antiquities._--A district known as Meath (Midhe), and
including the present county of Meath as well as Westmeath and Longford,
with parts of Cavan, Kildare and King's County, was formed by Tuathal
(c. 130) into a kingdom to serve as mensal land or personal estate of
the Ard Ri or over-king of Ireland. Kings of Meath reigned until 1173,
and the title was claimed as late as the 15th century by their
descendants, but at the date mentioned Hugh de Lacy obtained the
lordship of the country and was confirmed in it by Henry II. Meath thus
came into the English "Pale." But though it was declared a county in the
reign of Edward I. (1296), and though it came by descent into the
possession of the Crown in the person of Edward IV., it was long before
it was fully subdued and its boundaries clearly defined. In 1543
Westmeath was created a county apart from that of Meath, but as late as
1598 Meath was still regarded as a province by some, who included in it
the counties Westmeath, East Meath, Longford and Cavan. In the early
part of the 17th century it was at last established as a county, and no
longer considered as a fifth province of Ireland.

There are two ancient round towers, the one at Kells and the other in
the churchyard of Donaghmore, near Navan. By the river Boyne near Slane
there is an extensive ancient burial-place called Brugh. Here are some
twenty burial mounds, the largest of which is that of New Grange, a
domed tumulus erected above a circular chamber, which is entered by a
narrow passage enclosed by great upright blocks of stone, covered with
carvings. The mound is surrounded by remains of a stone circle, and the
whole forms one of the most remarkable extant erections of its kind.
Tara (q.v.) is famous in history, especially as the seat of a royal
palace referred to in the well-known lines of Thomas Moore. Monastic
buildings were very numerous in Meath, among the more important ruins
being those of Duleek, which is said to have been the first
ecclesiastical building in Ireland of stone and mortar; the extensive
remains of Bective Abbey; and those of Clonard, where also were a
cathedral and a famous college. Of the old fortresses, the castle of
Trim still presents an imposing appearance. There are many fine old
mansions.




MEAUX, a town of northern France, capital of an arrondissement in the
department of Seine-et-Marne, and chief town of the agricultural region
of Brie, 28 m. E.N.E. of Paris by rail. Pop. (1906), 11,089. The town
proper stands on an eminence on the right bank of the Marne; on the left
bank lies the old suburb of Le Marché, with which it is united by a
bridge of the 16th century. Two rows of picturesque mills of the same
period are built across the river. The cathedral of St Stephen dates
from the 12th to the 16th centuries, and was restored in the 19th
century. Of the two western towers, the completed one is that to the
north of the façade, the other being disfigured by an unsightly slate
roof. The building, which is 275 ft. long and 105 ft. high, consists of
a short nave, with aisles, a fine transept, a choir and a sanctuary. The
choir contains the statue and the tomb of Bossuet, bishop from 1681 to
1704, and the pulpit of the cathedral has been reconstructed with the
panels of that from which the "eagle of Meaux" used to preach. The
transept terminates at each end in a fine portal surmounted by a
rose-window. The episcopal palace (17th century) has several curious old
rooms; the buildings of the choir school are likewise of some
archaeological interest. A statue of General Raoult (1870) stands in one
of the squares.

Meaux is the centre of a considerable trade in cereals, wool, Brie
cheeses, and other farm-produce, while its mills provide much of the
flour with which Paris is supplied. Other industries are saw-milling,
metal-founding, distilling, the preparation of vermicelli and preserved
vegetables, and the manufacture of mustard, hosiery, plaster and
machinery. There are nursery-gardens in the vicinity. The Canal de
l'Ourcq, which surrounds the town, and the Marne furnish the means of
transport. Meaux is the seat of a bishopric dating from the 4th century,
and has among its public institutions a sub-prefecture, and tribunals of
first instance and of commerce.

In the Roman period Meaux was the capital of the Meldi, a small Gallic
tribe, and in the middle ages of the Brie. It formed part of the kingdom
of Austrasia, and afterwards belonged to the counts of Vermandois and
Champagne, the latter of whom established important markets on the left
bank of the Marne. Its communal charter, received from them, is dated
1179. A treaty signed at Meaux in 1229 after the Albigensian War sealed
the submission of Raymond VII., count of Toulouse. The town suffered
much during the Jacquerie, the peasants receiving a severe check there
in 1358; during the Hundred Years' War; and also during the Religious
Wars, in which it was an important Protestant centre. It was the first
town which opened its gates to Henry IV. in 1594. On the high-road for
invaders marching on Paris from the east of France, Meaux saw its
environs ravaged by the army of Lorraine in 1652, and was laid under
heavy requisitions in 1814, 1815 and 1870. In September 1567 Meaux was
the scene of an attempt made by the Protestants to seize the French king
Charles IX., and his mother Catherine de' Medici. The plot, which is
sometimes called the "enterprise of Meaux," failed, the king and queen
with their courtiers escaping to Paris. This conduct, however, on the
part of the Huguenots had doubtless some share in influencing Charles to
assent to the massacre of St Bartholomew.




MECCA (Arab. _Makkah_),[1] the chief town of the Hejaz in Arabia, and
the great holy city of Islam. It is situated two camel marches (the
resting-place being Bahra or Hadda), or about 45 m. almost due E., from
Jidda on the Red Sea. Thus on a rough estimate Mecca lies in 21° 25´ N.,
39° 50´ E. It is said in the Koran (_Sur._ xiv. 40) that Mecca lies in a
sterile valley, and the old geographers observe that the whole Haram or
sacred territory round the city is almost without cultivation or date
palms, while fruit trees, springs, wells, gardens and green valleys are
found immediately beyond. Mecca in fact lies in the heart of a mass of
rough hills, intersected by a labyrinth of narrow valleys and passes,
and projecting into the Tehama or low country on the Red Sea, in front
of the great mountain wall that divides the coast-lands from the central
plateau, though in turn they are themselves separated from the sea by a
second curtain of hills forming the western wall of the great Wadi Marr.
The inner mountain wall is pierced by only two great passes, and the
valleys descending from these embrace on both sides the Mecca hills.

Holding this position commanding two great routes between the lowlands
and inner Arabia, and situated in a narrow and barren valley incapable
of supporting an urban population, Mecca must have been from the first a
commercial centre.[2] In the palmy days of South Arabia it was probably
a station on the great incense route, and thus Ptolemy may have learned
the name, which he writes Makoraba. At all events, long before Mahomet
we find Mecca established in the twofold quality of a commercial centre
and a privileged holy place, surrounded by an inviolable territory (the
Haram), which was not the sanctuary of a single tribe but a place of
pilgrimage, where religious observances were associated with a series of
annual fairs at different points in the vicinity. Indeed in the
unsettled state of the country commerce was possible only under the
sanctions of religion, and through the provisions of the sacred truce
which prohibited war for four months of the year, three of these being
the month of pilgrimage, with those immediately preceding and following.
The first of the series of fairs in which the Meccans had an interest
was at Okaz on the easier road between Mecca and Taif, where there was
also a sanctuary, and from it the visitors moved on to points still
nearer Mecca (Majanna, and finally Dhul-Majaz, on the flank of Jebel
Kabkab behind Arafa) where further fairs were held,[3] culminating in
the special religious ceremonies of the great feast at 'Arafa, Quzah
(Mozdalifa), and Mecca itself. The system of intercalation in the lunar
calendar of the heathen Arabs was designed to secure that the feast
should always fall at the time when the hides, fruits and other
merchandise were ready for market,[4] and the Meccans, who knew how to
attract the Bedouins by hospitality, bought up these wares in exchange
for imported goods, and so became the leaders of the international trade
of Arabia. Their caravans traversed the length and breadth of the
peninsula. Syria, and especially Gaza, was their chief goal. The Syrian
caravan intercepted, on its return, at Badr (see MAHOMET) represented
capital to the value of £20,000, an enormous sum for those days.[5]

The victory of Mahommedanism made a vast change in the position of
Mecca. The merchant aristocracy became satraps or pensioners of a great
empire; but the seat of dominion was removed beyond the desert, and
though Mecca and the Hejaz strove for a time to maintain political as
well as religious predominance, the struggle was vain, and terminated on
the death of Ibn Zubair, the Meccan pretendant to the caliphate, when
the city was taken by Hajjaj (A.D. 692). The sanctuary and feast of
Mecca received, however, a new prestige from the victory of Islam.
Purged of elements obviously heathen, the Ka'ba became the holiest site,
and the pilgrimage the most sacred ritual observance of Mahommedanism,
drawing worshippers from so wide a circle that the confluence of the
petty traders of the desert was no longer the main feature of the holy
season. The pilgrimage retained its importance for the commercial
well-being of Mecca; to this day the Meccans live by the Hajj--letting
rooms, acting as guides and directors in the sacred ceremonies, as
contractors and touts for land and sea transport, as well as exploiting
the many benefactions that flow to the holy city; while the surrounding
Bedouins derive support from the camel-transport it demands and from the
subsidies by which they are engaged to protect or abstain from molesting
the pilgrim caravans. But the ancient "fairs of heathenism" were given
up, and the traffic of the pilgrim season, sanctioned by the Prophet in
_Sur._ ii. 194, was concentrated at Mina and Mecca, where most of the
pilgrims still have something to buy or sell, so that Mina, after the
sacrifice of the feast day, presents the aspect of a huge international
fancy fair.[6] In the middle ages this trade was much more important
than it is now. Ibn Jubair (ed. Wright, p. 118 seq.) in the 12th century
describes the mart of Mecca in the eight days following the feast as
full of gems, unguents, precious drugs, and all rare merchandise from
India, Irak, Khorasan, and every part of the Moslem world.

The hills east and west of Mecca, which are partly built over and rise
several hundred feet above the valley, so enclose the city that the
ancient walls only barred the valley at three points, where three gates
led into the town. In the time of Ibn Jubair the gates still stood
though the walls were ruined, but now the gates have only left their
names to quarters of the town. At the northern or upper end was the Bab
el Ma'la, or gate of the upper quarter, whence the road continues up the
valley towards Mina and Arafa as well as towards Zeima and the Nejd.
Beyond the gate, in a place called the Hajun, is the chief cemetery,
commonly called el Ma'la, and said to be the resting-place of many of
the companions of Mahomet. Here a cross-road, running over the hill to
join the main Medina road from the western gate, turns off to the west
by the pass of Kada, the point from which the troops of the Prophet
stormed the city (A.H. 8).[7] Here too the body of Ibn Zubair was hung
on a cross by Hajjaj. The lower or southern gate, at the Masfala
quarter, opened on the Yemen road, where the rain-water from Mecca flows
off into an open valley. Beyond, there are mountains on both sides; on
that to the east, commanding the town, is the great castle, a fortress
of considerable strength. The third or western gate, Bab el-Omra
(formerly also Bab el-Zahir, from a village of that name), lay almost
opposite the great mosque, and opened on a road leading westwards round
the southern spurs of the Red Mountain. This is the way to Wadi Fatima
and Medina, the Jidda road branching off from it to the left.
Considerable suburbs now lie outside the quarter named after this gate;
in the middle ages a pleasant country road led for some miles through
partly cultivated land with good wells, as far as the boundary of the
sacred territory and gathering place of the pilgrims at Tanim, near the
mosque of Ayesha. This is the spot on the Medina road now called the
Omra, from a ceremonial connected with it which will be mentioned below.

The length of the sinuous main axis of the city from the farthest
suburbs on the Medina road to the suburbs in the extreme north, now
frequented by Bedouins, is, according to Burckhardt, 3500 paces.[8]
About the middle of this line the longitudinal thoroughfares are pushed
aside by the vast courtyard and colonnades composing the great mosque,
which, with its spacious arcades surrounding the Ka'ba and other holy
places, and its seven minarets, forms the only prominent architectural
feature of the city. The mosque is enclosed by houses with windows
opening on the arcades and commanding a view of the Ka'ba. Immediately
beyond these, on the side facing Jebel Abu Kobais, a broad street runs
south-east and north-west across the valley. This is the Mas'a (sacred
course) between the eminences of Safa and Merwa, and has been from very
early times one of the most lively bazaars and the centre of Meccan
life. The other chief bazaars are also near the mosque in smaller
streets. The general aspect of the town is picturesque; the streets are
fairly spacious, though ill-kept and filthy; the houses are all of
stone, many of them well-built and four or five storeys high, with
terraced roofs and large projecting windows as in Jidda--a style of
building which has not varied materially since the 10th century
(Mukaddasi, p. 71), and gains in effect from the way in which the
dwellings run up the sides and spurs of the mountains. Of public
institutions there are baths, ribats, or hospices, for poor pilgrims
from India, Java, &c., a hospital and a public kitchen for the poor.

The mosque is at the same time the university hall, where between two
pilgrim seasons lectures are delivered on Mahommedan law, doctrine and
connected branches of science. A poorly provided public library is open
to the use of students. The madrassehs or buildings around the mosque,
originally intended as lodgings for students and professors, have long
been let out to rich pilgrims. The minor places of visitation for
pilgrims, such as the birthplaces of the prophet and his chief
followers, are not notable.[9] Both these and the court of the great
mosque lie beneath the general level of the city, the site having been
gradually raised by accumulated rubbish. The town in fact has little air
of antiquity; genuine Arab buildings do not last long, especially in a
valley periodically ravaged by tremendous floods when the tropical rains
burst on the surrounding hills. The history of Mecca is full of the
record of these inundations, unsuccessfully combated by the great dam
drawn across the valley by the caliph Omar (_Kutbeddin_, p. 76), and
later works of Mahdi.[10]

The fixed population of Mecca in 1878 was estimated by Assistant-Surgeon
'Abd el-Razzaq at 50,000 to 60,000; there is a large floating
population--and that not merely at the proper season of pilgrimage, the
pilgrims of one season often beginning to arrive before those of the
former season have all dispersed. At the height of the season the town
is much overcrowded, and the entire want of a drainage system is
severely felt. Fortunately good water is tolerably plentiful; for,
though the wells are mostly undrinkable, and even the famous Zamzam
water only available for medicinal or religious purposes, the
underground conduit from beyond Arafa, completed by Sultan Selim II. in
1571, supplies to the public fountains a sweet and light water,
containing, according to 'Abd el-Razzaq, a large amount of chlorides.
The water is said to be free to townsmen, but is sold to the pilgrims at
a rather high rate.[11]

Medieval writers celebrate the copious supplies, especially of fine
fruits, brought to the city from Taif and other fertile parts of Arabia.
These fruits are still famous; rice and other foreign products are
brought by sea to Jidda; mutton, milk and butter are plentifully
supplied from the desert.[12] The industries all centre in the
pilgrimage; the chief object of every Meccan--from the notables and
sheikhs, who use their influence to gain custom for the Jidda
speculators in the pilgrim traffic, down to the cicerones, pilgrim
brokers, lodging-house keepers, and mendicants at the holy places--being
to pillage the visitor in every possible way. The fanaticism of the
Meccan is an affair of the purse; the mongrel population (for the town
is by no means purely Arab) has exchanged the virtues of the Bedouin for
the worst corruptions of Eastern town life, without casting off the
ferocity of the desert, and it is hardly possible to find a worse
certificate of character than the three parallel gashes on each cheek,
called Tashrit, which are the customary mark of birth in the holy city.
The unspeakable vices of Mecca are a scandal to all Islam, and a
constant source of wonder to pious pilgrims.[13] The slave trade has
connexions with the pilgrimage which are not thoroughly clear; but under
cover of the pilgrimage a great deal of importation and exportation of
slaves goes on.

Since the fall of Ibn Zubair the political position of Mecca has always
been dependent on the movements of the greater Mahommedan world. In the
splendid times of the caliphs immense sums were lavished upon the
pilgrimage and the holy city; and conversely the decay of the central
authority of Islam brought with it a long period of faction, wars and
misery, in which the most notable episode was the sack of Mecca by the
Carmathians at the pilgrimage season of A.D. 930. The victors carried
off the "black stone," which was not restored for twenty-two years, and
then only for a great ransom, when it was plain that even the loss of
its palladium could not destroy the sacred character of the city. Under
the Fatimites Egyptian influence began to be strong in Mecca; it was
opposed by the sultans of Yemen, while native princes claiming descent
from the Prophet--the Hashimite amirs of Mecca, and after them the amirs
of the house of Qatada (since 1202)--attained to great authority and
aimed at independence; but soon after the final fall of the Abbasids the
Egyptian overlordship was definitely established by sultan Bibars (A.D.
1269). The Turkish conquest of Egypt transferred the supremacy to the
Ottoman sultans (1517), who treated Mecca with much favour, and during
the 16th century executed great works in the sanctuary and temple. The
Ottoman power, however, became gradually almost nominal, and that of the
amirs or sherifs increased in proportion, culminating under Ghalib,
whose accession dates from 1786. Then followed the wars of the Wahhabis
(see ARABIA and WAHHABIS) and the restoration of Turkish rule by the
troops of Mehemet 'Ali. By him the dignity of sherif was deprived of
much of its weight, and in 1827 a change of dynasty was effected by the
appointment of Ibn 'Aun. Afterwards Turkish authority again decayed.
Mecca is, however, officially the capital of a Turkish province, and has
a governor-general and a Turkish garrison, while Mahommedan law is
administered by a judge sent from Constantinople. But the real sovereign
of Mecca and the Hejaz is the sherif, who, as head of a princely family
claiming descent from the Prophet, holds a sort of feudal position. The
dignity of sherif (or grand sherif, as Europeans usually say for the
sake of distinction, since all the kin of the princely houses reckoning
descent from the Prophet are also named sherifs), although by no means a
religious pontificate, is highly respected owing to its traditional
descent in the line of Hasan, son of the fourth caliph 'Ali. From a
political point of view the sherif is the modern counterpart of the
ancient amirs of Mecca, who were named in the public prayers immediately
after the reigning caliph. When the great Mahommedan sultanates had
become too much occupied in internecine wars to maintain order in the
distant Hejaz, those branches of the Hassanids which from the beginning
of Islam had retained rural property in Arabia usurped power in the holy
cities and the adjacent Bedouin territories. About A.D. 960 they
established a sort of kingdom with Mecca as capital. The influence of
the princes of Mecca has varied from time to time, according to the
strength of the foreign protectorate in the Hejaz or in consequence of
feuds among the branches of the house; until about 1882 it was for most
purposes much greater than that of the Turks. The latter were strong
enough to hold the garrisoned towns, and thus the sultan was able within
certain limits--playing off one against the other the two rival branches
of the aristocracy, viz. the kin of Ghalib and the house of Ibn'Aun--to
assert the right of designating or removing the sherif, to whom in turn
he owed the possibility of maintaining, with the aid of considerable
pensions, the semblance of his much-prized lordship over the holy
cities. The grand sherif can muster a considerable force of freedmen and
clients, and his kin, holding wells and lands in various places through
the Hejaz, act as his deputies and administer the old Arabic customary
law to the Bedouin. To this influence the Hejaz owes what little of law
and order it enjoys. During the last quarter of the 19th century Turkish
influence became preponderant in western Arabia, and the railway from
Syria to the Hejaz tended to consolidate the sultan's supremacy. After
the sherifs, the principal family of Mecca is the house of Shaibah,
which holds the hereditary custodianship of the Ka'ba.

_The Great Mosque and the Ka'ba._--Long before Mahomet the chief
sanctuary of Mecca was the Ka'ba, a rude stone building without windows,
and having a door 7 ft. from the ground; and so named from its
resemblance to a monstrous _astragalus_ (die) of about 40 ft. cube,
though the shapeless structure is not really an exact cube nor even
exactly rectangular.[14] The Ka'ba has been rebuilt more than once since
Mahomet purged it of idols and adopted it as the chief sanctuary of
Islam, but the old form has been preserved, except in secondary
details;[15] so that the "Ancient House," as it is titled, is still
essentially a heathen temple, adapted to the worship of Islam by the
clumsy fiction that it was built by Abraham and Ishmael by divine
revelation as a temple of pure monotheism, and that it was only
temporarily perverted to idol worship from the time when 'Amr ibn Lohai
introduced the statue of Hobal from Syria[16] till the victory of Islam.
This fiction has involved the superinduction of a new mythology over the
old heathen ritual, which remains practically unchanged. Thus the chief
object of veneration is the black stone, which is fixed in the external
angle facing Safa. The building is not exactly oriented, but it may be
called the south-east corner. Its technical name is the black corner,
the others being named the Yemen (south-west), Syrian (north-west), and
Irak (north-east) corners, from the lands to which they approximately
point. The black stone is a small dark mass a span long, with an aspect
suggesting volcanic or meteoric origin, fixed at such a height that it
can be conveniently kissed by a person of middle size. It was broken by
fire in the siege of A.D. 683 (not, as many authors relate, by the
Carmathians), and the pieces are kept together by a silver setting. The
history of this heavenly stone, given by Gabriel to Abraham, does not
conceal the fact that it was originally a fetish, the most venerated of
a multitude of idols and sacred stones which stood all round the
sanctuary in the time of Mahomet. The Prophet destroyed the idols, but
he left the characteristic form of worship--the _tawaf_, or sevenfold
circuit of the sanctuary, the worshipper kissing or touching the objects
of his veneration--and besides the black stone he recognized the
so-called "southern" stone, the same presumably as that which is still
touched in the tawaf at the Yemen corner (_Muh. in Med._ pp. 336, 425).
The ceremony of the tawaf and the worship of stone fetishes was common
to Mecca with other ancient Arabian sanctuaries.[17] It was, as it still
is, a frequent religious exercise of the Meccans, and the first duty of
one who returned to the city or arrived there under a vow of pilgrimage;
and thus the outside of the Ka'ba was and is more important than the
inside. Islam did away with the worship of idols; what was lost in
interest by their suppression has been supplied by the invention of
spots consecrated by recollections of Abraham, Ishmael and Hagar, or
held to be acceptable places of prayer. Thus the space of ten spans
between the black stone and the door, which is on the east side, between
the black and Irak corners, and a man's height from the ground, is
called the _Multazam_, and here prayer should be offered after the tawaf
with outstretched arms and breast pressed against the house. On the
other side of the door, against the same wall, is a shallow trough,
which is said to mark the original site of the stone on which Abraham
stood to build the Ka'ba. Here the growth of the legend can be traced,
for the place is now called the "kneading-place" (Ma'jan), where the
cement for the Ka'ba was prepared. This name and story do not appear in
the older accounts. Once more, on the north side of the Ka'ba, there
projects a low semicircular wall of marble, with an opening at each end
between it and the walls of the house. The space within is paved with
mosaic, and is called the Hijr. It is included in the tawaf, and two
slabs of _verde antico_ within it are called the graves of Ishmael and
Hagar, and are places of acceptable prayer. Even the golden or gilded
_mizab_ (water-spout) that projects into the Hijr marks a place where
prayer is heard, and another such place is the part of the west wall
close to the Yemen corner.

The feeling of religious conservatism which has preserved the structural
rudeness of the Ka'ba did not prohibit costly surface decoration. In
Mahomet's time the outer walls were covered by a veil (or _kiswa_) of
striped Yemen cloth. The caliphs substituted a covering of figured
brocade, and the Egyptian government still sends with each pilgrim
caravan from Cairo a new kiswa of black brocade, adorned with a broad
band embroidered with golden inscriptions from the Koran, as well as a
richer curtain for the door.[18] The door of two leaves, with its posts
and lintel, is of silver gilt.

The interior of the Ka'ba is now opened but a few times every year for
the general public, which ascends by the portable staircase brought
forward for the purpose. Foreigners can obtain admission at any time for
a special fee. The modern descriptions, from observations made under
difficulties, are not very complete. Little change, however, seems to
have been made since the time of Ibn Jubair, who describes the floor and
walls as overlaid with richly variegated marbles, and the upper half of
the walls as plated with silver thickly gilt, while the roof was veiled
with coloured silk. Modern writers describe the place as windowless, but
Ibn Jubair mentions five windows of rich stained glass from Irak.
Between the three pillars of teak hung thirteen silver lamps. A chest in
the corner to the left of one entering contained Korans, and at the Irak
corner a space was cut off enclosing the stair that leads to the roof.
The door to this stair (called the door of mercy--Bab el-Rahma) was
plated with silver by the caliph Motawakkil. Here, in the time of Ibn
Jubair, the _Maqam_ or standing stone of Abraham was usually placed for
better security, but brought out on great occasions.[19]

The houses of ancient Mecca pressed close upon the Ka'ba, the noblest
families, who traced their descent from Kosai, the reputed founder of
the city, having their dwellings immediately round the sanctuary. To the
north of the Ka'ba was the Dar el-Nadwa, or place of assembly of the
Koreish. The multiplication of pilgrims after Islam soon made it
necessary to clear away the nearest dwellings and enlarge the place of
prayer around the Ancient House. Omar, Othman and Ibn Jubair had all a
share in this work, but the great founder of the mosque in its present
form, with its spacious area and deep colonnades, was the caliph Mahdi,
who spent enormous sums in bringing costly pillars from Egypt and Syria.
The work was still incomplete at his death in A.D. 785, and was finished
in less sumptuous style by his successor. Subsequent repairs and
additions, extending down to Turkish times, have left little of Mahdi's
work untouched, though a few of the pillars probably date from his days.
There are more than five hundred pillars in all, of very various style
and workmanship, and the enclosure--250 paces in length and 200 in
breadth, according to Burckhardt's measurement--is entered by nineteen
archways irregularly disposed.

After the Ka'ba the principal points of interest in the mosque are the
well Zamzam and the Maqam Ibrahim. The former is a deep shaft enclosed
in a massive vaulted building paved with marble, and, according to
Mahommedan tradition, is the source (corresponding to the Beer-lahai-roi
of Gen. xvi. 14) from which Hagar drew water for her son Ishmael. The
legend tells that the well was long covered up and rediscovered by 'Abd
al-Mot[t.]alib, the grandfather of the Prophet. Sacred wells are
familiar features of Semitic sanctuaries, and Islam, retaining the well,
made a quasi-biblical story for it, and endowed its tepid waters with
miraculous curative virtues. They are eagerly drunk by the pilgrims, or
when poured over the body are held to give a miraculous refreshment
after the fatigues of religious exercise; and the manufacture of bottles
or jars for carrying the water to distant countries is quite a trade.
Ibn Jubair mentions a curious superstition of the Meccans, who believed
that the water rose in the shaft at the full moon of the month Shaban.
On this occasion a great crowd, especially of young people, thronged
round the well with shouts of religious enthusiasm, while the servants
of the well dashed buckets of water over their heads. The Maqam of
Abraham is also connected with a relic of heathenism, the ancient holy
stone which once stood on the Ma'jan, and is said to bear the prints of
the patriarch's feet. The whole legend of this stone, which is full of
miraculous incidents, seems to have arisen from a misconception, the
Maqam Ibrahim in the Koran meaning the sanctuary itself; but the stone,
which is a block about 3 spans in height and 2 in breadth, and in shape
"like a potter's furnace" (Ibn Jubair), is certainly very ancient. No
one is now allowed to see it, though the box in which it lies can be
seen or touched through a grating in the little chapel that surrounds
it. In the middle ages it was sometimes shown, and Ibn Jubair describes
the pious enthusiasm with which he drank Zamzam water poured on the
footprints. It was covered with inscriptions in an unknown character,
one of which was copied by Fakihi in his history of Mecca. To judge by
the facsimile in Dozy's _Israeliten te Mekka_, the character is probably
essentially one with that of the Syrian Safa inscriptions, which
extended through the Nejd and into the Hejaz.[20]

  _Safa and Merwa._--In religious importance these two points or
  "hills," connected by the Mas'a, stand second only to the Ka'ba. Safa
  is an elevated platform surmounted by a triple arch, and approached by
  a flight of steps.[21] It lies south-east of the Ka'ba, facing the
  black corner, and 76 paces from the "Gate of Safa," which is
  architecturally the chief gate of the mosque. Merwa is a similar
  platform, formerly covered with a single arch, on the opposite side of
  the valley. It stands on a spur of the Red Mountain called Jebel
  Kuaykian. The course between these two sacred points is 493 paces
  long, and the religious ceremony called the "sa'y" consists in
  traversing it seven times, beginning and ending at Safa. The lowest
  part of the course, between the so-called green milestones, is done at
  a run. This ceremony, which, as we shall presently see, is part of the
  omra, is generally said to be performed in memory of Hagar, who ran to
  and fro between the two eminences vainly seeking water for her son.
  The observance, however, is certainly of pagan origin; and at one time
  there were idols on both the so-called hills (see especially Azraqi,
  pp. 74, 78).

  _The Ceremonies and the Pilgrimage._--Before Islam the Ka'ba was the
  local sanctuary of the Meccans, where they prayed and did sacrifice,
  where oaths were administered and hard cases submitted to divine
  sentence according to the immemorial custom of Semitic shrines. But,
  besides this, Mecca was already a place of pilgrimage. Pilgrimage with
  the ancient Arabs was the fulfilment of a vow, which appears to have
  generally terminated--at least on the part of the well-to-do--in a
  sacrificial feast. A vow of pilgrimage might be directed to other
  sanctuaries than Mecca--the technical word for it (_ihlal_) is
  applied, for example, to the pilgrimage to Manat (_Bakri_, p. 519). He
  who was under such a vow was bound by ceremonial observances of
  abstinence from certain acts (e.g. hunting) and sensual pleasures, and
  in particular was forbidden to shear or comb his hair till the
  fulfilment of the vow. This old Semitic usage has its close parallel
  in the vow of the Nazarite. It was not peculiarly connected with
  Mecca; at Taif, for example, it was customary on return to the city
  after an absence to present oneself at the sanctuary, and there shear
  the hair (_Muh. in Med._, p. 381). Pilgrimages to Mecca were not tied
  to a single time, but they were naturally associated with festive
  occasions, and especially with the great annual feast and market. The
  pilgrimage was so intimately connected with the well-being of Mecca,
  and had already such a hold on the Arabs round about, that Mahomet
  could not afford to sacrifice it to an abstract purity of religion,
  and thus the old usages were transplanted into Islam in the double
  form of the omra or vow of pilgrimage to Mecca, which can be
  discharged at any time, and the hajj or pilgrimage at the great annual
  feast. The latter closes with a visit to the Ka'ba, but its essential
  ceremonies lie outside Mecca, at the neighbouring shrines where the
  old Arabs gathered before the Meccan fair.

  The omra begins at some point outside the Haram (or holy territory),
  generally at Tanim, both for convenience sake and because Ayesha began
  the omra there in the year 10 of the Hegira. The pilgrim enters the
  Haram in the antique and scanty pilgrimage dress (ihram), consisting
  of two cloths wound round his person in a way prescribed by ritual.
  His devotion is expressed in shouts of "Labbeyka" (a word of obscure
  origin and meaning); he enters the great mosque, performs the tawaf
  and the sa'y[22] and then has his head shaved and resumes his common
  dress. This ceremony is now generally combined with the hajj, or is
  performed by every stranger or traveller when he enters Mecca, and the
  ihram (which involves the acts of abstinence already referred to) is
  assumed at a considerable distance from the city. But it is also
  proper during one's residence in the holy city to perform at least one
  omra from Tanim in connexion with a visit to the mosque of Ayesha
  there. The triviality of these rites is ill concealed by the legends
  of the sa'y of Hagar and of the tawaf being first performed by Adam in
  imitation of the circuit of the angels about the throne of God; the
  meaning of their ceremonies seems to have been almost a blank to the
  Arabs before Islam, whose religion had become a mere formal tradition.
  We do not even know to what deity the worship expressed in the tawaf
  was properly addressed. There is a tradition that the Ka'ba was a
  temple of Saturn (Shahrastani, p. 431); perhaps the most distinctive
  feature of the shrine may be sought in the sacred doves which still
  enjoy the protection of the sanctuary. These recall the sacred doves
  of Ascalon (Philo vi. 200 of Richter's ed.), and suggests
  Venus-worship as at least one element (cf. Herod i. 131, iii. 8; Ephr.
  Syr., _Op. Syr._ ii. 457).

  To the ordinary pilgrim the omra has become so much an episode of the
  hajj that it is described by some European pilgrims as a mere visit to
  the mosque of Ayesha; a better conception of its original significance
  is got from the Meccan feast of the seventh month (Rajab), graphically
  described by Ibn Jubair from his observations in A.D. 1184. Rajab was
  one of the ancient sacred months, and the feast, which extended
  through the whole month and was a joyful season of hospitality and
  thanksgiving, no doubt represents the ancient feasts of Mecca more
  exactly than the ceremonies of the hajj, in which old usage has been
  overlaid by traditions and glosses of Islam. The omra was performed by
  crowds from day to day, especially at new and full moon.[23] The new
  moon celebration was nocturnal; the road to Tanim, the Mas'a, and the
  mosque were brilliantly illuminated; and the appearing of the moon was
  greeted with noisy music. A genuine old Arab market was held, for the
  wild Bedouins of the Yemen mountains came in thousands to barter their
  cattle and fruits for clothing, and deemed that to absent themselves
  would bring drought and cattle plague in their homes. Though ignorant
  of the legal ritual and prayers, they performed the tawaf with
  enthusiasm, throwing themselves against the Ka'ba and clinging to its
  curtains as a child clings to its mother. They also made a point of
  entering the Ka'ba. The 29th of the month was the feast day of the
  Meccan women, when they and their little ones had the Ka'ba to
  themselves without the presence even of the Sheybas.

  The central and essential ceremonies of the hajj or greater pilgrimage
  are those of the day of Arafa, the 9th of the "pilgrimage month"
  (Dhu'l Hijja), the last of the Arab year; and every Moslem who is his
  own master, and can command the necessary means, is bound to join in
  these once in his life, or to have them fulfilled by a substitute on
  his behalf and at his expense. By them the pilgrim becomes as pure
  from sin as when he was born, and gains for the rest of his life the
  honourable title of hajj. Neglect of many other parts of the pilgrim
  ceremonial may be compensated by offerings, but to miss the "stand"
  (_woquf_) at Arafa is to miss the pilgrimage. Arafa or Arafat is a
  space, artificially limited, round a small isolated hill called the
  Hill of Mercy, a little way outside the holy territory, on the road
  from Mecca to Taif. One leaving Mecca after midday can easily reach
  the place on foot the same evening. The road is first northwards along
  the Mecca valley and then turns eastward. It leads through the
  straggling village of Mina, occupying a long narrow valley (Wadi
  Mina), two to three hours from Mecca, and thence by the mosque of
  Mozdalifa over a narrow pass opening out into the plain of Arafa,
  which is an expansion of the great Wadi Naman, through which the Taif
  road descends from Mount Kara. The lofty and rugged mountains of the
  Hodheyl tower over the plain on the north side and overshadow the
  little Hill of Mercy, which is one of those bosses of weathered
  granite so common in the Hejaz. Arafa lay quite near Dhul-Majaz,
  where, according to Arabian tradition, a great fair was held from the
  1st to the 8th of the pilgrimage month; and the ceremonies from which
  the hajj was derived were originally an appendix to this fair. Now, on
  the contrary, the pilgrim is expected to follow as closely as may be
  the movements of the prophet at his "farewell pilgrimage" in the year
  10 of the Hegira (A.D. 632). He therefore leaves Mecca in pilgrim garb
  on the 8th of Dhu'l Hijja, called the day of _tarwiya_ (an obscure and
  pre-Islamic name), and, strictly speaking, should spend the night at
  Mina. It is now, however, customary to go right on and encamp at once
  at Arafa. The night should be spent in devotion, but the coffee booths
  do a lively trade, and songs are as common as prayers. Next forenoon
  the pilgrim is free to move about, and towards midday he may if he
  please hear a sermon. In the afternoon the essential ceremony begins;
  it consists simply in "standing" on Arafa shouting "Labbeyka" and
  reciting prayers and texts till sunset. After the sun is down the vast
  assemblage breaks up, and a rush (technically _ifada_, _daf'_, _nafr_)
  is made in the utmost confusion to Mozdalifa, where the night prayer
  is said and the night spent. Before sunrise next morning (the 10th) a
  second "stand" like that on Arafa is made for a short time by
  torchlight round the mosque of Mozdalifa, but before the sun is fairly
  up all must be in motion in the second _ifada_ towards Mina. The day
  thus begun is the "day of sacrifice," and has four ceremonies--(1) to
  pelt with seven stones a cairn (_jamrat al 'aqaba_) at the eastern end
  of W. Mina, (2) to slay a victim at Mina and hold a sacrificial meal,
  part of the flesh being also dried and so preserved, or given to the
  poor,[24] (3) to be shaved and so terminate the _ihram_, (4) to make
  the third _ifada_, i.e. go to Mecca and perform the tawaf and sa'y
  (_'omrat al-ifada_), returning thereafter to Mina. The sacrifice and
  visit to Mecca may, however, be delayed till the 11th, 12th or 13th.
  These are the days of Mina, a fair and joyous feast, with no special
  ceremony except that each day the pilgrim is expected to throw seven
  stones at the _jamrat al 'aqaba_, and also at each of two similar
  cairns in the valley. The stones are thrown in the name of Allah, and
  are generally thought to be directed at the devil. This is, however, a
  custom older than Islam, and a tradition in Azraqi, p. 412, represents
  it as an act of worship to idols at Mina. As the stones are thrown on
  the days of the fair, it is not unlikely that they have something to
  do with the old Arab mode of closing a sale by the purchaser throwing
  a stone (Biruni, p. 328).[25] The pilgrims leave Mina on the 12th or
  13th, and the hajj is then over. (See further MAHOMMEDAN RELIGION.)

  The colourless character of these ceremonies is plainly due to the
  fact that they are nothing more than expurgated heathen rites. In
  Islam proper they have no _raison d'être_; the legends about Adam and
  Eve on Arafa, about Abraham's sacrifice of the ram at Thabii by Mina,
  imitated in the sacrifices of the pilgrimage, are clumsy
  afterthoughts, as appears from their variations and only partial
  acceptance. It is not so easy to get at the nature of the original
  rites, which Islam was careful to suppress. But we find mention of
  practices condemned by the orthodox, or forming no part of the Moslem
  ritual, which may be regarded as traces of an older ceremonial. Such
  are nocturnal illuminations at Mina (Ibn Batuta i. 396), Arafa and
  Mozdalifa (Ibn Jubair, 179), and tawafs performed by the ignorant at
  holy spots at Arafa not recognized by law (Snouck-Hurgronje p. 149
  sqq.). We know that the rites at Mozdalifa were originally connected
  with a holy hill bearing the name of the god Quzah (the Edomite Koze)
  whose bow is the rainbow, and there is reason to think that the
  _ifadas_ from Arafa and Quzah, which were not made as now after sunset
  and before sunrise, but when the sun rested on the tops of the
  mountains, were ceremonies of farewell and salutation to the sun-god.

  The statistics of the pilgrimage cannot be given with certainty and
  vary much from year to year. The quarantine office keeps a record of
  arrivals by sea at Jidda (66,000 for 1904); but to these must be added
  those travelling by land from Cairo, Damascus and Irak, the pilgrims
  who reach Medina from Yanbu and go on to Mecca, and those from all
  parts of the peninsula. Burckhardt in 1814 estimated the crowd at
  Arafa at 70,000, Burton in 1853 at 50,000, 'Abd el-Razzak in 1858 at
  60,000. This great assemblage is always a dangerous centre of
  infection, and the days of Mina especially, spent under circumstances
  originally adapted only for a Bedouin fair, with no provisions for
  proper cleanliness, and with the air full of the smell of putrefying
  offal and flesh drying in the sun, produce much sickness.

  LITERATURE.--Besides the Arabic geographers and cosmographers, we have
  Ibn 'Abd Rabbih's description of the mosque, early in the 10th century
  (_'Ikd Farid_, Cairo ed., iii. 362 sqq.), but above all the admirable
  record of Ibn Jubair (A.D. 1184), by far the best account extant of
  Mecca and the pilgrimage. It has been much pillaged by Ibn Batuta. The
  Arabic historians are largely occupied with fabulous matter as to
  Mecca before Islam; for these legends the reader may refer to C. de
  Perceval's _Essai_. How little confidence can be placed in the
  pre-Islamic history appears very clearly from the distorted accounts
  of Abraha's excursion against the Hejaz, which fell but a few years
  before the birth of the Prophet, and is the first event in Meccan
  history which has confirmation from other sources. See Nöldeke's
  version of Tabari, p. 204 sqq. For the period of the Prophet, Ibn
  Hisham and Wakidi are valuable sources in topography as well as
  history. Of the special histories and descriptions of Mecca published
  by Wüstenfeld (_Chroniken der Stadt Mekka_, 3 vols., 1857-1859, with
  an abstract in German, 1861), the most valuable is that of Azraqi. It
  has passed through the hands of several editors, but the oldest part
  goes back to the beginning of the 9th Christian century. Kutbeddin's
  history (vol. iii. of the _Chroniken_) goes down with the additions of
  his nephew to A.D. 1592.

  Of European descriptions of Mecca from personal observation the best
  is Burckhardt's _Travels in Arabia_ (cited above from the 8vo ed.,
  1829). _The Travels of Aly Bey_ (Badia, London, 1816) describe a visit
  in 1807; Burton's _Pilgrimage_ (3rd ed., 1879) often supplements
  Burckhardt; Von Maltzan's _Wallfahrt nach Mekka_ (1865) is lively but
  very slight. 'Abd el-Razzaq's report to the government of India on the
  pilgrimage of 1858 is specially directed to sanitary questions; C.
  Snouck-Hurgronje, _Mekka_ (2 vols., and a collection of photographs,
  The Hague, 1888-1889), gives a description of the Meccan sanctuary and
  of the public and private life of the Meccans as observed by the
  author during a sojourn in the holy city in 1884-1885 and a political
  history of Mecca from native sources from the Hegira till 1884. For
  the pilgrimage see particularly Snouck-Hurgronje, _Het Mekkaansche
  Feest_ (Leiden, 1880).     (W. R. S.)


FOOTNOTES:

  [1] A variant of the name Makkah is Bakkah (_Sur._ iii. 90; Bakri,
    155 seq.). For other names and honorific epithets of the city see
    Bakri, _ut supra_, Azraqi, p. 197, Yaqut iv. 617 seq. The lists are
    in part corrupt, and some of the names (Kutha and 'Arsh or 'Ursh,
    "the huts") are not properly names of the town as a whole.

  [2] Mecca, says one of its citizens, in Waqidi (Kremer's ed., p. 196,
    or _Muh. in Med._ p. 100), is a settlement formed for trade with
    Syria in summer and Abyssinia in winter, and cannot continue to exist
    if the trade is interrupted.

  [3] The details are variously related. See Biruni, p. 328 (E. T., p.
    324); Asma'i in Yaqut, iii. 705, iv. 416, 421; Azraqi, p. 129 seq.;
    Bakri, p. 661. Jebel Kabkab is a great mountain occupying the angle
    between W. Naman and the plain of Arafa. The peak is due north of
    Sheddad, the hamlet which Burckhardt (i. 115) calls Shedad. According
    to Azraqi, p. 80, the last shrine visited was that of the three trees
    of Uzza in W. Nakhla.

  [4] So we are told by Biruni, p. 62 (E. T., 73).

  [5] Waqidi, ed. Kremer, pp. 20, 21; _Muh. in Med._ p. 39.

  [6] The older fairs were not entirely deserted till the troubles of
    the last days of the Omayyads (Azraqi, p. 131).

  [7] This is the cross-road traversed by Burckhardt (i. 109), and
    described by him as cut through the rocks with much labour.

  [8] Istakhri gives the length of the city proper from north to south
    as 2 m., and the greatest breadth from the Jiyad quarter east of the
    great mosque across the valley and up the western slopes as
    two-thirds of the length.

  [9] For details as to the ancient quarters of Mecca, where the
    several families or septs lived apart, see Azraqi, 455 pp. seq., and
    compare Ya'qubi, ed. Juynboll, p. 100. The minor sacred places are
    described at length by Azraqi and Ibn Jubair. They are either
    connected with genuine memories of the Prophet and his times, or have
    spurious legends to conceal the fact that they were originally holy
    stones, wells, or the like, of heathen sanctity.

  [10] Baladhuri, in his chapter on the floods of Mecca (pp. 53 seq.),
    says that 'Omar built two dams.

  [11] The aqueduct is the successor of an older one associated with
    the names of Zobaida, wife of Harun al-Rashid, and other benefactors.
    But the old aqueduct was frequently out of repair, and seems to have
    played but a secondary part in the medieval water supply. Even the
    new aqueduct gave no adequate supply in Burckhardt's time.

  [12] In Ibn Jubair's time large supplies were brought from the Yemen
    mountains.

  [13] The corruption of manners in Mecca is no new thing. See the
    letter of the caliph Mahdi on the subject; Wüstenfeld, _Chron. Mek._,
    iv. 168.

  [14] The exact measurements (which, however, vary according to
    different authorities) are stated to be: sides 37 ft. 2 in. and 38
    ft. 4 in.; ends 31 ft. 7 in. and 29 ft.; height 35 ft.

  [15] The Ka'ba of Mahomet's time was the successor of an older
    building, said to have been destroyed by fire. It was constructed in
    the still usual rude style of Arabic masonry, with string courses of
    timber between the stones (like Solomon's Temple). The roof rested on
    six pillars; the door was raised above the ground and approached by a
    stair (probably on account of the floods which often swept the
    valley); and worshippers left their shoes under the stair before
    entering. During the first siege of Mecca (A.D. 683), the building
    was burned down, the Ibn Zubair reconstructed it on an enlarged scale
    and in better style of solid ashlar-work. After his death his most
    glaring innovations (the introduction of two doors on a level with
    the ground, and the extension of the building lengthwise to include
    the Hijr) were corrected by Hajjaj, under orders from the caliph, but
    the building retained its more solid structure. The roof now rested
    on three pillars, and the height was raised one-half. The Ka'ba was
    again entirely rebuilt after the flood of A.D. 1626, but since Hajjaj
    there seem to have been no structural changes.

  [16] Hobal was set up within the Temple over the pit that contained
    the sacred treasures. His chief function was connected with the
    sacred lot to which the Meccans were accustomed to betake themselves
    in all matters of difficulty.

  [17] See Ibn Hisham i. 54, Azraki p. 80 ('Uzza in Batn Marr); Yakut
    iii. 705 (Otheyda); Bar Hebraeus on Psalm xii. 9. Stones worshipped
    by circling round them bore the name _dawar_ or _duwar_ (Krehl, _Rel.
    d. Araber_, p. 69). The later Arabs not unnaturally viewed such
    cultus as imitated from that of Mecca (Yaqut iv. 622, cf. Dozy,
    _Israeliten te Mekka_, p. 125, who draws very perverse inferences).

  [18] The old _kiswa_ is removed on the 25th day of the month before
    the pilgrimage, and fragments of it are bought by the pilgrims as
    charms. Till the 10th day of the pilgrimage month the Ka'ba is bare.

  [19] Before Islam the Ka'ba was opened every Monday and Thursday; in
    the time of Ibn Jubair it was opened with considerable ceremony every
    Monday and Friday, and daily in the month Rajab. But, though prayer
    within the building is favoured by the example of the Prophet, it is
    not compulsory on the Moslem, and even in the time of Ibn Batuta the
    opportunities of entrance were reduced to Friday and the birthday of
    the Prophet.

  [20] See De Vogué, _Syrie centrale: inscr. sem._; Lady Anne Blunt
    _Pilgrimage of Nejd_, ii., and W. R. Smith, in the _Athenaeum_, March
    20, 1880.

  [21] Ibn Jubair speaks of fourteen steps, Ali Bey of four, Burckhardt
    of three. The surrounding ground no doubt has risen so that the old
    name "hill of Safa" is now inapplicable.

  [22] The latter perhaps was no part of the ancient omra; see
    Snouck-Hurgronje, _Het Mekkaansche Feest_ (1880) p. 115 sqq.

  [23] The 27th was also a great day, but this day was in commemoration
    of the rebuilding of the Ka'ba by Ibn Jubair.

  [24] The sacrifice is not indispensable except for those who can
    afford it and are combining the hajj with the omra.

  [25] On the similar pelting of the supposed graves of Abu Lahab and
    his wife (Ibn Jubair, p. 110) and of Abu Righal at Mughammas, see
    Nöldeke's translation of Tabari, 208.




MECHANICS. The subject of mechanics may be divided into two parts: (1)
theoretical or abstract mechanics, and (2) applied mechanics.


1. THEORETICAL MECHANICS

Historically theoretical mechanics began with the study of practical
contrivances such as the lever, and the name _mechanics_ (Gr. [Greek: ta
mêchanika]), which might more properly be restricted to the theory of
mechanisms, and which was indeed used in this narrower sense by Newton,
has clung to it, although the subject has long attained a far wider
scope. In recent times it has been proposed to adopt the term _dynamics_
(from Gr. [Greek: dynamis] force,) as including the whole science of the
action of force on bodies, whether at rest or in motion. The subject is
usually expounded under the two divisions of _statics_ and _kinetics_,
the former dealing with the conditions of rest or equilibrium and the
latter with the phenomena of motion as affected by force. To this latter
division the old name of _dynamics_ (in a restricted sense) is still
often applied. The mere geometrical description and analysis of various
types of motion, apart from the consideration of the forces concerned,
belongs to _kinematics_. This is sometimes discussed as a separate
theory, but for our present purposes it is more convenient to introduce
kinematical motions as they are required. We follow also the traditional
practice of dealing first with statics and then with kinetics. This is,
in the main, the historical order of development, and for purposes of
exposition it has many advantages. The laws of equilibrium are, it is
true, necessarily included as a particular case under those of motion;
but there is no real inconvenience in formulating as the basis of
statics a few provisional postulates which are afterwards seen to be
comprehended in a more general scheme.

The whole subject rests ultimately on the Newtonian laws of motion and
on some natural extensions of them. As these laws are discussed under a
separate heading (MOTION, LAWS OF), it is here only necessary to
indicate the standpoint from which the present article is written. It is
a purely empirical one. Guided by experience, we are able to frame
rules which enable us to say with more or less accuracy what will be the
consequences, or what were the antecedents, of a given state of things.
These rules are sometimes dignified by the name of "laws of nature," but
they have relation to our present state of knowledge and to the degree
of skill with which we have succeeded in giving more or less compact
expression to it. They are therefore liable to be modified from time to
time, or to be superseded by more convenient or more comprehensive modes
of statement. Again, we do not aim at anything so hopeless, or indeed so
useless, as a _complete_ description of any phenomenon. Some features
are naturally more important or more interesting to us than others; by
their relative simplicity and evident constancy they have the first hold
on our attention, whilst those which are apparently accidental and vary
from one occasion to another arc ignored, or postponed for later
examination. It follows that for the purposes of such description as is
possible some process of abstraction is inevitable if our statements are
to be simple and definite. Thus in studying the flight of a stone
through the air we replace the body in imagination by a mathematical
point endowed with a mass-coefficient. The size and shape, the
complicated spinning motion which it is seen to execute, the internal
strains and vibrations which doubtless take place, are all sacrificed in
the mental picture in order that attention may be concentrated on those
features of the phenomenon which are in the first place most interesting
to us. At a later stage in our subject the conception of the ideal rigid
body is introduced; this enables us to fill in some details which were
previously wanting, but others are still omitted. Again, the conception
of a force as concentrated in a mathematical line is as unreal as that
of a mass concentrated in a point, but it is a convenient fiction for
our purpose, owing to the simplicity which it lends to our statements.

The laws which are to be imposed on these ideal representations are in
the first instance largely at our choice. Any scheme of abstract
dynamics constructed in this way, provided it be self-consistent, is
mathematically legitimate; but from the physical point of view we
require that it should help us to picture the sequence of phenomena as
they actually occur. Its success or failure in this respect can only be
judged a posteriori by comparison of the results to which it leads with
the facts. It is to be noticed, moreover, that all available tests apply
only to the scheme as a whole; owing to the complexity of phenomena we
cannot submit any one of its postulates to verification apart from the
rest.

It is from this point of view that the question of relativity of motion,
which is often felt to be a stumbling-block on the very threshold of the
subject, is to be judged. By "motion" we mean of necessity motion
relative to some frame of reference which is conventionally spoken of as
"fixed." In the earlier stages of our subject this may be any rigid, or
apparently rigid, structure fixed relatively to the earth. If we meet
with phenomena which do not fit easily into this view, we have the
alternatives either to modify our assumed laws of motion, or to call to
our aid adventitious forces, or to examine whether the discrepancy can
be reconciled by the simpler expedient of a new basis of reference. It
is hardly necessary to say that the latter procedure has hitherto been
found to be adequate. As a first step we adopt a system of rectangular
axes whose origin is fixed in the earth, but whose directions are fixed
by relation to the stars; in the planetary theory the origin is
transferred to the sun, and afterwards to the mass-centre of the solar
system; and so on. At each step there is a gain in accuracy and
comprehensiveness; and the conviction is cherished that _some_ system of
rectangular axes exists with respect to which the Newtonian scheme holds
with all imaginable accuracy.

A similar account might be given of the conception of time as a
measurable quantity, but the remarks which it is necessary to make under
this head will find a place later.

  The following synopsis shows the scheme on which the treatment is
  based:--

  _Part 1.--Statics._

   1. Statics of a particle.
   2. Statics of a system of particles.
   3. Plane kinematics of a rigid body.
   4. Plane statics.
   5. Graphical statics.
   6. Theory of frames.
   7. Three-dimensional kinematics of a rigid body.
   8. Three-dimensional statics.
   9. Work.
  10. Statics of inextensible chains.
  11. Theory of mass-systems.

  _Part 2.--Kinetics._

  12. Rectilinear motion.
  13. General motion of a particle.
  14. Central forces. Hodograph.
  15. Kinetics of a system of discrete particles.
  16. Kinetics of a rigid body. Fundamental principles.
  17. Two-dimensional problems.
  18. Equations of motion in three dimensions.
  19. Free motion of a solid.
  20. Motion of a solid of revolution.
  21. Moving axes of reference.
  22. Equations of motion in generalized co-ordinates.
  23. Stability of equilibrium. Theory of vibrations.


PART I.--STATICS

§ 1. _Statics of a Particle._--By a _particle_ is meant a body whose
position can for the purpose in hand be sufficiently specified by a
mathematical point. It need not be "infinitely small," or even small
compared with ordinary standards; thus in astronomy such vast bodies as
the sun, the earth, and the other planets can for many purposes be
treated merely as points endowed with mass.

A _force_ is conceived as an effort having a certain direction and a
certain magnitude. It is therefore adequately represented, for
mathematical purposes, by a straight line AB drawn in the direction in
question, of length proportional (on any convenient scale) to the
magnitude of the force. In other words, a force is mathematically of the
nature of a "vector" (see VECTOR ANALYSIS, QUATERNIONS). In most
questions of pure statics we are concerned only with the _ratios_ of the
various forces which enter into the problem, so that it is indifferent
what _unit_ of force is adopted. For many purposes a gravitational
system of measurement is most natural; thus we speak of a force of so
many pounds or so many kilogrammes. The "absolute" system of measurement
will be referred to below in PART II., KINETICS. It is to be remembered
that all "force" is of the nature of a push or a pull, and that
according to the accepted terminology of modern mechanics such phrases
as "force of inertia," "accelerating force," "moving force," once
classical, are proscribed. This rigorous limitation of the meaning of
the word is of comparatively recent origin, and it is perhaps to be
regretted that some more technical term has not been devised, but the
convention must now be regarded as established.

[Illustration: FIG. 1.]

The fundamental postulate of this part of our subject is that the two
forces acting on a particle may be compounded by the "parallelogram
rule." Thus, if the two forces P,Q be represented by the lines OA, OB,
they can be replaced by a single force R represented by the diagonal OC
of the parallelogram determined by OA, OB. This is of course a physical
assumption whose propriety is justified solely by experience. We shall
see later that it is implied in Newton's statement of his Second Law of
motion. In modern language, forces are compounded by "vector-addition";
thus, if we draw in succession vectors [->HK], [->KL] to represent P, Q,
the force R is represented by the vector [->HL] which is the "geometric
sum" of [->HK], [->KL].

By successive applications of the above rule any number of forces acting
on a particle may be replaced by a single force which is the vector-sum
of the given forces: this single force is called the _resultant_. Thus
if [->AB], [->BC], [->CD] ..., [->HK] be vectors representing the given
forces, the resultant will be given by [->AK]. It will be understood
that the figure ABCD ... K need not be confined to one plane.

[Illustration: FIG. 2.]

If, in particular, the point K coincides with A, so that the resultant
vanishes, the given system of forces is said to be in _equilibrium_--i.e.
the particle could remain permanently at rest under its action. This is
the proposition known as the _polygon of forces_. In the particular case
of three forces it reduces to the _triangle of forces_, viz. "If three
forces acting on a particle are represented as to magnitude and direction
by the sides of a triangle taken in order, they are in equilibrium."

A sort of converse proposition is frequently useful, viz. if three
forces acting on a particle be in equilibrium, and any triangle be
constructed whose sides are respectively parallel to the forces, the
magnitudes of the forces will be to one another as the corresponding
sides of the triangle. This follows from the fact that all such
triangles are necessarily similar.

[Illustration: FIG. 3.]

  As a simple example of the geometrical method of treating statical
  problems we may consider the equilibrium of a particle on a "rough"
  inclined plane. The usual empirical law of sliding friction is that
  the mutual action between two plane surfaces in contact, or between a
  particle and a curve or surface, cannot make with the normal an angle
  exceeding a certain limit [lambda] called the _angle of friction_. If
  the conditions of equilibrium require an obliquity greater than this,
  sliding will take place. The precise value of [lambda] will vary with
  the nature and condition of the surfaces in contact. In the case of a
  body simply resting on an inclined plane, the reaction must of course
  be vertical, for equilibrium, and the slope [alpha] of the plane must
  therefore not exceed [lambda]. For this reason [lambda] is also known
  as the _angle of repose_. If [alpha] > [lambda], a force P must be
  applied in order to maintain equilibrium; let [theta] be the
  inclination of P to the plane, as shown in the left-hand diagram. The
  relations between this force P, the gravity W of the body, and the
  reaction S of the plane are then determined by a triangle of forces
  HKL. Since the inclination of S to the normal cannot exceed [lambda]
  on either side, the value of P must lie between two limits which are
  represented by L1H, L2H, in the right-hand diagram. Denoting these
  limits by P1, P2, we have

    P1/W = L1H/HK = sin ([alpha] - [lambda])/cos ([theta] + [lambda]),
    P2/W = L2H/HK = sin ([alpha] + [lambda])/cos ([theta] - [lambda]).

  It appears, moreover, that if [theta] be varied P will be least when
  L1H is at right angles to KL1, in which case P1 = W sin ([alpha] -
  [lambda]), corresponding to [theta] = -[lambda].

[Illustration: FIG. 4.]

Just as two or more forces can be combined into a single resultant, so a
single force may be _resolved_ into _components_ acting in assigned
directions. Thus a force can be uniquely resolved into two components
acting in two assigned directions in the same plane with it by an
inversion of the parallelogram construction of fig. 1. If, as is usually
most convenient, the two assigned directions are at right angles, the
two components of a force P will be P cos [theta], P sin [theta], where
[theta] is the inclination of P to the direction of the former
component. This leads to formulae for the analytical reduction of a
system of coplanar forces acting on a particle. Adopting rectangular
axes Ox, Oy, in the plane of the forces, and distinguishing the various
forces of the system by suffixes, we can replace the system by two
forces X, Y, in the direction of co-ordinate axes; viz.--

  X = P1 cos [theta]1 + P2 cos [theta]2 + ... = [Sigma](P cos [theta]),  }
  Y = P1 sin [theta]1 + P2 sin [theta]2 + ... = [Sigma](P sin [theta]).  }  (1)

These two forces X, Y, may be combined into a single resultant R making
an angle [phi] with Ox, provided

  X = R cos [phi], Y = R sin [phi],  (2)

whence

  R² = X² + Y², tan [phi] = Y/X.  (3)

For equilibrium we must have R = 0, which requires X = 0, Y = 0; in
words, the sum of the components of the system must be zero for each of
two perpendicular directions in the plane.

[Illustration: FIG. 5.]

A similar procedure applies to a three-dimensional system. Thus if, O
being the origin, [->OH] represent any force P of the system, the planes
drawn through H parallel to the co-ordinate planes will enclose with the
latter a parallelepiped, and it is evident that [->OH] is the geometric
sum of [->OA], [->AN], [->NH], or [->OA], [->OB], [->OC], in the figure.
Hence P is equivalent to three forces Pl, Pm, Pn acting along Ox, Oy,
Oz, respectively, where l, m, n, are the "direction-ratios" of [->OH].
The whole system can be reduced in this way to three forces

  X = [Sigma] (Pl), Y = [Sigma] (Pm), Z = [Sigma] (Pn),  (4)

acting along the co-ordinate axes. These can again be combined into a
single resultant R acting in the direction ([lambda], [mu], [nu]),
provided

  X = R[lambda], Y = R[mu], Z = R[nu].  (5)

If the axes are rectangular, the direction-ratios become
direction-cosines, so that [lambda]² + [mu]² + [nu]² = 1, whence

  R² = X² + Y² + Z².  (6)

The conditions of equilibrium are X = 0, Y = 0, Z = 0.

§ 2. _Statics of a System of Particles._--We assume that the mutual
forces between the pairs of particles, whatever their nature, are
subject to the "Law of Action and Reaction" (Newton's Third Law); i.e.
the force exerted by a particle A on a particle B, and the force exerted
by B on A, are equal and opposite in the line AB. The problem of
determining the possible configurations of equilibrium of a system of
particles subject to extraneous forces which are known functions of the
positions of the particles, and to internal forces which are known
functions of the distances of the pairs of particles between which they
act, is in general determinate. For if n be the number of particles, the
3n conditions of equilibrium (three for each particle) are equal in
number to the 3n Cartesian (or other) co-ordinates of the particles,
which are to be found. If the system be subject to frictionless
constraints, e.g. if some of the particles be constrained to lie on
smooth surfaces, or if pairs of particles be connected by inextensible
strings, then for each geometrical relation thus introduced we have an
unknown reaction (e.g. the pressure of the smooth surface, or the
tension of the string), so that the problem is still determinate.

[Illustration: FIG. 6.]

[Illustration: FIG. 7.]

  The case of the _funicular polygon_ will be of use to us later. A
  number of particles attached at various points of a string are acted
  on by given extraneous forces P1, P2, P3 ... respectively. The
  relation between the three forces acting on any particle, viz. the
  extraneous force and the tensions in the two adjacent portions of the
  string can be exhibited by means of a triangle of forces; and if the
  successive triangles be drawn to the same scale they can be fitted
  together so as to constitute a single _force-diagram_, as shown in
  fig. 6. This diagram consists of a polygon whose successive sides
  represent the given forces P1, P2, P3 ..., and of a series of lines
  connecting the vertices with a point O. These latter lines measure the
  tensions in the successive portions of string. As a special, but very
  important case, the forces P1, P2, P3 ... may be parallel, e.g. they
  may be the weights of the several particles. The polygon of forces is
  then made up of segments of a vertical line. We note that the tensions
  have now the same horizontal projection (represented by the dotted
  line in fig. 7). It is further of interest to note that if the weights
  be all equal, and at equal horizontal intervals, the vertices of the
  funicular will lie on a parabola whose axis is vertical. To prove this
  statement, let A, B, C, D ... be successive vertices, and let H, K ...
  be the middle points of AC, BD ...; then BH, CK ... will be vertical
  by the hypothesis, and since the geometric sum of [->BA], [->BC] is
  represented by 2[->BH], the tension in BA: tension in BC: weight at B

  as BA: BC: 2BH.

  [Illustration: FIG. 8.]

  The tensions in the successive portions of the string are therefore
  proportional to the respective lengths, and the lines BH, CK ... are
  all equal. Hence AD, BC are parallel and are bisected by the same
  vertical line; and a parabola with vertical axis can therefore be
  described through A, B, C, D. The same holds for the four points B, C,
  D, E and so on; but since a parabola is uniquely determined by the
  direction of its axis and by three points on the curve, the successive
  parabolas ABCD, BCDE, CDEF ... must be coincident.

§ 3. _Plane Kinematics of a Rigid Body._--The ideal _rigid body_ is one
in which the distance between any two points is invariable. For the
present we confine ourselves to the consideration of displacements in
two dimensions, so that the body is adequately represented by a thin
lamina or plate.

[Illustration: FIG. 9.]

The position of a lamina movable in its own plane is determinate when we
know the positions of any two points A, B of it. Since the four
co-ordinates (Cartesian or other) of these two points are connected by
the relation which expresses the invariability of the length AB, it is
plain that virtually three independent elements are required and suffice
to specify the position of the lamina. For instance, the lamina may in
general be fixed by connecting any three points of it by rigid links to
three fixed points in its plane. The three independent elements may be
chosen in a variety of ways (e.g. they may be the lengths of the three
links in the above example). They may be called (in a generalized sense)
the _co-ordinates_ of the lamina. The lamina when perfectly free to move
in its own plane is said to have _three degrees of freedom_.

[Illustration: FIG. 10.]

By a theorem due to M. Chasles any displacement whatever of the lamina
in its own plane is equivalent to a rotation about some finite or
infinitely distant point J. For suppose that in consequence of the
displacement a point of the lamina is brought from A to B, whilst the
point of the lamina which was originally at B is brought to C. Since AB,
BC, are two different positions of the same line in the lamina they are
equal, and it is evident that the rotation could have been effected by a
rotation about J, the centre of the circle ABC, through an angle AJB. As
a special case the three points A, B, C may be in a straight line; J is
then at infinity and the displacement is equivalent to a pure
_translation_, since every point of the lamina is now displaced parallel
to AB through a space equal to AB.

[Illustration: FIG. 11.]

Next, consider any continuous motion of the lamina. The latter may be
brought from any one of its positions to a neighbouring one by a
rotation about the proper centre. The limiting position J of this
centre, when the two positions are taken infinitely close to one
another, is called the _instantaneous centre_. If P, P´ be consecutive
positions of the same point, and [delta][theta] the corresponding angle
of rotation, then ultimately PP´ is at right angles to JP and equal to
JP·[delta][theta]. The instantaneous centre will have a certain locus in
space, and a certain locus in the lamina. These two loci are called
_pole-curves_ or _centrodes_, and are sometimes distinguished as the
_space-centrode_ and the _body-centrode_, respectively. In the
continuous motion in question the latter curve rolls without slipping on
the former (M. Chasles). Consider in fact any series of successive
positions 1, 2, 3... of the lamina (fig. 11); and let J12, J23, J34...
be the positions in space of the centres of the rotations by which the
lamina can be brought from the first position to the second, from the
second to the third, and so on. Further, in the position 1, let J12,
J´23, J´34 ... be the points of the lamina which have become the
successive centres of rotation. The given series of positions will be
assumed in succession if we imagine the lamina to rotate first about J12
until J´23 comes into coincidence with J23, then about J23 until J´34
comes into coincidence with J34, and so on. This is equivalent to
imagining the polygon J12 J´23 J´34 ..., supposed fixed in the lamina,
to roll on the polygon J12 J23 J34 ..., which is supposed fixed in
space. By imagining the successive positions to be taken infinitely
close to one another we derive the theorem stated. The particular case
where both centrodes are circles is specially important in mechanism.

[Illustration: FIG. 12.]

  The theory may be illustrated by the case of "three-bar motion." Let
  ABCD be any quadrilateral formed of jointed links. If, AB being held
  fixed, the quadrilateral be slightly deformed, it is obvious that the
  instantaneous centre J will be at the intersection of the straight
  lines AD, BC, since the displacements of the points D, C are
  necessarily at right angles to AD, BC, respectively. Hence these
  displacements are proportional to JD, JC, and therefore to DD´ CC´,
  where C´D´ is any line drawn parallel to CD, meeting BC, AD in C´, D´,
  respectively. The determination of the centrodes in three-bar motion
  is in general complicated, but in one case, that of the "crossed
  parallelogram" (fig. 13), they assume simple forms. We then have AB =
  DC and AD = BC, and from the symmetries of the figure it is plain that

    AJ + JB = CJ + JD = AD.

  Hence the locus of J relative to AB, and the locus relative to CD are
  equal ellipses of which A, B and C, D are respectively the foci. It
  may be noticed that the lamina in fig. 9 is not, strictly speaking,
  fixed, but admits of infinitesimal displacement, whenever the
  directions of the three links are concurrent (or parallel).

[Illustration: FIG. 13.]

The matter may of course be treated analytically, but we shall only
require the formula for infinitely small displacements. If the origin of
rectangular axes fixed in the lamina be shifted through a space whose
projections on the original directions of the axes are [lambda], [mu],
and if the axes are simultaneously turned through an angle [epsilon],
the co-ordinates of a point of the lamina, relative to the original
axes, are changed from x, y to [lambda] + x cos [epsilon] - y sin
[epsilon], [mu] + x sin [epsilon] + y cos [epsilon], or [lambda] + x -
y[epsilon], [mu] + x[epsilon] + y, ultimately. Hence the component
displacements are ultimately

  [delta]x = [lambda] - y[epsilon], [delta]y = [mu] + x[epsilon]  (1)

If we equate these to zero we get the co-ordinates of the instantaneous
centre.

§ 4. _Plane Statics._--The statics of a rigid body rests on the
following two assumptions:--

(i) A force may be supposed to be applied indifferently at any point in
its line of action. In other words, a force is of the nature of a
"bound" or "localized" vector; it is regarded as resident in a certain
line, but has no special reference to any particular point of the line.

(ii) Two forces in intersecting lines may be replaced by a force which
is their geometric sum, acting through the intersection. The theory of
parallel forces is included as a limiting case. For if O, A, B be any
three points, and m, n any scalar quantities, we have in vectors

  m · [->OA] + n·[->OB] = (m + n) [->OC],  (1)

provided

  m · [->CA] + n·[->CB] = 0.  (2)

Hence if forces P, Q act in OA, OB, the resultant R will pass through C,
provided

  m = P/OA, n = Q/OB;

also

  R = P·OC/OA + Q·OC/OB,  (3)

and

  P·AC : Q·CB = OA : OB.  (4)

These formulae give a means of constructing the resultant by means of
any transversal AB cutting the lines of action. If we now imagine the
point O to recede to infinity, the forces P, Q and the resultant R are
parallel, and we have

  R = P + Q, P·AC = Q·CB.  (5)

[Illustration: FIG. 14.]

When P, Q have opposite signs the point C divides AB externally on the
side of the greater force. The investigation fails when P + Q = 0, since
it leads to an infinitely small resultant acting in an infinitely
distant line. A combination of two equal, parallel, but oppositely
directed forces cannot in fact be replaced by anything simpler, and must
therefore be recognized as an independent entity in statics. It was
called by L. Poinsot, who first systematically investigated its
properties, a _couple_.

We now restrict ourselves for the present to the systems of forces in
one plane. By successive applications of (ii) any such coplanar system
can in general be reduced to a _single resultant_ acting in a definite
line. As exceptional cases the system may reduce to a couple, or it may
be in equilibrium.

[Illustration: FIG. 15.]

The _moment_ of a force about a point O is the product of the force into
the perpendicular drawn to its line of action from O, this perpendicular
being reckoned positive or negative according as O lies on one side or
other of the line of action. If we mark off a segment AB along the line
of action so as to represent the force completely, the moment is
represented as to magnitude by twice the area of the triangle OAB, and
the usual convention as to sign is that the area is to be reckoned
positive or negative according as the letters O, A, B, occur in
"counter-clockwise" or "clockwise" order.

[Illustration: FIG. 16.]

The sum of the moments of two forces about any point O is equal to the
moment of their resultant (P. Varignon, 1687). Let AB, AC (fig. 16)
represent the two forces, AD their resultant; we have to prove that the
sum of the triangles OAB, OAC is equal to the triangle OAD, regard being
had to signs. Since the side OA is common, we have to prove that the sum
of the perpendiculars from B and C on OA is equal to the perpendicular
from D on OA, these perpendiculars being reckoned positive or negative
according as they lie to the right or left of AO. Regarded as a
statement concerning the orthogonal projections of the vectors [->AB]
and [->AC] (or BD), and of their sum [->AD], on a line perpendicular to
AO, this is obvious.

It is now evident that in the process of reduction of a coplanar system
no change is made at any stage either in the sum of the projections of
the forces on any line or in the sum of their moments about any point.
It follows that the single resultant to which the system in general
reduces is uniquely determinate, i.e. it acts in a definite line and has
a definite magnitude and sense. Again it is necessary and sufficient for
equilibrium that the sum of the projections of the forces on each of two
perpendicular directions should vanish, and (moreover) that the sum of
the moments about some one point should be zero. The fact that three
independent conditions must hold for equilibrium is important. The
conditions may of course be expressed in different (but equivalent)
forms; e.g. the sum of the moments of the forces about each of the three
points which are not collinear must be zero.

[Illustration: FIG. 17.]

The particular case of three forces is of interest. If they are not all
parallel they must be concurrent, and their vector-sum must be zero.
Thus three forces acting perpendicular to the sides of a triangle at the
middle points will be in equilibrium provided they are proportional to
the respective sides, and act all inwards or all outwards. This result
is easily extended to the case of a polygon of any number of sides; it
has an important application in hydrostatics.

  Again, suppose we have a bar AB resting with its ends on two smooth
  inclined planes which face each other. Let G be the centre of gravity
  (§ 11), and let AG = a, GB = b. Let [alpha], [beta] be the
  inclinations of the planes, and [theta] the angle which the bar makes
  with the vertical. The position of equilibrium is determined by the
  consideration that the reactions at A and B, which are by hypothesis
  normal to the planes, must meet at a point J on the vertical through
  G. Hence

    JG/a = sin ([theta] - [alpha])/sin [alpha], JG/b = sin ([theta] + [beta])/sin [beta],

  whence

                  a cot [alpha] - b cot [beta]
    cot [theta] = ----------------------------.  (6)
                            a + b

  If the bar is uniform we have a = b, and

    cot [theta] = ½ (cot [alpha] - cot [beta]).  (7)

  The problem of a rod suspended by strings attached to two points of it
  is virtually identical, the tensions of the strings taking the place
  of the reactions of the planes.

[Illustration: FIG. 18.]

Just as a system of forces is in general equivalent to a single force,
so a given force can conversely be replaced by combinations of other
forces, in various ways. For instance, a given force (and consequently a
system of forces) can be replaced in one and only one way by three
forces acting in three assigned straight lines, provided these lines be
not concurrent or parallel. Thus if the three lines form a triangle ABC,
and if the given force F meet BC in H, then F can be resolved into two
components acting in HA, BC, respectively. And the force in HA can be
resolved into two components acting in BC, CA, respectively. A simple
graphical construction is indicated in fig. 19, where the dotted lines
are parallel. As an example, any system of forces acting on the lamina
in fig. 9 is balanced by three determinate tensions (or thrusts) in the
three links, provided the directions of the latter are not concurrent.

[Illustration: FIG. 19.]

  If P, Q, R, be any three forces acting along BC, CA, AB, respectively,
  the line of action of the resultant is determined by the consideration
  that the sum of the moments about any point on it must vanish. Hence
  in "trilinear" co-ordinates, with ABC as fundamental triangle, its
  equation is P[alpha] + Q[beta] + R[gamma] = 0. If P : Q : R = a : b :
  c, where a, b, c are the lengths of the sides, this becomes the "line
  at infinity," and the forces reduce to a couple.

[Illustration: FIG. 20.]

The sum of the moments of the two forces of a couple is the same about
any point in the plane. Thus in the figure the sum of the moments about
O is P·OA - P·OB or P·AB, which is independent of the position of O.
This sum is called the _moment of the couple_; it must of course have
the proper sign attributed to it. It easily follows that any two couples
of the same moment are equivalent, and that any number of couples can be
replaced by a single couple whose moment is the sum of their moments.
Since a couple is for our purposes sufficiently represented by its
moment, it has been proposed to substitute the name _torque_ (or
twisting effort), as free from the suggestion of any special pair of
forces.

A system of forces represented completely by the sides of a plane
polygon taken in order is equivalent to a couple whose moment is
represented by twice the area of the polygon; this is proved by taking
moments about any point. If the polygon intersects itself, care must be
taken to attribute to the different parts of the area their proper
signs.

[Illustration: FIG. 21.]

Again, any coplanar system of forces can be replaced by a single force R
acting at any assigned point O, together with a couple G. The force R is
the geometric sum of the given forces, and the moment (G) of the couple
is equal to the sum of the moments of the given forces about O. The
value of G will in general vary with the position of O, and will vanish
when O lies on the line of action of the single resultant.

[Illustration: FIG. 22.]

The formal analytical reduction of a system of coplanar forces is as
follows. Let (x1, y1), (x2, y2), ... be the rectangular co-ordinates of
any points A1, A2, ... on the lines of action of the respective forces.
The force at A1 may be replaced by its components X1, Y1, parallel to
the co-ordinate axes; that at A2 by its components X2, Y2, and so on.
Introducing at O two equal and opposite forces ±X1 in Ox, we see that X1
at A1 may be replaced by an equal and parallel force at O together with
a couple -y1X1. Similarly the force Y1 at A1 may be replaced by a force
Y1 at O together with a couple x1Y1. The forces X1, Y1, at O can thus be
transferred to O provided we introduce a couple x1Y1 - y1X1. Treating
the remaining forces in the same way we get a force X1 + X2 + ... or
[Sigma](X) along Ox, a force Y1 + Y2 + ... or [Sigma](Y) along Oy, and a
couple (x1Y1 - y1X1) + (x2Y2 - y2X2) + ... or [Sigma](xY - yX). The
three conditions of equilibrium are therefore

  [Sigma](X) = 0, [Sigma](Y) = 0, [Sigma](xY - yX) = 0.  (8)

If O´ be a point whose co-ordinates are ([xi], [eta]), the moment of the
couple when the forces are transferred to O´ as a new origin will be
[Sigma]{(x - [xi]) Y - (y - [eta]) X}. This vanishes, i.e. the system
reduces to a single resultant through O´, provided

  -[xi]·[Sigma](Y) + [eta]·[Sigma](X) + [Sigma](xY - yX) = 0.  (9)

If [xi], [eta] be regarded as current co-ordinates, this is the equation
of the line of action of the single resultant to which the system is in
general reducible.

If the forces are all parallel, making say an angle [theta] with Ox, we
may write X1 = P1 cos [theta], Y1 = P1 sin [theta], X2 = P2 cos [theta],
Y2 = P2 sin [theta], .... The equation (9) then becomes

  {[Sigma](xP) - [xi]·[Sigma](P)} sin [theta] - {[Sigma](yP) - [eta]·[Sigma](P)} cos [theta] = 0.  (10)

If the forces P1, P2, ... be turned in the same sense through the same
angle about the respective points A1, A2, ... so as to remain parallel,
the value of [theta] is alone altered, and the resultant [Sigma](P)
passes always through the point

         [Sigma](xP)         [Sigma](yP)
  [|x] = -----------, [|y] = -----------,  (11)
          [Sigma](P)          [Sigma](P)

which is determined solely by the configuration of the points A1, A2,
... and by the ratios P1: P2: ... of the forces acting at them
respectively. This point is called the _centre_ of the given system of
parallel forces; it is finite and determinate unless [Sigma](P) = 0. A
geometrical proof of this theorem, which is not restricted to a
two-dimensional system, is given later (§ 11). It contains the theory of
the _centre of gravity_ as ordinarily understood. For if we have an
assemblage of particles whose mutual distances are small compared with
the dimensions of the earth, the forces of gravity on them constitute a
system of sensibly parallel forces, sensibly proportional to the
respective masses. If now the assemblage be brought into any other
position relative to the earth, without alteration of the mutual
distances, this is equivalent to a rotation of the directions of the
forces relatively to the assemblage, the ratios of the forces remaining
unaltered. Hence there is a certain point, fixed relatively to the
assemblage, through which the resultant of gravitational action always
passes; this resultant is moreover equal to the sum of the forces on the
several particles.

[Illustration: FIG. 23.]

  The theorem that any coplanar system of forces can be reduced to a
  force acting through any assigned point, together with a couple, has
  an important illustration in the theory of the distribution of
  shearing stress and bending moment in a horizontal beam, or other
  structure, subject to vertical extraneous forces. If we consider any
  vertical section P, the forces exerted across the section by the
  portion of the structure on one side on the portion on the other may
  be reduced to a vertical force F at P and a couple M. The force
  measures the _shearing stress_, and the couple the _bending moment_ at
  P; we will reckon these quantities positive when the senses are as
  indicated in the figure.

  If the remaining forces acting on the portion of the structure on
  either side of P are known, then resolving vertically we find F, and
  taking moments about P we find M. Again if PQ be any segment of the
  beam which is free from load, Q lying to the right of P, we find

    F_P = F_Q, M_P - M_Q = -F·PQ;  (12)

  hence F is constant between the loads, whilst M decreases as we travel
  to the right, with a constant gradient -F. If PQ be a short segment
  containing an isolated load W, we have

    F_Q - F_P = -W, M_Q = M_P;  (13)

  hence F is discontinuous at a concentrated load, diminishing by an
  amount equal to the load as we pass the loaded point to the right,
  whilst M is continuous. Accordingly the graph of F for any system of
  isolated loads will consist of a series of horizontal lines, whilst
  that of M will be a continuous polygon.

  [Illustration: FIG. 24.]

  To pass to the case of continuous loads, let x be measured
  horizontally along the beam to the right. The load on an element
  [delta]x of the beam may be represented by w[delta]x, where w is in
  general a function of x. The equations (12) are now replaced by

    [delta]F = -w[delta]x, [delta]M = -F[delta]x,

  whence
                   _                          _
                  / Q                        / Q
    F_Q - F_P = - |   w dx,    M_Q - M_P = - |   F dx.  (14)
                 _/P                        _/P

  The latter relation shows that the bending moment varies as the area
  cut off by the ordinate in the graph of F. In the case of uniform load
  we have

    F = -wx + A, M = ½wx² - Ax + B,  (15)

  where the arbitrary constants A,B are to be determined by the
  conditions of the special problem, e.g. the conditions at the ends of
  the beam. The graph of F is a straight line; that of M is a parabola
  with vertical axis. In all cases the graphs due to different
  distributions of load may be superposed. The figure shows the case of
  a uniform heavy beam supported at its ends.

[Illustration: FIG. 25.]

[Illustration: FIG. 26.]

§ 5. _Graphical Statics._--A graphical method of reducing a plane system
of forces was introduced by C. Culmann (1864). It involves the
construction of two figures, a _force-diagram_ and a _funicular
polygon_. The force-diagram is constructed by placing end to end a
series of vectors representing the given forces in magnitude and
direction, and joining the vertices of the polygon thus formed to an
arbitrary _pole_ O. The funicular or link polygon has its vertices on
the lines of action of the given forces, and its sides respectively
parallel to the lines drawn from O in the force-diagram; in particular,
the two sides meeting in any vertex are respectively parallel to the
lines drawn from O to the ends of that side of the force-polygon which
represents the corresponding force. The relations will be understood
from the annexed diagram, where corresponding lines in the force-diagram
(to the right) and the funicular (to the left) are numbered similarly.
The sides of the force-polygon may in the first instance be arranged in
any order; the force-diagram can then be completed in a doubly infinite
number of ways, owing to the arbitrary position of O; and for each
force-diagram a simply infinite number of funiculars can be drawn. The
two diagrams being supposed constructed, it is seen that each of the
given systems of forces can be replaced by two components acting in the
sides of the funicular which meet at the corresponding vertex, and that
the magnitudes of these components will be given by the corresponding
triangle of forces in the force-diagram; thus the force 1 in the figure
is equivalent to two forces represented by 01 and 12. When this process
of replacement is complete, each terminated side of the funicular is the
seat of two forces which neutralize one another, and there remain only
two uncompensated forces, viz., those resident in the first and last
sides of the funicular. If these sides intersect, the resultant acts
through the intersection, and its magnitude and direction are given by
the line joining the first and last sides of the force-polygon (see fig.
26, where the resultant of the four given forces is denoted by R). As a
special case it may happen that the force-polygon is closed, i.e. its
first and last points coincide; the first and last sides of the
funicular will then be parallel (unless they coincide), and the two
uncompensated forces form a couple. If, however, the first and last
sides of the funicular coincide, the two outstanding forces neutralize
one another, and we have equilibrium. Hence the necessary and sufficient
conditions of equilibrium are that the force-polygon and the funicular
should both be closed. This is illustrated by fig. 26 if we imagine the
force R, reversed, to be included in the system of given forces.

It is evident that a system of jointed bars having the shape of the
funicular polygon would be in equilibrium under the action of the given
forces, supposed applied to the joints; moreover any bar in which the
stress is of the nature of a tension (as distinguished from a thrust)
might be replaced by a string. This is the origin of the names
"link-polygon" and "funicular" (cf. § 2).

  If funiculars be drawn for two positions O, O´ of the pole in the
  force-diagram, their corresponding sides will intersect on a straight
  line parallel to OO´. This is essentially a theorem of projective
  geometry, but the following statical proof is interesting. Let AB
  (fig. 27) be any side of the force-polygon, and construct the
  corresponding portions of the two diagrams, first with O and then with
  O´ as pole. The force corresponding to AB may be replaced by the two
  components marked x, y; and a force corresponding to BA may be
  represented by the two components marked x´, y´. Hence the forces x,
  y, x´, y´ are in equilibrium. Now x, x´ have a resultant through H,
  represented in magnitude and direction by OO´, whilst y, y´ have a
  resultant through K represented in magnitude and direction by O´O.
  Hence HK must be parallel to OO´. This theorem enables us, when one
  funicular has been drawn, to construct any other without further
  reference to the force-diagram.

  [Illustration: FIG. 27.]

  The complete figures obtained by drawing first the force-diagrams of a
  system of forces in equilibrium with two distinct poles O, O´, and
  secondly the corresponding funiculars, have various interesting
  relations. In the first place, each of these figures may be conceived
  as an orthogonal projection of a closed plane-faced polyhedron. As
  regards the former figure this is evident at once; viz. the polyhedron
  consists of two pyramids with vertices represented by O, O´, and a
  common base whose perimeter is represented by the force-polygon (only
  one of these is shown in fig. 28). As regards the funicular diagram,
  let LM be the line on which the pairs of corresponding sides of the
  two polygons meet, and through it draw any two planes [omega],
  [omega]´. Through the vertices A, B, C, ... and A´, B´, C´, ... of the
  two funiculars draw normals to the plane of the diagram, to meet
  [omega] and [omega]´ respectively. The points thus obtained are
  evidently the vertices of a polyhedron with plane faces.

  [Illustration: FIG. 28.]

  [Illustration: FIG. 29.]

  To every line in either of the original figures corresponds of course
  a parallel line in the other; moreover, it is seen that concurrent
  lines in either figure correspond to lines forming a closed polygon in
  the other. Two plane figures so related are called _reciprocal_, since
  the properties of the first figure in relation to the second are the
  same as those of the second with respect to the first. A still simpler
  instance of reciprocal figures is supplied by the case of concurrent
  forces in equilibrium (fig. 29). The theory of these reciprocal
  figures was first studied by J. Clerk Maxwell, who showed amongst
  other things that a reciprocal can always be drawn to any figure which
  is the orthogonal projection of a plane-faced polyhedron. If in fact
  we take the pole of each face of such a polyhedron with respect to a
  paraboloid of revolution, these poles will be the vertices of a second
  polyhedron whose edges are the "conjugate lines" of those of the
  former. If we project both polyhedra orthogonally on a plane
  perpendicular to the axis of the paraboloid, we obtain two figures
  which are reciprocal, except that corresponding lines are orthogonal
  instead of parallel. Another proof will be indicated later (§ 8) in
  connexion with the properties of the linear complex. It is convenient
  to have a notation which shall put in evidence the reciprocal
  character. For this purpose we may designate the points in one figure
  by letters A, B, C, ... and the corresponding polygons in the other
  figure by the same letters; a line joining two points A, B in one
  figure will then correspond to the side common to the two polygons A,
  B in the other. This notation was employed by R. H. Bow in connexion
  with the theory of frames (§ 6, and see also APPLIED MECHANICS below)
  where reciprocal diagrams are frequently of use (cf. DIAGRAM).

  When the given forces are all parallel, the force-polygon consists of
  a series of segments of a straight line. This case has important
  practical applications; for instance we may use the method to find the
  pressures on the supports of a beam loaded in any given manner. Thus
  if AB, BC, CD represent the given loads, in the force-diagram, we
  construct the sides corresponding to OA, OB, OC, OD in the funicular;
  we then draw the _closing line_ of the funicular polygon, and a
  parallel OE to it in the force diagram. The segments DE, EA then
  represent the upward pressures of the two supports on the beam, which
  pressures together with the given loads constitute a system of forces
  in equilibrium. The pressures of the beam on the supports are of
  course represented by ED, AE. The two diagrams are portions of
  reciprocal figures, so that Bow's notation is applicable.

  [Illustration: FIG. 30.]

  [Illustration: FIG. 31.]

  A graphical method can also be applied to find the moment of a force,
  or of a system of forces, about any assigned point P. Let F be a force
  represented by AB in the force-diagram. Draw a parallel through P to
  meet the sides of the funicular which correspond to OA, OB in the
  points H, K. If R be the intersection of these sides, the triangles
  OAB, RHK are similar, and if the perpendiculars OM, RN be drawn we
  have

    HK·OM = AB·RN = F·RN,

  which is the moment of F about P. If the given forces are all parallel
  (say vertical) OM is the same for all, and the moments of the several
  forces about P are represented on a certain scale by the lengths
  intercepted by the successive pairs of sides on the vertical through
  P. Moreover, the moments are compounded by adding (geometrically) the
  corresponding lengths HK. Hence if a system of vertical forces be in
  equilibrium, so that the funicular polygon is closed, the length which
  this polygon intercepts on the vertical through any point P gives the
  sum of the moments about P of all the forces on one side of this
  vertical. For instance, in the case of a beam in equilibrium under any
  given loads and the reactions at the supports, we get a graphical
  representation of the distribution of bending moment over the beam.
  The construction in fig. 30 can easily be adjusted so that the closing
  line shall be horizontal; and the figure then becomes identical with
  the bending-moment diagram of § 4. If we wish to study the effects of
  a movable load, or system of loads, in different positions on the
  beam, it is only necessary to shift the lines of action of the
  pressures of the supports relatively to the funicular, keeping them at
  the same, distance apart; the only change is then in the position of
  the closing line of the funicular. It may be remarked that since this
  line joins homologous points of two "similar" rows it will envelope a
  parabola.

The "centre" (§ 4) of a system of parallel forces of given magnitudes,
acting at given points, is easily determined graphically. We have only
to construct the line of action of the resultant for each of two
arbitrary directions of the forces; the intersection of the two lines
gives the point required. The construction is neatest if the two
arbitrary directions are taken at right angles to one another.

§ 6. _Theory of Frames._--A _frame_ is a structure made up of pieces, or
_members_, each of which has two _joints_ connecting it with other
members. In a two-dimensional frame, each joint may be conceived as
consisting of a small cylindrical pin fitting accurately and smoothly
into holes drilled through the members which it connects. This
supposition is a somewhat ideal one, and is often only roughly
approximated to in practice. We shall suppose, in the first instance,
that extraneous forces act on the frame at the joints only, i.e. on the
pins.

On this assumption, the reactions on any member at its two joints must
be equal and opposite. This combination of equal and opposite forces is
called the _stress_ in the member; it may be a _tension_ or a _thrust_.
For diagrammatic purposes each member is sufficiently represented by a
straight line terminating at the two joints; these lines will be
referred to as the _bars_ of the frame.

[Illustration: FIG. 32.]

In structural applications a frame must be _stiff_, or _rigid_, i.e. it
must be incapable of deformation without alteration of length in at
least one of its bars. It is said to be _just rigid_ if it ceases to be
rigid when any one of its bars is removed. A frame which has more bars
than are essential for rigidity may be called _over-rigid_; such a frame
is in general self-stressed, i.e. it is in a state of stress
independently of the action of extraneous forces. A plane frame of n
joints which is just rigid (as regards deformation in its own plane) has
2n - 3 bars, for if one bar be held fixed the 2(n - 2) co-ordinates of
the remaining n - 2 joints must just be determined by the lengths of the
remaining bars. The total number of bars is therefore 2(n - 2) + 1. When
a plane frame which is just rigid is subject to a given system of
equilibrating extraneous forces (in its own plane) acting on the joints,
the stresses in the bars are in general uniquely determinate. For the
conditions of equilibrium of the forces on each pin furnish 2n
equations, viz. two for each point, which are linear in respect of the
stresses and the extraneous forces. This system of equations must
involve the three conditions of equilibrium of the extraneous forces
which are already identically satisfied, by hypothesis; there remain
therefore 2n - 3 independent relations to determine the 2n - 3 unknown
stresses. A frame of n joints and 2n - 3 bars may of course fail to be
rigid owing to some parts being over-stiff whilst others are deformable;
in such a case it will be found that the statical equations, apart from
the three identical relations imposed by the equilibrium of the
extraneous forces, are not all independent but are equivalent to less
than 2n - 3 relations. Another exceptional case, known as the _critical
case_, will be noticed later (§ 9).

A plane frame which can be built up from a single bar by successive
steps, at each of which a new joint is introduced by two new bars
meeting there, is called a _simple_ frame; it is obviously just rigid.
The stresses produced by extraneous forces in a simple frame can be
found by considering the equilibrium of the various joints in a proper
succession; and if the graphical method be employed the various polygons
of force can be combined into a single force-diagram. This procedure was
introduced by W. J. M. Rankine and J. Clerk Maxwell (1864). It may be
noticed that if we take an arbitrary pole in the force-diagram, and draw
a corresponding funicular in the skeleton diagram which represents the
frame together with the lines of action of the extraneous forces, we
obtain two complete reciprocal figures, in Maxwell's sense. It is
accordingly convenient to use Bow's notation (§ 5), and to distinguish
the several compartments of the frame-diagram by letters. See fig. 33,
where the successive triangles in the diagram of forces may be
constructed in the order XYZ, ZXA, AZB. The class of "simple" frames
includes many of the frameworks used in the construction of roofs,
lattice girders and suspension bridges; a number of examples will be
found in the article BRIDGES. By examining the senses in which the
respective forces act at each joint we can ascertain which members are
in tension and which are in thrust; in fig. 33 this is indicated by the
directions of the arrowheads.

[Illustration: FIG. 33.]

[Illustration: FIG. 34.]

When a frame, though just rigid, is not "simple" in the above sense, the
preceding method must be replaced, or supplemented, by one or other of
various artifices. In some cases the _method of sections_ is sufficient
for the purpose. If an ideal section be drawn across the frame, the
extraneous forces on either side must be in equilibrium with the forces
in the bars cut across; and if the section can be drawn so as to cut
only three bars, the forces in these can be found, since the problem
reduces to that of resolving a given force into three components acting
in three given lines (§ 4). The "critical case" where the directions of
the three bars are concurrent is of course excluded. Another method,
always available, will be explained under "Work" (§ 9).

  When extraneous forces act on the bars themselves the stress in each
  bar no longer consists of a simple longitudinal tension or thrust. To
  find the reactions at the joints we may proceed as follows. Each
  extraneous force W acting on a bar may be replaced (in an infinite
  number of ways) by two components P, Q in lines through the centres of
  the pins at the extremities. In practice the forces W are usually
  vertical, and the components P, Q are then conveniently taken to be
  vertical also. We first alter the problem by transferring the forces
  P, Q to the pins. The stresses in the bars, in the problem as thus
  modified, may be supposed found by the preceding methods; it remains
  to infer from the results thus obtained the reactions in the original
  form of the problem. To find the pressure exerted by a bar AB on the
  pin A we compound with the force in AB given by the diagram a force
  equal to P. Conversely, to find the pressure of the pin A on the bar
  AB we must compound with the force given by the diagram a force equal
  and opposite to P. This question arises in practice in the theory of
  "three-jointed" structures; for the purpose in hand such a structure
  is sufficiently represented by two bars AB, BC. The right-hand figure
  represents a portion of the force-diagram; in particular [->ZX]
  represents the pressure of AB on B in the modified problem where the
  loads W1 and W2 on the two bars are replaced by loads P1, Q1, and P2,
  Q2 respectively, acting on the pins. Compounding with this [->XV],
  which represents Q1, we get the actual pressure [->ZV] exerted by AB
  on B. The directions and magnitudes of the reactions at A and C are
  then easily ascertained. On account of its practical importance
  several other graphical solutions of this problem have been devised.

[Illustration: FIG. 35.]

§ 7. _Three-dimensional Kinematics of a Rigid Body._--The position of a
rigid body is determined when we know the positions of three points A,
B, C of it which are not collinear, for the position of any other point
P is then determined by the three distances PA, PB, PC. The nine
co-ordinates (Cartesian or other) of A, B, C are subject to the three
relations which express the invariability of the distances BC, CA, AB,
and are therefore equivalent to six independent quantities. Hence a
rigid body not constrained in any way is said to have six degrees of
freedom. Conversely, any six geometrical relations restrict the body in
general to one or other of a series of definite positions, none of which
can be departed from without violating the conditions in question. For
instance, the position of a theodolite is fixed by the fact that its
rounded feet rest in contact with six given plane surfaces. Again, a
rigid three-dimensional frame can be rigidly fixed relatively to the
earth by means of six links.

[Illustration: FIG. 36.]

[Illustration: FIG. 37.]

  The six independent quantities, or "co-ordinates," which serve to
  specify the position of a rigid body in space may of course be chosen
  in an endless variety of ways. We may, for instance, employ the three
  Cartesian co-ordinates of a particular point O of the body, and three
  angular co-ordinates which express the orientation of the body with
  respect to O. Thus in fig. 36, if OA, OB, OC be three mutually
  perpendicular lines in the solid, we may denote by [theta] the angle
  which OC makes with a fixed direction OZ, by [psi] the azimuth of the
  plane ZOC measured from some fixed plane through OZ, and by [phi] the
  inclination of the plane COA to the plane ZOC. In fig. 36 these
  various lines and planes are represented by their intersections with a
  unit sphere having O as centre. This very useful, although
  unsymmetrical, system of angular co-ordinates was introduced by L.
  Euler. It is exemplified in "Cardan's suspension," as used in
  connexion with a compass-bowl or a gyroscope. Thus in the gyroscope
  the "flywheel" (represented by the globe in fig. 37) can turn about a
  diameter OC of a ring which is itself free to turn about a diametral
  axis OX at right angles to the former; this axis is carried by a
  second ring which is free to turn about a fixed diameter OZ, which is
  at right angles to OX.

[Illustration: FIG. 10.]

We proceed to sketch the theory of the finite displacements of a rigid
body. It was shown by Euler (1776) that any displacement in which one
point O of the body is fixed is equivalent to a pure _rotation_ about
some axis through O. Imagine two spheres of equal radius with O as their
common centre, one fixed in the body and moving with it, the other fixed
in space. In any displacement about O as a fixed point, the former
sphere slides over the latter, as in a "ball-and-socket" joint. Suppose
that as the result of the displacement a point of the moving sphere is
brought from A to B, whilst the point which was at B is brought to C
(cf. fig. 10). Let J be the pole of the circle ABC (usually a "small
circle" of the fixed sphere), and join JA, JB, JC, AB, BC by
great-circle arcs. The spherical isosceles triangles AJB, BJC are
congruent, and we see that AB can be brought into the position BC by a
rotation about the axis OJ through an angle AJB.

[Illustration: FIG. 38.]

[Illustration: FIG. 39.]

It is convenient to distinguish the two senses in which rotation may
take place about an axis OA by opposite signs. We shall reckon a
rotation as positive when it is related to the direction from O to A as
the direction of rotation is related to that of translation in a
right-handed screw. Thus a negative rotation about OA may be regarded as
a positive rotation about OA´, the prolongation of AO. Now suppose that
a body receives first a positive rotation [alpha] about OA, and secondly
a positive rotation [beta] about OB; and let A, B be the intersections
of these axes with a sphere described about O as centre. If we construct
the spherical triangles ABC, ABC´ (fig. 38), having in each case the
angles at A and B equal to ½[alpha] and ½[beta] respectively, it is
evident that the first rotation will bring a point from C to C´ and that
the second will bring it back to C; the result is therefore equivalent
to a rotation about OC. We note also that if the given rotations had
been effected in the inverse order, the axis of the resultant rotation
would have been OC´, so that finite rotations do not obey the
"commutative law." To find the angle of the equivalent rotation, in the
actual case, suppose that the second rotation (about OB) brings a point
from A to A´. The spherical triangles ABC, A´BC (fig. 39) are
"symmetrically equal," and the angle of the resultant rotation, viz.
ACA´, is 2[pi] - 2C. This is equivalent to a negative rotation 2C about
OC, whence the theorem that the effect of three successive positive
rotations 2A, 2B, 2C about OA, OB, OC, respectively, is to leave the
body in its original position, provided the circuit ABC is left-handed
as seen from O. This theorem is due to O. Rodrigues (1840). The
composition of finite rotations about parallel axes is a particular case
of the preceding; the radius of the sphere is now infinite, and the
triangles are plane.

In any continuous motion of a solid about a fixed point O, the limiting
position of the axis of the rotation by which the body can be brought
from any one of its positions to a consecutive one is called the
_instantaneous axis_. This axis traces out a certain cone in the body,
and a certain cone in space, and the continuous motion in question may
be represented as consisting in a rolling of the former cone on the
latter. The proof is similar to that of the corresponding theorem of
plane kinematics (§ 3).

It follows from Euler's theorem that the most general displacement of a
rigid body may be effected by a pure translation which brings any one
point of it to its final position O, followed by a pure rotation about
some axis through O. Those planes in the body which are perpendicular to
this axis obviously remain parallel to their original positions. Hence,
if [sigma], [sigma]´ denote the initial and final positions of any
figure in one of these planes, the displacement could evidently have
been effected by (1) a translation perpendicular to the planes in
question, bringing [sigma] into some position [sigma]´´ in the plane of
[sigma]´, and (2) a rotation about a normal to the planes, bringing
[sigma]´´ into coincidence with [sigma] (§ 3). In other words, the most
general displacement is equivalent to a translation parallel to a
certain axis combined with a rotation about that axis; i.e. it may be
described as a _twist_ about a certain _screw_. In particular cases, of
course, the translation, or the rotation, may vanish.

  The preceding theorem, which is due to Michel Chasles (1830), may be
  proved in various other interesting ways. Thus if a point of the body
  be displaced from A to B, whilst the point which was at B is displaced
  to C, and that which was at C to D, the four points A, B, C, D lie on
  a helix whose axis is the common perpendicular to the bisectors of the
  angles ABC, BCD. This is the axis of the required screw; the amount of
  the translation is measured by the projection of AB or BC or CD on the
  axis; and the angle of rotation is given by the inclination of the
  aforesaid bisectors. This construction was given by M. W. Crofton.
  Again, H. Wiener and W. Burnside have employed the _half-turn_ (i.e. a
  rotation through two right angles) as the fundamental operation. This
  has the advantage that it is completely specified by the axis of the
  rotation, the sense being immaterial. Successive half-turns about
  parallel axes a, b are equivalent to a translation measured by double
  the distance between these axes in the direction from a to b.
  Successive half-turns about intersecting axes a, b are equivalent to a
  rotation about the common perpendicular to a, b at their intersection,
  of amount equal to twice the acute angle between them, in the
  direction from a to b. Successive half-turns about two skew axes a, b
  are equivalent to a twist about a screw whose axis is the common
  perpendicular to a, b, the translation being double the shortest
  distance, and the angle of rotation being twice the acute angle
  between a, b, in the direction from a to b. It is easily shown that
  any displacement whatever is equivalent to two half-turns and
  therefore to a screw.

[Illustration: FIG. 16.]

In mechanics we are specially concerned with the theory of infinitesimal
displacements. This is included in the preceding, but it is simpler in
that the various operations are commutative. An infinitely small
rotation about any axis is conveniently represented geometrically by a
length AB measures along the axis and proportional to the angle of
rotation, with the convention that the direction from A to B shall be
related to the rotation as is the direction of translation to that of
rotation in a right-handed screw. The consequent displacement of any
point P will then be at right angles to the plane PAB, its amount will
be represented by double the area of the triangle PAB, and its sense
will depend on the cyclical order of the letters P, A, B. If AB, AC
represent infinitesimal rotations about intersecting axes, the
consequent displacement of any point O in the plane BAC will be at right
angles to this plane, and will be represented by twice the sum of the
areas OAB, OAC, taken with proper signs. It follows by analogy with the
theory of moments (§ 4) that the resultant rotation will be represented
by AD, the vector-sum of AB, AC (see fig. 16). It is easily inferred as
a limiting case, or proved directly, that two infinitesimal rotations
[alpha], [beta] about parallel axes are equivalent to a rotation [alpha]
+ [beta] about a parallel axis in the same plane with the two former,
and dividing a common perpendicular AB in a point C so that AC/CB =
[beta]/[alpha]. If the rotations are equal and opposite, so that [alpha]
+ [beta] = 0, the point C is at infinity, and the effect is a
translation perpendicular to the plane of the two given axes, of amount
[alpha]·AB. It thus appears that an infinitesimal rotation is of the
nature of a "localized vector," and is subject in all respects to the
same mathematical laws as a force, conceived as acting on a rigid body.
Moreover, that an infinitesimal translation is analogous to a couple and
follows the same laws. These results are due to Poinsot.

The analytical treatment of small displacements is as follows. We first
suppose that one point O of the body is fixed, and take this as the
origin of a "right-handed" system of rectangular co-ordinates; i.e. the
positive directions of the axes are assumed to be so arranged that a
positive rotation of 90° about Ox would bring Oy into the position of
Oz, and so on. The displacement will consist of an infinitesimal
rotation [epsilon] about some axis through O, whose direction-cosines
are, say, l, m, n. From the equivalence of a small rotation to a
localized vector it follows that the rotation [epsilon] will be
equivalent to rotations [xi], [eta], [zeta] about Ox, Oy, Oz,
respectively, provided

  [xi] = l[epsilon], [eta] = m[epsilon], [zeta] = n[epsilon],  (1)

and we note that

  [xi]² + [eta]² + [zeta]² = [epsilon]².  (2)

  Thus in the case of fig. 36 it may be required to connect the
  infinitesimal rotations [xi], [eta], [zeta] about OA, OB, OC with the
  variations of the angular co-ordinates [theta], [psi], [phi]. The
  displacement of the point C of the body is made up of [delta][theta]
  tangential to the meridian ZC and sin [theta] [delta][psi]
  perpendicular to the plane of this meridian. Hence, resolving along
  the tangents to the arcs BC, CA, respectively, we have

    [xi] = [delta][theta] sin [phi] - sin [theta] [delta][psi] cos [phi],
    [eta] = [delta][theta] cos [phi] + sin [theta] [delta][psi] sin [phi].  (3)

  Again, consider the point of the solid which was initially at A´ in
  the figure. This is displaced relatively to A´ through a space
  [delta][psi] perpendicular to the plane of the meridian, whilst A´
  itself is displaced through a space cos [theta] [delta][psi] in the
  same direction. Hence

    [zeta] = [delta][phi] + cos [theta] [delta][psi].  (4)

[Illustration: FIG. 40.]

To find the component displacements of a point P of the body, whose
co-ordinates are x, y, z, we draw PL normal to the plane yOz, and LH, LK
perpendicular to Oy, Oz, respectively. The displacement of P parallel to
Ox is the same as that of L, which is made up of [eta]z and -[zeta]y. In
this way we obtain the formulae

  [delta]x = [eta]z - [zeta]y, [delta]y = [zeta]x - [xi]z, [delta]z = [xi]y - [eta]x.      (5)

The most general case is derived from this by adding the component
displacements [lambda], [mu], [nu] (say) of the point which was at O;
thus

  [delta]x = [lambda] + [eta]z  - [zeta]y,  \
  [delta]y = [mu]     + [zeta]x - [xi]z,     >  (6)
  [delta]z = [nu]     + [xi]y   - [eta]x.   /

The displacement is thus expressed in terms of the six independent
quantities [xi], [eta], [zeta], [lambda], [mu], [nu]. The points whose
displacements are in the direction of the resultant axis of rotation are
determined by [delta]x:[delta]y:[delta]z = [xi]:[eta]:[zeta], or

  ([lambda] + [eta]z - [zeta]y)/([xi] = [mu] + [zeta]x - [xi]z)/[eta] = ([nu] + [xi]y - [eta]x)/[zeta].  (7)

These are the equations of a straight line, and the displacement is in
fact equivalent to a twist about a screw having this line as axis. The
translation parallel to this axis is

  l[delta]x + m[delta]y + n[delta]z = ([lambda][xi] + [mu][eta] + [nu][zeta])/[epsilon].  (8)

The linear magnitude which measures the ratio of translation to rotation
in a screw is called the _pitch_. In the present case the pitch is

  ([lambda][xi] + [mu][eta] + [nu][zeta])/([xi]² + [eta]² + [zeta]²).  (9)

Since [xi]² + [eta]² + [zeta]², or [epsilon]², is necessarily an
absolute invariant for all transformations of the (rectangular)
co-ordinate axes, we infer that [lambda][xi] + [mu][eta] + [nu][zeta] is
also an absolute invariant. When the latter invariant, but not the
former, vanishes, the displacement is equivalent to a pure rotation.

  If the small displacements of a rigid body be subject to one
  constraint, e.g. if a point of the body be restricted to lie on a
  given surface, the mathematical expression of this fact leads to a
  homogeneous linear equation between the infinitesimals [xi], [eta],
  [zeta], [lambda], [mu], [nu], say

    A[xi] + B[eta] + C[zeta] + F[lambda] + G[mu] + H[nu] = 0.  (10)

  The quantities [xi], [eta], [zeta], [lambda], [mu], [nu] are no longer
  independent, and the body has now only five degrees of freedom. Every
  additional constraint introduces an additional equation of the type
  (10) and reduces the number of degrees of freedom by one. In Sir R. S.
  Ball's _Theory of Screws_ an analysis is made of the possible
  displacements of a body which has respectively two, three, four, five
  degrees of freedom. We will briefly notice the case of two degrees,
  which involves an interesting generalization of the method (already
  explained) of compounding rotations about intersecting axes. We assume
  that the body receives arbitrary twists about two given screws, and
  it is required to determine the character of the resultant
  displacement. We examine first the case where the axes of the two
  screws are at right angles and intersect. We take these as axes of x
  and y; then if [xi], [eta] be the component rotations about them, we
  have

    [lambda] = h[xi], [mu] = k[eta], [nu] = 0,  (11)

  where h, k, are the pitches of the two given screws. The equations (7)
  of the axis of the resultant screw then reduce to

    x/[xi] = y/[eta], z([xi]² + [eta]²) = (k - h)[xi][eta].  (12)

  Hence, whatever the ratio [xi] : [eta], the axis of the resultant
  screw lies on the conoidal surface

   z(x² + y²) = cxy,  (13)

  where c = ½(k - h). The co-ordinates of any point on (13) may be
  written

   x = r cos [theta], y = r sin [theta], z = c sin 2[theta];  (14)

  hence if we imagine a curve of sines to be traced on a circular
  cylinder so that the circumference just includes two complete
  undulations, a straight line cutting the axis of the cylinder at right
  angles and meeting this curve will generate the surface. This is
  called a _cylindroid_. Again, the pitch of the resultant screw is

    p = ([lambda][xi] + [mu][eta])/([xi]² + [eta]²) = h cos² [theta] + k sin² [theta].  (15)

  [Illustration: From Sir Robert S. Ball's _Theory of Screws_.

  FIG. 41.]

  The distribution of pitch among the various screws has therefore a
  simple relation to the _pitch-conic_

    hx² + ky² = const;  (16)

  viz. the pitch of any screw varies inversely as the square of that
  diameter of the conic which is parallel to its axis. It is to be
  noticed that the parameter c of the cylindroid is unaltered if the two
  pitches h, k be increased by equal amounts; the only change is that
  all the pitches are increased by the same amount. It remains to show
  that a system of screws of the above type can be constructed so as to
  contain any two given screws whatever. In the first place, a
  cylindroid can be constructed so as to have its axis coincident with
  the common perpendicular to the axes of the two given screws and to
  satisfy three other conditions, for the position of the centre, the
  parameter, and the orientation about the axis are still at our
  disposal. Hence we can adjust these so that the surface shall contain
  the axes of the two given screws as generators, and that the
  difference of the corresponding pitches shall have the proper value.
  It follows that when a body has two degrees of freedom it can twist
  about any one of a singly infinite system of screws whose axes lie on
  a certain cylindroid. In particular cases the cylindroid may
  degenerate into a plane, the pitches being then all equal.

§ 8. _Three-dimensional Statics._--A system of parallel forces can be
combined two and two until they are replaced by a single resultant equal
to their sum, acting in a certain line. As special cases, the system may
reduce to a couple, or it may be in equilibrium.

In general, however, a three-dimensional system of forces cannot be
replaced by a single resultant force. But it may be reduced to simpler
elements in a variety of ways. For example, it may be reduced to two
forces in perpendicular skew lines. For consider any plane, and let each
force, at its intersection with the plane, be resolved into two
components, one (P) normal to the plane, the other (Q) in the plane. The
assemblage of parallel forces P can be replaced in general by a single
force, and the coplanar system of forces Q by another single force.

If the plane in question be chosen perpendicular to the direction of the
vector-sum of the given forces, the vector-sum of the components Q is
zero, and these components are therefore equivalent to a couple (§ 4).
Hence any three-dimensional system can be reduced to a single force R
acting in a certain line, together with a couple G in a plane
perpendicular to the line. This theorem was first given by L. Poinsot,
and the line of action of R was called by him the _central axis_ of the
system. The combination of a force and a couple in a perpendicular plane
is termed by Sir R. S. Ball a _wrench_. Its type, as distinguished from
its absolute magnitude, may be specified by a screw whose axis is the
line of action of R, and whose pitch is the ratio G/R.

[Illustration: FIG. 42.]

  The case of two forces may be specially noticed. Let AB be the
  shortest distance between the lines of action, and let AA´, BB´ (fig.
  42) represent the forces. Let [alpha], [beta] be the angles which AA´,
  BB´ make with the direction of the vector-sum, on opposite sides.
  Divide AB in O, so that

    AA´·cos [alpha]·AO = BB´·cos [beta]·OB,  (1)

  and draw OC parallel to the vector-sum. Resolving AA´, BB´ each into
  two components parallel and perpendicular to OC, we see that the
  former components have a single resultant in OC, of amount

    R = AA´ cos [alpha] + BB´ cos [beta],  (2)

  whilst the latter components form a couple of moment

    G = AA´·AB·sin [alpha] = BB´·AB·sin [beta].  (3)

  Conversely it is seen that any wrench can be replaced in an infinite
  number of ways by two forces, and that the line of action of one of
  these may be chosen quite arbitrarily. Also, we find from (2) and (3)
  that

    G·R = AA´·BB´·AB·sin ([alpha] + [beta]).  (4)

  The right-hand expression is six times the volume of the tetrahedron
  of which the lines AA´, BB´ representing the forces are opposite
  edges; and we infer that, in whatever way the wrench be resolved into
  two forces, the volume of this tetrahedron is invariable.

To define the _moment_ of a force _about an axis_ HK, we project the
force orthogonally on a plane perpendicular to HK and take the moment of
the projection about the intersection of HK with the plane (see § 4).
Some convention as to sign is necessary; we shall reckon the moment to
be positive when the tendency of the force is right-handed as regards
the direction from H to K. Since two concurrent forces and their
resultant obviously project into two concurrent forces and their
resultant, we see that the sum of the moments of two concurrent forces
about any axis HK is equal to the moment of their resultant. Parallel
forces may be included in this statement as a limiting case. Hence, in
whatever way one system of forces is by successive steps replaced by
another, no change is made in the sum of the moments about any assigned
axis. By means of this theorem we can show that the previous reduction
of any system to a wrench is unique.

From the analogy of couples to translations which was pointed out in §
7, we may infer that a couple is sufficiently represented by a "free"
(or non-localized) vector perpendicular to its plane. The length of the
vector must be proportional to the moment of the couple, and its sense
must be such that the sum of the moments of the two forces of the couple
about it is positive. In particular, we infer that couples of the same
moment in parallel planes are equivalent; and that couples in any two
planes may be compounded by geometrical addition of the corresponding
vectors. Independent statical proofs are of course easily given. Thus,
let the plane of the paper be perpendicular to the planes of two
couples, and therefore perpendicular to the line of intersection of
these planes. By § 4, each couple can be replaced by two forces ± P
(fig. 43) perpendicular to the plane of the paper, and so that one force
of each couple is in the line of intersection (B); the arms (AB, BC)
will then be proportional to the respective moments. The two forces at B
will cancel, and we are left with a couple of moment P · AC in the plane
AC. If we draw three vectors to represent these three couples, they will
be perpendicular and proportional to the respective sides of the
triangle ABC; hence the third vector is the geometric sum of the other
two. Since, in this proof the magnitude of P is arbitrary, It follows
incidentally that couples of the same moment in parallel planes, e.g.
planes parallel to AC, are equivalent.

[Illustration: FIG. 43.]

[Illustration: FIG. 44.]

Hence a couple of moment G, whose axis has the direction (l, m, n)
relative to a right-handed system of rectangular axes, is equivalent to
three couples lG, mG, nG in the co-ordinate planes. The analytical
reduction of a three-dimensional system can now be conducted as follows.
Let (x1, y1, z1) be the co-ordinates of a point P1 on the line of action
of one of the forces, whose components are (say) X1, Y1, Z1. Draw P1H
normal to the plane zOx, and HK perpendicular to Oz. In KH introduce two
equal and opposite forces ± X1. The force X1 at P1 with -X1 in KH forms
a couple about Oz, of moment -y1X1. Next, introduce along Ox two equal
and opposite forces ±X1. The force X1 in KH with -X1 in Ox forms a
couple about Oy, of moment z1X1. Hence the force X1 can be transferred
from P1 to O, provided we introduce couples of moments z1X1 about Oy and
-y1X1, about Oz. Dealing in the same way with the forces Y1, Z1 at P1,
we find that all three components of the force at P1 can be transferred
to O, provided we introduce three couples L1, M1, N1 about Ox, Oy, Oz
respectively, viz.

  L1 = y1Z1 - z1Y1,  M1 = z1X1 - x1Z1,  N1 = x1Y1 - y1X1.  (5)

It is seen that L1, M1, N1 are the moments of the original force at P1
about the co-ordinate axes. Summing up for all the forces of the given
system, we obtain a force R at O, whose components are

  X = [Sigma](X_r),  Y = [Sigma](Y_r),  Z = [Sigma](Z_r),  (6)

and a couple G whose components are

  L = [Sigma](L_r),  M = [Sigma](M_r),  N = [Sigma](N_r),  (7)

where r= 1, 2, 3 ... Since R² = X² + Y² + Z², G² = L² + M² + N², it is
necessary and sufficient for equilibrium that the six quantities X, Y,
Z, L, M, N, should all vanish. In words: the sum of the projections of
the forces on each of the co-ordinate axes must vanish; and, the sum of
the moments of the forces about each of these axes must vanish.

If any other point O´, whose co-ordinates are x, y, z, be chosen in
place of O, as the point to which the forces are transferred, we have to
write x1 - x, y1 - y, z1 - z for x1, y1, z1, and so on, in the preceding
process. The components of the resultant force R are unaltered, but the
new components of couple are found to be

  L´ = L - yZ + zY,  \
  M´ = M - zX + xZ,   >  (8)
  N´ = N - xY + yX.  /

By properly choosing O´ we can make the plane of the couple
perpendicular to the resultant force. The conditions for this are L´ :
M´ : N´ = X : Y : Z, or

  L - yZ + zY   M - zX + xZ   N - xY + yX
  ----------- = ----------- = -----------  (9)
       X             Y             Z

These are the equations of the central axis. Since the moment of the
resultant couple is now

        X         Y        Z       LX + MY + NZ
  G´ = --- L´ +  --- M´ + --- N´ = ------------,  (10)
        R         R        R             R

the pitch of the equivalent wrench is

  (LX + MY + NZ)/(X² + Y² + Z²).

It appears that X² + Y² + Z² and LX + MY + NZ are absolute invariants
(cf. § 7). When the latter invariant, but not the former, vanishes, the
system reduces to a single force.

The analogy between the mathematical relations of infinitely small
displacements on the one hand and those of force-systems on the other
enables us immediately to convert any theorem in the one subject into a
theorem in the other. For example, we can assert without further proof
that any infinitely small displacement may be resolved into two
rotations, and that the axis of one of these can be chosen arbitrarily.
Again, that wrenches of arbitrary amounts about two given screws
compound into a wrench the locus of whose axis is a cylindroid.

  The mathematical properties of a twist or of a wrench have been the
  subject of many remarkable investigations, which are, however, of
  secondary importance from a physical point of view. In the
  "Null-System" of A. F. Möbius (1790-1868), a line such that the moment
  of a given wrench about it is zero is called a _null-line_. The triply
  infinite system of null-lines form what is called in line-geometry a
  "complex." As regards the configuration of this complex, consider a
  line whose shortest distance from the central axis is r, and whose
  inclination to the central axis is [theta]. The moment of the
  resultant force R of the wrench about this line is - Rr sin [theta],
  and that of the couple G is G cos [theta]. Hence the line will be a
  null-line provided

    tan [theta] = k/r,  (11)

  where k is the pitch of the wrench. The null-lines which are at a
  given distance r from a point O of the central axis will therefore
  form one system of generators of a hyperboloid of revolution; and by
  varying r we get a series of such hyperboloids with a common centre
  and axis. By moving O along the central axis we obtain the whole
  complex of null-lines. It appears also from (11) that the null-lines
  whose distance from the central axis is r are tangent lines to a
  system of helices of slope tan^-1 (r/k); and it is to be noticed that
  these helices are left-handed if the given wrench is right-handed, and
  vice versa.

  Since the given wrench can be replaced by a force acting through any
  assigned point P, and a couple, the locus of the null-lines through P
  is a plane, viz. a plane perpendicular to the vector which represents
  the couple. The complex is therefore of the type called "linear" (in
  relation to the degree of this locus). The plane in question is called
  the _null-plane_ of P. If the null-plane of P pass through Q, the
  null-plane of Q will pass through P, since PQ is a null-line. Again,
  any plane [omega] is the locus of a system of null-lines meeting in a
  point, called the _null-point_ of [omega]. If a plane revolve about a
  fixed straight line p in it, its null-point describes another straight
  line p´, which is called the _conjugate line_ of p. We have seen that
  the wrench may be replaced by two forces, one of which may act in any
  arbitrary line p. It is now evident that the second force must act in
  the conjugate line p´, since every line meeting p, p´ is a null-line.
  Again, since the shortest distance between any two conjugate lines
  cuts the central axis at right angles, the orthogonal projections of
  two conjugate lines on a plane perpendicular to the central axis will
  be parallel (fig. 42). This property was employed by L. Cremona to
  prove the existence under certain conditions of "reciprocal figures"
  in a plane (§ 5). If we take any polyhedron with plane faces, the
  null-planes of its vertices with respect to a given wrench will form
  another polyhedron, and the edges of the latter will be conjugate (in
  the above sense) to those of the former. Projecting orthogonally on a
  plane perpendicular to the central axis we obtain two reciprocal
  figures.

  In the analogous theory of infinitely small displacements of a solid,
  a "null-line" is a line such that the lengthwise displacement of any
  point on it is zero.

  Since a wrench is defined by six independent quantities, it can in
  general be replaced by any system of forces which involves six
  adjustable elements. For instance, it can in general be replaced by
  six forces acting in six given lines, e.g. in the six edges of a given
  tetrahedron. An exception to the general statement occurs when the six
  lines are such that they are possible lines of action of a system of
  six forces in equilibrium; they are then said to be _in involution_.
  The theory of forces in involution has been studied by A. Cayley, J.
  J. Sylvester and others. We have seen that a rigid structure may in
  general be rigidly connected with the earth by six links, and it now
  appears that any system of forces acting on the structure can in
  general be balanced by six determinate forces exerted by the links.
  If, however, the links are in involution, these forces become infinite
  or indeterminate. There is a corresponding kinematic peculiarity, in
  that the connexion is now not strictly rigid, an infinitely small
  relative displacement being possible. See § 9.

When parallel forces of given magnitudes act at given points, the
resultant acts through a definite point, or _centre of parallel forces_,
which is independent of the special direction of the forces. If P_r be
the force at (x_r, y_r, z_r), acting in the direction (l, m, n), the
formulae (6) and (7) reduce to

  X = [Sigma](P).l,  Y = [Sigma](P).m,  Z = [Sigma](P).n,  (12)

and

  L = [Sigma](P)·(n[|y] - m[|z]), M = [Sigma](P)·(l[|z] - n[|x]), N = [Sigma](P)·(m[|x] - l[|y]),  (13)

provided

         [Sigma](Px)         [Sigma](Py)         [Sigma](Pz)
  [|x] = -----------, [|y] = -----------, [|z] = -----------.  (14)
         [Sigma](P)          [Sigma](P)          [Sigma](P)

These are the same as if we had a single force [Sigma](P) acting at the
point ([|x], [|y], [|z]), which is the same for all directions (l, m,
n). We can hence derive the theory of the centre of gravity, as in § 4.
An exceptional case occurs when [Sigma](P) = 0.

  If we imagine a rigid body to be acted on at given points by forces of
  given magnitudes in directions (not all parallel) which are fixed in
  space, then as the body is turned about the resultant wrench will
  assume different configurations in the body, and will in certain
  positions reduce to a single force. The investigation of such
  questions forms the subject of "Astatics," which has been cultivated
  by Möbius, Minding, G. Darboux and others. As it has no physical
  bearing it is passed over here.

[Illustration: FIG. 45.]

§ 9. _Work._--The _work_ done by a force acting on a particle, in any
infinitely small displacement, is defined as the product of the force
into the orthogonal projection of the displacement on the direction of
the force; i.e. it is equal to F·[delta]s cos [theta], where F is the
force, [delta]s the displacement, and [theta] is the angle between the
directions of F and [delta]s. In the language of vector analysis (q.v.)
it is the "scalar product" of the vector representing the force and the
displacement. In the same way, the work done by a force acting on a
rigid body in any infinitely small displacement of the body is the
scalar product of the force into the displacement of any point on the
line of action. This product is the same whatever point on the line of
action be taken, since the lengthwise components of the displacements of
any two points A, B on a line AB are equal, to the first order of small
quantities. To see this, let A´, B´ be the displaced positions of A, B,
and let [phi] be the infinitely small angle between AB and A´B´. Then if
[alpha], [beta] be the orthogonal projections of A´, B´ on AB, we have

  A[alpha] - B[beta] = AB - [alpha][beta] = AB(1 - cos [phi]) = ½AB·[phi]²,

ultimately. Since this is of the second order, the products F·A[alpha]
and F·B[beta] are ultimately equal.

[Illustration: FIG. 46.]

[Illustration: FIG. 47.]

The total work done by two concurrent forces acting on a particle, or on
a rigid body, in any infinitely small displacement, is equal to the work
of their resultant. Let AB, AC (fig. 46) represent the forces, AD their
resultant, and let AH be the direction of the displacement [delta]s of
the point A. The proposition follows at once from the fact that the sum
of orthogonal projections of [->AB], [->AC] on AH is equal to the
projection of [->AD]. It is to be noticed that AH need not be in the
same plane with AB, AC.

It follows from the preceding statements that any two systems of forces
which are statically equivalent, according to the principles of §§ 4, 8,
will (to the first order of small quantities) do the same amount of work
in any infinitely small displacement of a rigid body to which they may
be applied. It is also evident that the total work done in two or more
successive infinitely small displacements is equal to the work done in
the resultant displacement.

The work of a couple in any infinitely small rotation of a rigid body
about an axis perpendicular to the plane of the couple is equal to the
product of the moment of the couple into the angle of rotation, proper
conventions as to sign being observed. Let the couple consist of two
forces P, P (fig. 47) in the plane of the paper, and let J be the point
where this plane is met by the axis of rotation. Draw JBA perpendicular
to the lines of action, and let [epsilon] be the angle of rotation. The
work of the couple is

  P·JA·[epsilon] - P·JB·[epsilon] = P·AB·[epsilon] = G[epsilon],

if G be the moment of the couple.

The analytical calculation of the work done by a system of forces in any
infinitesimal displacement is as follows. For a two-dimensional system
we have, in the notation of §§ 3, 4,

  [Sigma](X[delta]x + Y[delta]y) = [Sigma]{X([lambda] - y[epsilon]) + Y([mu] + x[epsilon])}
                                 = [Sigma](X)·[lambda] + [Sigma](Y)·[mu] + [Sigma](xY - yX)[epsilon]
                                 = X[lambda] + Y[mu] + N[epsilon].      (1)

Again, for a three-dimensional system, in the notation of §§ 7, 8,

  [Sigma](X[delta]x + Y[delta]y + Z[delta]z)
  = [Sigma]{(X([lambda] + [eta]z - [zeta]y) + Y([mu] + [zeta]x - [xi]x) + Z([nu] + [xi]y - [eta]x)}
  = [Sigma](X)·[lambda] + [Sigma](Y)·[mu] + [Sigma](Z)·[nu] + [Sigma](yZ - zY)·[xi]
     + [Sigma](zX - xZ)·[eta] + [Sigma](xY - yX)·[zeta]
  = X[lambda] + Y[mu] + Z[nu] + L[xi] + M[eta] + N[zeta].  (2)

This expression gives the work done by a given wrench when the body
receives a given infinitely small twist; it must of course be an
absolute invariant for all transformations of rectangular axes. The
first three terms express the work done by the components of a force (X,
Y, Z) acting at O, and the remaining three terms express the work of a
couple (L, M, N).

[Illustration: FIG. 48.]

  The work done by a wrench about a given screw, when the body twists
  about a second given screw, may be calculated directly as follows. In
  fig. 48 let R, G be the force and couple of the wrench,
  [epsilon],[tau] the rotation and translation in the twist. Let the
  axes of the wrench and the twist be inclined at an angle [theta], and
  let h be the shortest distance between them. The displacement of the
  point H in the figure, resolved in the direction of R, is [tau] cos
  [theta] - [epsilon]h sin [theta]. The work is therefore

    R([tau] cos [theta] - [epsilon]h sin [theta]) + G cos [theta]
      = R[epsilon]{(p + p´) cos [theta] - h sin [theta]},  (3)

  if G = pR, [tau] = p´[epsilon], i.e. p, p´ are the pitches of the two
  screws. The factor (p + p´) cos[theta] - h sin[theta] is called the
  _virtual coefficient_ of the two screws which define the types of the
  wrench and twist, respectively.

  A screw is determined by its axis and its pitch, and therefore
  involves five Independent elements. These may be, for instance, the
  five ratios [xi]:[eta]:[zeta]:[lambda]:[mu]:[nu] of the six quantities
  which specify an infinitesimal twist about the screw. If the twist is
  a pure rotation, these quantities are subject to the relation

    [lambda][xi] + [mu][eta] + [nu][zeta] = 0.  (4)

  In the analytical investigations of line geometry, these six
  quantities, supposed subject to the relation (4), are used to specify
  a line, and are called the six "co-ordinates" of the line; they are of
  course equivalent to only four independent quantities. If a line is a
  null-line with respect to the wrench (X, Y, Z, L, M, N), the work done
  in an infinitely small rotation about it is zero, and its co-ordinates
  are accordingly subject to the further relation

    L[xi] + M[eta] + N[zeta] + X[lambda] + Y[mu] + Z[nu] = 0,  (5)

  where the coefficients are constant. This is the equation of a "linear
  complex" (cf. § 8).

  Two screws are _reciprocal_ when a wrench about one does no work on a
  body which twists about the other. The condition for this is

    [lambda][xi]´ + [mu][eta]´ + [nu][zeta]´ + [lambda]´[xi] + [mu]´[eta] + [nu]´[zeta] = 0,  (6)

  if the screws be defined by the ratios [xi] : [eta] : [zeta] :
  [lambda] : [mu] : [nu] and [xi]´ : [eta]´ : [zeta]´ : [lambda]´ :
  [mu]´ : [nu]´, respectively. The theory of the screw-systems which are
  reciprocal to one, two, three, four given screws respectively has been
  investigated by Sir R. S. Ball.

Considering a rigid body in any given position, we may contemplate the
whole group of infinitesimal displacements which might be given to it.
If the extraneous forces are in equilibrium the total work which they
would perform in any such displacement would be zero, since they reduce
to a zero force and a zero couple. This is (in part) the celebrated
principle of _virtual velocities_, now often described as the principle
of _virtual work_, enunciated by John Bernoulli (1667-1748). The word
"virtual" is used because the displacements in question are not regarded
as actually taking place, the body being in fact at rest. The
"velocities" referred to are the velocities of the various points of the
body in any imagined motion of the body through the position in
question; they obviously bear to one another the same ratios as the
corresponding infinitesimal displacements. Conversely, we can show that
if the virtual work of the extraneous forces be zero for every
infinitesimal displacement of the body as rigid, these forces must be in
equilibrium. For by giving the body (in imagination) a displacement of
translation we learn that the sum of the resolved parts of the forces in
any assigned direction is zero, and by giving it a displacement of pure
rotation we learn that the sum of the moments about any assigned axis is
zero. The same thing follows of course from the analytical expression
(2) for the virtual work. If this vanishes for all values of [lambda],
[mu], [nu], [xi], [eta], [zeta] we must have X, Y, Z, L, M, N = 0, which
are the conditions of equilibrium.

The principle can of course be extended to any system of particles or
rigid bodies, connected together in any way, provided we take into
account the internal stresses, or reactions, between the various parts.
Each such reaction consists of two equal and opposite forces, both of
which may contribute to the equation of virtual work.

The proper significance of the principle of virtual work, and of its
converse, will appear more clearly when we come to kinetics (§ 16); for
the present it may be regarded merely as a compact and (for many
purposes) highly convenient summary of the laws of equilibrium. Its
special value lies in this, that by a suitable adjustment of the
hypothetical displacements we are often enabled to eliminate unknown
reactions. For example, in the case of a particle lying on a smooth
curve, or on a smooth surface, if it be displaced along the curve, or on
the surface, the virtual work of the normal component of the pressure
may be ignored, since it is of the second order. Again, if two bodies
are connected by a string or rod, and if the hypothetical displacements
be adjusted so that the distance between the points of attachment is
unaltered, the corresponding stress may be ignored. This is evident from
fig. 45; if AB, A´B´ represent the two positions of a string, and T be
the tension, the virtual work of the two forces ±T at A, B is T(A[alpha]
- B[beta]), which was shown to be of the second order. Again, the normal
pressure between two surfaces disappears from the equation, provided the
displacements be such that one of these surfaces merely slides
relatively to the other. It is evident, in the first place, that in any
displacement common to the two surfaces, the work of the two equal and
opposite normal pressures will cancel; moreover if, one of the surfaces
being fixed, an infinitely small displacement shifts the point of
contact from A to B, and if A´ be the new position of that point of the
sliding body which was at A, the projection of AA´ on the normal at A is
of the second order. It is to be noticed, in this case, that the
tangential reaction (if any) between the two surfaces is not eliminated.
Again, if the displacements be such that one curved surface rolls
without sliding on another, the reaction, whether normal or tangential,
at the point of contact may be ignored. For the virtual work of two
equal and opposite forces will cancel in any displacement which is
common to the two surfaces; whilst, if one surface be fixed, the
displacement of that point of the rolling surface which was in contact
with the other is of the second order. We are thus able to imagine a
great variety of mechanical systems to which the principle of virtual
work can be applied without any regard to the internal stresses,
provided the hypothetical displacements be such that none of the
connexions of the system are violated.

If the system be subject to gravity, the corresponding part of the
virtual work can be calculated from the displacement of the centre of
gravity. If W1, W2, ... be the weights of a system of particles, whose
depths below a fixed horizontal plane of reference are z1, z2, ...,
respectively, the virtual work of gravity is

  W1[delta]·z1 + W2[delta]z2 + ... = [delta](W1z1 + W2z2 + ...)  (7)
                                   = (W1 + W2 + ...) [delta][|z],

where [|z] is the depth of the centre of gravity (see § 8 (14) and § 11
(6)). This expression is the same as if the whole mass were concentrated
at the centre of gravity, and displaced with this point. An important
conclusion is that in any displacement of a system of bodies in
equilibrium, such that the virtual work of all forces except gravity may
be ignored, the depth of the centre of gravity is "stationary."

The question as to stability of equilibrium belongs essentially to
kinetics; but we may state by anticipation that in cases where gravity
is the only force which does work, the equilibrium of a body or system
of bodies is stable only if the depth of the centre of gravity be a
maximum.

[Illustration: FIG. 49.]

  Consider, for instance, the case of a bar resting with its ends on two
  smooth inclines (fig. 18). If the bar be displaced in a vertical plane
  so that its ends slide on the two inclines, the instantaneous centre
  is at the point J. The displacement of G is at right angles to JG;
  this shows that for equilibrium JG must be vertical. Again, the locus
  of G is an arc of an ellipse whose centre is in the intersection of
  the planes; since this arc is convex upwards the equilibrium is
  unstable. A general criterion for the case of a rigid body movable in
  two dimensions, with one degree of freedom, can be obtained as
  follows. We have seen (§ 3) that the sequence of possible positions is
  obtained if we imagine the "body-centrode" to roll on the
  "space-centrode." For equilibrium, the altitude of the centre of
  gravity G must be stationary; hence G must lie in the same vertical
  line with the point of contact J of the two curves. Further, it is
  known from the theory of "roulettes" that the locus of G will be
  concave or convex upwards according as

    cos[phi]     1       1
    -------- = ----- + ------,  (8)
       h       [rho]   [rho]´

  where [rho], [rho]´ are the radii of curvature of the two curves at J,
  [phi] is the inclination of the common tangent at J to the horizontal,
  and h is the height of G above J. The signs of [rho], [rho]´ are to be
  taken positive when the curvatures are as in the standard case shown
  in fig. 49. Hence for stability the upper sign must obtain in (8). The
  same criterion may be arrived at in a more intuitive manner as
  follows. If the body be supposed to roll (say to the right) until the
  curves touch at J´, and if JJ´ = [delta]s, the angle through which the
  upper figure rotates is [delta]s/[rho] + [delta]s/[rho]´, and the
  horizontal displacement of G is equal to the product of this
  expression into h. If this displacement be less than the horizontal
  projection of JJ´, viz. [delta]s cos[phi], the vertical through the
  new position of G will fall to the left of J´ and gravity will tend to
  restore the body to its former position. It is here assumed that the
  remaining forces acting on the body in its displaced position have
  zero moment about J´; this is evidently the case, for instance, in the
  problem of "rocking stones."

The principle of virtual work is specially convenient in the theory of
frames (§ 6), since the reactions at smooth joints and the stresses in
inextensible bars may be left out of account. In particular, in the case
of a frame which is just rigid, the principle enables us to find the
stress in any one bar independently of the rest. If we imagine the bar
in question to be removed, equilibrium will still persist if we
introduce two equal and opposite forces S, of suitable magnitude, at the
joints which it connected. In any infinitely small deformation of the
frame as thus modified, the virtual work of the forces S, together with
that of the original extraneous forces, must vanish; this determines S.

  As a simple example, take the case of a light frame, whose bars form
  the slides of a rhombus ABCD with the diagonal BD, suspended from A
  and carrying a weight W at C; and let it be required to find the
  stress in BD. If we remove the bar BD, and apply two equal and
  opposite forces S at B and D, the equation is

    W·[delta](2l cos[theta]) + 2S·[delta](l sin [theta]) = 0,

  where l is the length of a side of the rhombus, and [theta] its
  inclination to the vertical. Hence

    S = W tan [theta] = W·BD/AC.  (8)

  [Illustration: FIG. 50.]

  The method is specially appropriate when the frame, although just
  rigid, is not "simple" in the sense of § 6, and when accordingly the
  method of reciprocal figures is not immediately available. To avoid
  the intricate trigonometrical calculations which would often be
  necessary, graphical devices have been introduced by H. Müller-Breslau
  and others. For this purpose the infinitesimal displacements of the
  various joints are replaced by finite lengths proportional to them,
  and therefore proportional to the velocities of the joints in some
  imagined motion of the deformable frame through its actual
  configuration; this is really (it may be remarked) a reversion to the
  original notion of "virtual velocities." Let J be the instantaneous
  centre for any bar CD (fig. 12), and let s1, s2 represent the virtual
  velocities of C, D. If these lines be turned through a right angle in
  the same sense, they take up positions such as CC´, DD´, where C´, D´
  are on JC, JD, respectively, and C´D´ is parallel to CD. Further, if
  F1 (fig. 51) be any force acting on the joint C, its virtual work will
  be equal to the moment of F1 about C´; the equation of virtual work is
  thus transformed into an equation of moments.

  [Illustration: FIG. 12.]

  [Illustration: FIG. 51.]

  [Illustration: FIG. 52.]

  Consider, for example, a frame whose sides form the six sides of a
  hexagon ABCDEF and the three diagonals AD, BE, CF; and suppose that it
  is required to find the stress in CF due to a given system of
  extraneous forces in equilibrium, acting on the joints. Imagine the
  bar CF to be removed, and consider a deformation in which AB is fixed.
  The instantaneous centre of CD will be at the intersection of AD, BC,
  and if C´D´ be drawn parallel to CD, the lines CC´, DD´ may be taken
  to represent the virtual velocities of C, D turned each through a
  right angle. Moreover, if we draw D´E´ parallel to DE, and E´F´
  parallel to EF, the lines CC´, DD´, EE´, FF´ will represent on the
  same scale the virtual velocities of the points C, D, E, F,
  respectively, turned each through a right angle. The equation of
  virtual work is then formed by taking moments about C´, D´, E´, F´ of
  the extraneous forces which act at C, D, E, F, respectively. Amongst
  these forces we must include the two equal and opposite forces S which
  take the place of the stress in the removed bar FC.

  The above method lends itself naturally to the investigation of the
  _critical forms_ of a frame whose general structure is given. We have
  seen that the stresses produced by an equilibrating system of
  extraneous forces in a frame which is just rigid, according to the
  criterion of § 6, are in general uniquely determinate; in particular,
  when there are no extraneous forces the bars are in general free from
  stress. It may however happen that owing to some special relation
  between the lengths of the bars the frame admits of an infinitesimal
  deformation. The simplest case is that of a frame of three bars, when
  the three joints A, B, C fall into a straight line; a small
  displacement of the joint B at right angles to AC would involve
  changes in the lengths of AB, BC which are only of the second order of
  small quantities. Another example is shown in fig. 53. The graphical
  method leads at once to the detection of such cases. Thus in the
  hexagonal frame of fig. 52, if an infinitesimal deformation is
  possible without removing the bar CF, the instantaneous centre of CF
  (when AB is fixed) will be at the intersection of AF and BC, and since
  CC´, FF´ represent the virtual velocities of the points C, F, turned
  each through a right angle, C´F´ must be parallel to CF. Conversely,
  if this condition be satisfied, an infinitesimal deformation is
  possible. The result may be generalized into the statement that a
  frame has a critical form whenever a frame of the same structure can
  be designed with corresponding bars parallel, but without complete
  geometric similarity. In the case of fig. 52 it may be shown that an
  equivalent condition is that the six points A, B, C, D, E, F should
  lie on a conic (M. W. Crofton). This is fulfilled when the opposite
  sides of the hexagon are parallel, and (as a still more special case)
  when the hexagon is regular.

  [Illustration: FIG. 53.]

  When a frame has a critical form it may be in a state of stress
  independently of the action of extraneous forces; moreover, the
  stresses due to extraneous forces are indeterminate, and may be
  infinite. For suppose as before that one of the bars is removed. If
  there are no extraneous forces the equation of virtual work reduces to
  S·[delta]s = 0, where S is the stress in the removed bar, and [delta]s
  is the change in the distance between the joints which it connected.
  In a critical form we have [delta]s = 0, and the equation is satisfied
  by an arbitrary value of S; a consistent system of stresses in the
  remaining bars can then be found by preceding rules. Again, when
  extraneous forces P act on the joints, the equation is

    [Sigma](P·[delta]p) + S·[delta]s = 0,

  where [delta]p is the displacement of any joint in the direction of
  the corresponding force P. If [Sigma](P·[delta]p) = 0, the stresses
  are merely indeterminate as before; but if [Sigma] (P·[delta]p) does
  not vanish, the equation cannot be satisfied by any finite value of S,
  since [delta]s = 0. This means that, if the material of the frame were
  absolutely unyielding, no finite stresses in the bars would enable it
  to withstand the extraneous forces. With actual materials, the frame
  would yield elastically, until its configuration is no longer
  "critical." The stresses in the bars would then be comparatively very
  great, although finite. The use of frames which approximate to a
  critical form is of course to be avoided in practice.

  A brief reference must suffice to the theory of three dimensional
  frames. This is important from a technical point of view, since all
  structures are practically three-dimensional. We may note that a frame
  of n joints which is just rigid must have 3n - 6 bars; and that the
  stresses produced in such a frame by a given system of extraneous
  forces in equilibrium are statically determinate, subject to the
  exception of "critical forms."

§ 10. _Statics of Inextensible Chains._--The theory of bodies or
structures which are deformable in their smallest parts belongs properly
to elasticity (q.v.). The case of inextensible strings or chains is,
however, so simple that it is generally included in expositions of pure
statics.

It is assumed that the form can be sufficiently represented by a plane
curve, that the stress (tension) at any point P of the curve, between
the two portions which meet there, is in the direction of the tangent at
P, and that the forces on any linear element [delta]s must satisfy the
conditions of equilibrium laid down in § 1. It follows that the forces
on any finite portion will satisfy the conditions of equilibrium which
apply to the case of a rigid body (§ 4).

[Illustration: FIG. 54.]

We will suppose in the first instance that the curve is plane. It is
often convenient to resolve the forces on an element PQ (= [delta]s) in
the directions of the tangent and normal respectively. If T, T +
[delta]T be the tensions at P, Q, and [delta][psi] be the angle between
the directions of the curve at these points, the components of the
tensions along the tangent at P give (T + [delta]T) cos [psi] - T, or
[delta]T, ultimately; whilst for the component along the normal at P we
have (T + [delta]T) sin [delta][psi], or T[delta][psi], or
T[delta]s/[rho], where [rho] is the radius of curvature.

Suppose, for example, that we have a light string stretched over a
smooth curve; and let R[delta]s denote the normal pressure (outwards
from the centre of curvature) on [delta]s. The two resolutions give
[delta]T = 0, T[delta][psi] = R[delta]s, or

  T = const., R = T/[rho].  (1)

The tension is constant, and the pressure per unit length varies as the
curvature.

Next suppose that the curve is "rough"; and let F[delta]s be the
tangential force of friction on [delta]s. We have [delta]T ± F[delta]s =
0, T[delta][psi] = R[delta]s, where the upper or lower sign is to be
taken according to the sense in which F acts. We assume that in
limiting equilibrium we have F = [mu]R, everywhere, where [mu] is the
coefficient of friction. If the string be on the point of slipping in
the direction in which [psi] increases, the lower sign is to be taken;
hence [delta]T = F[delta]s = [mu]T[delta][psi], whence

  T = T0 e^([mu][psi]),  (2)

if T0 be the tension corresponding to [psi] = 0. This illustrates the
resistance to dragging of a rope coiled round a post; e.g. if we put
[mu] = .3, [psi] = 2[pi], we find for the change of tension in one turn
T/T0 = 6.5. In two turns this ratio is squared, and so on.

Again, take the case of a string under gravity, in contact with a smooth
curve in a vertical plane. Let [psi] denote the inclination to the
horizontal, and w [delta]s the weight of an element [delta]s. The
tangential and normal components of w[delta]s are -s sin [psi] and
-w [delta]s cos [psi]. Hence

  [delta]T = w [delta]s sin [psi],  T [delta][psi] = w [delta]s cos [psi] + R[delta]s.  (3)

If we take rectangular axes Ox, Oy, of which Oy is drawn vertically
upwards, we have [delta]y = sin[psi] [delta]s, whence [delta]T =
w[delta]y. If the string be uniform, w is constant, and

  T = wy + const. = w(y - y0),  (4)

say; hence the tension varies as the height above some fixed level (y0).
The pressure is then given by the formula

        d[psi]
  R = T ------ - w cos [psi].  (5)
          ds

In the case of a chain hanging freely under gravity it is usually
convenient to formulate the conditions of equilibrium of a finite
portion PQ. The forces on this reduce to three, viz. the weight of PQ
and the tensions at P, Q. Hence these three forces will be concurrent,
and their ratios will be given by a triangle of forces. In particular,
if we consider a length AP beginning at the lowest point A, then
resolving horizontally and vertically we have

  T cos [psi] = T0, T sin [psi] = W,  (6)

where T0 is the tension at A, and W is the weight of PA. The former
equation expresses that the horizontal tension is constant.

[Illustration: FIG. 55.]

If the chain be uniform we have W = ws, where s is the arc AP: hence ws
= T0 tan[psi]. If we write T0 = wa, so that a is the length of a portion
of the chain whose weight would equal the horizontal tension, this
becomes

  s = a tan [psi].  (7)

This is the "intrinsic" equation of the curve. If the axes of x and y be
taken horizontal and vertical (upwards), we derive

  x = a log (sec [psi] + tan [psi]), y = a sec [psi].  (8)

Eliminating [psi] we obtain the Cartesian equation

              x
  y = a cosh ---  (9)
              a

of the _common catenary_, as it is called (fig. 56). The omission of the
additive arbitrary constants of integration in (8) is equivalent to a
special choice of the origin O of co-ordinates; viz. O is at a distance
a vertically below the lowest point ([psi] = 0) of the curve. The
horizontal line through O is called the _directrix_. The relations

  s = a sinh x/a,  y² = a² + s²,  T = T0 sec [psi] = wy,  (10)

[Illustration: FIG. 56.]

which are involved in the preceding formulae are also noteworthy. It is
a classical problem in the calculus of variations to deduce the equation
(9) from the condition that the depth of the centre of gravity of a
chain of given length hanging between fixed points must be stationary (§
9). The length a is called the _parameter_ of the catenary; it
determines the scale of the curve, all catenaries being geometrically
similar. If weights be suspended from various points of a hanging chain,
the intervening portions will form arcs of equal catenaries, since the
horizontal tension (wa) is the same for all. Again, if a chain pass over
a perfectly smooth peg, the catenaries in which it hangs on the two
sides, though usually of different parameters, will have the same
directrix, since by (10) y is the same for both at the peg.

  As an example of the use of the formulae we may determine the maximum
  span for a wire of given material. The condition is that the tension
  must not exceed the weight of a certain length [lambda] of the wire.
  At the ends we shall have y = [lambda], or

                       x
    [lambda] = a cosh ---,  (11)
                       a

  and the problem is to make x a maximum for variations of a.
  Differentiating (11) we find that, if dx/da = 0,

     x        x
    --- tanh --- = 1.  (12)
     a        a

  It is easily seen graphically, or from a table of hyperbolic tangents,
  that the equation u tanh u = 1 has only one positive root (u = 1.200);
  the span is therefore

    2x = 2au = 2[lambda]/sinh u = 1.326[lambda],

  and the length of wire is

    2s = 2[lambda]/u = 1.667 [lambda].

  The tangents at the ends meet on the directrix, and their inclination
  to the horizontal is 56° 30´.

  [Illustration: FIG. 57.]

  The relation between the sag, the tension, and the span of a wire
  (e.g. a telegraph wire) stretched nearly straight between two points
  A, B at the same level is determined most simply from first
  principles. If T be the tension, W the total weight, k the sag in the
  middle, and [psi] the inclination to the horizontal at A or B, we have
  2T[psi] = W, AB = 2[rho][psi], approximately, where [rho] is the
  radius of curvature. Since 2k[rho] = (½AB)², ultimately, we have

    k = (1/8)W·AB/T.  (13)

  The same formula applies if A, B be at different levels, provided k be
  the sag, measured vertically, half way between A and B.

In relation to the theory of suspension bridges the case where the
weight of any portion of the chain varies as its horizontal projection
is of interest. The vertical through the centre of gravity of the arc AP
(see fig. 55) will then bisect its horizontal projection AN; hence if PS
be the tangent at P we shall have AS = SN. This property is
characteristic of a parabola whose axis is vertical. If we take A as
origin and AN as axis of x, the weight of AP may be denoted by wx, where
w is the weight per unit length at A. Since PNS is a triangle of forces
for the portion AP of the chain, we have wx/T0 = PN/NS, or

  y = w·x²/2T0,  (14)

which is the equation of the parabola in question. The result might of
course have been inferred from the theory of the parabolic funicular in
§ 2.

  Finally, we may refer to the _catenary of uniform strength_, where the
  cross-section of the wire (or cable) is supposed to vary as the
  tension. Hence w, the weight per foot, varies as T, and we may write
  T = w[lambda], where [lambda] is a constant length. Resolving along
  the normal the forces on an element [delta]s, we find T[delta][psi] =
  w[delta]s cos[psi], whence

          ds
    p = ------ = [lambda] sec [psi].  (15)
        d[psi]

  From this we derive

                                               x
    x = [lambda][psi], y = [lambda] log sec --------,  (16)
                                            [lambda]

  where the directions of x and y are horizontal and vertical, and the
  origin is taken at the lowest point. The curve (fig. 58) has two
  vertical asymptotes x = ± ½[pi][lambda]; this shows that however the
  thickness of a cable be adjusted there is a limit [pi][lambda] to the
  horizontal span, where [lambda] depends on the tensile strength of the
  material. For a uniform catenary the limit was found above to be
  1.326[lambda].

[Illustration: FIG. 58.]

For investigations relating to the equilibrium of a string in three
dimensions we must refer to the textbooks. In the case of a string
stretched over a smooth surface, but in other respects free from
extraneous force, the tensions at the ends of a small element [delta]s
must be balanced by the normal reaction of the surface. It follows that
the osculating plane of the curve formed by the string must contain the
normal to the surface, i.e. the curve must be a "geodesic," and that the
normal pressure per unit length must vary as the principal curvature of
the curve.

§ 11. _Theory of Mass-Systems._--This is a purely geometrical subject.
We consider a system of points P1, P2 ..., P_n, with which are
associated certain coefficients m1, m2, ... m_n, respectively. In the
application to mechanics these coefficients are the masses of particles
situate at the respective points, and are therefore all positive. We
shall make this supposition in what follows, but it should be remarked
that hardly any difference is made in the theory if some of the
coefficients have a different sign from the rest, except in the special
case where [Sigma](m) = 0. This has a certain interest in magnetism.

In a given mass-system there exists one and only one point G such that

  [Sigma](m·[->GP]) = 0.  (1)

For, take any point O, and construct the vector

           [Sigma](m·[->OP])
  [->OG] = -----------------.  (2)
               [Sigma](m)

Then

  [Sigma](m·[->GP]) = [Sigma]{m([->GO] + [->OP])} = [Sigma](m)·[->GO] + [Sigma](m)·[->OP] = 0.  (3)

Also there cannot be a distinct point G´ such that [Sigma](m·G´P) = 0,
for we should have, by subtraction,

  [Sigma]{m([->GP] + [->PG´])} = 0, or [Sigma](m)·GG´ = 0;  (4)

i.e. G´ must coincide with G. The point G determined by (1) is called
the _mass-centre_ or _centre of inertia_ of the given system. It is
easily seen that, in the process of determining the mass-centre, any
group of particles may be replaced by a single particle whose mass is
equal to that of the group, situate at the mass-centre of the group.

If through P1, P2, ... P_n we draw any system of parallel planes meeting
a straight line OX in the points M1, M2 ... M_n, the collinear vectors
[->OM1], [->OM2] ... [->OM_n] may be called the "projections" of
[->OP1], [->OP2], ... [->OP_n] on OX. Let these projections be denoted
algebraically by x1, x2, ... x_n, the sign being positive or negative
according as the direction is that of OX or the reverse. Since the
projection of a vector-sum is the sum of the projections of the several
vectors, the equation (2) gives

         [Sigma](mx)
  [|x] = -----------,  (5)
         [Sigma](m)

if [|x] be the projection of [->OG]. Hence if the Cartesian co-ordinates
of P1, P2, ... P_n relative to any axes, rectangular or oblique be (x1,
y1, z1), (x2, y2, z2), ..., (x_n, y_n, z_n), the mass-centre ([|x],
[|y], [|z]) is determined by the formulae

         [Sigma](mx)         [Sigma](my)         [Sigma](mz)
  [|x] = -----------, [|y] = -----------, [|z] = -----------.  (6)
         [Sigma](m)          [Sigma](m)          [Sigma](m)

If we write x = [|x] + [xi], y = [|y] + [eta], z = [|z] + [zeta], so
that [xi], [eta], [zeta] denote co-ordinates relative to the mass-centre
G, we have from (6)

  [Sigma](m[xi]) = 0, [Sigma](m[eta]) = 0, [Sigma](m[zeta]) = 0.  (7)

  One or two special cases may be noticed. If three masses [alpha],
  [beta], [gamma] be situate at the vertices of a triangle ABC, the
  mass-centre of [beta] and [gamma] is at a point A´ in BC, such that
  [beta]·BA´ = [gamma]·A´C. The mass-centre (G) of [alpha], [beta],
  [gamma] will then divide AA´ so that [alpha]·AG = ([beta] + [gamma])
  GA´. It is easily proved that

    [alpha] : [beta] : [gamma] = [Delta]BGA : [Delta]GCA : [Delta]GAB;

  also, by giving suitable values (positive or negative) to the ratios
  [alpha] : [beta] : [gamma] we can make G assume any assigned position
  in the plane ABC. We have here the origin of the "barycentric
  co-ordinates" of Möbius, now usually known as "areal" co-ordinates. If
  [alpha] + [beta] + [gamma] = 0, G is at infinity; if [alpha] = [beta]
  = [gamma], G is at the intersection of the median lines of the
  triangle; if [alpha] : [beta] : [gamma] = a : b : c, G is at the
  centre of the inscribed circle. Again, if G be the mass-centre of four
  particles [alpha], [beta], [gamma], [delta] situate at the vertices of
  a tetrahedron ABCD, we find

    [alpha] : [beta] : [gamma] : [delta] = tet^n GBCD : tet^n GCDA : tet^n GDAB : tet^n GABC,

  and by suitable determination of the ratios on the left hand we can
  make G assume any assigned position in space. If [alpha] + [beta] +
  [gamma] + [delta] = O, G is at infinity; if [alpha] = [beta] = [gamma]
  = [delta], G bisects the lines joining the middle points of opposite
  edges of the tetrahedron ABCD; if [alpha] : [beta] : [gamma] : [delta]
  = [Delta]BCD : [Delta]CDA : [Delta]DAB : [Delta]ABC, G is at the
  centre of the inscribed sphere.

  If we have a continuous distribution of matter, instead of a system of
  discrete particles, the summations in (6) are to be replaced by
  integrations. Examples will be found in textbooks of the calculus and
  of analytical statics. As particular cases: the mass-centre of a
  uniform thin triangular plate coincides with that of three equal
  particles at the corners; and that of a uniform solid tetrahedron
  coincides with that of four equal particles at the vertices. Again,
  the mass-centre of a uniform solid right circular cone divides the
  axis in the ratio 3 : 1; that of a uniform solid hemisphere divides
  the axial radius in the ratio 3 : 5.

  It is easily seen from (6) that if the configuration of a system of
  particles be altered by "homogeneous strain" (see ELASTICITY) the new
  position of the mass-centre will be at that point of the strained
  figure which corresponds to the original mass-centre.

The formula (2) shows that a system of concurrent forces represented by
m1·[->OP1], m2·[->OP2], ... m_n·[->OP_n] will have a resultant
represented hy [Sigma](m)·[->OG]. If we imagine O to recede to infinity
in any direction we learn that a system of parallel forces proportional
to m1, m2,... m_n, acting at P1, P2 ... P_n have a resultant
proportional to [Sigma](m) which acts always through a point G fixed
relatively to the given mass-system. This contains the theory of the
"centre of gravity" (§§ 4, 9). We may note also that if P1, P2, ... P_n,
and P1´, P2´, ... P_n´ represent two configurations of the series of
particles, then

  [Sigma](m·[->PP´]) = Sigma(m)·[->GG´],  (8)

where G, G´ are the two positions of the mass-centre. The forces
m1·[->P1P1´], m2·[->P2P2´], ... m_n·[->P_nP_n´], considered as localized
vectors, do not, however, as a rule reduce to a single resultant.

We proceed to the theory of the _plane_, _axial_ and _polar quadratic
moments_ of the system. The axial moments have alone a dynamical
significance, but the others are useful as subsidiary conceptions. If
h1, h2, ... h_n be the perpendicular distances of the particles from any
fixed plane, the sum [Sigma](mh²) is the quadratic moment with respect
to the plane. If p1, p2, ... p_n be the perpendicular distances from any
given axis, the sum [Sigma](mp²) is the quadratic moment with respect to
the axis; it is also called the _moment of inertia_ about the axis. If
r1, r2, ... r_n be the distances from a fixed point, the sum
[Sigma](mr²) is the quadratic moment with respect to that point (or
pole). If we divide any of the above quadratic moments by the total
mass [Sigma](m), the result is called the _mean square_ of the distances
of the particles from the respective plane, axis or pole. In the case of
an axial moment, the square root of the resulting mean square is called
the _radius of gyration_ of the system about the axis in question. If we
take rectangular axes through any point O, the quadratic moments with
respect to the co-ordinate planes are

  I_x = [Sigma](mx²), I_y = [Sigma](my²), I_z = [Sigma](mz²);  (9)

those with respect to the co-ordinate axes are

  I_yz = [Sigma]{m(y² + z²)}, I_zx = [Sigma]{m(z² + x²)},
  I_xy = [Sigma]{m(x² + y²)};  (10)

whilst the polar quadratic moment with respect to O is

  I0 = [Sigma]{m(x² + y² + z²)}.  (11)

We note that

  I_yz = I_y + I_z, I_zx = I_z + I_x, I_xy = I_x + I_y,  (12)

and

  I0 = I_x + I_y + I_z = ½(I_yz + I_zx + I_xy).  (13)

  In the case of continuous distributions of matter the summations in
  (9), (10), (11) are of course to be replaced by integrations. For a
  uniform thin circular plate, we find, taking the origin at its centre,
  and the axis of z normal to its plane, I0 = ½Ma², where M is the mass
  and a the radius. Since I_x = I_y, I_z = 0, we deduce I_zx = ½Ma²,
  I_xy = ½Ma²; hence the value of the squared radius of gyration is for
  a diameter ¼a², and for the axis of symmetry ½a². Again, for a uniform
  solid sphere having its centre at the origin we find I0 = (3/5)Ma²,
  I_x = I_y = I_z = (1/5)Ma², I_yz = I_zx = l_xy = (3/5)Ma²; i.e. the
  square of the radius of gyration with respect to a diameter is
  (2/5)a². The method of homogeneous strain can be applied to deduce the
  corresponding results for an ellipsoid of semi-axes a, b, c. If the
  co-ordinate axes coincide with the principal axes, we find I_x =
  (1/5)Ma², I_y = (1/5)Mb², I_z = (1/5)Mc², whence I_yz = (1/5)M (b² +
  c²), &c.

If [phi](x, y, z) be any homogeneous quadratic function of x, y, z, we
have

  [Sigma]{m[phi](x, y, z)} = [Sigma] {m[phi]([|x] + [xi], [|y] + [eta], [|z] + [zeta])}
    = [Sigma] {m[phi](x, y, z)} + [Sigma]{m[phi]([xi], [eta], [zeta])},  (14)

since the terms which are bilinear in respect to [|x], [|y], [|z], and
[xi], [eta], [zeta] vanish, in virtue of the relations (7). Thus

  I_x = I[xi] + [Sigma](m)x²,  (15)

  I_yz = I[eta][zeta] + [Sigma](m)·(y² + z²),  (16)

with similar relations, and

  I_O = I_G + [Sigma](m)·OG².  (17)

The formula (16) expresses that the squared radius of gyration about any
axis (Ox) exceeds the squared radius of gyration about a parallel axis
through G by the square of the distance between the two axes. The
formula (17) is due to J. L. Lagrange; it may be written

  [Sigma](m·OP²)   [Sigma](m·GP²)
  -------------- = -------------- + OG²,  (18)
    [Sigma](m)       [Sigma](m)

and expresses that the mean square of the distances of the particles
from O exceeds the mean square of the distances from G by OG². The
mass-centre is accordingly that point the mean square of whose distances
from the several particles is least. If in (18) we make O coincide with
P1, P2, ... P_n in succession, we obtain

  0 + m2·P1P2² + ... + mn·P1P_n² = [Sigma](m·GP²) + [Sigma](m)·GP1²,   \
  m1·P2P1² + 0 + ... + mn·P2P_n² = [Sigma](m·GP²) + [Sigma](m)·GP2²,    >  (19)
         ...         ...          ...            ...         ...        |
  m1·P_nP1² + m2·P_nP2² + ... + 0 = [Sigma](m·GP²) + [Sigma](m)·GP_n². /

If we multiply these equations by m1, m2 ... m_n, respectively, and add,
we find

  [Sigma][Sigma](m_r m_s·P_r P_s²) = [Sigma](m)·[Sigma](m·GP²),  (20)

provided the summation [Sigma][Sigma] on the left hand be understood to
include each pair of particles once only. This theorem, also due to
Lagrange, enables us to express the mean square of the distances of the
particles from the centre of mass in terms of the masses and mutual
distances. For instance, considering four equal particles at the
vertices of a regular tetrahedron, we can infer that the radius R of the
circumscribing sphere is given by R² = (3/8)a², if a be the length of an
edge.

Another type of quadratic moment is supplied by the _deviation-moments_,
or _products of inertia_ of a distribution of matter. Thus the sum
[Sigma](m·yz) is called the "product of inertia" with respect to the
planes y = 0, z = 0. This may be expressed In terms of the product of
inertia with respect to parallel planes through G by means of the
formula (14); viz.:--

  [Sigma](m·yz) = [Sigma](m·[eta][zeta]) + [Sigma](m)·yz  (21)

The quadratic moments with respect to different planes through a fixed
point O are related to one another as follows. The moment with respect
to the plane

  [lambda]x + [mu]y + [nu]z = 0,  (22)

where [lambda], [mu], [nu] are direction-cosines, is

  [Sigma]{(m([lambda]x + [mu]y + [nu]z)²} = [Sigma](mx²)·[lambda]² + [Sigma](my²)·[mu]² + [Sigma](mz²)·[nu]²
    + 2[Sigma](myz)·[mu][nu] + 2[Sigma](mzx)·[nu][lambda] + 2[Sigma](mxy)·[lambda][mu],  (23)

and therefore varies as the square of the perpendicular drawn from O to
a tangent plane of a certain quadric surface, the tangent plane in
question being parallel to (22). If the co-ordinate axes coincide with
the principal axes of this quadric, we shall have

  [Sigma](myz) = 0, [Sigma](mzx) = 0, [Sigma](mxy) = 0;  (24)

and if we write

  [Sigma](mx²) = Ma², [Sigma](my²) = Mb², [Sigma](mz²) = Mc²,  (25)

where M = [Sigma](m), the quadratic moment becomes M(a²[lambda]² +
b²[mu]² + c²[nu]²), or Mp², where p is the distance of the origin from
that tangent plane of the ellipsoid

   x²    y²    z²
  --- + --- + --- = 1,  (26)
   a²    b²    c²

which is parallel to (22). It appears from (24) that through any
assigned point O three rectangular axes can be drawn such that the
product of inertia with respect to each pair of co-ordinate planes
vanishes; these are called the _principal axes of inertia_ at O. The
ellipsoid (26) was first employed by J. Binet (1811), and may be called
"Binet's Ellipsoid" for the point O. Evidently the quadratic moment for
a variable plane through O will have a "stationary" value when, and only
when, the plane coincides with a principal plane of (26). It may further
be shown that if Binet's ellipsoid be referred to any system of
conjugate diameters as co-ordinate axes, its equation will be

  x´²   y´²   z´²
  --- + --- + --- = 1,  (27)
  a´²   b´²   c´²

provided

  [Sigma](mx´²) = Ma´², [Sigma](my´²) Mb´², [Sigma](mz´²) = Mc´²;

also that

  [Sigma](my´z´) = 0, [Sigma](mz´x´) = 0, [Sigma](mx´y´) = 0.  (28)

Let us now take as co-ordinate axes the principal axes of inertia at the
mass-centre G. If a, b, c be the semi-axes of the Binet's ellipsoid of
G, the quadratic moment with respect to the plane [lambda]x + [mu]y +
[nu]z = 0 will be M(a²[lambda]² + b²[mu]² + c²[nu]²), and that with
respect to a parallel plane

  [lambda]x + [mu]y + [nu]z = p  (29)

will be M(a²[lambda]² + b²[mu]² + c²[nu]² + p²), by (15). This will have
a given value Mk², provided

  p² = (k² - a²)[lambda]² + (k² - b²)[mu]² + (k² - c²)[nu]².  (30)

Hence the planes of constant quadratic moment Mk² will envelop the
quadric

     x²        y²         z²
  ------- + ------- +  ------- = 1,  (31)
  k² - a²   k² - b²    k² - c²

and the quadrics corresponding to different values of k² will be
confocal. If we write

  k² = a² + b² + c² + [theta],
  b² + c² = [alpha]², c² + a² = [beta]², a² + b² = [gamma]²  (32)

the equation (31) becomes

          x²                  y²                   z²
  ------------------ + ----------------- + ------------------ = 1  (33)
  [alpha]² + [theta]   [beta]² + [theta]   [gamma]² + [theta]

for different values of [theta] this represents a system of quadrics
confocal with the ellipsoid

     x²         y²        z²
  -------- + ------- + -------- = 1,  (34)
  [alpha]²   [beta]²   [gamma]²

which we shall meet with presently as the "ellipsoid of gyration" at G.
Now consider the tangent plane [omega] at any point P of a confocal, the
tangent plane [omega]´ at an adjacent point N´, and a plane [omega]´´
through P parallel to [omega]´. The distance between the planes [omega]´
and [omega]´´ will be of the second order of small quantities, and the
quadratic moments with respect to [omega]´ and [omega]´´ will therefore
be equal, to the first order. Since the quadratic moments with respect
to [omega] and [omega]´ are equal, it follows that [omega] is a plane of
stationary quadratic moment at P, and therefore a principal plane of
inertia at P. In other words, the principal axes of inertia at P arc the
normals to the three confocals of the system (33) which pass through P.
Moreover if x, y, z be the co-ordinates of P, (33) is an equation to
find the corresponding values of [theta]; and if [theta]1, [theta]2,
[theta]3 be the roots we find

  [theta]1 + [theta]2 + [theta]3 = r² - [alpha]² - [beta]² -[gamma]²,  (35)

where r² = x² + y² + z². The squares of the radii of gyration about the
principal axes at P may be denoted by k2² + k3², k3² + k1², k1² + k2²;
hence by (32) and (35) they are r² - [theta]1, r² - [theta]2, r² -
[theta]3, respectively.

To find the relations between the moments of inertia about different
axes through any assigned point O, we take O as origin. Since the square
of the distance of a point (x, y, z) from the axis

     x         y     z
  -------- = ---- = ----  (36)
  [lambda]   [mu]   [nu]

is x² + y² + z² - ([lambda]x + [mu]y + [nu]z)², the moment of inertia
about this axis is

  I = [Sigma][m{([lambda]² + [mu]² + [nu]²)(x² + y² + z²) - ([lambda]x + [mu]y + [nu]z)²}]
    = A[lambda]² + B[mu]² + C[nu]² - 2F[mu][nu] - 2G[nu][lambda] - 2H[lambda][mu],  (37)

provided

  A = [Sigma]{m(y² + z²)}, B = [Sigma]{m(z² + x²)}, C = [Sigma]{m(x² + y²)},
  F = [Sigma](myz), G = [Sigma](mzx), H = [Sigma](mxy);  (38)

i.e. A, B, C are the moments of inertia about the co-ordinate axes, and
F, G, H are the products of inertia with respect to the pairs of
co-ordinate planes. If we construct the quadric

  Ax² + By² + Cz² - 2Fyz - 2Gzx - 2Hxy = M[epsilon]^4  (39)

where [epsilon] is an arbitrary linear magnitude, the intercept r which
it makes on a radius drawn in the direction [lambda], [mu], [nu] is
found by putting x, y, z = [lambda]r, [mu]r, [nu]r. Hence, by comparison
with (37),

  I = M[epsilon]^4/r².  (40)

The moment of inertia about any radius of the quadric (39) therefore
varies inversely as the square of the length of this radius. When
referred to its principal axes, the equation of the quadric takes the
form

  Ax² + By² + Cz² = M[epsilon]^4.  (41)

The directions of these axes are determined by the property (24), and
therefore coincide with those of the principal axes of inertia at O, as
already defined in connexion with the theory of plane quadratic moments.
The new A, B, C are called the _principal moments of inertia_ at O.
Since they are essentially positive the quadric is an ellipsoid; it is
called the _momental ellipsoid_ at O. Since, by (12), B + C > A, &c.,
the sum of the two lesser principal moments must exceed the greatest
principal moment. A limitation is thus imposed on the possible forms of
the momental ellipsoid; e.g. in the case of symmetry about an axis it
appears that the ratio of the polar to the equatorial diameter of the
ellipsoid cannot be less than 1/[root]2.

If we write A = M[alpha]², B = M[beta]², C = M[gamma]², the formula
(37), when referred to the principal axes at O, becomes

  I = M([alpha]²[lambda]² + [beta]²[mu]² + [gamma]²[nu]²) = Mp²,  (42)

if p denotes the perpendicular drawn from O in the direction ([lambda],
[mu], [nu]) to a tangent plane of the ellipsoid

     x²         y²        z²
  -------- + ------- + -------- = 1  (43)
  [alpha]²   [beta]²   [gamma]²

This is called the _ellipsoid of gyration_ at O; it was introduced into
the theory by J. MacCullagh. The ellipsoids (41) and (43) are reciprocal
polars with respect to a sphere having O as centre.

If A = B = C, the momental ellipsoid becomes a sphere; all axes through
O are then principal axes, and the moment of inertia is the same for
each. The mass-system is then said to possess kinetic symmetry about O.

  If all the masses lie in a plane (z = 0) we have, in the notation of
  (25), c² = 0, and therefore A = Mb², B = Ma², C = M(a² + b²), so that
  the equation of the momental ellipsoid takes the form

    b²x² + a²y² + (a² + b²)z² = [epsilon]^4.  (44)

  The section of this by the plane z = 0 is similar to

     x²     y²
    ---- + ---- = 1,  (45)
     a²     b²

  which may be called the _momental ellipse_ at O. It possesses the
  property that the radius of gyration about any diameter is half the
  distance between the two tangents which are parallel to that diameter.
  In the case of a uniform triangular plate it may be shown that the
  momental ellipse at G is concentric, similar and similarly situated

  to the ellipse which touches the sides of the triangle at their middle
  points.

  [Illustration: FIG. 59.]

  [Illustration: FIG. 60.]

  The graphical methods of determining the moment of inertia of a plane
  system of particles with respect to any line in its plane may be
  briefly noticed. It appears from § 5 (fig. 31) that the linear moment
  of each particle about the line may be found by means of a funicular
  polygon. If we replace the mass of each particle by its moment, as
  thus found, we can in like manner obtain the quadratic moment of the
  system with respect to the line. For if the line in question be the
  axis of y, the first process gives us the values of mx, and the second
  the value of [Sigma](mx·x) or [Sigma](mx²). The construction of a
  second funicular may be dispensed with by the employment of a
  planimeter, as follows. In fig. 59 p is the line with respect to which
  moments are to be taken, and the masses of the respective particles
  are indicated by the corresponding segments of a line in the
  force-diagram, drawn parallel to p. The funicular ZABCD ...
  corresponding to any pole O is constructed for a system of forces
  acting parallel to p through the positions of the particles and
  proportional to the respective masses; and its successive sides are
  produced to meet p in the points H, K, L, M, ... As explained in § 5,
  the moment of the first particle is represented on a certain scale by
  HK, that of the second by KL, and so on. The quadratic moment of the
  first particle will then be represented by twice the area AHK, that of
  the second by twice the area BKL, and so on. The quadratic moment of
  the whole system is therefore represented by twice the area AHEDCBA.
  Since a quadratic moment is essentially positive, the various areas
  are to taken positive in all cases. If k be the radius of gyration
  about p we find

    k² = 2 × area AHEDCBA × ON ÷ [alpha][beta],

  where [alpha][beta] is the line in the force-diagram which represents
  the sum of the masses, and ON is the distance of the pole O from this
  line. If some of the particles lie on one side of p and some on the
  other, the quadratic moment of each set may be found, and the results
  added. This is illustrated in fig. 60, where the total quadratic
  moment is represented by the sum of the shaded areas. It is seen that
  for a given direction of p this moment is least when p passes through
  the intersection X of the first and last sides of the funicular; i.e.
  when p goes through the mass-centre of the given system; cf. equation
  (15).


PART II.--KINETICS

§ 12. _Rectilinear Motion._--Let x denote the distance OP of a moving
point P at time t from a fixed origin O on the line of motion, this
distance being reckoned positive or negative according as it lies to one
side or the other of O. At time t + [delta]t let the point be at Q, and
let OQ = x + [delta]x. The _mean velocity_ of the point in the interval
[delta]t is [delta]x/[delta]t. The limiting value of this when [delta]t
is infinitely small, viz. dx/dt, is adopted as the definition of the
_velocity_ at the instant t. Again, let u be the velocity at time t, u +
[delta]u that at time t + [delta]t. The mean rate of increase of
velocity, or the _mean acceleration_, in the interval [delta]t is then
[delta]u/[delta]t. The limiting value of this when [delta]t is
infinitely small, viz., du/dt, is adopted as the definition of the
_acceleration_ at the instant t. Since u = dx/dt, the acceleration is
also denoted by d²x/dt². It is often convenient to use the "fluxional"
notation for differential coefficients with respect to time; thus the
velocity may be represented by [.x] and the acceleration by [.u] or
[:x]. There is another formula for the acceleration, in which u is
regarded as a function of the position; thus du/dt = (du/dx)(dx/dt) =
u(du/dx). The relation between x and t in any particular case may be
illustrated by means of a curve constructed with t as abscissa and x as
ordinate. This is called the _curve of positions_ or _space-time curve_;
its gradient represents the velocity. Such curves are often traced
mechanically in acoustical and other experiments. A, curve with t as
abscissa and u as ordinate is called the _curve of velocities_ or
_velocity-time curve_. Its gradient represents the acceleration, and the
area ([int]udt) included between any two ordinates represents the space
described in the interval between the corresponding instants (see fig.
62).

So far nothing has been said about the measurement of time. From the
purely kinematic point of view, the t of our formulae may be any
continuous independent variable, suggested (it may be) by some physical
process. But from the dynamical standpoint it is obvious that equations
which represent the facts correctly on one system of time-measurement
might become seriously defective on another. It is found that for almost
all purposes a system of measurement based ultimately on the earth's
rotation is perfectly adequate. It is only when we come to consider such
delicate questions as the influence of tidal friction that other
standards become necessary.

The most important conception in kinetics is that of "inertia." It is a
matter of ordinary observation that different bodies acted on by the
same force, or what is judged to be the same force, undergo different
changes of velocity in equal times. In our ideal representation of
natural phenomena this is allowed for by endowing each material particle
with a suitable _mass_ or _inertia-coefficient_ m. The product _mu_ of
the mass into the velocity is called the _momentum_ or (in Newton's
phrase) the _quantity of motion_. On the Newtonian system the motion of
a particle entirely uninfluenced by other bodies, when referred to a
suitable base, would be rectilinear, with constant velocity. If the
velocity changes, this is attributed to the action of force; and if we
agree to measure the force (X) by the rate of change of momentum which
it produces, we have the equation

   d
  --- (mu) = X.  (1)
  dt

From this point of view the equation is a mere truism, its real
importance resting on the fact that by attributing suitable values to
the masses m, and by making simple assumptions as to the value of X in
each case, we are able to frame adequate representations of whole
classes of phenomena as they actually occur. The question remains, of
course, as to how far the measurement of force here implied is
practically consistent with the gravitational method usually adopted in
statics; this will be referred to presently.

The practical unit or standard of mass must, from the nature of the
case, be the mass of some particular body, e.g. the imperial pound, or
the kilogramme. In the "C.G.S." system a subdivision of the latter, viz.
the gramme, is adopted, and is associated with the centimetre as the
unit of length, and the mean solar second as the unit of time. The unit
of force implied in (1) is that which produces unit momentum in unit
time. On the C.G.S. system it is that force which acting on one gramme
for one second produces a velocity of one centimetre per second; this
unit is known as the _dyne_. Units of this kind are called _absolute_ on
account of their fundamental and invariable character as contrasted with
gravitational units, which (as we shall see presently) vary somewhat
with the locality at which the measurements are supposed to be made.

If we integrate the equation (1) with respect to t between the limits t,
t´ we obtain
               _
              / t´
  mu´- mu =   |   X dt.  (2)
             _/ t

The time-integral on the right hand is called the _impulse_ of the force
on the interval t´ - t. The statement that the increase of momentum is
equal to the impulse is (it maybe remarked) equivalent to Newton's own
formulation of his Second Law. The form (1) is deduced from it by
putting t´- t = [delta]t, and taking [delta]t to be infinitely small. In
problems of impact we have to deal with cases of practically
instantaneous impulse, where a very great and rapidly varying force
produces an appreciable change of momentum in an exceedingly minute
interval of time.

In the case of a constant force, the acceleration [.u] or [:x] is,
according to (1), constant, and we have

  d²x
  --- = [alpha],  (3)
  dt²

say, the general solution of which is

  x = ½[alpha]t² + At + B.  (4)

The "arbitrary constants" A, B enable us to represent the circumstances
of any particular case; thus if the velocity [.x] and the position x be
given for any one value of t, we have two conditions to determine A, B.
The curve of positions corresponding to (4) is a parabola, and that of
velocities is a straight line. We may take it as an experimental result,
although the best evidence is indirect, that a particle falling freely
under gravity experiences a constant acceleration which at the same
place is the same for all bodies. This acceleration is denoted by g; its
value at Greenwich is about 981 centimetre-second units, or 32.2 feet
per second. It increases somewhat with the latitude, the extreme
variation from the equator to the pole being about ½%. We infer that on
our reckoning the force of gravity on a mass m is to be measured by mg,
the momentum produced per second when this force acts alone. Since this
is proportional to the mass, the relative masses to be attributed to
various bodies can be determined practically by means of the balance. We
learn also that on account of the variation of g with the locality a
gravitational system of force-measurement is inapplicable when more than
a moderate degree of accuracy is desired.

[Illustration: FIG. 61.]

We take next the case of a particle attracted towards a fixed point O in
the line of motion with a force varying as the distance from that point.
If [mu] be the acceleration at unit distance, the equation of motion
becomes

  d²x
  --- = -[mu]x,  (5)
  dt²

the solution of which may be written in either of the forms

  x = A cos [sigma]t + B sin [sigma]t,  x = a cos ([sigma]t + [epsilon]),  (6)

where [sigma]= [root][mu], and the two constants A, B or a, [epsilon]
are arbitrary. The particle oscillates between the two positions x = ±a,
and the same point is passed through in the same direction with the same
velocity at equal intervals of time 2[pi]/[sigma]. The type of motion
represented by (6) is of fundamental importance in the theory of
vibrations (§ 23); it is called a _simple-harmonic_ or (shortly) a
_simple_ vibration. If we imagine a point Q to describe a circle of
radius a with the angular velocity [sigma], its orthogonal projection P
on a fixed diameter AA´ will execute a vibration of this character. The
angle [sigma]t + [epsilon] (or AOQ) is called the _phase_; the arbitrary
elements a, [epsilon] are called the _amplitude_ and _epoch_ (or initial
phase), respectively. In the case of very rapid vibrations it is usual
to specify, not the _period_ (2[pi]/[sigma]), but its reciprocal the
_frequency_, i.e. the number of complete vibrations per unit time. Fig.
62 shows the curves of position and velocity; they both have the form of
the "curve of sines." The numbers correspond to an amplitude of 10
centimetres and a period of two seconds.

The vertical oscillations of a weight which hangs from a fixed point by
a spiral spring come under this case. If M be the mass, and x the
vertical displacement from the position of equilibrium, the equation of
motion is of the form

    d²x
  M --- = - Kx,  (7)
    dt²

provided the inertia of the spring itself be neglected. This becomes
identical with (5) if we put [mu] = K/M; and the period is therefore
2[pi][root](M/K), the same for all amplitudes. The period is increased
by an increase of the mass M, and diminished by an increase in the
stiffness (K) of the spring. If c be the statical increase of length
which is produced by the gravity of the mass M, we have Kc = Mg, and the
period is 2[pi][root](c/g).

[Illustration: FIG. 62.]

The small oscillations of a simple pendulum in a vertical plane also
come under equation (5). According to the principles of § 13, the
horizontal motion of the bob is affected only by the horizontal
component of the force acting upon it. If the inclination of the string
to the vertical does not exceed a few degrees, the vertical displacement
of the particle is of the second order, so that the vertical
acceleration may be neglected, and the tension of the string may be
equated to the gravity mg of the particle. Hence if l be the length of
the string, and x the horizontal displacement of the bob from the
equilibrium position, the horizontal component of gravity is mgx/l,
whence

  d²x     gx
  --- = - ---,  (8)
  dt²      l

The motion is therefore simple-harmonic, of period [tau] =
2[pi][root](l/g). This indicates an experimental method of determining g
with considerable accuracy, using the formula g = 4[pi]²l/[tau]².

  In the case of a repulsive force varying as the distance from the
  origin, the equation of motion is of the type

    d²x
    --- = [mu]x,  (9)
    dt²

  the solution of which is

    x = A e^(nt) + B e^(-nt),  (10)

  where n = [root][mu]. Unless the initial conditions be adjusted so as
  to make A = 0 exactly, x will ultimately increase indefinitely with t.
  The position x = 0 is one of equilibrium, but it is unstable. This
  applies to the inverted pendulum, with [mu] = g/l, but the equation
  (9) is then only approximate, and the solution therefore only serves
  to represent the initial stages of a motion in the neighbourhood of
  the position of unstable equilibrium.

In acoustics we meet with the case where a body is urged towards a fixed
point by a force varying as the distance, and is also acted upon by an
"extraneous" or "disturbing" force which is a given function of the
time. The most important case is where this function is simple-harmonic,
so that the equation (5) is replaced by

  d²x
  --- + [mu]x = f cos ([sigma]1t + [alpha]),  (11)
  dt²

where [sigma]1 is prescribed. A particular solution is

             f
  x = ---------------- cos ([sigma]1t + [alpha]).  (12)
      [mu] - [sigma]1²

This represents a _forced oscillation_ whose period 2[pi]/[sigma]1,
coincides with that of the disturbing force; and the phase agrees with
that of the force, or is opposed to it, according as [sigma]1² < or > [mu];
i.e. according as the imposed period is greater or less than the natural
period 2[pi]/[root][mu]. The solution fails when the two periods agree
exactly; the formula (12) is then replaced by

           ft
  x  = ---------- sin ([sigma]1t + [alpha]),  (13)
       2 [sigma]1

which represents a vibration of continually increasing amplitude. Since
the equation (12) is in practice generally only an approximation (as in
the case of the pendulum), this solution can only be accepted as a
representation of the initial stages of the forced oscillation. To
obtain the complete solution of (11) we must of course superpose the
free vibration (6) with its arbitrary constants in order to obtain a
complete representation of the most general motion consequent on
arbitrary initial conditions.

[Illustration: FIG. 63.]

  A simple mechanical illustration is afforded by the pendulum. If the
  point of suspension have an imposed simple vibration [xi] = a cos
  [sigma]t in a horizontal line, the equation of small motion of the bob
  is

                x - [xi]
    m[:x] = -mg --------,
                   l

  or

           gx    [xi]
    [:x] + --- = ----.  (14)
            l      l

  This is the same as if the point of suspension were fixed, and a
  horizontal disturbing force mg[xi]/l were to act on the bob. The
  difference of phase of the forced vibration in the two cases is
  illustrated and explained in the annexed fig. 63, where the pendulum
  virtually oscillates about C as a fixed point of suspension. This
  illustration was given by T. Young in connexion with the kinetic
  theory of the tides, where the same point arises.

  We may notice also the case of an attractive force varying inversely
  as the square of the distance from the origin. If [mu] be the
  acceleration at unit distance, we have

      du      [mu]
    u --- = - ----  (15)
      dx       x²

  whence

         2[mu]
    u² = ----- + C.  (16)
           x

  In the case of a particle falling directly towards the earth from rest
  at a very great distance we have C = 0 and, by Newton's Law of
  Gravitation, [mu]/a² = g, where a is the earth's radius. The deviation
  of the earth's figure from sphericity, and the variation of g with
  latitude, are here ignored. We find that the velocity with which the
  particle would arrive at the earth's surface (x = a) is [root](2ga).
  If we take as rough values a = 21 × 10^6 feet, g = 32 foot-second
  units, we get a velocity of 36,500 feet, or about seven miles, per
  second. If the particles start from rest at a finite distance c, we
  have in (16), C = - 2[mu]/c, and therefore

    dx             / / 2[mu](c - x) \
    --  = u = -   / (  ------------- ),  (17)
    dt          \/   \     cx       /

  the minus sign indicating motion towards the origin. If we put x = c
  cos² ½[phi], we find

           c^(3/2)
    t = ------------- ([phi] + sin [phi]),  (18)
        [root](8[mu])

  no additive constant being necessary if t be reckoned from the instant
  of starting, when [phi] = 0. The time t of reaching the origin ([phi]
  = [pi]) is

         [pi] c^(3/2)
    t1 = -------------.  (19)
         [root](8[mu])

  This may be compared with the period of revolution in a circular orbit
  of radius c about the same centre of force, viz.
  2[pi]c^(3/2)/[root][mu](§ 14). We learn that if the orbital motion of
  a planet, or a satellite, were arrested, the body would fall into the
  sun, or into its primary, in the fraction 0.1768 of its actual
  periodic time. Thus the moon would reach the earth in about five days.
  It may be noticed that if the scales of x and t be properly adjusted,
  the curve of positions in the present problem is the portion of a
  cycloid extending from a vertex to a cusp.

In any case of rectilinear motion, if we integrate both sides of the
equation

     du
  mu -- = X,  (20)
     dx

which is equivalent to (1), with respect to x between the limits x0, x1,
we obtain
                     _
                    / x1
  ½ mu1² - ½ mu0² = |    X dx.  (21)
                   _/ x0

We recognize the right-hand member as the _work_ done by the force X on
the particle as the latter moves from the position x0 to the position
x1. If we construct a curve with x as abscissa and X as ordinate, this
work is represented, as in J. Watt's "indicator-diagram," by the area
cut off by the ordinates x = x0, x = x1. The product ½mu² is called the
_kinetic energy_ of the particle, and the equation (21) is therefore
equivalent to the statement that the increment of the kinetic energy is
equal to the work done on the particle. If the force X be always the
same in the same position, the particle may be regarded as moving in a
certain invariable "field of force." The work which would have to be
supplied by other forces, extraneous to the field, in order to bring the
particle from rest in some standard position P0 to rest in any assigned
position P, will depend only on the position of P; it is called the
_statical_ or _potential energy_ of the particle with respect to the
field, in the position P. Denoting this by V, we have [delta]V -
X[delta]x = 0, whence

        dV
  X = - --,  (22)
        dx

The equation (21) may now be written

  ½ mu1² + V1 = ½ mu0² + V0,  (23)

which asserts that when no extraneous forces act the sum of the kinetic
and potential energies is constant. Thus in the case of a weight hanging
by a spiral spring the work required to increase the length by x is V =
[int 0 to x] Kxdx = ½Kx², whence ½Mu² + ½Kx² = const., as is easily
verified from preceding results. It is easily seen that the effect of
extraneous forces will be to increase the sum of the kinetic and
potential energies by an amount equal to the work done by them. If this
amount be negative the sum in question is diminished by a corresponding
amount. It appears then that this sum is a measure of the total capacity
for doing work against extraneous resistances which the particle
possesses in virtue of its motion and its position; this is in fact the
origin of the term "energy." The product mv² had been called by G. W.
Leibnitz the "vis viva"; the name "energy" was substituted by T. Young;
finally the name "actual energy" was appropriated to the expression ½mv²
by W. J. M. Rankine.

  The laws which regulate the resistance of a medium such as air to the
  motion of bodies through it are only imperfectly known. We may briefly
  notice the case of resistance varying as the square of the velocity,
  which is mathematically simple. If the positive direction of x be
  downwards, the equation of motion of a falling particle will be of the
  form

    du
    -- = g - ku²;  (24)
    dt

  this shows that the velocity u will send asymptotically to a certain
  limit V (called the _terminal velocity_) such that kV² = g. The
  solution is

               gt        V²           gt
    u = V tanh ---,  x = --- log cosh ---,  (25)
                V         g            V

  if the particle start from rest in the position x = 0 at the instant t
  = 0. In the case of a particle projected vertically upwards we have

    du
    -- = -g - ku²,  (26)
    dt

  the positive direction being now upwards. This leads to

            u           u0     gt        V²      V² + u0²
    tan^-1 --- = tan^-1 ---  - ---,  x = --- log --------,  (27)
            V            V      V        2g      V² + u²

  where u0 is the velocity of projection. The particle comes to rest
  when

         V         u0       V²       /    u0² \
    t = --- tan^-1 ---, x = --- log ( 1 + ---  ).  (28)
         g          V       2g       \    V²  /

  For small velocities the resistance of the air is more nearly
  proportional to the first power of the velocity. The effect of forces
  of this type on small vibratory motions may be investigated as
  follows. The equation (5) when modified by the introduction of a
  frictional term becomes

    [:x] = -[mu]x - k [.x].  (29)

  If k² < 4[mu] the solution is

    x = a e^{-t/[tau]} cos ([sigma]t + [epsilon]),  (30)

  where

    [tau] = 2/k,  [sigma] = [root]([mu] - ¼k²),  (31)

  and the constants a, [epsilon] are arbitrary. This may be described as
  a simple harmonic oscillation whose amplitude diminishes
  asymptotically to zero according to the law e^(-t/[tau]). The constant
  [tau] is called the _modulus of decay_ of the oscillations; if it is
  large compared with 2[pi]/[sigma] the effect of friction on the period
  is of the second order of small quantities and may in general be
  ignored. We have seen that a true simple-harmonic vibration may be
  regarded as the orthogonal projection of uniform circular motion; it
  was pointed out by P. G. Tait that a similar representation of the
  type (30) is obtained if we replace the circle by an equiangular
  spiral described, with a constant angular velocity about the pole, in
  the direction of diminishing radius vector. When k² > 4[mu], the
  solution of (29) is, in real form,

    x = a1 e^(-t/[tau]1) + a2 e^(-t/[tau]2),  (32)

  where

    1/[tau]1, 1/[tau]2 = ½k ± [root](¼k² - [mu]).  (33)

  The body now passes once (at most) through its equilibrium position,
  and the vibration is therefore styled _aperiodic_.

  To find the forced oscillation due to a periodic force we have

    [:x] + k[.x] + [mu]x = f cos ([sigma]1t + [epsilon]).  (34)

  The solution is

         f
    x = --- cos ([sigma]1t + [epsilon] - [epsilon]1),   (35)
         R

  provided
                                                                  k[sigma]1
    R = {([mu] - [sigma]1²)² + k²[sigma]1²}^½, tan[epsilon]1 = ----------------.  (36)
                                                               [mu] - [sigma]1²

  Hence the phase of the vibration lags behind that of the force by the
  amount [epsilon]1, which lies between 0 and ½[pi] or between ½[pi] and
  [pi], according as [sigma]1² <> [mu]. If the friction be comparatively
  slight the amplitude is greatest when the imposed period coincides
  with the free period, being then equal to f/k[sigma]1, and therefore
  very great compared with that due to a slowly varying force of the
  same average intensity. We have here, in principle, the explanation of
  the phenomenon of "resonance" in acoustics. The abnormal amplitude is
  greater, and is restricted to a narrower range of frequency, the
  smaller the friction. For a complete solution of (34) we must of
  course superpose the free vibration (30); but owing to the factor
  e^(-t/[tau]) the influence of the initial conditions gradually
  disappears.

For purposes of mathematical treatment a force which produces a finite
change of velocity in a time too short to be appreciated is regarded as
infinitely great, and the time of action as infinitely short. The whole
effect is summed up in the value of the instantaneous impulse, which is
the time-integral of the force. Thus if an instantaneous impulse [xi]
changes the velocity of a mass m from u to u´ we have

  mu´- mu = [xi].  (37)

The effect of ordinary finite forces during the infinitely short
duration of this impulse is of course ignored.

We may apply this to the theory of impact. If two masses m1, m2 moving
in the same straight line impinge, with the result that the velocities
are changed from u1, u2, to u1´, u2´, then, since the impulses on the
two bodies must be equal and opposite, the total momentum is unchanged,
i.e.

  m1u1´ + m2u2´ = m1u1 + m2u2.  (38)

The complete determination of the result of a collision under given
circumstances is not a matter of abstract dynamics alone, but requires
some auxiliary assumption. If we assume that there is no loss of
apparent kinetic energy we have also

  m1u1² + m2u2´² = m1u1² + m2u2².  (39)

Hence, and from (38),

  u2´ - u1´ = -(u2 - u1),  (40)

i.e. the relative velocity of the two bodies is reversed in direction,
but unaltered in magnitude. This appears to be the case very
approximately with steel or glass balls; generally, however, there is
some appreciable loss of apparent energy; this is accounted for by
vibrations produced in the balls and imperfect elasticity of the
materials. The usual empirical assumption is that

  u2´ - u1´ = -e(u2 - u1),  (41)

where e is a proper fraction which is constant for the same two bodies.
It follows from the formula § 15 (10) for the internal kinetic energy of
a system of particles that as a result of the impact this energy is
diminished by the amount

              m1m2
  ½(1 - e²) ------- (u1 - u2)².  (42)
            m1 + m2

The further theoretical discussion of the subject belongs to ELASTICITY.

This is perhaps the most suitable place for a few remarks on the theory
of "dimensions." (See also UNITS, DIMENSIONS OF.) In any absolute system
of dynamical measurement the fundamental units are those of mass, length
and time; we may denote them by the symbols M, L, T, respectively. They
may be chosen quite arbitrarily, e.g. on the C.G.S. system they are the
gramme, centimetre and second. All other units are derived from these.
Thus the unit of velocity is that of a point describing the unit of
length in the unit of time; it may be denoted by LT^-1, this symbol
indicating that the magnitude of the unit in question varies directly as
the unit of length and inversely as the unit of time. The unit of
acceleration is the acceleration of a point which gains unit velocity in
unit time; it is accordingly denoted by LT^-2. The unit of momentum is
MLT^-1; the unit force generates unit momentum in unit time and is
therefore denoted by MLT^-2. The unit of work on the same principles is
ML²T^-2, and it is to be noticed that this is identical with the unit of
kinetic energy. Some of these derivative units have special names
assigned to them; thus on the C.G.S. system the unit of force is called
the _dyne_, and the unit of work or energy the _erg_. The number which
expresses a physical quantity of any particular kind will of course vary
inversely as the magnitude of the corresponding unit. In any general
dynamical equation the dimensions of each term in the fundamental units
must be the same, for a change of units would otherwise alter the
various terms in different ratios. This principle is often useful as a
check on the accuracy of an equation.

  The theory of dimensions often enables us to forecast, to some extent,
  the manner in which the magnitudes involved in any particular problem
  will enter into the result. Thus, assuming that the period of a small
  oscillation of a given pendulum at a given place is a definite
  quantity, we see that it must vary as [root](l/g). For it can only
  depend on the mass m of the bob, the length l of the string, and the
  value of g at the place in question; and the above expression is the
  only combination of these symbols whose dimensions are those of a
  time, simply. Again, the time of falling from a distance a into a
  given centre of force varying inversely as the square of the distance
  will depend only on a and on the constant [mu] of equation (15). The
  dimensions of [mu]/x² are those of an acceleration; hence the
  dimensions of [mu] are L³T^-2. Assuming that the time in question
  varies as a^x[mu]^y, whose dimensions are L^(x + 3y)T^(-2y), we must
  have x + 3y = 0, -2y = 1, so that the time of falling will vary as
  a^(3/2)/[root][mu], in agreement with (19).

  The argument appears in a more demonstrative form in the theory of
  "similar" systems, or (more precisely) of the similar motion of
  similar systems. Thus, considering the equations

    d²x     [mu]  d²x´     [mu]´
    --- = - ----, ---- = - -----,   (43)
    dt²      x²   dt´²      x´²

  which refer to two particles falling independently into two distinct
  centres of force, it is obvious that it is possible to have x in a
  constant ratio to x´, and t in a constant ratio to t´, provided that

     x     x´   [mu]   [mu]´
    --- : --- = ---- : -----,   (44)
     t²   t´²    x²     x´²

  and that there is a suitable correspondence between the initial
  conditions. The relation (44) is equivalent to

             x^(3/2)   x´^(3/2)
    t : t´ = ------- : --------,  (45)
             [mu]^½    [mu]´^½

  where x, x´ are any two corresponding distances; e.g. they may be the
  initial distances, both particles being supposed to start from rest.
  The consideration of dimensions was introduced by J. B. Fourier (1822)
  in connexion with the conduction of heat.

[Illustration: FIG. 64.]

§ 13. _General Motion of a Particle._--Let P, Q be the positions of a
moving point at times t, t + [delta]t respectively. A vector [->OU]
drawn parallel to PQ, of length proportional to PQ/[delta]t on any
convenient scale, will represent the _mean velocity_ in the interval
[delta]t, i.e. a point moving with a constant velocity having the
magnitude and direction indicated by this vector would experience the
same resultant displacement [->PQ] in the same time. As [delta]t is
indefinitely diminished, the vector [->OU] will tend to a definite limit
[->OV]; this is adopted as the definition of the _velocity_ of the
moving point at the instant t. Obviously [->OV] is parallel to the
tangent to the path at P, and its magnitude is ds/dt, where s is the
arc. If we project [->OV] on the co-ordinate axes (rectangular or
oblique) in the usual manner, the projections u, v, w are called the
_component velocities_ parallel to the axes. If x, y, z be the
co-ordinates of P it is easily proved that

      dx      dy      dz
  u = --, v = --, w = --.  (1)
      dt      dt      dt

The momentum of a particle is the vector obtained by multiplying the
velocity by the mass m. The _impulse_ of a force in any infinitely small
interval of time [delta]t is the product of the force into [delta]t; it
is to be regarded as a vector. The total impulse in any finite interval
of time is the integral of the impulses corresponding to the
infinitesimal elements [delta]t into which the interval may be
subdivided; the summation of which the integral is the limit is of
course to be understood in the vectorial sense.

Newton's Second Law asserts that change of momentum is equal to the
impulse; this is a statement as to equality of vectors and so implies
identity of direction as well as of magnitude. If X, Y, Z are the
components of force, then considering the changes in an infinitely short
time [delta]t we have, by projection on the co-ordinate axes,
[delta](mu) = X[delta]t, and so on, or

    du        dv        dw
  m -- = X, m -- = Y, m -- = Z.  (2)
    dt        dt        dt

For example, the path of a particle projected anyhow under gravity will
obviously be confined to the vertical plane through the initial
direction of motion. Taking this as the plane xy, with the axis of x
drawn horizontally, and that of y vertically upwards, we have X = 0, Y =
-mg; so that

  d²x      d²y
  --- = 0, --- = -g.  (3)
  dt²      dt²

The solution is

  x = At + B, y = -½ gt² + Ct + D.  (4)

If the initial values of x, y, [.x], [.y] are given, we have four
conditions to determine the four arbitrary constants A, B, C, D. Thus if
the particle start at time t = 0 from the origin, with the component
velocities u0, v0, we have

  x = u0t, y = v0t - ½ gt².  (5)

Eliminating t we have the equation of the path, viz.

      v0      gx²
  y = --- x - ---.  (6)
      u0      2u²

This is a parabola with vertical axis, of latus-rectum 2u0²/g. The range
on a horizontal plane through O is got by putting y = 0, viz. it is
2u0v0/g. we denote the resultant velocity at any instant by [.s] we have

  [.s]² = [.x]² + [.y]² = [.s]0² - 2gy.  (7)

Another important example is that of a particle subject to an
acceleration which is directed always towards a fixed point O and is
proportional to the distance from O. The motion will evidently be in one
plane, which we take as the plane z = 0. If [mu] be the acceleration at
unit distance, the component accelerations parallel to axes of x and y
through O as origin will be -[mu]x, -[mu]y, whence

  d²x           d²y
  --- = -[mu]x, --- = - [mu]y.  (8)
  dt²           dt²

The solution is

  x = A cos nt + B sin nt, y = C cos nt + D sin nt,  (9)

where n = [root][mu]. If P be the initial position of the particle, we
may conveniently take OP as axis of x, and draw Oy parallel to the
direction of motion at P. If OP = a, and [.s]0 be the velocity at P, we
have, initially, x = a, y = 0, [.x] = 0, [.y] = [.s]0 whence

  x = a cos nt, y = b sin nt,  (10)

if b = [.s]0/n. The path is therefore an ellipse of which a, b are
conjugate semi-diameters, and is described in the period
2[pi]/[root][mu]; moreover, the velocity at any point P is equal to
[root][mu]·OD, where OD is the semi-diameter conjugate to OP. This type
of motion is called _elliptic harmonic_. If the co-ordinate axes are the
principal axes of the ellipse, the angle nt in (10) is identical with
the "excentric angle." The motion of the bob of a "spherical pendulum,"
i.e. a simple pendulum whose oscillations are not confined to one
vertical plane, is of this character, provided the extreme inclination
of the string to the vertical be small. The acceleration is towards the
vertical through the point of suspension, and is equal to gr/l,
approximately, if r denote distance from this vertical. Hence the path
is approximately an ellipse, and the period is 2[pi] [root](l/g).

[Illustration: FIG. 65.]

  The above problem is identical with that of the oscillation of a
  particle in a smooth spherical bowl, in the neighbourhood of the
  lowest point. If the bowl has any other shape, the axes Ox, Oy may be
  taken tangential to the lines of curvature at the lowest point O; the
  equations of small motion then are

    d²x        x     d²y        y
    --- = -g ------, --- = -g ------,  (11)
    dt²      [rho]1  dt²      [rho]2

  where [rho]1, [rho]2, are the principal radii of curvature at O. The
  motion is therefore the resultant of two simple vibrations in
  perpendicular directions, of periods 2[pi] [root]([rho]1/g),
  2[pi] [root]([rho]2/g). The circumstances are realized in "Blackburn's
  pendulum," which consists of a weight P hanging from a point C of a
  string ACB whose ends A, B are fixed. If E be the point in which the
  line of the string meets AB, we have [rho]1 = CP, [rho]2 = EP. Many
  contrivances for actually drawing the resulting curves have been
  devised.

[Illustration: FIG. 66.]

It is sometimes convenient to resolve the accelerations in directions
having a more intrinsic relation to the path. Thus, in a plane path, let
P, Q be two consecutive positions, corresponding to the times t, t +
[delta]t; and let the normals at P, Q meet in C, making an angle
[delta][psi]. Let v (= [.s]) be the velocity at P, v + [delta]v that at
Q. In the time [delta]t the velocity parallel to the tangent at P
changes from v to v + [delta]v, ultimately, and the tangential
acceleration at P is therefore dv/dt or [:s]. Again, the velocity
parallel to the normal at P changes from 0 to v[delta][psi], ultimately,
so that the normal acceleration is v d[psi]/dt. Since

  dv   dv ds     dv    d[psi]     d[psi] ds     v²
  -- = -- -- = v --, v ------ = v ------ -- = -----,  (12)
  dt   ds dt     ds      dt         ds   dt   [rho]

where [rho] is the radius of curvature of the path at P, the tangential
and normal accelerations are also expressed by v dv/ds and v²/[rho],
respectively. Take, for example, the case of a particle moving on a
smooth curve in a vertical plane, under the action of gravity and the
pressure R of the curve. If the axes of x and y be drawn horizontal and
vertical (upwards), and if [psi] be the inclination of the tangent to
the horizontal, we have

     dv                         dy    mv²
  mv -- = - mg sin [psi] = - mg --,  ----- = - mg cos [psi] + R.  (13)
     ds                         ds   [rho]

The former equation gives

  v² = C - 2gy,  (14)

and the latter then determines R.

  In the case of the pendulum the tension of the string takes the place
  of the pressure of the curve. If l be the length of the string, [psi]
  its inclination to the downward vertical, we have [delta]s =
  l[delta][psi], so that v = ld[psi]/dt. The tangential resolution then
  gives

    d²[psi]
  l ------- = - g sin [psi].  (15)
      dt²

  If we multiply by 2d[psi]/dt and integrate, we obtain

     / d[psi]\²   2g
    ( ------  ) = --- cos [psi] + const.,  (16)
     \  dt   /     l

  which is seen to be equivalent to (14). If the pendulum oscillate
  between the limits [psi] = ±[alpha], we have

     /[delta][psi]\²   2g                              4g
    ( ------------ ) = --- (cos [psi] - cos [alpha]) = --- (sin² ½[alpha] - sin² ½[psi]);  (17)
     \   dt       /     l                               l

  and, putting sin ½[psi] = sin ½[alpha]. sin [phi], we find for the
  period ([tau]) of a complete oscillation

                _½[pi]                         _½[pi]
               /    dt                 / l    /                  d[phi]
    [tau] = 4  |   ------ d[phi] = 4  / --- · |   ------------------------------------
              _/0  d[phi]           \/   g   _/0  [root](1 - sin² ½[alpha]·sin² [phi])

                                    / l
                             = 4   / ---·F1(sin ½[alpha]),  (18)
                                 \/   g

  in the notation of elliptic integrals. The function F1 (sin [beta])
  was tabulated by A. M. Legendre for values of [beta] ranging from 0°
  to 90°. The following table gives the period, for various amplitudes
  [alpha], in terms of that of oscillation in an infinitely small arc
  [viz. 2[pi] [root](l/g)] as unit.

    +--------------+----------++--------------+----------+
    | [alpha]/[pi] |  [tau]   || [alpha]/[pi] |  [tau]   |
    +--------------+----------++--------------+----------+
    |      .1      |  1.0062  ||      .6      |  1.2817  |
    |      .2      |  1.0253  ||      .7      |  1.4283  |
    |      .3      |  1.0585  ||      .8      |  1.6551  |
    |      .4      |  1.1087  ||      .9      |  2.0724  |
    |      .5      |  1.1804  ||     1.0      |   [oo]   |
    +--------------+----------++--------------+----------+

  The value of [tau] can also be obtained as an infinite series, by
  expanding the integrand in (18) by the binomial theorem, and
  integrating term by term. Thus

                    / l     /     1²                 1²·3²                     \
    [tau] = 2[pi]  / --- · ( 1 + --- sin² ½[alpha] + ----- sin^4 ½[alpha] + ... ).  (19)
                 \/   g     \     2²                 2²·4²                     /

  If [alpha] be small, an approximation (usually sufficient) is

    [tau] = 2[pi] [root](l/g)·(1 + (1/16)[alpha]²).

  In the extreme case of [alpha] = [pi], the equation (17) is
  immediately integrable; thus the time from the lowest position is

    t = [root](l/g)·log tan (¼[pi] + ¼[psi]).  (20)

  This becomes infinite for [psi] = [pi], showing that the pendulum only
  tends asymptotically to the highest position.

  [Illustration: FIG. 67.]

  The variation of period with amplitude was at one time a hindrance to
  the accurate performance of pendulum clocks, since the errors produced
  are cumulative. It was therefore sought to replace the circular
  pendulum by some other contrivance free from this defect. The equation
  of motion of a particle in any smooth path is

    d²s
    --- = -g sin [psi],  (21)
    dt²

  where [psi] is the inclination of the tangent to the horizontal. If
  sin [psi] were accurately and not merely approximately proportional to
  the arc s, say

    s = k sin [psi],  (22)

  the equation (21) would assume the same form as § 12 (5). The motion
  along the arc would then be accurately simple-harmonic, and the period
  2[pi][root](k/g) would be the same for all amplitudes. Now equation
  (22) is the intrinsic equation of a cycloid; viz. the curve is that
  traced by a point on the circumference of a circle of radius ¼k which
  rolls on the under side of a horizontal straight line. Since the
  evolute of a cycloid is an equal cycloid the object is attained by
  means of two metal cheeks, having the form of the evolute near the
  cusp, on which the string wraps itself alternately as the pendulum
  swings. The device has long been abandoned, the difficulty being met
  in other ways, but the problem, originally investigated by C. Huygens,
  is important in the history of mathematics.

The component accelerations of a point describing a tortuous curve, in
the directions of the tangent, the principal normal, and the binormal,
respectively, are found as follows. If [->OV], [->OV´] be vectors
representing the velocities at two consecutive points P, P´ of the path,
the plane VOV´ is ultimately parallel to the osculating plane of the
path at P; the resultant acceleration is therefore in the osculating
plane. Also, the projections of [->VV´] on OV and on a perpendicular to
OV in the plane VOV´ are [delta]v and v[delta][epsilon], where
[delta][epsilon] is the angle between the directions of the tangents at
P, P´. Since [delta][epsilon] = [delta]s/[rho], where [delta]s = PP´ =
v[delta]t and [rho] is the radius of principal curvature at P, the
component accelerations along the tangent and principal normal are dv/dt
and vd[epsilon]/dt, respectively, or vdv/ds and v²/[rho]. For example,
if a particle moves on a smooth surface, under no forces except the
reaction of the surface, v is constant, and the principal normal to the
path will coincide with the normal to the surface. Hence the path is a
"geodesic" on the surface.

If we resolve along the tangent to the path (whether plane or tortuous),
the equation of motion of a particle may be written

     dv
  mv -- = [T],  (23)
     ds

where [T] is the tangential component of the force. Integrating with
respect to s we find
                     _
                    / s1
  ½ mv1² - ½ mv0² = |     [T] ds;  (24)
                   _/ s0

i.e. the increase of kinetic energy between any two positions is equal
to the work done by the forces. The result follows also from the
Cartesian equations (2); viz. we have

  m([.x][:x] + [.y][:y] + [.z][:z]) = X[.x] + Y[.y] + Z[.z],  (25)

whence, on integration with respect to t,

                               _
                              /
  ½m([.x]² + [.y]² + [.z]²) = |(X[.x] + Y[.y] + Z[.z]) dt + const.
                             _/
                               _
                              /
                            = |(X dx + Y dy + Z dz) + const.  (26)
                             _/

If the axes be rectangular, this has the same interpretation as (24).

Suppose now that we have a constant field of force; i.e. the force
acting on the particle is always the same at the same place. The work
which must be done by forces extraneous to the field in order to bring
the particle from rest in some standard position A to rest in any other
position P will not necessarily be the same for all paths between A and
P. If it is different for different paths, then by bringing the particle
from A to P by one path, and back again from P to A by another, we might
secure a gain of work, and the process could be repeated indefinitely.
If the work required is the same for all paths between A and P, and
therefore zero for a closed circuit, the field is said to be
_conservative_. In this case the work required to bring the particle
from rest at A to rest at P is called the _potential energy_ of the
particle in the position P; we denote it by V. If PP´ be a linear
element [delta]s drawn in any direction from P, and S be the force due
to the field, resolved in the direction PP´, we have [delta]V =
-S[delta]s or

      [dP]V
  S = -----.  (27)
      [dP]s

In particular, by taking PP´ parallel to each of the (rectangular)
co-ordinate axes in succession, we find

      [dP]V      [dP]V      [dP]V
  X = -----, Y = -----, Z = -----.  (28)
      [dP]x      [dP]y      [dP]z

The equation (24) or (26) now gives

  ½ mv1² + V1 = ½ mv0² + V0;  (29)

i.e. the sum of the kinetic and potential energies is constant when no
work is done by extraneous forces. For example, if the field be that due
to gravity we have V = fmgdy = mgy + const., if the axis of y be drawn
vertically upwards; hence

  ½ mv² + mgy = const.  (30)

This applies to motion on a smooth curve, as well as to the free motion
of a projectile; cf. (7), (14). Again, in the case of a force Kr towards
O, where r denotes distance from O we have V = [int] Kr dr = ½Kr² +
const., whence

  ½ mv² + ½ Kr² = const.  (31)

It has been seen that the orbit is in this case an ellipse; also that if
we put [mu] = K/m the velocity at any point P is v = [root][mu]·OD,
where OD is the semi-diameter conjugate to OP. Hence (31) is consistent
with the known property of the ellipse that OP² + OD² is constant.

  The forms assumed by the dynamical equations when the axes of
  reference are themselves in motion will be considered in § 21. At
  present we take only the case where the rectangular axes Ox, Oy rotate
  in their own plane, with angular velocity [omega] about Oz, which is
  fixed. In the interval [delta]t the projections of the line joining
  the origin to any point (x, y, z) on the directions of the co-ordinate
  axes at time t are changed from x, y, z to (x + [delta]x) cos
  [omega][delta]t - (y + [delta]y) sin [omega][delta]t, (x + [delta]x)
  sin [omega][delta]t + (y + [delta]y) cos [omega][delta]t, z
  respectively. Hence the component velocities parallel to the
  instantaneous positions of the co-ordinate axes at time t are

    u = [.x] - [omega]y, v = [.y] + [omega]z, [omega] = [.z].  (32)

  In the same way we find that the component accelerations are

    [.u] - [omega]v, [.v] + [omega]u, [.omega].  (33)

  Hence if [omega] be constant the equations of motion take the forms

    m([:x] - 2[omega][.y] - [omega]²[.x]) = X, m([:y] + 2[omega][.x] - [omega]²y) = Y, m[:z] = Z.  (34)

  These become identical with the equations of motion relative to fixed
  axes provided we introduce a fictitious force m[omega]²r acting
  outwards from the axis of z, where r = [root](x² + y²), and a second
  fictitious force 2m[omega]v at right angles to the path, where v is
  the component of the relative velocity parallel to the plane xy. The
  former force is called by French writers the _force centrifuge
  ordinaire_, and the latter the _force centrifuge composée_, or _force
  de Coriolis_. As an application of (34) we may take the case of a
  symmetrical Blackburn's pendulum hanging from a horizontal bar which
  is made to rotate about a vertical axis half-way between the points
  of attachment of the upper string. The equations of small motion are
  then of the type

    [:x] - 2[omega][.y] - [omega]²x = -p²x, [:y] + 2[omega][.x] - [omega]²y = -q²y.  (35)

  This is satisfied by

    [:x] = A cos ([sigma]t + [epsilon]), y = B sin ([sigma]t + [epsilon]),  (36)

  provided

    ([sigma]² + [omega]² - p²)A + 2[sigma][omega]B = 0, \  (37)
    2[sigma][omega]A + ([sigma]² + [omega]² - q²)B = 0. /

  Eliminating the ratio A : B we have

    ([sigma]² + [omega]² - p²)([sigma]² + [omega]² - q²) - 4[sigma]²[omega]² = 0.  (38)

  It is easily proved that the roots of this quadratic in [sigma]² are
  always real, and that they are moreover both positive unless [omega]²
  lies between p² and q². The ratio B/A is determined in each case by
  either of the equations (37); hence each root of the quadratic gives a
  solution of the type (36), with two arbitrary constants A, [epsilon].
  Since the equations (35) are linear, these two solutions are to be
  superposed. If the quadratic (38) has a negative root, the
  trigonometrical functions in (36) are to be replaced by real
  exponentials, and the position x = 0, y = 0 is unstable. This occurs
  only when the period (2[pi]/[omega]) of revolution of the arm lies
  between the two periods (2[pi]/p, 2[pi]/q) of oscillation when the arm
  is fixed.

§ 14. _Central Forces. Hodograph._--The motion of a particle subject to
a force which passes always through a fixed point O is necessarily in a
plane orbit. For its investigation we require two equations; these may
be obtained in a variety of forms.

Since the impulse of the force in any element of time [delta]t has zero
moment about O, the same will be true of the additional momentum
generated. Hence the moment of the momentum (considered as a localized
vector) about O will be constant. In symbols, if v be the velocity and p
the perpendicular from O to the tangent to the path,

  pv = h,  (1)

where h is a constant. If [delta]s be an element of the path, p[delta]s
is twice the area enclosed by [delta]s and the radii drawn to its
extremities from O. Hence if [delta]A be this area, we have [delta]A = ½
p[delta]s = ½ h[delta]t, or

  dA
  -- = ½h.  (2)
  dt

Hence equal areas are swept over by the radius vector in equal times.

If P be the acceleration towards O, we have

    dv      dr
  v -- = -P --,  (3)
    ds      ds

since dr/ds is the cosine of the angle between the directions of r and
[delta]s. We will suppose that P is a function of r only; then
integrating (3) we find
            _
           /
  ½ v² = - | P dr + const.,  (4)
          _/

which is recognized as the equation of energy. Combining this with (1)
we have
              _
  h²         /
  -- = C - 2 | P dr,  (5)
  p²        _/

which completely determines the path except as to its orientation with
respect to O.

If the law of attraction be that of the inverse square of the distance,
we have P = [mu]/r², and

  h²        2[mu]
  --  = C + -----.  (6)
  p²        [tau]

Now in a conic whose focus is at O we have

   l     2    1
  --- = -- ± ---,  (7)
   p²    r    a

where l is half the latus-rectum, a is half the major axis, and the
upper or lower sign is to be taken according as the conic is an ellipse
or hyperbola. In the intermediate case of the parabola we have a = [oo]
and the last term disappears. The equations (6) and (7) are identified
by putting

  l = h²/[mu], a = ± [mu]/C.  (8)

Since

       h²        / 2     1 \
  v² = -- = [mu]( --- ± --- ),  (9)
       p²        \ r     a /

it appears that the orbit is an ellipse, parabola or hyperbola,
according as v² is less than, equal to, or greater than 2[mu]/r. Now it
appears from (6) that 2[mu]/r is the square of the velocity which would
be acquired by a particle falling from rest at infinity to the distance
r. Hence the character of the orbit depends on whether the velocity at
any point is less than, equal to, or greater than the _velocity from
infinity_, as it is called. In an elliptic orbit the area [pi]ab is
swept over in the time

      [pi]ab   2[pi]a^(3/2)
  r = ------ = ------------,  (10)
        ½h      [root][mu]

since h = [mu]^½ l^½ = [mu]^½ ba^-½ by (8).

  The converse problem, to determine the law of force under which a
  given orbit can be described about a given pole, is solved by
  differentiating (5) with respect to r; thus

        h² dp
    P = -----.  (11)
        p³ dr

  In the case of an ellipse described about the centre as pole we have

    a²b²
    ---- = a² + b² - r²;  (12)
     p²

  hence P = [mu]r, if [mu] = h²/a²b². This merely shows that a
  particular ellipse may be described under the law of the direct
  distance provided the circumstances of projection be suitably
  adjusted. But since an ellipse can always be constructed with a given
  centre so as to touch a given line at a given point, and to have a
  given value of ab (= h/[root][mu]) we infer that the orbit will be
  elliptic whatever the initial circumstances. Also the period is
  2[pi]ab/h = 2[pi]/[root][mu], as previously found.

  Again, in the equiangular spiral we have p = r sin[alpha], and
  therefore P = [mu]/r³, if [mu] = h²/sin²[alpha]. But since an
  equiangular spiral having a given pole is completely determined by a
  given point and a given tangent, this type of orbit is not a general
  one for the law of the inverse cube. In order that the spiral may be
  described it is necessary that the velocity of projection should be
  adjusted to make h = [root][mu]·sin[alpha]. Similarly, in the case of
  a circle with the pole on the circumference we have p² = r²/2a, P =
  [mu]/r^5, if [mu] = 8h²a²; but this orbit is not a general one for the
  law of the inverse fifth power.

[Illustration: FIG. 68.]

In astronomical and other investigations relating to central forces it
is often convenient to use polar co-ordinates with the centre of force
as pole. Let P, Q be the positions of a moving point at times t, t +
[delta]t, and write OP = r, OQ = r + [delta]r, [angle]POQ =
[delta][theta], O being any fixed origin. If u, v be the component
velocities at P along and perpendicular to OP (in the direction of
[theta] increasing), we have

          [delta]r   dr           r[delta][theta]     d[theta]
  u = lim.-------- = --, v = lim. --------------- = r --------.  (13)
          [delta]t   dt              [delta]t            dt

Again, the velocities parallel and perpendicular to OP change in the
time [delta]t from u, v to u - v[delta][theta], v + u[delta][theta],
ultimately. The component accelerations at P in these directions are
therefore

  du     d[theta]   d²r      /d[theta]\²      \
  -- - v -------- = --- - r ( -------- ),     |
  dt        dt      dt²      \  dt    /       |
                                               >  (14)
  dv     d[theta]     1   d   /   d[theta]\   |
  -- + u -------- =  --- --- ( r² -------- ), |
  dt        dt        r  dt   \      dt   /   /

respectively.

In the case of a central force, with O as pole, the transverse
acceleration vanishes, so that

  r²d[theta]/dt = h,  (15)

where h is constant; this shows (again) that the radius vector sweeps
over equal areas in equal times. The radial resolution gives

  d²r      /d[theta]\²
  --- - r ( -------- ) = -P,  (16)
  dt²      \   dt   /

where P, as before, denotes the acceleration towards O. If in this we
put r = 1/u, and eliminate t by means of (15), we obtain the general
differential equation of central orbits, viz.

     d²u           P
  --------- + u = ----.  (17)
  d[theta]²       h²u²

  If, for example, the law be that of the inverse square, we have P =
  [mu]u², and the solution is of the form

         [mu]
    u = ------ {1 + e cos ([theta] - [alpha])},  (18)
          h²

  where e, [alpha] are arbitrary constants. This is recognized as the
  polar equation of a conic referred to the focus, the half latus-rectum
  being h²/[mu].

  The law of the inverse cube P = [mu]u³ is interesting by way of
  contrast. The orbits may be divided into two classes according as h²
  <> [mu], i.e. according as the transverse velocity (hu) is greater or
  less than the velocity [root]([mu]·u) appropriate to a circular orbit
  at the same distance. In the former case the equation (17) takes the
  form

       d²u
    -------- + m²u = 0,  (19)
    d[theta]²

  the solution of which is

    au = sin m ([theta] - [alpha]).  (20)

  The orbit has therefore two asymptotes, inclined at an angle [pi]/m.
  In the latter case the differential equation is of the form

       d²u
    --------- = m²u,  (21)
    d[theta]²

  so that

    u = A e^(m[theta]) + B e^(-m[theta])  (22)

  If A, B have the same sign, this is equivalent to

    au = cosh m[theta],  (23)

  if the origin of [theta] be suitably adjusted; hence r has a maximum
  value [alpha], and the particle ultimately approaches the pole
  asymptotically by an infinite number of convolutions. If A, B have
  opposite signs the form is

    au = sinh m[theta],  (24)

  this has an asymptote parallel to [theta] = 0, but the path near the
  origin has the same general form as in the case of (23). If A or B
  vanish we have an equiangular spiral, and the velocity at infinity is
  zero. In the critical case of h² = [mu], we have d²u/d[theta]² = 0,
  and

    u = A[theta] + B;  (25)

  the orbit is therefore a "reciprocal spiral," except in the special
  case of A = 0, when it is a circle. It will be seen that unless the
  conditions be exactly adjusted for a circular orbit the particle will
  either recede to infinity or approach the pole asymptotically. This
  problem was investigated by R. Cotes (1682-1716), and the various
  curves obtained arc known as _Coles's spirals_.

A point on a central orbit where the radial velocity (dr/dt) vanishes is
called an _apse_, and the corresponding radius is called an _apse-line_.
If the force is always the same at the same distance any apse-line will
divide the orbit symmetrically, as is seen by imagining the velocity at
the apse to be reversed. It follows that the angle between successive
apse-lines is constant; it is called the _apsidal angle_ of the orbit.

If in a central orbit the velocity is equal to the velocity from
infinity, we have, from (5),
          _
  h²     / [oo]
  -- = 2 |    P dr;  (26)
  p²    _/ r

this determines the form of the critical orbit, as it is called. If P =
[mu]/r^[n], its polar equation is

  r^m cos m[theta] = a^m,  (27)

where m = ½(3 - n), except in the case n = 3, when the orbit is an
equiangular spiral. The case n = 2 gives the parabola as before.

  If we eliminate d[theta]/dt between (15) and (16) we obtain

    d²r   h²
    --- - -- = -P = -f(r),
    dt²   r³

  say. We may apply this to the investigation of the stability of a
  circular orbit. Assuming that r = a + x, where x is small, we have,
  approximately,

    d²x   h²  /     3x\
    --- - -- ( 1 - --  ) = -f(a) - xf´(a).
    dt²   r³  \     a /

  Hence if h and a be connected by the relation h² = a³f(a) proper to a
  circular orbit, we have
           _               _
    d²x   |          3      |
    --- + | f´(a) + --- f(a)| x = 0.  (28)
    dt²   |_         a     _|

  If the coefficient of x be positive the variations of x are
  simple-harmonic, and x can remain permanently small; the circular
  orbit is then said to be stable. The condition for this may be written
        _      _
     d |        |
    -- | a³f(a) | > 0,  (29)
    da |_      _|

  i.e. the intensity of the force in the region for which r = a, nearly,
  must diminish with increasing distance less rapidly than according to
  the law of the inverse cube. Again, the half-period of x is
  [pi]/sqrt[f´(a) + 3^{-1}f(a)], and since the angular velocity in the
  orbit is h/a², approximately, the apsidal angle is, ultimately,
               _                _
            / |       f(a)       |
    [pi]   /  |  --------------- |,  (30)
         \/   |_ af´(a) + 3f(a) _|

  or, in the case of f(a) = [mu]/r^n, [pi]/[root](3 - n). This is in
  agreement with the known results for n = 2, n = -1.

  We have seen that under the law of the inverse square all finite
  orbits are elliptical. The question presents itself whether there
  then is any other law of force, giving a finite velocity from
  infinity, under which all finite orbits are necessarily closed curves.
  If this is the case, the apsidal angle must evidently be commensurable
  with [pi], and since it cannot vary discontinuously the apsidal angle
  in a nearly circular orbit must be constant. Equating the expression
  (30) to [pi]/m, we find that f(a) = C/a^n, where n = 3 - m². The
  force must therefore vary as a power of the distance, and n must be
  less than 3. Moreover, the case n = 2 is the only one in which the
  critical orbit (27) can be regarded as the limiting form of a closed
  curve. Hence the only law of force which satisfies the conditions is
  that of the inverse square.

At the beginning of § 13 the velocity of a moving point P was
represented by a vector [->OV] drawn from a fixed origin O. The locus of
the point V is called the _hodograph_ (q.v.); and it appears that the
velocity of the point V along the hodograph represents in magnitude and
in direction the acceleration in the original orbit. Thus in the case of
a plane orbit, if v be the velocity of P, [psi] the inclination of the
direction of motion to some fixed direction, the polar co-ordinates of V
may be taken to be v, [psi]; hence the velocities of V along and
perpendicular to OV will be dv/dt and vd[psi]/dt. These expressions
therefore give the tangential and normal accelerations of P; cf. § 13
(12).

[Illustration: FIG. 69.]

  In the motion of a projectile under gravity the hodograph is a
  vertical line described with constant velocity. In elliptic harmonic
  motion the velocity of P is parallel and proportional to the
  semi-diameter CD which is conjugate to the radius CP; the hodograph is
  therefore an ellipse similar to the actual orbit. In the case of a
  central orbit described under the law of the inverse square we have v
  = h/SY = h. SZ/b², where S is the centre of force, SY is the
  perpendicular to the tangent at P, and Z is the point where YS meets
  the auxiliary circle again. Hence the hodograph is similar and
  similarly situated to the locus of Z (the auxiliary circle) turned
  about S through a right angle. This applies to an elliptic or
  hyperbolic orbit; the case of the parabolic orbit may be examined
  separately or treated as a limiting case. The annexed fig. 70 exhibits
  the various cases, with the hodograph in its proper orientation. The
  pole O of the hodograph is inside on or outside the circle, according
  as the orbit is an ellipse, parabola or hyperbola. In any case of a
  central orbit the hodograph (when turned through a right angle) is
  similar and similarly situated to the "reciprocal polar" of the orbit
  with respect to the centre of force. Thus for a circular orbit with
  the centre of force at an excentric point, the hodograph is a conic
  with the pole as focus. In the case of a particle oscillating under
  gravity on a smooth cycloid from rest at the cusp the hodograph is a
  circle through the pole, described with constant velocity.

§ 15. _Kinetics of a System of Discrete Particles._--The momenta of the
several particles constitute a system of localized vectors which, for
purposes of resolving and taking moments, may be reduced like a system
of forces in statics (§ 8). Thus taking any point O as base, we have
first a _linear momentum_ whose components referred to rectangular axes
through O are

  [Sigma](m[.x]), [Sigma](m[.y]), [Sigma](m[.z]);  (1)

its representative vector is the same whatever point O be chosen.
Secondly, we have an _angular momentum_ whose components are

  [Sigma]{m(y[.z] - z[.y])}, [Sigma]{m(z[.x] - xz[.z])}, [Sigma]{m(x[.y] - y[.x])},  (2)

these being the sums of the moments of the momenta of the several
particles about the respective axes. This is subject to the same
relations as a couple in statics; it may be represented by a vector
which will, however, in general vary with the position of O.

The linear momentum is the same as if the whole mass were concentrated
at the centre of mass G, and endowed with the velocity of this point.
This follows at once from equation (8) of § 11, if we imagine the two
configurations of the system there referred to to be those corresponding
to the instants t, t + [delta]t. Thus

   __  /   [->PP]  \      __     [->GG´]
  \   ( m·--------  )  = \   (m)·--------.  (3)
  /__  \  [delta]t /     /__     [delta]t

Analytically we have

                    d                           d[|x]
  [Sigma](m[.x]) = --- [Sigma](mx) = [Sigma](m)·-----.  (4)
                   dt                            dt

with two similar formulae.

[Illustration: FIG. 70.]

Again, if the instantaneous position of G be taken as base, the angular
momentum of the absolute motion is the same as the angular momentum of
the motion relative to G. For the velocity of a particle m at P may be
replaced by two components one of which (v) is identical in magnitude
and direction with the velocity of G, whilst the other (v) is the
velocity relative to G. The aggregate of the components mv of momentum
is equivalent to a single localized vector [Sigma](m)·v in a line
through G, and has therefore zero moment about any axis through G; hence
in taking moments about such an axis we need only regard the velocities
relative to G. In symbols, we have

                                          /  d[|z]     d[|y]\
  [Sigma]{m(y[.z] - z[.y])} = [Sigma](m)·( y ----- - z ----- ) + [Sigma]{m([eta][zeta] - [.zeta][eta])}.  (5)
                                          \    dt        dt /

since [Sigma](m[xi]) = 0, [Sigma](m[xi]) = 0, and so on, the notation
being as in § 11. This expresses that the moment of momentum about any
fixed axis (e.g. Ox) is equal to the moment of momentum of the motion
relative to G about a parallel axis through G, together with the moment
of momentum of the whole mass supposed concentrated at G and moving with
this point. If in (5) we make O coincide with the instantaneous position
of G, we have [|x], [|y], [|z] = 0, and the theorem follows.

[Illustration: FIG. 71.]

Finally, the rates of change of the components of the angular momentum
of the motion relative to G referred to G as a moving base, are equal to
the rates of change of the corresponding components of angular momentum
relative to a fixed base coincident with the instantaneous position of
G. For let G´ be a consecutive position of G. At the instant t +
[delta]t the momenta of the system are equivalent to a linear momentum
represented by a localized vector [Sigma](m)·(v + [delta]v) in a line
through G´ tangential to the path of G´, together with a certain angular
momentum. Now the moment of this localized vector with respect to any
axis through G is zero, to the first order of [delta]t, since the
perpendicular distance of G from the tangent line at G´ is of the order
([delta]t)². Analytically we have from (5),

   d                                            /  d[|z]²     d²[|y] \     d
  --- [Sigma] {m (y[.z] - z[.y])} = [Sigma](m)·( y ------ - z ------- ) + --- [Sigma] {m([eta][zeta - [zeta][.eta])}  (6)
  dt                                            \    dt²        dt²  /    dt

If we put x, y, z = 0, the theorem is proved as regards axes parallel to
Ox.

Next consider the kinetic energy of the system. If from a fixed point O
we draw vectors [->OV1], [->OV2] to represent the velocities of the
several particles m1, m2 ..., and if we construct the vector


           [Sigma](m·[->OV])
  [->OK] = -----------------  (7)
               [Sigma](m)

this will represent the velocity of the mass-centre, by (3). We find,
exactly as in the proof of Lagrange's First Theorem (§ 11), that

  ½[Sigma](m·OV²) = ½[Sigma](m)·OK² + ½[Sigma](m·KV²);  (8)

i.e. the total kinetic energy is equal to the kinetic energy of the
whole mass supposed concentrated at G and moving with this point,
together with the kinetic energy of the motion relative to G. The latter
may be called the _internal kinetic energy_ of the system. Analytically
we have
                                                    _                                 _
                                                   |  /d[|x]\²    /d[|y]\²    /d[|z]\  |
  ½[Sigma]{m([.x]² + [.y]² + [.z]²)} = ½[Sigma](m)·| ( ----- ) + ( ----- ) + ( ----- ) |
                                                   |_ \ dt  /     \ dt  /     \ dt  / _|

                                     + ½[Sigma] {m([zeta]² + [.eta]² + [zeta]²)}.  (9)

There is also an analogue to Lagrange's Second Theorem, viz.

                      [Sigma][Sigma] (m_p m_q·V_p V_q²)
  ½[Sigma](m·KV²) = ½ ---------------------------------  (10)
                                   [Sigma]m

which expresses the internal kinetic energy in terms of the relative
velocities of the several pairs of particles. This formula is due to
Möbius.

The preceding theorems are purely kinematical. We have now to consider
the effect of the forces acting on the particles. These may be divided
into two categories; we have first, the _extraneous forces_ exerted on
the various particles from without, and, secondly, the mutual or
_internal forces_ between the various pairs of particles. It is assumed
that these latter are subject to the law of equality of action and
reaction. If the equations of motion of each particle be formed
separately, each such internal force will appear twice over, with
opposite signs for its components, viz. as affecting the motion of each
of the two particles between which it acts. The full working out is in
general difficult, the comparatively simple problem of "three bodies,"
for instance, in gravitational astronomy being still unsolved, but some
general theorems can be formulated.

The first of these may be called the _Principle of Linear Momentum_. If
there are no extraneous forces, the resultant linear momentum is
constant in every respect. For consider any two particles at P and Q,
acting on one another with equal and opposite forces in the line PQ. In
the time [delta]t a certain impulse is given to the first particle in
the direction (say) from P to Q, whilst an equal and opposite impulse is
given to the second in the direction from Q to P. Since these impulses
produce equal and opposite momenta in the two particles, the resultant
linear momentum of the system is unaltered. If extraneous forces act, it
is seen in like manner that the resultant linear momentum of the system
is in any given time modified by the geometric addition of the total
impulse of the extraneous forces. It follows, by the preceding kinematic
theory, that the mass-centre G of the system will move exactly as if the
whole mass were concentrated there and were acted on by the extraneous
forces applied parallel to their original directions. For example, the
mass-centre of a system free from extraneous force will describe a
straight line with constant velocity. Again, the mass-centre of a chain
of particles connected by strings, projected anyhow under gravity, will
describe a parabola.

The second general result is the _Principle of Angular Momentum_. If
there are no extraneous forces, the moment of momentum about any fixed
axis is constant. For in time [delta]t the mutual action between two
particles at P and Q produces equal and opposite momenta in the line PQ,
and these will have equal and opposite moments about the fixed axis. If
extraneous forces act, the total angular momentum about any fixed axis
is in time [delta]t increased by the total extraneous impulse about that
axis. The kinematical relations above explained now lead to the
conclusion that in calculating the effect of extraneous forces in an
infinitely short time [delta]t we may take moments about an axis passing
through the instantaneous position of G exactly as if G were fixed;
moreover, the result will be the same whether in this process we employ
the true velocities of the particles or merely their velocities relative
to G. If there are no extraneous forces, or if the extraneous forces
have zero moment about any axis through G, the vector which represents
the resultant angular momentum relative to G is constant in every
respect. A plane through G perpendicular to this vector has a fixed
direction in space, and is called the _invariable plane_; it may
sometimes be conveniently used as a plane of reference.

  For example, if we have two particles connected by a string, the
  invariable plane passes through the string, and if [omega] be the
  angular velocity in this plane, the angular momentum relative to G is

    m1[omega]1r1·r1 + m2[omega]r2·r2 = (m1r1² + m2r2²)[omega],

  where r1, r2 are the distances of m1, m2 from their mass-centre G.
  Hence if the extraneous forces (e.g. gravity) have zero moment about
  G, [omega] will be constant. Again, the tension R of the string is
  given by

                        m1m2
    R = m1[omega]²r1 = ------- [omega]²a,
                       m1 + m2

  where a = r1 + r2. Also by (10) the internal kinetic energy is

       m1m2
    ½ ------- [omega]²a².
      m1 + m2

The increase of the kinetic energy of the system in any interval of time
will of course be equal to the total work done by all the forces acting
on the particles. In many questions relating to systems of discrete
particles the internal force R_pq (which we will reckon positive when
attractive) between any two particles m_p, m_q is a function only of the
distance r_pq between them. In this case the work done by the internal
forces will be represented by
            _
           /
  -[Sigma] | R_(pg) dr_(pq),
          _/

when the summation includes every pair of particles, and each integral
is to be taken between the proper limits. If we write
               _
              /
  V = [Sigma] | R_(pq) dr_(pq),  (11)
             _/

when r_pq ranges from its value in some standard configuration A of the
system to its value in any other configuration P, it is plain that V
represents the work which would have to be done in order to bring the
system from rest in the configuration A to rest in the configuration P.
Hence V is a definite function of the configuration P; it is called the
_internal potential energy_. If T denote the kinetic energy, we may say
then that the sum T + V is in any interval of time increased by an
amount equal to the work done by the extraneous forces. In particular,
if there are no extraneous forces T + V is constant. Again, if some of
the extraneous forces are due to a conservative field of force, the work
which they do may be reckoned as a diminution of the potential energy
relative to the field as in § 13.

§ 16. _Kinetics of a Rigid Body. Fundamental Principles._--When we pass
from the consideration of discrete particles to that of continuous
distributions of matter, we require some physical postulate over and
above what is contained in the Laws of Motion, in their original
formulation. This additional postulate may be introduced under various
forms. One plan is to assume that any body whatever may be treated as if
it were composed of material particles, i.e. mathematical points endowed
with inertia coefficients, separated by finite intervals, and acting on
one another with forces in the lines joining them subject to the law of
equality of action and reaction. In the case of a rigid body we must
suppose that those forces adjust themselves so as to preserve the mutual
distances of the various particles unaltered. On this basis we can
predicate the principles of linear and angular momentum, as in § 15.

An alternative procedure is to adopt the principle first formally
enunciated by J. Le R. d'Alembert and since known by his name. If x, y,
z be the rectangular co-ordinates of a mass-element m, the expressions
m[:x], m[:y], m[:z] must be equal to the components of the total force
on m, these forces being partly extraneous and partly forces exerted on
m by other mass-elements of the system. Hence (m[:x], m[:y], m[:z]) is
called the actual or _effective_ force on m. According to d'Alembert's
formulation, the extraneous forces together with the _effective forces
reversed_ fulfil the statical conditions of equilibrium. In other words,
the whole assemblage of effective forces is statically equivalent to the
extraneous forces. This leads, by the principles of § 8, to the
equations

  [Sigma](m[:x]) = X, [Sigma](m[:y]) = Y, [Sigma](m[:z]) = Z,                               \
                                                                                             >  (1)
  [Sigma]{m(y[:z] - z[:y]) = L, [Sigma]{m(z[:x] - x[:z]) = M, [Sigma]{m(x[:y] - y[:x]) = N, /

where (X, Y, Z) and (L, M, N) are the force--and couple--constituents of
the system of extraneous forces, referred to O as base, and the
summations extend over all the mass-elements of the system. These
equations may be written

   d                       d                       d
  --- [Sigma](m[.x]) = X, --- [Sigma](m[.y]) = Y, --- [Sigma](m[.z]) = Z,                             \
  dt                      dt                      dt                                                   |                                                                   } (2)
                                                                                                        >  (2)
   d                                 d                               d                                 |
  --- [Sigma]{m(y[.z] - z[.y]) = L, --- [Sigma]{m(z[.x]-x[.z]) = M, --- [Sigma]{m(x[.y] - y[.x]) = N, /
  dt                                dt                              dt

and so express that the rate of change of the linear momentum in any
fixed direction (e.g. that of Ox) is equal to the total extraneous force
in that direction, and that the rate of change of the angular momentum
about any fixed axis is equal to the moment of the extraneous forces
about that axis. If we integrate with respect to t between fixed limits,
we obtain the principles of linear and angular momentum in the form
previously given. Hence, whichever form of postulate we adopt, we are
led to the principles of linear and angular momentum, which form in fact
the basis of all our subsequent work. It is to be noticed that the
preceding statements are not intended to be restricted to rigid bodies;
they are assumed to hold for all material systems whatever. The peculiar
status of rigid bodies is that the principles in question are in most
cases sufficient for the complete determination of the motion, the
dynamical equations (1 or 2) being equal in number to the degrees of
freedom (six) of a rigid solid, whereas in cases where the freedom is
greater we have to invoke the aid of other supplementary physical
hypotheses (cf. ELASTICITY; HYDROMECHANICS).

The increase of the kinetic energy of a rigid body in any interval of
time is equal to the work done by the extraneous forces acting on the
body. This is an immediate consequence of the fundamental postulate, in
either of the forms above stated, since the internal forces do on the
whole no work. The statement may be extended to a system of rigid
bodies, provided the mutual reactions consist of the stresses in
inextensible links, or the pressures between smooth surfaces, or the
reactions at rolling contacts (§ 9).

§ 17. _Two-dimensional Problems._--In the case of rotation about a fixed
axis, the principles take a very simple form. The position of the body
is specified by a single co-ordinate, viz. the angle [theta] through
which some plane passing through the axis and fixed in the body has
turned from a standard position in space. Then d[theta]/dt, = [omega]
say, is the _angular velocity_ of the body. The angular momentum of a
particle m at a distance r from the axis is m[omega]r·r, and the total
angular momentum is [Sigma](mr²)·[omega], or I[omega], if I denote the
moment of inertia (§ 11) about the axis. Hence if N be the moment of the
extraneous forces about the axis, we have

   d
  --- (I[omega]) = N.  (1)
  dt

This may be compared with the equation of rectilinear motion of a
particle, viz. d/dt·(Mu) = X; it shows that I measures the inertia of
the body as regards rotation, just as M measures its inertia as regards
translation. If N = 0, [omega] is constant.

[Illustration: FIG. 72.]

[Illustration: FIG. 73.]

  As a first example, suppose we have a flywheel free to rotate about a
  horizontal axis, and that a weight m hangs by a vertical string from
  the circumferences of an axle of radius b (fig. 72). Neglecting
  frictional resistance we have, if R be the tension of the string,

    I[.omega] = Rb, m[.u] = mg - R,

  whence
                  mb²
    b[.omega] = -------  (2)
                1 + mb²

  This gives the acceleration of m as modified by the inertia of the
  wheel.

  A "compound pendulum" is a body of any form which is free to rotate
  about a fixed horizontal axis, the only extraneous force (other than
  the pressures of the axis) being that of gravity. If M be the total
  mass, k the radius of gyration (§ 11) about the axis, we have

     d   /    d[theta]\
    --- ( Mk² -------- )  = -Mgh sin [theta],  (3)
    dt   \       dt   /

  where [theta] is the angle which the plane containing the axis and the
  centre of gravity G makes with the vertical, and h is the distance of
  G from the axis. This coincides with the equation of motion of a
  simple pendulum [§ 13 (15)] of length l, provided l = k²/h. The plane
  of the diagram (fig. 73) is supposed to be a plane through G
  perpendicular to the axis, which it meets in O. If we produce OG to P,
  making OP = l, the point P is called the _centre of oscillation_; the
  bob of a simple pendulum of length OP suspended from O will keep step
  with the motion of P, if properly started. If [kappa] be the radius of
  gyration about a parallel axis through G, we have k² = [kappa]² + h²
  by § 11 (16), and therefore l = h + [kappa]²/h, whence

    GO·GP = [kappa]².  (4)

  This shows that if the body were swung from a parallel axis through P
  the new centre of oscillation would be at O. For different parallel
  axes, the period of a small oscillation varies as [root]l, or
  [root](GO + OP); this is least, subject to the condition (4), when GO
  = GP = [kappa]. The reciprocal relation between the centres of
  suspension and oscillation is the basis of Kater's method of
  determining g experimentally. A pendulum is constructed with two
  parallel knife-edges as nearly as possible in the same plane with G,
  the position of one of them being adjustable. If it could be arranged
  that the period of a small oscillation should be exactly the same
  about either edge, the two knife-edges would in general occupy the
  positions of conjugate centres of suspension and oscillation; and the
  distances between them would be the length l of the equivalent simple
  pendulum. For if h1 + [kappa]²/h1 = h2 + [kappa]²/h2, then unless h1 =
  h2, we must have [kappa]² = h1h2, l = h1 + h2. Exact equality of the
  two observed periods ([tau]1, [tau]2, say) cannot of course be secured
  in practice, and a modification is necessary. If we write l1 = h1 +
  [kappa]²/h1, l2 = h2 + [kappa]²/h2, we find, on elimination of
  [kappa],

      l1 + l2     l1 - l2
    ½ ------- + ½ ------- = 1,
      h1 + h2     h1 - h2

  whence

    4[pi]²   ½ ([tau]1² + [tau]2²)   ½ ([tau]1² - [tau]2²)
    ------ = --------------------- + ---------------------  (5)
      g             h1 + h2                  h1 - h2

  The distance h1 + h2, which occurs in the first term on the right hand
  can be measured directly. For the second term we require the values of
  h1, h2 separately, but if [tau]1, [tau]2 are nearly equal whilst h1,
  h2 are distinctly unequal this term will be relatively small, so that
  an approximate knowledge of h1, h2 is sufficient.

  As a final example we may note the arrangement, often employed in
  physical measurements, where a body performs small oscillations about
  a vertical axis through its mass-centre G, under the influence of a
  couple whose moment varies as the angle of rotation from the
  equilibrium position. The equation of motion is of the type

    I[:theta] = -K[theta],  (6)

  and the period is therefore [tau] = 2[pi][root](I/K). If by the
  attachment of another body of known moment of inertia I´, the period
  is altered from [tau] to [tau]´, we have [tau]´ = 2[pi][root][(I +
  I´)/K]. We are thus enabled to determine both I and K, viz.

    I/I´ = [tau]²/([tau]´² - [tau]²), K = 4[pi]²[tau]²I/([tau]´² - [tau]²).  (7)

  The couple may be due to the earth's magnetism, or to the torsion of
  a suspending wire, or to a "bifilar" suspension. In the latter case,
  the body hangs by two vertical threads of equal length l in a plane
  through G. The motion being assumed to be small, the tensions of the
  two strings may be taken to have their statical values Mgb/(a + b),
  Mga/(a + b), where a, b are the distances of G from the two threads.
  When the body is twisted through an angle [theta] the threads make
  angles a[theta]/l, b[theta]/l with the vertical, and the moment of the
  tensions about the vertical through G is accordingly -K[theta], where
  K = M gab/l.

For the determination of the motion it has only been necessary to use
one of the dynamical equations. The remaining equations serve to
determine the reactions of the rotating body on its bearings. Suppose,
for example, that there are no extraneous forces. Take rectangular axes,
of which Oz coincides with the axis of rotation. The angular velocity
being constant, the effective force on a particle m at a distance r from
Oz is m[omega]²r towards this axis, and its components are accordingly
-[omega]²mx, -[omega]²my, O. Since the reactions on the bearings must be
statically equivalent to the whole system of effective forces, they will
reduce to a force (X Y Z) at O and a couple (L M N) given by

  X = -[omega]²[Sigma](mx) = -[omega]²[Sigma](m)[|x], Y = -[omega]²[Sigma](my) = -[omega]²[Sigma](m)[|y], Z = 0,

                         L = [omega]²[Sigma](myz), M = -[omega]²[Sigma](mzx), N = 0, (8)


where [|x], [|y] refer to the mass-centre G. The reactions do not
therefore reduce to a single force at O unless [Sigma](myz) = 0,
[Sigma](msx) = 0, i.e. unless the axis of rotation be a principal axis
of inertia (§ 11) at O. In order that the force may vanish we must also
have x, y = 0, i.e. the mass-centre must lie in the axis of rotation.
These considerations are important in the "balancing" of machinery. We
note further that if a body be free to turn about a fixed point O, there
are three mutually perpendicular lines through this point about which it
can rotate steadily, without further constraint. The theory of principal
or "permanent" axes was first investigated from this point of view by J.
A. Segner (1755). The origin of the name "deviation moment" sometimes
applied to a product of inertia is also now apparent.

[Illustration: FIG. 74.]

Proceeding to the general motion of a rigid body in two dimensions we
may take as the three co-ordinates of the body the rectangular Cartesian
co-ordinates x, y of the mass-centre G and the angle [theta] through
which the body has turned from some standard position. The components of
linear momentum are then M[.x], M[.y], and the angular momentum relative
to G as base is I[.theta], where M is the mass and I the moment of
inertia about G. If the extraneous forces be reduced to a force (X, Y)
at G and a couple N, we have

  M[:x] = X, M[:y] = Y, I[:theta] = N.  (9)

If the extraneous forces have zero moment about G the angular velocity
[.theta] is constant. Thus a circular disk projected under gravity in a
vertical plane spins with constant angular velocity, whilst its centre
describes a parabola.

  We may apply the equations (9) to the case of a solid of revolution
  rolling with its axis horizontal on a plane of inclination [alpha]. If
  the axis of x be taken parallel to the slope of the plane, with x
  increasing downwards, we have

    M[:x] = Mg sin [alpha] - F, 0 = Mg cos [alpha] - R, M[kappa]²[:theta] = Fa  (10)

  where [kappa] is the radius of gyration about the axis of symmetry, a
  is the constant distance of G from the plane, and R, F are the normal
  and tangential components of the reaction of the plane, as shown in
  fig. 74. We have also the kinematical relation [.x] = a[.theta]. Hence

                 a²                                                [kappa]²
    [:x] = ------------- g sin [alpha], R = Mg cos [alpha], F = ------------- Mg sin [alpha].  (11)
           [kappa]² + a²                                        [kappa]² + a²

  The acceleration of G is therefore less than in the case of
  frictionless sliding in the ratio a²/([kappa]² + a²). For a
  homogeneous sphere this ratio is 5/7, for a uniform circular cylinder
  or disk 2/3, for a circular hoop or a thin cylindrical shell ½.

The equation of energy for a rigid body has already been stated (in
effect) as a corollary from fundamental assumptions. It may also be
deduced from the principles of linear and angular momentum as embodied
in the equations (9). We have

  M([.x][:x] + [.y][:]y) + l[.theta][:theta] + X[.x] + Y[.y] + N[.theta],  (12)

whence, integrating with respect to t,

  ½ M([.x]² + [.y]²) + ½I[.theta]² = [int](X dx + Y dy + Nd[theta]) + const.  (13)

The left-hand side is the kinetic energy of the whole mass, supposed
concentrated at G and moving with this point, together with the kinetic
energy of the motion relative to G (§ 15); and the right-hand member
represents the integral work done by the extraneous forces in the
successive infinitesimal displacements into which the motion may be
resolved.

[Illustration: FIG. 75.]

  The formula (13) may be easily verified in the case of the compound
  pendulum, or of the solid rolling down an incline. As another example,
  suppose we have a circular cylinder whose mass-centre is at an
  excentric point, rolling on a horizontal plane. This includes the case
  of a compound pendulum in which the knife-edge is replaced by a
  cylindrical pin. If [alpha] be the radius of the cylinder, h the
  distance of G from its axis (O), [kappa] the radius of gyration about
  a longitudinal axis through G, and [theta] the inclination of OG to
  the vertical, the kinetic energy is 1/2M[kappa]²[.theta]² +
  ½M·CG²·[.theta]², by § 3, since the body is turning about the line of
  contact (C) as instantaneous axis, and the potential energy is--Mgh
  cos [theta]. The equation of energy is therefore

    ½ M([kappa]² + [alpha]² + h² - 2 ah cos [theta]) [.theta]² - Mgh cos [theta] - const.  (14)

Whenever, as in the preceding examples, a body or a system of bodies, is
subject to constraints which leave it virtually only one degree of
freedom, the equation of energy is sufficient for the complete
determination of the motion. If q be any variable co-ordinate defining
the position or (in the case of a system of bodies) the configuration,
the velocity of each particle at any instant will be proportional to
[.q], and the total kinetic energy may be expressed in the form ½A[.q]²,
where A is in general a function of q [cf. equation (14)]. This
coefficient A is called the coefficient of inertia, or the reduced
inertia of the system, referred to the co-ordinate q.

[Illustration: FIG. 76.]

  Thus in the case of a railway truck travelling with velocity u the
  kinetic energy is ½(M + m[kappa]²/[alpha]²)u², where M is the total
  mass, [alpha] the radius and [kappa] the radius of gyration of each
  wheel, and m is the sum of the masses of the wheels; the reduced
  inertia is therefore M + m[kappa]²/[alpha]². Again, take the system
  composed of the flywheel, connecting rod, and piston of a
  steam-engine. We have here a limiting case of three-bar motion (§ 3),
  and the instantaneous centre J of the connecting-rod PQ will have the
  position shown in the figure. The velocities of P and Q will be in the
  ratio of JP to JQ, or OR to OQ; the velocity of the piston is
  therefore y[.theta], where y = OR. Hence if, for simplicity, we
  neglect the inertia of the connecting-rod, the kinetic energy will be
  ½(I + My²)[.theta]², where I is the moment of inertia of the flywheel,
  and M is the mass of the piston. The effect of the mass of the piston
  is therefore to increase the apparent moment of inertia of the
  flywheel by the variable amount My². If, on the other hand, we take OP
  (= x) as our variable, the kinetic energy is 1/2(M + I/y²)[.x]². We
  may also say, therefore, that the effect of the flywheel is to
  increase the apparent mass of the piston by the amount I/y²; this
  becomes infinite at the "dead-points" where the crank is in line with
  the connecting-rod.

If the system be "conservative," we have

  ½ Aq² + V = const.,  (15)

where V is the potential energy. If we differentiate this with respect
to t, and divide out by [.q], we obtain

            dA      dV
  A[:q] + ½ -- q² + -- = 0  (16)
            dq      dq

as the equation of motion of the system with the unknown reactions (if
any) eliminated. For equilibrium this must be satisfied by [.q] = O;
this requires that dV/dq = 0, i.e. the potential energy must be
"stationary." To examine the effect of a small disturbance from
equilibrium we put V = f(q), and write q = q0 + [eta], where q0 is a
root of f´(q0) = 0 and [eta] is small. Neglecting terms of the second
order in [eta] we have dV/dq = f´(q) = f´´(q0)·[eta], and the equation
(16) reduces to

  A[:eta] + f´´(q0)[eta] = 0,  (17)

where A may be supposed to be constant and to have the value
corresponding to q = q0. Hence if f´´(q0) > 0, i.e. if V is a minimum in
the configuration of equilibrium, the variation of [eta] is
simple-harmonic, and the period is 2[pi][root][A/f´´(q0)]. This depends
only on the constitution of the system, whereas the amplitude and epoch
will vary with the initial circumstances. If f´´(q0) < 0, the solution
of (17) will involve real exponentials, and [eta] will in general
increase until the neglect of the terms of the second order is no longer
justified. The configuration q = q0, is then unstable.

  As an example of the method, we may take the problem to which equation
  (14) relates. If we differentiate, and divide by [theta], and retain
  only the terms of the first order in [theta], we obtain

    {x² + (h - [alpha])²} [:theta] + gh[theta] = 0,  (18)

  as the equation of small oscillations about the position [theta] = 0.
  The length of the equivalent simple pendulum is {[kappa]² + (h -
  [alpha])²}/h.

The equations which express the change of motion (in two dimensions) due
to an instantaneous impulse are of the forms

  M(u´- u) = [xi], M([nu]´ - [nu]) = [eta], I([omega]´ - [omega]) = [nu].  (19)

[Illustration: FIG. 77.]

Here u´, [nu]´ are the values of the component velocities of G just
before, and u, [nu] their values just after, the impulse, whilst
[omega]´, [omega] denote the corresponding angular velocities. Further,
[xi], [eta] are the time-integrals of the forces parallel to the
co-ordinate axes, and [nu] is the time-integral of their moment about G.
Suppose, for example, that a rigid lamina at rest, but free to move, is
struck by an instantaneous impulse F in a given line. Evidently G will
begin to move parallel to the line of F; let its initial velocity be u´,
and let [omega]´ be the initial angular velocity. Then Mu´ = F,
I[omega]´ = F·GP, where GP is the perpendicular from G to the line of F.
If PG be produced to any point C, the initial velocity of the point C of
the lamina will be

  u´ - [omega]´·GC = (F/M)·(I - GC·CP/[kappa]²),

where [kappa]² is the radius of gyration about G. The initial centre of
rotation will therefore be at C, provided GC·GP = [kappa]². If this
condition be satisfied there would be no impulsive reaction at C even if
this point were fixed. The point P is therefore called the _centre of
percussion_ for the axis at C. It will be noted that the relation
between C and P is the same as that which connects the centres of
suspension and oscillation in the compound pendulum.

§ 18. _Equations of Motion in Three Dimensions._--It was proved in § 7
that a body moving about a fixed point O can be brought from its
position at time t to its position at time t + [delta]t by an
infinitesimal rotation [epsilon] about some axis through O; and the
limiting position of this axis, when [delta]t is infinitely small, was
called the "instantaneous axis." The limiting value of the ratio
[epsilon]/[delta]t is called the _angular velocity_ of the body; we
denote it by [omega]. If [xi], [eta], [zeta] are the components of
[epsilon] about rectangular co-ordinate axes through O, the limiting
values of [xi]/[delta]t, [eta]/[delta]t, [zeta]/[delta]t are called the
_component angular velocities_; we denote them by p, q, r. If l, m, n be
the direction-cosines of the instantaneous axis we have

  p = l[omega], q = m[omega], r = n[omega],  (1)
  p² + q² + r² = [omega]².  (2)

If we draw a vector OJ to represent the angular velocity, then J traces
out a certain curve in the body, called the _polhode_, and a certain
curve in space, called the _herpolhode_. The cones generated by the
instantaneous axis in the body and in space are called the polhode and
herpolhode cones, respectively; in the actual motion the former cone
rolls on the latter (§ 7).

[Illustration: FIG. 78.]

  The special case where both cones are right circular and [omega] is
  constant is important in astronomy and also in mechanism (theory of
  bevel wheels). The "precession of the equinoxes" is due to the fact
  that the earth performs a motion of this kind about its centre, and
  the whole class of such motions has therefore been termed
  _precessional_. In fig. 78, which shows the various cases, OZ is the
  axis of the fixed and OC that of the rolling cone, and J is the point
  of contact of the polhode and herpolhode, which are of course both
  circles. If [alpha]be the semi-angle of the rolling cone, [beta] the
  constant inclination of OC to OZ, and [.psi] the angular velocity with
  which the plane ZOC revolves about OZ, then, considering the velocity
  of a point in OC at unit distance from O, we have

    [omega] sin [alpha] = ±[.psi] sin [beta],  (3)

  where the lower sign belongs to the third case. The earth's
  precessional motion is of this latter type, the angles being [alpha] =
  .0087´´, [beta] = 23° 28´.

If m be the mass of a particle at P, and PN the perpendicular to the
instantaneous axis, the kinetic energy T is given by

  2T = [Sigma] {m([omega]·PN)²} = [omega]²·[Sigma](m·PN²) = I[omega]²,  (4)

where I is the moment of inertia about the instantaneous axis. With the
same notation for moments and products of inertia as in § 11 (38), we
have

  I = Al² + Bm² + Cn² - 2Fmn - 2Gnl - 2Hlm,

and therefore by (1),

  2T = Ap² + Bq² + Cr² - 2Fqr - 2Grp - 2Hpq.  (5)

Again, if x, y, z be the co-ordinates of P, the component velocities of
m are

  qz - ry, rx - pz, py - qx,  (6)

by § 7 (5); hence, if [lambda], [mu], [nu] be now used to denote the
component angular momenta about the co-ordinate axes, we have [lambda] =
[Sigma][m(py - qx)y - m(rx - pz)z], with two similar formulae, or

                           [dP]T  \
  [lambda] = Ap - Hq - Gr= -----,  |
                           [dP]p   |
                                   |
                         [dP]T     |
  [mu] = -Hp + Bq - Fr = -----,     >  (7)
                         [dP]q     |
                                   |
                         [dP]T     |
  [nu] = -Gp - Fq + Cr = -----.    |
                         [dP]r    /

If the co-ordinate axes be taken to coincide with the principal axes of
inertia at O, at the instant under consideration, we have the simpler
formulae

  2T = Ap² + Bq² + Cr²,  (8)

  [lambda] = Ap, [mu] = Bq, [nu] = Cr.  (9)

It is to be carefully noticed that the axis of resultant angular
momentum about O does not in general coincide with the instantaneous
axis of rotation. The relation between these axes may be expressed by
means of the momental ellipsoid at O. The equation of the latter,
referred to its principal axes, being as in § 11 (41), the co-ordinates
of the point J where it is met by the instantaneous axis are
proportional to p, q, r, and the direction-cosines of the normal at J
are therefore proportional to Ap, Bq, Cr, or [lambda], [mu], [nu]. The
axis of resultant angular momentum is therefore normal to the tangent
plane at J, and does not coincide with OJ unless the latter be a
principal axis. Again, if [Gamma] be the resultant angular momentum, so
that

  [lambda]² + [mu]² + [nu]² = [Gamma]²,  (10)

the length of the perpendicular OH on the tangent plane at J is

         Ap         p             Bq         q             Cr         r             2T       [rho]
  OH = ------- · -------[rho] + ------- · -------[rho] + ------- · -------[rho] = ------- · -------,  (11)
       [Gamma]   [omega]        [Gamma]   [omega]        [Gamma]   [omega]        [Gamma]   [omega]

where [rho] = OJ. This relation will be of use to us presently (§ 19).

The motion of a rigid body in the most general case may be specified by
means of the component velocities u, v, w of any point O of it which is
taken as base, and the component angular velocities p, q, r. The
component velocities of any point whose co-ordinates relative to O are
x, y, z are then

  u + qz - ry, v + rx - pz, w + py - qx  (12)

by § 7 (6). It is usually convenient to take as our base-point the
mass-centre of the body. In this case the kinetic energy is given by

  2T = M0(u² + v² + w²) + Ap² + Bq² + Cr² - 2Fqr - 2Grp - 2Hpg,  (13)

where M0 is the mass, and A, B, C, F, G, H are the moments and products
of inertia with respect to the mass-centre; cf. § 15 (9).

The components [xi], [eta], [zeta] of linear momentum are

               [dP]T                [dP]T                 [dP]T
  [xi] = M0u = -----, [eta] = M0v = -----, [zeta] = M0w = -----,  (14)
               [dP]u                [dP]v                 [dP]w

whilst those of the relative angular momentum are given by (7). The
preceding formulae are sufficient for the treatment of instantaneous
impulses. Thus if an impulse ([xi], [eta], [zeta], [lambda], [mu], [nu])
change the motion from (u, v, w, p, q, r) to (u´, v´, w´, p´, q´, r´) we
have

  M0(u´- u) = [xi], M0(v´- v) = [eta], M0(w´- w) = [zeta], \
                                                            >  (15)
  A(p´ - p) = [lambda], B(q´- q) = [mu], C(r´- r) = [nu],  /

where, for simplicity, the co-ordinate axes are supposed to coincide
with the principal axes at the mass-centre. Hence the change of kinetic
energy is

  T´- T = [xi] · ½(u + u´) + [eta] · ½(v + v´) + [zeta] · ½(w + w´),
        + [lambda] · ½(p + p´) + [mu] · ½(q + q´) + [nu] · ½(r + r´).  (16)

The factors of [xi], [eta], [zeta], [lambda], [mu], [nu] on the
right-hand side are proportional to the constituents of a possible
infinitesimal displacement of the solid, and the whole expression is
proportional (on the same scale) to the work done by the given system of
impulsive forces in such a displacement. As in § 9 this must be equal to
the total work done in such a displacement by the several forces,
whatever they are, which make up the impulse. We are thus led to the
following statement: the change of kinetic energy due to any system of
impulsive forces is equal to the sum of the products of the several
forces into the semi-sum of the initial and final velocities of their
respective points of application, resolved in the directions of the
forces. Thus in the problem of fig. 77 the kinetic energy generated is
½M([kappa]² + Cq²)[omega]´², if C be the instantaneous centre; this is
seen to be equal to ½F·[omega]´·CP, where [omega]´·CP represents the
initial velocity of P.

The equations of continuous motion of a solid are obtained by
substituting the values of [xi], [eta], [zeta], [lambda], [mu], [nu]
from (14) and (7) in the general equations

  d[xi]      d[eta]      d[zeta]        \
  ----- = X, ------ = Y, ------- = Z,   |
   dt         dt           dt           |
                                         >  (17)
  d[lambda]      d[mu]       d[nu]      |
  --------- = L, ----- = M,  ----- = N, |
      dt           dt         dt        /

where (X, Y, Z, L, M, N) denotes the system of extraneous forces
referred (like the momenta) to the mass-centre as base, the co-ordinate
axes being of course fixed in direction. The resulting equations are not
as a rule easy of application, owing to the fact that the moments and
products of inertia A, B, C, F, G, H are not constants but vary in
consequence of the changing orientation of the body with respect to the
co-ordinate axes.

[Illustration: FIG. 79.]

  An exception occurs, however, in the case of a solid which is
  kinetically symmetrical (§ 11) about the mass-centre, e.g. a uniform
  sphere. The equations then take the forms

    M0[.u] = X, M0[.v] = Y, M0[.w] = Z,
    C[.p] = L, C[.q] = M, C[.r] = N,  (18)

  where C is the constant moment of inertia about any axis through the
  mass-centre. Take, for example, the case of a sphere rolling on a
  plane; and let the axes Ox, Oy be drawn through the centre parallel to
  the plane, so that the equation of the latter is z = -a. We will
  suppose that the extraneous forces consist of a known force (X, Y, Z)
  at the centre, and of the reactions (F1, F2, R) at the point of
  contact. Hence

    M0[.u] = X + F1, M0[.v] = Y + F2, 0 = Z + R,  \
     C[.p] = F2a, C[.q] = -F1a, C[.r] = 0.        /  (19)

  The last equation shows that the angular velocity about the normal to
  the plane is constant. Again, since the point of the sphere which is
  in contact with the plane is instantaneously at rest, we have the
  geometrical relations

    u + qa = 0, v + pa = 0, w = 0,  (20)

  by (12). Eliminating p, q, we get

    (M0 + Ca^-2)[.u] = X, (M0 + Ca^-2)[.v] = Y.  (21)

  The acceleration of the centre is therefore the same as if the plane
  were smooth and the mass of the sphere were increased by C/[alpha]².
  Thus the centre of a sphere rolling under gravity on a plane of
  inclination a describes a parabola with an acceleration

    g sin [alpha]/(1 + C/Ma²)

  parallel to the lines of greatest slope.

  Take next the case of a sphere rolling on a fixed spherical surface.
  Let a be the radius of the rolling sphere, c that of the spherical
  surface which is the locus of its centre, and let x, y, z be the
  co-ordinates of this centre relative to axes through O, the centre of
  the fixed sphere. If the only extraneous forces are the reactions (P,
  Q, R) at the point of contact, we have

    M0[:x] = P, M0[.y] = Q, M0[:z] = R,                            \
                                                                   |
          a                     a                     a             >  (22)
    Cp = ---(yR - zQ), C[.q] = ---(zP - xR), C[.r] = ---(xQ - yP), |
          c                     c                     c            /

  the standard case being that where the rolling sphere is outside the
  fixed surface. The opposite case is obtained by reversing the sign of
  a. We have also the geometrical relations

    [.x] = (a/c)(qz - ry), [.y] = (a/c)(rx - pz), [.z] = (a/c)(py - gx),  (23)

  If we eliminate P, Q, R from (22), the resulting equations are
  integrable with respect to t; thus

          M0a                                 M0a
    p = - ---(y[.z] - z[.y]) + [alpha], q = - ---(z[.x] - x[.z]) + [beta],
          Cc                                  Cc

          M0a
    r = - ---(x[.y] - y[.x]) + [gamma],  (24)
          Cc

  where [alpha], [beta], [gamma] are arbitrary constants. Substituting
  in (23) we find

     /    M0a²\         a                        /    M0a²\         a
    ( 1 + ---- )[.x] = ---([beta]z - [gamma]y), ( 1 + ---- )[.y] = ---([gamma]x - [alpha]z),
     \     C  /         c                        \     C  /         c

     /    M0a²\         a
    ( 1 + ---- )[.z] = ---([alpha]y - [beta]x).  (25)
     \     C  /         c

  Hence [alpha][.x] + [beta][.y] + [gamma][.z] = 0, or

    [alpha]x + [beta]y + [gamma]z = const.;  (26)

  which shows that the centre of the rolling sphere describes a circle.
  If the axis of z be taken normal to the plane of this circle we have
  [alpha] = 0, [beta] = 0, and

     /    M0a²\                 a      /     M0a²\                a
    ( 1 + ---- )[.x] = -[gamma]--- y, ( 1 + ----- )[.y] = [gamma]--- x.  (27)
     \     C  /                 c      \      C  /                c

  The solution of these equations is of the type

    x = b cos ([sigma][tau] + [epsilon]), y = b sin ([sigma][iota] + [epsilon]),  (28)

  where b, [epsilon] are arbitrary, and

             [gamma]a/c
    [sigma]= ----------  (29)
             1 + M0a²/C

  The circle is described with the constant angular velocity [sigma].

  When the gravity of the rolling sphere is to be taken into account the
  preceding method is not in general convenient, unless the whole motion
  of G is small. As an example of this latter type, suppose that a
  sphere is placed on the highest point of a fixed sphere and set
  spinning about the vertical diameter with the angular velocity n; it
  will appear that under a certain condition the motion of G consequent
  on a slight disturbance will be oscillatory. If Oz be drawn vertically
  upwards, then in the beginning of the disturbed motion the quantities
  x, y, p, q, P, Q will all be small. Hence, omitting terms of the
  second order, we find

    M0[:x] = P, M0[.y] = Q, R = M0g,                              \
                                                                   >  (30)
    C[.p] = -(M0ga/c)y + aQ, C[.q] = (M0ga/c)x - aP, C[.r] = 0.   /

  The last equation shows that the component r of the angular velocity
  retains (to the first order) the constant value n. The geometrical
  relations reduce to

    [.x] = aq - (na/c)y, [.y] = -ap + (na/c)x.  (31)

  Eliminating p, g, P, Q, we obtain the equations

  (C + M0a²)[:x] + (Cna/c)y - (M0ga²/c)x = 0,     }
  (C + M0a²)[:y] - (Cna/c)x - (M0ga²/c)y = 0,     }          (32)

  which are both contained in
     _                                _
    |          d²      Cna  d    M0ga² |
    |(C + M0a²)--- - i --- --- - ----- | (x + iy) = 0.  (33)
    |_         dt²      c  dt      c  _|


  This has two solutions of the type x + iy = [alpha]e^{i([sigma]t +
  [epsilon])}, where [alpha], [epsilon] are arbitrary, and [sigma] is a
  root of the quadratic

    (C + M0a²)[sigma]² - (Cna/c)[sigma] + M0ga²/c = 0.  (34)

  If

    n² > (4Mgc/C) (1 + M0a²/C),  (35)

  both roots are real, and have the same sign as n. The motion of G then
  consists of two superposed circular vibrations of the type

    x = [alpha] cos ([sigma]t + [epsilon]), y = [alpha] sin ([sigma]t + [epsilon]),  (36)

  in each of which the direction of revolution is the same as that of
  the initial spin of the sphere. It follows therefore that the original
  position is stable provided the spin n exceed the limit defined by
  (35). The case of a sphere spinning about a vertical axis at the
  lowest point of a spherical bowl is obtained by reversing the signs of
  [alpha] and c. It appears that this position is always stable.

  It is to be remarked, however, that in the first form of the problem
  the stability above investigated is practically of a limited or
  temporary kind. The slightest frictional forces--such as the
  resistance of the air--even if they act in lines through the centre of
  the rolling sphere, and so do not directly affect its angular
  momentum, will cause the centre gradually to descend in an
  ever-widening spiral path.

§ 19. _Free Motion of a Solid._--Before proceeding to further problems
of motion under extraneous forces it is convenient to investigate the
free motion of a solid relative to its mass-centre O, in the most
general case. This is the same as the motion about a fixed point under
the action of extraneous forces which have zero moment about that point.
The question was first discussed by Euler (1750); the geometrical
representation to be given is due to Poinsot (1851).

The kinetic energy T of the motion relative to O will be constant. Now T
= ½I[omega]², where [omega] is the angular velocity and I is the moment
of inertia about the instantaneous axis. If [rho] be the radius-vector
OJ of the momental ellipsoid

  Ax² + By² + Cz² = M[epsilon]^4  (1)

drawn in the direction of the instantaneous axis, we have I =
M[epsilon]^4/[rho]² (§ 11); hence [omega] varies as [rho]. The locus of
J may therefore be taken as the "polhode" (§ 18). Again, the vector
which represents the angular momentum with respect to O will be constant
in every respect. We have seen (§ 18) that this vector coincides in
direction with the perpendicular OH to the tangent plane of the momental
ellipsoid at J; also that

         2T       [rho]
  OH = ------- · -------,  (2)
       [Gamma]   [omega]

where [Gamma] is the resultant angular momentum about O. Since [omega]
varies as [rho], it follows that OH is constant, and the tangent plane
at J is therefore fixed in space. The motion of the body relative to O
is therefore completely represented if we imagine the momental ellipsoid
at O to roll without sliding on a plane fixed in space, with an angular
velocity proportional at each instant to the radius-vector of the point
of contact. The fixed plane is parallel to the invariable plane at O,
and the line OH is called the _invariable line_. The trace of the point
of contact J on the fixed plane is the "herpolhode."

If p, q, r be the component angular velocities about the principal axes
at O, we have

  (A²p² + B²q² + C²r²)/[Gamma]² = (Ap² + Bq² + Cr²)/2T,  (3)

each side being in fact equal to unity. At a point on the polhode cone x
: y : z = p : q : r, and the equation of this cone is therefore

     /    [Gamma]²\         /    [Gamma]²\         /    [Gamma]²\
  A²( 1 - -------- )x² + B²( 1 - -------- )y² + C²( 1 - -------- )z² = 0.  (4)
     \       2AT  /         \       2BT  /         \       2CT  /

Since 2AT - [Gamma]² = B (A - B)q² + C(A - C)r², it appears that if A >
B > C the coefficient of x² in (4) is positive, that of z² is negative,
whilst that of y² is positive or negative according as 2BT <> [Gamma]².
Hence the polhode cone surrounds the axis of greatest or least moment
according as 2BT <> [Gamma]². In the critical case of 2BT = [Gamma]² it
breaks up into two planes through the axis of mean moment (Oy). The
herpolhode curve in the fixed plane is obviously confined between two
concentric circles which it alternately touches; it is not in general a
re-entrant curve. It has been shown by De Sparre that, owing to the
limitation imposed on the possible forms of the momental ellipsoid by
the relation B + C > A, the curve has no points of inflexion. The
invariable line OH describes another cone in the body, called the
_invariable cone_. At any point of this we have x : y : z = Ap. Bq : Cr,
and the equation is therefore

   /    [Gamma]²\       /    [Gamma]²\       /    [Gamma]²\
  ( 1 - -------- )x² + ( 1 - -------- )y² + ( 1 - -------- )z² = 0.  (5)
   \       2AT  /       \       2BT  /       \       2CT  /

[Illustration: FIG. 80.]

The signs of the coefficients follow the same rule as in the case of
(4). The possible forms of the invariable cone are indicated in fig. 80
by means of the intersections with a concentric spherical surface. In
the critical case of 2BT = [Gamma]² the cone degenerates into two
planes. It appears that if the body be sightly disturbed from a state of
rotation about the principal axis of greatest or least moment, the
invariable cone will closely surround this axis, which will therefore
never deviate far from the invariable line. If, on the other hand, the
body be slightly disturbed from a state of rotation about the mean axis
a wide deviation will take place. Hence a rotation about the axis of
greatest or least moment is reckoned as stable, a rotation about the
mean axis as unstable. The question is greatly simplified when two of
the principal moments are equal, say A = B. The polhode and herpolhode
cones are then right circular, and the motion is "precessional"
according to the definition of § 18. If [alpha] be the inclination of
the instantaneous axis to the axis of symmetry, [beta] the inclination
of the latter axis to the invariable line, we have

  [Gamma] cos [beta] = C [omega] cos [alpha], [Gamma] sin [beta] = A [omega] sin [alpha],  (6)

whence

                A
  tan [beta] = --- tan [alpha].  (7)
                C

[Illustration: FIG. 81.]

Hence [beta] <> [alpha], and the circumstances are therefore those of
the first or second case in fig. 78, according as A <> C. If [psi] be
the rate at which the plane HOJ revolves about OH, we have

          sin [alpha]           C cos [alpha]
  [psi] = ----------- [omega] = ------------- [omega],  (8)
          sin [beta]            A cos [beta]

by § 18 (3). Also if [.chi] be the rate at which J describes the
polhode, we have [.psi] sin ([beta]-[alpha]) = [.chi] sin [beta], whence

           sin([alpha] - [beta])
  [.chi] = --------------------- [omega].  (9)
                 sin[alpha]

If the instantaneous axis only deviate slightly from the axis of
symmetry the angles [alpha], [beta] are small, and [.chi] = (A -
C)A·[omega]; the instantaneous axis therefore completes its revolution
in the body in the period

   2[pi]   A - C
  ------ = ----- [omega].  (10)
  [.chi]     A

  In the case of the earth it is inferred from the independent
  phenomenon of luni-solar precession that (C - A)/A = .00313. Hence if
  the earth's axis of rotation deviates slightly from the axis of
  figure, it should describe a cone about the latter in 320 sidereal
  days. This would cause a periodic variation in the latitude of any
  place on the earth's surface, as determined by astronomical methods.
  There appears to be evidence of a slight periodic variation of
  latitude, but the period would seem to be about fourteen months. The
  discrepancy is attributed to a defect of rigidity in the earth. The
  phenomenon is known as the _Eulerian nutation_, since it is supposed
  to come under the free rotations first discussed by Euler.

§ 20. _Motion of a Solid of Revolution._--In the case of a solid of
revolution, or (more generally) whenever there is kinetic symmetry about
an axis through the mass-centre, or through a fixed point O, a number
of interesting problems can be treated almost directly from first
principles. It frequently happens that the extraneous forces have zero
moment about the axis of symmetry, as e.g. in the case of the flywheel
of a gyroscope if we neglect the friction at the bearings. The angular
velocity (r) about this axis is then constant. For we have seen that r
is constant when there are no extraneous forces; and r is evidently not
affected by an instantaneous impulse which leaves the angular momentum
Cr, about the axis of symmetry, unaltered. And a continuous force may be
regarded as the limit of a succession of infinitesimal instantaneous
impulses.

[Illustration: FIG. 82.]

  Suppose, for example, that a flywheel is rotating with angular
  velocity n about its axis, which is (say) horizontal, and that this
  axis is made to rotate with the angular velocity [psi] in the
  horizontal plane. The components of angular momentum about the axis of
  the flywheel and about the vertical will be Cn and A [psi]
  respectively, where A is the moment of inertia about any axis through
  the mass-centre (or through the fixed point O) perpendicular to that
  of symmetry. If [->OK] be the vector representing the former component
  at time t, the vector which represents it at time t + [delta]t will be
  [->OK´], equal to [->OK] in magnitude and making with it an angle
  [delta][psi]. Hence [->KK´] ( = Cn [delta][psi]) will represent the
  change in this component due to the extraneous forces. Hence, so far
  as this component is concerned, the extraneous forces must supply a
  couple of moment Cn[.psi] in a vertical plane through the axis of the
  flywheel. If this couple be absent, the axis will be tilted out of the
  horizontal plane in such a sense that the direction of the spin n
  approximates to that of the azimuthal rotation [.psi]. The remaining
  constituent of the extraneous forces is a couple A[:psi] about the
  vertical; this vanishes if [.psi] is constant. If the axis of the
  flywheel make an angle [theta] with the vertical, it is seen in like
  manner that the required couple in the vertical plane through the axis
  is Cn sin [theta] [.psi]. This matter can be strikingly illustrated
  with an ordinary gyroscope, e.g. by making the larger movable ring in
  fig. 37 rotate about its vertical diameter.

[Illustration: FIG. 83.]

If the direction of the axis of kinetic symmetry be specified by means
of the angular co-ordinates [theta], [psi] of § 7, then considering the
component velocities of the point C in fig. 83, which are [.theta] and
sin [theta][.psi] along and perpendicular to the meridian ZC, we see
that the component angular velocities about the lines OA´, OB´ are -sin
[theta] [.psi] and [.theta] respectively. Hence if the principal moments
of inertia at O be A, A, C, and if n be the constant angular velocity
about the axis OC, the kinetic energy is given by

  2T = A ([.theta]² + sin² [theta][.psi]²) + Cn².  (1)

Again, the components of angular momentum about OC, OA´ are Cn, -A sin
[theta] [.psi], and therefore the angular momentum ([mu], say) about OZ
is

  [mu] = A sin² [theta][.psi] + Cn cos [theta].  (2)

We can hence deduce the condition of steady precessional motion in a
top. A solid of revolution is supposed to be free to turn about a fixed
point O on its axis of symmetry, its mass-centre G being in this axis at
a distance h from O. In fig. 83 OZ is supposed to be vertical, and OC is
the axis of the solid drawn in the direction OG. If [theta] is constant
the points C, A´ will in time [delta]t come to positions C´´, A´´ such
that CC´´ = sin [theta] [delta][psi], A´A´´ = cos [theta] [delta][psi],
and the angular momentum about OB´ will become Cn sin [theta]
[delta][psi] - A sin [theta] [.psi] · cos [theta] [delta][psi]. Equating
this to Mgh sin [theta] [delta]t, and dividing out by sin [theta], we
obtain

  A cos [theta] [.psi]² - Cn[.psi] + Mgh = 0,  (3)

as the condition in question. For given values of n and [theta] we have
two possible values of [.psi] provided n exceed a certain limit. With a
very rapid spin, or (more precisely) with Cn large in comparison with
[root](4AMgh cos [theta]), one value of [.psi] is small and the other
large, viz. the two values are Mgh/Cn and Cn/A cos [theta]
approximately. The absence of g from the latter expression indicates
that the circumstances of the rapid precession are very nearly those of
a free Eulerian rotation (§ 19), gravity playing only a subordinate
part.

[Illustration: FIG. 84.]

  Again, take the case of a circular disk rolling in steady motion on a
  horizontal plane. The centre O of the disk is supposed to describe a
  horizontal circle of radius c with the constant angular velocity
  [.psi], whilst its plane preserves a constant inclination [theta] to
  the horizontal. The components of the reaction of the horizontal lane
  will be Mc[.psi]² at right angles to the tangent line at the point of
  contact and Mg vertically upwards, and the moment of these about the
  horizontal diameter of the disk, which corresponds to OB´ in fig. 83,
  is Mc[.psi]². [alpha] sin [theta] - Mg[alpha] cos [theta], where
  [alpha] is the radius of the disk. Equating this to the rate of
  increase of the angular momentum about OB´, investigated as above, we
  find

     /             a             \                a²
    ( C + Ma² + A --- cos [theta] ) [.psi]² = Mg --- cot [theta],  (4)
     \             c             /                c

  where use has been made of the obvious relation n[alpha] = c[.psi]. If
  c and [theta] be given this formula determines the value of [psi] for
  which the motion will be steady.

In the case of the top, the equation of energy and the condition of
constant angular momentum ([mu]) about the vertical OZ are sufficient to
determine the motion of the axis. Thus, we have

  ½A ([.theta]² + sin² [theta][.psi]²) + ½Cn² + Mgh cos [theta] = const.,  (5)

  A sin² [theta][.psi] + [nu] cos [theta] = [mu],  (6)

where [nu] is written for Cn. From these [.psi] may be eliminated, and
on differentiating the resulting equation with respect to t we obtain

              ([mu] - [nu] cos [theta])([mu] cos [theta] - [nu])
  A[:theta] - -------------------------------------------------- - Mgh sin [theta] = 0.  (7)
                                  A sin³ [theta]

If we put [:theta] = 0 we get the condition of steady precessional
motion in a form equivalent to (3). To find the small oscillation about
a state of steady precession in which the axis makes a constant angle
[alpha] with the vertical, we write [theta] = [alpha] + [chi], and
neglect terms of the second order in [chi]. The result is of the form

  [:chi] + [sigma]²[chi] = 0,  (8)

where

  [sigma]² = {([mu] - [nu] cos [alpha])² + 2([mu] - [nu] cos [alpha])([mu] cos [alpha] - [nu])
     cos [alpha] + ([mu] cos [alpha] - [nu])²} / A² sin^4 [alpha].  (9)

When [nu] is large we have, for the "slow" precession [sigma] = [nu]/A,
and for the "rapid" precession [sigma] = A/[nu] cos [alpha] = [.psi],
approximately. Further, on examining the small variation in [.psi], it
appears that in a slightly disturbed slow precession the motion of any
point of the axis consists of a rapid circular vibration superposed on
the steady precession, so that the resultant path has a trochoidal
character. This is a type of motion commonly observed in a top spun in
the ordinary way, although the successive undulations of the trochoid
may be too small to be easily observed. In a slightly disturbed rapid
precession the superposed vibration is elliptic-harmonic, with a period
equal to that of the precession itself. The ratio of the axes of the
ellipse is sec [alpha], the longer axis being in the plane of [theta].
The result is that the axis of the top describes a circular cone about a
fixed line making a small angle with the vertical. This is, in fact, the
"invariable line" of the free Eulerian rotation with which (as already
remarked) we are here virtually concerned. For the more general
discussion of the motion of a top see GYROSCOPE.

§ 21. _Moving Axes of Reference._--For the more general treatment of the
kinetics of a rigid body it is usually convenient to adopt a system of
moving axes. In order that the moments and products of inertia with
respect to these axes may be constant, it is in general necessary to
suppose them fixed in the solid.

We will assume for the present that the origin O is fixed. The moving
axes Ox, Oy, Oz form a rigid frame of reference whose motion at time t
may be specified by the three component angular velocities p, q, r. The
components of angular momentum about Ox, Oy, Oz will be denoted as usual
by [lambda], [mu], [nu]. Now consider a system of fixed axes Ox´, Oy´,
Oz´ chosen so as to coincide at the instant t with the moving system Ox,
Oy, Oz. At the instant t + [delta]t, Ox, Oy, Oz will no longer coincide
with Ox´, Oy´, Oz´; in particular they will make with Ox´ angles whose
cosines are, to the first order, 1, -r[delta]t, q[delta]t, respectively.
Hence the altered angular momentum about Ox´ will be [lambda] +
[delta][lambda] + ([mu] + [delta][mu]) (-r[delta]t) + ([nu] +
[delta][nu]) q[delta]t. If L, M, N be the moments of the extraneous
forces about Ox, Oy, Oz this must be equal to [lambda] + L[delta]t.
Hence, and by symmetry, we obtain

  d[lambda]                       \
  --------- - r[nu] + q[nu] = L,  |
     dt                           |
                                  |
  d[mu]                           |
  ----- - p[nu] + r[lanbda] = M,   >  (1)
   dt                             |
                                  |
  d[nu]                           |
  ----- - q[lambda] + p[nu] = N.  |
   dt                             /

These equations are applicable to any dynamical system whatever. If we
now apply them to the case of a rigid body moving about a fixed point O,
and make Ox, Oy, Oz coincide with the principal axes of inertia at O, we
have [lambda], [mu], [nu] = Ap, Bq, Cr, whence

    dp                    \
  A -- - (B - C) qr = L,  |
    dt                    |
                          |
    dq                    |
  B -- - (C - A) rp = M,   >  (2)
    dt                    |
                          |
    dr                    |
  C -- - (A - B) pq = N.  |
    dt                    /

If we multiply these by p, q, r and add, we get

   d
  --- · ½(Ap² + Bq² + Cr²) = Lp + Mq + Nr,  (3)
  dt

which is (virtually) the equation of energy.

As a first application of the equations (2) take the case of a solid
constrained to rotate with constant angular velocity [omega] about a
fixed axis (l, m, n). Since p, q, r are then constant, the requisite
constraining couple is

  L = (C - B) mn[omega]², M = (A - C) nl[omega]², N = (B - A) lm[omega]².  (4)

If we reverse the signs, we get the "centrifugal couple" exerted by the
solid on its bearings. This couple vanishes when the axis of rotation is
a principal axis at O, and in no other case (cf. § 17).

If in (2) we put, L, M, N = O we get the case of free rotation; thus

    dp                \
  A -- = (B - C) qr,  |
    dt                |
                      |
    dq                |
  B -- = (C - A) rp,   >  (5)
    dt                |
                      |
    dr                |
  C -- = (A - B) pq.  |
    dt                /

These equations are due to Euler, with whom the conception of moving
axes, and the application to the problem of free rotation, originated.
If we multiply them by p, q, r, respectively, or again by Ap, Bq, Cr
respectively, and add, we verify that the expressions Ap² + Bq² + Cr²
and A²p² + B²q² + C²r² are both constant. The former is, in fact, equal
to 2T, and the latter to [Gamma]², where T is the kinetic energy and
[Gamma] the resultant angular momentum.

  To complete the solution of (2) a third integral is required; this
  involves in general the use of elliptic functions. The problem has
  been the subject of numerous memoirs; we will here notice only the
  form of solution given by Rueb (1834), and at a later period by G.
  Kirchhoff (1875), If we write
           _
          / [phi]   d[phi]
    u  =  |      ------------, [Delta][phi] = [root](1 - k² sin² [phi]),
         _/ 0    [Delta][phi]

  we have, in the notation of elliptic functions, [phi] = am u. If we
  assume

    p = p0 cos am ([sigma]t + [epsilon]), q = q0sin am ([sigma]t + [epsilon]),
    r = r0[Delta] am ([sigma]t + [epsilon]),  (7)

  we find

             [sigma]p0            [sigma]q0              k²[sigma]r0
    [.p] = - --------- qr, [.q] = --------- rp, [.r] = - ----------- pq.  (8)
               q0r0                 r0p0                    p0q0

  Hence (5) will be satisfied, provided

    -[sigma]p0   B - C  [sigma]q0   C - A  -k²[sigma]r0   A - B
    ---------- = -----, --------- = -----, ------------ = -----.  (9)
       q0r0        A      r0p0        B        p0q0         C

  These equations, together with the arbitrary initial values of p, q,
  r, determine the six constants which we have denoted by p0, q0, r0,
  k², [sigma], [epsilon]. We will suppose that A > B > C. From the form
  of the polhode curves referred to in § 19 it appears that the angular
  velocity q about the axis of mean moment must vanish periodically. If
  we adopt one of these epochs as the origin of t, we have [epsilon] =
  0, and p0, r0 will become identical with the initial values of p, r.
  The conditions (9) then lead to

          A(A - C)                 (A - C)(B - C)           A(A - B)   p0²
    q0² = -------- p0², [sigma]² = -------------- r0², k² = -------- · ---.  (10)
          B(B - C)                       AB                 C(B - C)   r0²

  For a real solution we must have k² < 1, which is equivalent to 2BT > [Gamma]². If the initial
  conditions are such as to make 2BT < [Gamma]², we must interchange the
  forms of p and r in (7). In the present case the instantaneous axis
  returns to its initial position in the body whenever [phi] increases
  by 2[pi], i.e. whenever t increases by 4K/[sigma], when K is the
  "complete" elliptic integral of the first kind with respect to the
  modulus k.

  The elliptic functions degenerate into simpler forms when k² = 0 or k²
  = 1. The former case arises when two of the principal moments are
  equal; this has been sufficiently dealt with in § 19. If k² = 1, we
  must have 2BT = [Gamma]². We have seen that the alternative 2BT <>
  [Gamma]² determines whether the polhode cone surrounds the principal
  axis of least or greatest moment. The case of 2BT = [Gamma]², exactly,
  is therefore a critical case; it may be shown that the instantaneous
  axis either coincides permanently with the axis of mean moment or
  approaches it asymptotically.

When the origin of the moving axes is also in motion with a velocity
whose components are u, v, w, the dynamical equations are

  d[xi]                         d[eta]                         d[zeta]
  ----- - r[eta] + q[zeta] = X, ------ - p[zeta] - r[chi] = Y, ------- - q[chi] + p[eta] = Z,  (11)
   dt                             dt                              dt

  d[lambda]                                         d[mu]                                           \
  --------- - r[mu] + q[nu] - w[eta] + v[zeta] = L, ----- - p[nu] + r[lambda]- u[zeta] + w[xi] = M, |
     dt                                              dt                                             |
                                                                                                     >  (12)
  d[nu]                                                                                             |
  ----- - q[lambda] + p[mu] - v[xi] + u[eta] = N.                                                   /
   dt

To prove these, we may take fixed axes O´x´, O´y´, O´z´ coincident with
the moving axes at time t, and compare the linear and angular momenta
[xi] + [delta][xi], [eta] + [delta][eta], [zeta] + [delta][zeta],
[lambda] + [delta][lambda], [mu] + [delta][mu], [nu] + [delta][nu]
relative to the new position of the axes, Ox, Oy, Oz at time t +
[delta]t with the original momenta [xi], [eta], [zeta], [lambda], [mu],
[nu] relative to O´x´, O´y´, O´z´ at time t. As in the case of (2), the
equations are applicable to any dynamical system whatever. If the moving
origin coincide always with the mass-centre, we have [xi], [eta], [zeta]
= M0u, M0v, M0w, where M0 is the total mass, and the equations simplify.

When, in any problem, the values of u, v, w, p, q, r have been
determined as functions of t, it still remains to connect the moving
axes with some fixed frame of reference. It will be sufficient to take
the case of motion about a fixed point O; the angular co-ordinates
[theta], [phi], [psi] of Euler may then be used for the purpose.
Referring to fig. 36 we see that the angular velocities p, q, r of the
moving lines, OA, OB, OC about their instantaneous positions are

  p = [.theta] sin [phi] - sin [theta] cos [phi][.psi],  \
  q = [.theta] cos [phi] + sin [theta] sin [phi][.psi],   >  (13)
  r = [.phi] + cos [theta][.psi],                        /

by § 7 (3), (4). If OA, OB, OC be principal axes of inertia of a solid,
and if A, B, C denote the corresponding moments of inertia, the kinetic
energy is given by

  2T = A([.theta] sin [phi] - sin [theta] cos [phi][.psi])²   \
     + B([.theta] cos [phi] + sin [theta] sin [theta][psi])²   >  (14)
     + C([.phi] + cos [theta][.psi])².                        /

If A = B this reduces to

  2T = A([.theta]² + sin² [theta][.psi]²) + C([.phi] + cos [theta][.psi])²;  (15)

cf. § 20 (1).

§ 22. _Equations of Motion in Generalized Co-ordinates._--Suppose we
have a dynamical system composed of a finite number of material
particles or rigid bodies, whether free or constrained in any way, which
are subject to mutual forces and also to the action of any given
extraneous forces. The configuration of such a system can be completely
specified by means of a certain number (n) of independent quantities,
called the generalized co-ordinates of the system. These co-ordinates
may be chosen in an endless variety of ways, but their number is
determinate, and expresses the number of _degrees of freedom_ of the
system. We denote these co-ordinates by q1, q2, ... q_n. It is implied
in the above description of the system that the Cartesian co-ordinates
x, y, z of any particle of the system are known functions of the q's,
varying in form (of course) from particle to particle. Hence the kinetic
energy T is given by

        __
  2T = \   {m([.x]² + [.y]² + [.z]²)}
       /__

     = a11[.q]1² + a22[.q]2² + ... + 2a12[.q]1[.q]2 + ...,  (1)

where
              _                                              _
          __ |   {  / [dP]x \²    / [dP]y \²    / [dP]z \² }  |                    \
  a_rr = \   | m { ( ------- ) + ( ------- ) + ( ------- ) }  |,                   |
         /__ |_  {  \[dP]q_r/     \[dP]q_r/     \[dP]q_r/  } _|                    |
              _                                                         _           >  (2)
          __ |    / [dP]x   [dP]x     [dP]y   [dP]y     [dP]z   [dP]z \  |         |
  a_rs = \   | m ( ------- ------- + ------- ------- + ------- ------- ) | = a_sr. |
         /__ |_   \[dP]q_r [dP]q_s   [dP]q_r [dP]q_s   [dP]q_r [dP]q_s/ _|         /

Thus T is expressed as a homogeneous quadratic function of the
quantities [.q]1, [.q]2, ... [.q]_n, which are called the _generalized
components of velocity_. The coefficients a_rr, a_rs are called the
coefficients of inertia; they are not in general constants, being
functions of the q's and so variable with the configuration. Again, If
(X, Y, Z) be the force on m, the work done in an infinitesimal change of
configuration is

  [Sigma](X[delta]x + Y[delta]y + Z[delta]z) = Q1[delta]q1 + Q2[delta]q2 + ... + Q_n[delta]q_n,  (3)

where

                /  [dP]x      [dP]y      [dP]z  \
  Q_r = [Sigma]( X------- + Y------- + Z-------  ).  (4)
                \ [dP]q_r    [dP]q_r    [dP]q_r /

The quantities Q_r are called the _generalized components of force_.

The equations of motion of m being

  m[:x] = X, m[:y] = Y, m[:z] = Z,  (5)

we have
          _                                              _
      __ |    /     [dP]x         [dP]y         [dP]z  \  |
     \   | m ( [:x]------- + [:y]------- + [:z]-------  ) | = Q_r.  (6)
     /__ |_   \    [dP]q_r       [dP]q_r       [dP]q_r / _|

Now

          [dP]x         [dP]x               [dP]x
  [.x] = ------[.q]1 + ------[.q]2 + ... + -------[.q]_n,  (7)
         [dP]q1        [dP]q2              [dP]q_n

whence

   [dP][.x]      [dP]x
  ----------  = -------.  (8)
  [dP][.q]_r    [dP]q_r

Also

  d   / [dP]x \      [dP]²x               [dP]²x                      [dP]²x               [dP]x
  -- ( ------- ) = ------------[.q]1 + -------------[.q]2 + ... + --------------[.q]_r = --------.  (9)
  dt  \[dP]q_r/   [dP]q1[dP]q_r        [dP]q2[dP]q_r              [dP]q_n[dP]q_r          [dP]q_r

Hence

       [dP]x     d  /     [dP]x \         d  / [dP]x \     d  /     [dP][.x] \        [dP][.x]
  [:x]------- = ---( [.x]------- ) - [.x]---( ------- ) = ---( [.x]---------- ) - [.x]--------.  (10)
      [dP]q_r   dt  \    [dP]q_r/        dt  \[dP]q_r/    dt  \    [dP][.q]_r/        [dP]q_r

By these and the similar transformations relating to y and z the
equation (6) takes the form

   d   /   [dP]T  \    [dP]T
  --- ( ---------- ) - ------ = Q_r.  (11)
  dt   \[dP][.q]_r/   [dP]q_r

If we put r = 1, 2, ... n in succession, we get the n independent
equations of motion of the system. These equations are due to Lagrange,
with whom indeed the first conception, as well as the establishment, of
a general dynamical method applicable to all systems whatever appears to
have originated. The above proof was given by Sir W. R. Hamilton (1835).
Lagrange's own proof will be found under DYNAMICS, § _Analytical_. In a
conservative system free from extraneous force we have

  [Sigma](X [delta]x + Y [delta]y + Z [delta]z) = -[delta]V,  (12)

where V is the potential energy. Hence

           [dP]V
  Q_r = - -------,  (13)
          [dP]q_r

and

   d   /   [dP]T  \    [dP]T      [dP]V
  --- ( ---------- ) - ----- = - -------.  (14)
  dt   \[dP][.q]_r/    Vq_r      [dP]q_r

If we imagine any given state of motion ([.q]1, [.q]2 ... [.q]_n)
through the configuration (q1, q2, ... q_n) to be generated
instantaneously from rest by the action of suitable impulsive forces, we
find on integrating (11) with respect to t over the infinitely short
duration of the impulse

     [dP]T
  ---------- = Q_r´,  (15)
  [dP][.q]_r

where Q_r´ is the time integral of Q_r and so represents a _generalized
component of impulse_. By an obvious analogy, the expressions
[dP]T/[dP][.q]_r may be called the _generalized components of momentum_;
they are usually denoted by p_r thus

  p_r = [dP]T/[dP][.q]_r = a_(1r)[.q]1 + a_(2r)[.q]2 + ... + a_(nr)[.q]_n.  (16)

Since T is a homogeneous quadratic function of the velocities [.q]1,
[.q]2, ... [.q]_n, we have

         [dP]T            [dP]T                  [dP]T
  2T = ---------[.q]1 + ---------[.q]2 + ... + ----------[.q]_n = p1[.q]2 + p2[.q]2 + ... + p_n[.q]_n.  (17)
       [dP][.q]1        [dP][.q]2              [dP][.q]_n

Hence

   dT
  2-- = [.p]1[.q]1 + [.p]2[.q]2 + ... [.p]_n[.q]_n                                              \
   dt                                                                                           |
                                                                                                |
      + [.p]1[:q]1 + [.p]2[:q]2 + ... + [.p]_n[:q]_n                                            |
                                                                                                |
         /  [dP]T       \           /  [dP]T       \                 /   [dP]T        \         |
      = ( --------- + Q1 ) [.q]1 + ( --------- + Q2 ) [.q]2 + ... + ( ---------- + Q_n )[.q]_n   >  (18)
         \[dP][.q]1     /           \[dP][.q]2     /                 \[dP][.q]_n      /         |
                                                                                                |
          [dP]T            [dP]T                [dP]T                                           |
      + ---------[:q]1 + ---------[:q]2 + ... ----------[:q]_n                                  |
        [dP][.q]1        [dP][.q]2            [dP][.q]_n                                        |
                                                                                                |
        dT                                                                                      |
      = -- + Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n,                                               /
        dt

or

  dT
  -- = Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n.  (19)
  dt

This equation expresses that the kinetic energy is increasing at a rate
equal to that at which work is being done by the forces. In the case of
a conservative system free from extraneous force it becomes the equation
of energy

   d
  --- (T + V) = 0, or T + V = const.,  (20)
  dt

in virtue of (13).

  As a first application of Lagrange's formula (11) we may form the
  equations of motion of a particle in spherical polar co-ordinates. Let
  r be the distance of a point P from a fixed origin O, [theta] the
  angle which OP makes with a fixed direction OZ, [psi] the azimuth of
  the plane ZOP relative to some fixed plane through OZ. The
  displacements of P due to small variations of these co-ordinates are
  [dP]r along OP, r [delta][theta] perpendicular to OP in the plane ZOP,
  and r sin [theta] [delta][psi] perpendicular to this plane. The
  component velocities in these directions are therefore [.r],
  r[.theta], r sin [theta][.psi], and if m be the mass of a moving
  particle at P we have

    2T = m([.r]² + r²[.theta]² + r² sin² [theta][.psi]²).  (21)

  Hence the formula (11) gives

    m([:r] - r[.theta]² - r sin² [theta][.psi]²) = R,                  \
                                                                       |
     d                                                                 |
    ---(mr²[.theta]) - mr² · sin [theta] cos [theta][.psi]² = [Theta],  >  (22)
    dt                                                                 |
                                                                       |
     d                                                                 |
    ---(mr² sin² [theta][.psi]) = [Psi].                               /
    dt

  The quantities R, [Theta], [Psi] are the coefficients in the
  expression R [delta]r + [Theta] [delta][theta] + [Psi] [delta][psi]
  for the work done in an infinitely small displacement; viz. R is the
  radial component of force, [Theta] is the moment about a line through
  O perpendicular to the plane ZOP, and [Psi] is the moment about OZ. In
  the case of the spherical pendulum we have r = l, [Theta] = - mgl sin
  [theta], [Psi] = 0, if OZ be drawn vertically downwards, and therefore

                                                   g               \
    [:theta] - sin [theta] cos [theta][.psi]² = - --- sin [theta], |
                                                   l                >  (23)
                                                                   |
    sin² [theta][.psi] = h,                                        /


  where h is a constant. The latter equation expresses that the angular
  momentum ml² sin² [theta][.psi] about the vertical OZ is constant. By
  elimination of [.psi] we obtain

                                                   g
    [:theta] - h² cos² [theta] / sin^3[theta] = - --- sin [theta].  (24)
                                                   l

  If the particle describes a horizontal circle of angular radius
  [alpha] with constant angular velocity [Omega], we have [.omega] = 0,
  h = [Omega]² sin [alpha], and therefore

                g
    [Omega]² = --- cos [alpha],  (25)
                l

  as is otherwise evident from the elementary theory of uniform circular
  motion. To investigate the small oscillations about this state of
  steady motion we write [theta] = [alpha] + [chi] in (24) and neglect
  terms of the second order in [chi]. We find, after some reductions,

    [:chi] + (1 + 3 cos² [alpha]) [Omega]²[chi] = 0;  (26)

  this shows that the variation of [chi] is simple-harmonic, with the
  period

    2[pi]/[root](1 + 3 cos² [alpha])·[Omega]

  As regards the most general motion of a spherical pendulum, it is
  obvious that a particle moving under gravity on a smooth sphere cannot
  pass through the highest or lowest point unless it describes a
  vertical circle. In all other cases there must be an upper and a lower
  limit to the altitude. Again, a vertical plane passing through O and a
  point where the motion is horizontal is evidently a plane of symmetry
  as regards the path. Hence the path will be confined between two
  horizontal circles which it touches alternately, and the direction of
  motion is never horizontal except at these circles. In the case of
  disturbed steady motion, just considered, these circles are nearly
  coincident. When both are near the lowest point the horizontal
  projection of the path is approximately an ellipse, as shown in § 13;
  a closer investigation shows that the ellipse is to be regarded as
  revolving about its centre with the angular velocity 2/3 ab[Omega]/l²,
  where a, b are the semi-axes.

  To apply the equations (11) to the case of the top we start with the
  expression (15) of § 21 for the kinetic energy, the simplified form
  (1) of § 20 being for the present purpose inadmissible, since it is
  essential that the generalized co-ordinates employed should be
  competent to specify the position of every particle. If [lambda],
  [mu], [nu] be the components of momentum, we have

                   [dP]T                                                                  \
    [lambda]=  ------------ = A[.theta],                                                  |
               [dP][.theta]                                                               |
                                                                                          |
             [dP]T                                                                        |
    [mu] = ---------- = A sin² [theta][.psi] + C([.phi] + cos [theta][.psi]) cos [theta],  >  (27)
           [dP][.psi]                                                                     |
                                                                                          |
             [dP]T                                                                        |
    [nu] = ---------- = C ([.theta] + cos [theta][.psi]).                                 /
           [dP][.phi]

  The meaning of these quantities is easily recognized; thus [lambda] is
  the angular momentum about a horizontal axis normal to the plane of
  [theta], [mu] is the angular momentum about the vertical OZ, and [nu]
  is the angular momentum about the axis of symmetry. If M be the total
  mass, the potential energy is V = Mgh cos [theta], if OZ be drawn
  vertically upwards. Hence the equations (11) become

    A[:theta] - A sin [theta] cos [theta][.psi]² + C([.phi] + cos [theta][.psi]) [.psi] sin [theta] = Mgh sin [theta], \
              d/dt · {A sin² [theta][.psi] + C([.phi] + cos [theta][.psi]) cos [theta]} = 0,                            >  (28)
              d/dt · {C([.phi] + cos [theta][.psi])} = 0,                                                              /

  of which the last two express the constancy of the momenta [mu], [nu].
  Hence

    A[:theta] - A sin [theta] cos [theta][.psi]² + [nu] sin [theta][.psi] = Mgh sin [theta],  \  (29)
    A sin² [theta][.psi] + [nu] cos [theta] = [mu].                                           /

  If we eliminate [.psi] we obtain the equation (7) of § 20. The theory
  of disturbed precessional motion there outlined does not give a
  convenient view of the oscillations of the axis about the vertical
  position. If [theta] be small the equations (29) may be written

                                    [nu]²- 4AMgh         \
    [:theta] - [theta][.omega]² = - ------------[theta],  >  (30)
                                         4A²             |
    [theta]²[.omega] = const.,                           /

  where

                      [nu]
    [omega] = [psi] - ---- t.  (31)
                       2A

  Since [theta], [omega] are the polar co-ordinates (in a horizontal
  plane) of a point on the axis of symmetry, relative to an initial line
  which revolves with constant angular velocity [nu]/2A, we see by
  comparison with § 14 (15) (16) that the motion of such a point will be
  elliptic-harmonic superposed on a uniform rotation [nu]/2A, provided
  [nu]² > 4AMgh. This gives (in essentials) the theory of the
  "gyroscopic pendulum."

§ 23. _Stability of Equilibrium. Theory of Vibrations._--If, in a
conservative system, the configuration (q1, q2, ... q_n) be one of
equilibrium, the equations (14) of § 22 must be satisfied by [.q]1,
[.q]2 ... [.q]_n = 0, whence

  [dP]V / [dP]q_r = 0.  (1)

A necessary and sufficient condition of equilibrium is therefore that
the value of the potential energy should be stationary for infinitesimal
variations of the co-ordinates. If, further, V be a minimum, the
equilibrium is necessarily stable, as was shown by P. G. L. Dirichlet
(1846). In the motion consequent on any slight disturbance the total
energy T + V is constant, and since T is essentially positive it follows
that V can never exceed its equilibrium value by more than a slight
amount, depending on the energy of the disturbance. This implies, on the
present hypothesis, that there is an upper limit to the deviation of
each co-ordinate from its equilibrium value; moreover, this limit
diminishes indefinitely with the energy of the original disturbance. No
such simple proof is available to show without qualification that the
above condition is _necessary_. If, however, we recognize the existence
of dissipative forces called into play by any motion whatever of the
system, the conclusion can be drawn as follows. However slight these
forces may be, the total energy T + V must continually diminish so long
as the velocities [.q]1, [.q]2, ... [.q]_n differ from zero. Hence if
the system be started from rest in a configuration for which V is less
than in the equilibrium configuration considered, this quantity must
still further decrease (since T cannot be negative), and it is evident
that either the system will finally come to rest in some other
equilibrium configuration, or V will in the long run diminish
indefinitely. This argument is due to Lord Kelvin and P. G. Tait (1879).

In discussing the small oscillations of a system about a configuration
of stable equilibrium it is convenient so to choose the generalized
cc-ordinates q1, q2, ... q_n that they shall vanish in the configuration
in question. The potential energy is then given with sufficient
approximation by an expression of the form

  2V = c11q1² + c22q2² + ... + 2c12q1q2 + ...,  (2)

a constant term being irrelevant, and the terms of the first order being
absent since the equilibrium value of V is stationary. The coefficients
c_rr, c_rs are called _coefficients of stability_. We may further treat
the coefficients of inertia a_rr, a_rs of § 22 (1) as constants. The
Lagrangian equations of motion are then of the type

  a_(1r)[:q]1 + a_(2r)[:q]2 + ... + a_(nr)[:q]_n + c_(1r)q1 + c_(2r)q2 + ... + c_(nr)q_n = Q_r,  (3)

where Q_r now stands for a component of extraneous force. In a _free
oscillation_ we have Q1, Q2, ... Q_n = 0, and if we assume

  q_r = A_r e^(i[sigma]^t),  (4)

we obtain n equations of the type

  (c_(1r) - [sigma]²a_(1r)) A1 + (c_(2r) - [sigma]²a_(2r)) A2 + ... + (c_(nr) - [sigma]²a_nr) A_n = 0.  (5)

Eliminating the n - 1 ratios A1 : A2 : ... : A_n we obtain the
determinantal equation

  [Delta]([sigma]²) = 0,      (6)

where

  [Delta]([sigma]²) = | c11 - [sigma]²a11,      c21 - [sigma]²a21,       ..., C_(n1) - [sigma]²a_(nl) |
                      | c12 - [sigma]²a12,      c22 - [sigma]²a22,       ..., C_(n2) - [sigma]²a_(n2) |
                      |           .                       .              ...               .          |
                      |           .                       .              ...               .          |  (7)
                      |           .                       .              ...               .          |
                      | c_(1n) - [sigma]²a{1n}, c_(2n) - [sigma]²a_(2n), ..., C_(nn) - [sigma]²a_(nn) |

The quadratic expression for T is essentially positive, and the same
holds with regard to V in virtue of the assumed stability. It may be
shown algebraically that under these conditions the n roots of the above
equation in [sigma]² are all real and positive. For any particular root,
the equations (5) determine the ratios of the quantities A1, A2, ...
A_n, the absolute values being alone arbitrary; these quantities are in
fact proportional to the minors of any one row in the determinate
[Delta]([sigma]²). By combining the solutions corresponding to a pair of
equal and opposite values of [sigma] we obtain a solution in real form:

  q_r = C_(a_r) cos ([sigma]t + [epsilon]),  (8)

where a1, a2 ... a_r are a determinate series of quantities having to
one another the above-mentioned ratios, whilst the constants C,
[epsilon] are arbitrary. This solution, taken by itself, represents a
motion in which each particle of the system (since its displacements
parallel to Cartesian co-ordinate axes are linear functions of the q's)
executes a simple vibration of period 2[pi]/[sigma]. The amplitudes of
oscillation of the various particles have definite ratios to one
another, and the phases are in agreement, the absolute amplitude
(depending on C) and the phase-constant ([epsilon]) being alone
arbitrary. A vibration of this character is called a _normal mode_ of
vibration of the system; the number n of such modes is equal to that of
the degrees of freedom possessed by the system. These statements require
some modification when two or more of the roots of the equation (6) are
equal. In the case of a multiple root the minors of [Delta]([sigma]²)
all vanish, and the basis for the determination of the quantities a_r
disappears. Two or more normal modes then become to some extent
indeterminate, and elliptic vibrations of the individual particles are
possible. An example is furnished by the spherical pendulum (§ 13).

[Illustration: FIG. 85.]

  As an example of the method of determination of the normal modes we
  may take the "double pendulum." A mass M hangs from a fixed point by a
  string of length a, and a second mass m hangs from M by a string of
  length b. For simplicity we will suppose that the motion is confined
  to one vertical plane. If [theta], [phi] be the inclinations of the
  two strings to the vertical, we have, approximately,

    2T = Ma²[.theta]² + m(a[.theta] + b[.psi])²  \  (9)
    2V = Mga[theta]² + mg(a[theta]² + b[psi]²).  /

  The equations (3) take the forms

    a[:theta] + [mu]b[:phi] + g[theta] = 0,  \  (10)
    a[:theta] + b[:phi] + g[phi] = 0.        /

  where [mu] = m/(M + m). Hence

    ([sigma]² - g/a)a[theta] + [mu][sigma]²b[phi] = 0,  \  (11)
    [sigma]²a[theta] + ([sigma]² - g/b)b[phi] = 0.      /

  The frequency equation is therefore

    ([sigma]² - g/a)([sigma]² - g/b) - [mu][sigma]^4 = 0.  (12)

  The roots of this quadratic in [sigma]² are easily seen to be real and
  positive. If M be large compared with m, [mu] is small, and the roots
  are g/a and g/b, approximately. In the normal mode corresponding to
  the former root, M swings almost like the bob of a simple pendulum of
  length a, being comparatively uninfluenced by the presence of m,
  whilst m executes a "forced" vibration (§ 12) of the corresponding
  period. In the second mode, M is nearly at rest [as appears from the
  second of equations (11)], whilst m swings almost like the bob of a
  simple pendulum of length b. Whatever the ratio M/m, the two values of
  [sigma]² can never be exactly equal, but they are approximately equal
  if a, b are nearly equal and [mu] is very small. A curious phenomenon
  is then to be observed; the motion of each particle, being made up (in
  general) of two superposed simple vibrations of nearly equal period,
  is seen to fluctuate greatly in extent, and if the amplitudes be equal
  we have periods of approximate rest, as in the case of "beats" in
  acoustics. The vibration then appears to be transferred alternately
  from m to M at regular intervals. If, on the other hand, M is small
  compared with m, [mu] is nearly equal to unity, and the roots of (12)
  are [sigma]² = g/(a + b) and [sigma]² = mg/M·(a + b)/ab,
  approximately. The former root makes [theta] = [phi], nearly; in the
  corresponding normal mode m oscillates like the bob of a simple
  pendulum of length a + b. In the second mode a[theta] + b[phi] = 0,
  nearly, so that m is approximately at rest. The oscillation of M then
  resembles that of a particle at a distance a from one end of a string
  of length a + b fixed at the ends and subject to a tension mg.

The motion of the system consequent on arbitrary initial conditions may
be obtained by superposition of the n normal modes with suitable
amplitudes and phases. We have then

  q_r = [alpha]_r[theta] + [alpha]_r´[theta]´ + [alpha]_r´´[theta]´´ + ...,  (13)

where

  [theta] = C cos ([sigma]t + [epsilon]), [theta]´
    = C´ cos ([sigma]´t + [epsilon]), [theta]´´
    = C´´ cos([sigma]´´t + [epsilon]), ...  (14)

provided [sigma]², [sigma]´², [sigma]´´², ... are the n roots of (6).
The coefficients of [theta], [theta]´, [theta]´´, ... in (13) satisfy
the _conjugate_ or _orthogonal_ relations

  a11[alpha]1[alpha]1´ + a22[alpha]2[alpha]2´ + ... + a12([alpha]1[alpha]2´ + [alpha]2[alpha]1´) + ... = 0,  (15)
  c11[alpha]1[alpha]1´ + c22[alpha]2[alpha]2´ + ... + c12([alpha]1[alpha]2´ + [alpha]2[alpha]1´) + ... = 0,  (16)

provided the symbols [alpha]_r, [alpha]_r´ correspond to two distinct
roots [sigma]², [sigma]´² of (6). To prove these relations, we replace
the symbols A1, A2, ... A_n in (5) by [alpha]1, [alpha]2, ... [alpha]_n
respectively, multiply the resulting equations by a´1, a´2, ... a´_n, in
order, and add. The result, owing to its symmetry, must still hold if we
interchange accented and unaccented Greek letters, and by comparison we
deduce (15) and (16), provided [sigma]² and [sigma]´² are unequal. The
actual determination of C, C´, C´´, ... and [epsilon], [epsilon]´,
[epsilon]´´, ... in terms of the initial conditions is as follows. If we
write

  C cos [epsilon] = H, -C sin [epsilon] = K,  (17)

we must have

  [alpha]_rH + [alpha]_r´H´ + [alpha]_r´´H´´ + ...                         = [q_r]0,    \  (18)
  [sigma][alpha]_rH + [sigma]´[alpha]_r´H´ + [sigma]´´[alpha]_r´´H´´ + ... = [[.q]_r]0, /

where the zero suffix indicates initial values. These equations can be
at once solved for H, H´, H´´, ... and K, K´, K´´, ... by means of the
orthogonal relations (15).

By a suitable choice of the generalized co-ordinates it is possible to
reduce T and V simultaneously to sums of squares. The transformation is
in fact effected by the assumption (13), in virtue of the relations (15)
(16), and we may write

  2T = a[.theta]² + a´[.theta]´² + a´´[.theta]´´² + ..., \  (19)
  2V = c[theta]² + c´[theta]´² + c´´[theta]´´² + ....    /

The new co-ordinates [theta], [theta]´, [theta]´´ ... are called the
_normal_ co-ordinates of the system; in a normal mode of vibration one
of these varies alone. The physical characteristics of a normal mode are
that an impulse of a particular normal type generates an initial
velocity of that type only, and that a constant extraneous force of a
particular normal type maintains a displacement of that type only. The
normal modes are further distinguished by an important "stationary"
property, as regards the frequency. If we imagine the system reduced by
frictionless constraints to one degree of freedom, so that the
co-ordinates [theta], [theta]´, [theta]´´, ... have prescribed ratios to
one another, we have, from (19),

             c[theta]² + c´[theta]´² = c´´[theta]´´² + ...
  [sigma]² = ---------------------------------------------,  (20)
             a[theta]² + a´[theta]´² + a´´[theta]´´² + ...

This shows that the value of [sigma]² for the constrained mode is
intermediate to the greatest and least of the values c/a, c´/a´,
c´´/a´´, ... proper to the several normal modes. Also that if the
constrained mode differs little from a normal mode of free vibration
(e.g. if [theta]´, [theta]´´, ... are small compared with [theta]), the
change in the frequency is of the second order. This property can often
be utilized to estimate the frequency of the gravest normal mode of a
system, by means of an assumed approximate type, when the exact
determination would be difficult. It also appears that an estimate thus
obtained is necessarily too high.

From another point of view it is easily recognized that the equations
(5) are exactly those to which we are led in the ordinary process of
finding the stationary values of the function

  V (q1, q2, ... q_n)
  ------------------------,
  T (q1, q2, ... q_n)

where the denominator stands for the same homogeneous quadratic function
of the q's that T is for the [.q]'s. It is easy to construct in this
connexion a proof that the n values of [sigma]² are all real and
positive.

  The case of three degrees of freedom is instructive on account of the
  geometrical analogies. With a view to these we may write

    2T= a[.x]² + b[.y]² + c[.z]² + 2f[.y][.z] + 2g[.z][.x] + 2h[.x][.y],  \  (21)
    2V = Ax² + By² + Cz² + 2Fyz + 2Gzx + 2Hxy.                            /

  It is obvious that the ratio

    V (x, y, z)
    -----------  (22)
    T (x, y, z)

  must have a least value, which is moreover positive, since the
  numerator and denominator are both essentially positive. Denoting this
  value by [sigma]1², we have

    Ax1 + Hy1 + Gz1 = [sigma]1²(ax1 + hy1 + [dP]gz1), \
    Hx1 + By1 + Fz1 = [sigma]1²(hx1 + by1 + fz1),      >  (23)
    Gx1 + Fy1 + Cz1 = [sigma]1²(gx1 + fy1 + cz1),     /

  provided x1 : y1 : z1 be the corresponding values of the ratios x:y:z.
  Again, the expression (22) will also have a least value when the
  ratios x : y : z are subject to the condition

       [dP]V      [dP]V      [dP]V
    x1 ----- + y1 ----- + z1 ----- = 0;  (24)
       [dP]x      [dP]y      [dP]z

  and if this be denoted by [sigma]2² we have a second system of
  equations similar to (23). The remaining value [sigma]2² is the value
  of (22) when x : y : z arc chosen so as to satisfy (24) and

       [dP]V      [dP]V      [dP]V
    x2 ----- + y2 ----- + z2 ----- = 0  (25)
       [dP]x      [dP]y      [dP]z

  The problem is identical with that of finding the common conjugate
  diameters of the ellipsoids T(x, y, z) = const., V(x, y, z) = const.
  If in (21) we imagine that x, y, z denote infinitesimal rotations of a
  solid free to turn about a fixed point in a given field of force, it
  appears that the three normal modes consist each of a rotation about
  one of the three diameters aforesaid, and that the values of [sigma]
  are proportional to the ratios of the lengths of corresponding
  diameters of the two quadrics.

We proceed to the _forced vibrations_ of the system. The typical case is
where the extraneous forces are of the simple-harmonic type cos
([sigma]t + [epsilon]); the most general law of variation with time can
be derived from this by superposition, in virtue of Fourier's theorem.
Analytically, it is convenient to put Q_r, equal to e^(i[sigma]^t)
multiplied by a complex coefficient; owing to the linearity of the
equations the factor e^(i[sigma]^t) will run through them all, and need
not always be exhibited. For a system of one degree of freedom we have

  a[:q] + cq = Q,  (26)

and therefore on the present supposition as to the nature of Q

            Q
  q = -------------.  (27)
      c - [sigma]²a

This solution has been discussed to some extent in § 12, in connexion
with the forced oscillations of a pendulum. We may note further that
when [sigma] is small the displacement q has the "equilibrium value"
Q/c, the same as would be produced by a steady force equal to the
instantaneous value of the actual force, the inertia of the system being
inoperative. On the other hand, when [sigma]² is great q tends to the
value -Q/[sigma]²a, the same as if the potential energy were ignored.
When there are n degrees of freedom we have from (3)

  (c_(1r) - [sigma]² a_(2r)) q1 + (c²_r - [sigma]² a_(2r)) q2 + ... + (c_(nr) - [sigma]² a_(nr)) q_n = Qr,  (28)

and therefore

  [Delta]([sigma]²)·q_r = a_(1r)Q1 + a_(2r)Q2 + ... + a_(nr)Q_n,  (29)

where a_(1r), a_(2r), ... a_(nr) are the minors of the rth row of the
determinant (7). Every particle of the system executes in general a
simple vibration of the imposed period 2[pi]/[sigma], and all the
particles pass simultaneously through their equilibrium positions. The
amplitude becomes very great when [sigma]² approximates to a root of
(6), i.e. when the imposed period nearly coincides with one of the free
periods. Since a_(rs) = a_(sr), the coefficient of Q_s in the expression
for q_r is identical with that of Q_r in the expression for q_s. Various
important "reciprocal theorems" formulated by H. Helmholtz and Lord
Rayleigh are founded on this relation. Free vibrations must of course be
superposed on the forced vibrations given by (29) in order to obtain the
complete solution of the dynamical equations.

In practice the vibrations of a system are more or less affected by
dissipative forces. In order to obtain at all events a qualitative
representation of these it is usual to introduce into the equations
frictional terms proportional to the velocities. Thus in the case of one
degree of freedom we have, in place of (26),

  a[:q] + b[.q] + cq = Q,  (30)

where a, b, c are positive. The solution of this has been sufficiently
discussed in § 12. In the case of multiple freedom, the equations of
small motion when modified by the introduction of terms proportional to
the velocities are of the type

   d     [dP]T                                                       [dP]V
  --- ---------- + B_(1r)[.q]1 + B_(2r)[.q]2 + ... + B_(nr)[.q]_n + ------- = Q_r  (31)
  dt  [dP][.q]_r                                                    [dP]q_r

If we put

  b_(rs) = b_(sr) = ½[B_(rs) + B_(sr)], [beta]_(rs) = -[beta]_(sr) = ½[B_(rs) - B_(sr)],  (32)

this may be written

   d   [dP]T        [dP]F                                                                       [dP]V
  --- --------- + ---------- + [beta]_(1r)[.q]1 + [beta]_(2r)[.q]2 + ... + [beta]_(nr)[.q]_r + -------  (33)
  dt [dP][.q]_r   [dP][.q]_r                                                                   [dP]q_r

provided

  2F = b11[.q]1² + b22[.q]2² + ... + 2b12[.q]1[.q]2 + ...  (34)

The terms due to F in (33) are such as would arise from frictional
resistances proportional to the absolute velocities of the particles, or
to mutual forces of resistance proportional to the relative velocities;
they are therefore classed as _frictional_ or _dissipative_ forces. The
terms affected with the coefficients [beta]_(rs) on the other hand are
such as occur in "cyclic" systems with latent motion (DYNAMICS, §
_Analytical_); they are called the _gyrostatic terms_. If we multiply
(33) by [.q]_r and sum with respect to r from 1 to n, we obtain, in
virtue of the relations [beta]_(rs) = -[beta]_(sr), [beta]_(rr) = 0,
   d
  ---(T + V) = 2F + Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n.  (35)
  dt

This shows that mechanical energy is lost at the rate 2F per unit time.
The function F is therefore called by Lord Rayleigh the _dissipation
function_.

If we omit the gyrostatic terms, and write q_r = C_re^([lambda]t), we
find, for a free vibration,

  [a_(1r)[lambda]² + b_(1r)[lambda] + c_(1r)] C1 + [a_(2r)[lambda]² + b_(2r)[lambda] + c_(2r)] C2 + ...
    + [a_(nr)[lambda]² + b_(nr)[lambda] + c_(nr)] C_n = 0.  (36)

This leads to a determinantal equation in [lambda] whose 2n roots are
either real and negative, or complex with negative real parts, on the
present hypothesis that the functions T, V, F are all essentially
positive. If we combine the solutions corresponding to a pair of
conjugate complex roots, we obtain, in real form,

  q_r = C[alpha]_re^(-t/[tau]) cos ([sigma]t + [epsilon] - [epsilon]_r),  (37)

where [sigma], [tau], [alpha]_r, [epsilon]_r are determined by the
constitution of the system, whilst C, [epsilon] are arbitrary, and
independent of r. The n formulae of this type represent a normal mode of
free vibration: the individual particles revolve as a rule in elliptic
orbits which gradually contract according to the law indicated by the
exponential factor. If the friction be relatively small, all the normal
modes are of this character, and unless two or more values of [sigma]
are nearly equal the elliptic orbits are very elongated. The effect of
friction on the period is moreover of the second order.

In a forced vibration of e^(i[sigma]t) the variation of each co-ordinate
is simple-harmonic, with the prescribed period, but there is a
retardation of phase as compared with the force. If the friction be
small the amplitude becomes relatively very great if the imposed period
approximate to a free period. The validity of the "reciprocal theorems"
of Helmholtz and Lord Rayleigh, already referred to, is not affected by
frictional forces of the kind here considered.

  The most important applications of the theory of vibrations are to the
  case of continuous systems such as strings, bars, membranes, plates,
  columns of air, where the number of degrees of freedom is infinite.
  The series of equations of the type (3) is then replaced by a single
  linear partial differential equation, or by a set of two or three such
  equations, according to the number of dependent variables. These
  variables represent the whole assemblage of generalized co-ordinates
  q_r; they are continuous functions of the independent variables x, y,
  z whose range of variation corresponds to that of the index r, and of
  t. For example, in a one-dimensional system such as a string or a bar,
  we have one dependent variable, and two independent variables x and t.
  To determine the free oscillations we assume a time factor
  e^(i[sigma]t); the equations then become linear differential equations
  between the dependent variables of the problem and the independent
  variables x, or x, y, or x, y, z as the case may be. If the range of
  the independent variable or variables is unlimited, the value of
  [sigma] is at our disposal, and the solution gives us the laws of
  wave-propagation (see WAVE). If, on the other hand, the body is
  finite, certain terminal conditions have to be satisfied. These limit
  the admissible values of [sigma], which are in general determined by
  a transcendental equation corresponding to the determinantal equation
  (6).

  Numerous examples of this procedure, and of the corresponding
  treatment of forced oscillations, present themselves in theoretical
  acoustics. It must suffice here to consider the small oscillations of
  a chain hanging vertically from a fixed extremity. If x be measured
  upwards from the lower end, the horizontal component of the tension P
  at any point will be P[delta]y/[delta]x, approximately, if y denote
  the lateral displacement. Hence, forming the equation of motion of a
  mass-element, [rho][delta]x, we have

    [rho][delta]x·[:y] = [delta]P·([dP]y/[dP]x).  (38)

  Neglecting the vertical acceleration we have P = g[rho]x, whence

    [dP]²y     [dP]   /  [dP]y \
    ------ = g ----- ( x -----  ).  (39)
    [dP]t²     [dP]x  \  [dP]x /

  Assuming that y varies as e^(i[sigma]t) we have

    [dP]   /  [dP]y \
    ----- ( x -----  ) + ky = 0  (40)
    [dP]x  \  [dP]x /

  provided k = [sigma]²/g. The solution of (40) which is finite for x =
  0 is readily obtained in the form of a series, thus

           /    kx    k²x²      \
    y = C ( 1 - -- +  ---- - ... ) = CJ0(z),  (41)
           \    1²    1²2²      /

  in the notation of Bessel's functions, if z² = 4kx. Since y must
  vanish at the upper end (x = l), the admissible values of [sigma] are
  determined by

    [sigma]² = gz²/4l, J0(z) = 0.  (42)

  The function J0(z) has been tabulated; its lower roots are given by

    z/[pi]= .7655, 1.7571, 2.7546,...,

  approximately, where the numbers tend to the form s - ¼. The frequency
  of the gravest mode is to that of a uniform bar in the ratio .9815
  That this ratio should be less than unity agrees with the theory of
  "constrained types" already given. In the higher normal modes there
  are nodes or points of rest (y = 0); thus in the second mode there is
  a node at a distance .190l from the lower end.

  AUTHORITIES.--For indications as to the earlier history of the subject
  see W. W. R. Ball, _Short Account of the History of Mathematics_; M.
  Cantor, _Geschichte der Mathematik_ (Leipzig, 1880 ... ); J. Cox,
  _Mechanics_ (Cambridge, 1904); E. Mach, _Die Mechanik in ihrer
  Entwickelung_ (4th ed., Leipzig, 1901; Eng. trans.). Of the classical
  treatises which have had a notable influence on the development of the
  subject, and which may still be consulted with advantage, we may note
  particularly, Sir I. Newton, _Philosophiae naturalis Principia
  Mathematica_ (1st ed., London, 1687); J. L. Lagrange, _Mécanique
  analytique_ (2nd ed., Paris, 1811-1815); P. S. Laplace, _Mécanique
  céleste_ (Paris, 1799-1825); A. F. Möbius, _Lehrbuch der Statik_
  (Leipzig, 1837), and _Mechanik des Himmels_; L. Poinsot, _Éléments de
  statique_ (Paris, 1804), and _Théorie nouvelle de la rotation des
  corps_ (Paris, 1834).

  Of the more recent general treatises we may mention Sir W. Thomson
  (Lord Kelvin) and P. G. Tait, _Natural Philosophy_ (2nd ed.,
  Cambridge, 1879-1883); E. J. Routh, _Analytical Statics_ (2nd ed.,
  Cambridge, 1896), _Dynamics of a Particle_ (Cambridge, 1898), _Rigid
  Dynamics_ (6th ed., Cambridge 1905); G. Minchin, _Statics_ (4th ed.,
  Oxford, 1888); A. E. H. Love, _Theoretical Mechanics_ (2nd ed.,
  Cambridge, 1909); A. G. Webster, _Dynamics of Particles_, &c. (1904);
  E. T. Whittaker, _Analytical Dynamics_ (Cambridge, 1904); L. Arnal,
  _Traitê de mécanique_ (1888-1898); P. Appell, _Mécanique rationelle_
  (Paris, vols. i. and ii., 2nd ed., 1902 and 1904; vol. iii., 1st ed.,
  1896); G. Kirchhoff, _Vorlesungen über Mechanik_ (Leipzig, 1896); H.
  Helmholtz, _Vorlesungen über theoretische Physik_, vol. i. (Leipzig,
  1898); J. Somoff, _Theoretische Mechanik_ (Leipzig, 1878-1879).

  The literature of graphical statics and its technical applications is
  very extensive. We may mention K. Culmann, _Graphische Statik_ (2nd
  ed., Zürich, 1895); A. Föppl, _Technische Mechanik_, vol. ii.
  (Leipzig, 1900); L. Henneberg, _Statik des starren Systems_
  (Darmstadt, 1886); M. Lévy, _La statique graphique_ (2nd ed., Paris,
  1886-1888); H. Müller-Breslau, _Graphische Statik_ (3rd ed., Berlin,
  1901). Sir R. S. Ball's highly original investigations in kinematics
  and dynamics were published in collected form under the title _Theory
  of Screws_ (Cambridge, 1900).

  Detailed accounts of the developments of the various branches of the
  subject from the beginning of the 19th century to the present time,
  with full bibliographical references, are given in the fourth volume
  (edited by Professor F. Klein) of the _Encyclopädie der mathematischen
  Wissenschaften_ (Leipzig). There is a French translation of this work.
  (See also DYNAMICS.)     (H. Lb.)


II.--APPLIED MECHANICS[1]

§ 1. The practical application of mechanics may be divided into two
classes, according as the assemblages of material objects to which they
relate are intended to remain fixed or to move relatively to each
other--the former class being comprehended under the term "Theory of
Structures" and the latter under the term "Theory of Machines."


PART I.--OUTLINE OF THE THEORY OF STRUCTURES

  § 2. _Support of Structures._--Every structure, as a whole, is
  maintained in equilibrium by the joint action of its own _weight_, of
  the _external load_ or pressure applied to it from without and tending
  to displace it, and of the _resistance_ of the material which supports
  it. A structure is supported either by resting on the solid crust of
  the earth, as buildings do, or by floating in a fluid, as ships do in
  water and balloons in air. The principles of the support of a floating
  structure form an important part of Hydromechanics (q.v.). The
  principles of the support, as a whole, of a structure resting on the
  land, are so far identical with those which regulate the equilibrium
  and stability of the several parts of that structure that the only
  principle which seems to require special mention here is one which
  comprehends in one statement the power both of liquids and of loose
  earth to support structures. This was first demonstrated in a paper
  "On the Stability of Loose Earth," read to the Royal Society on the
  19th of June 1856 (Phil. _Trans._ 1856), as follows:--

  Let E represent the weight of the portion of a horizontal stratum of
  earth which is displaced by the foundation of a structure, S the
  utmost weight of that structure consistently with the power of the
  earth to resist displacement, [phi] the angle of repose of the earth;
  then

     S     /1 + sin[phi]\²
    --- = ( ------------ ).
     E     \1 - sin[phi]/

  To apply this to liquids [phi] must be made zero, and then S/E = 1, as
  is well known. For a proof of this expression see Rankine's _Applied
  Mechanics_, 17th ed., p. 219.

  § 3. _Composition of a Structure, and Connexion of its Pieces._--A
  structure is composed of _pieces_,--such as the stones of a building
  in masonry, the beams of a timber framework, the bars, plates and
  bolts of an iron bridge. Those pieces are connected at their joints or
  surfaces of mutual contact, either by simple pressure and friction (as
  in masonry with moist mortar or without mortar), by pressure and
  adhesion (as in masonry with cement or with hardened mortar, and
  timber with glue), or by the resistance of _fastenings_ of different
  kinds, whether made by means of the form of the joint (as dovetails,
  notches, mortices and tenons) or by separate fastening pieces (as
  trenails, pins, spikes, nails, holdfasts, screws, bolts, rivets,
  hoops, straps and sockets.)

  § 4. _Stability, Stiffness and Strength._--A structure may be damaged
  or destroyed in three ways:--first, by displacement of its pieces from
  their proper positions relatively to each other or to the earth;
  secondly by disfigurement of one or more of those pieces, owing to
  their being unable to preserve their proper shapes under the pressures
  to which they are subjected; thirdly, by _breaking_ of one or more of
  those pieces. The power of resisting displacement constitutes
  stability, the power of each piece to resist disfigurement is its
  _stiffness_; and its power to resist breaking, its _strength_.

  § 5. _Conditions of Stability._--The principles of the stability of a
  structure can be to a certain extent investigated independently of the
  stiffness and strength, by assuming, in the first instance, that each
  piece has strength sufficient to be safe against being broken, and
  stiffness sufficient to prevent its being disfigured to an extent
  inconsistent with the purposes of the structure, by the greatest
  forces which are to be applied to it. The condition that each piece of
  the structure is to be maintained in equilibrium by having its gross
  load, consisting of its own weight and of the external pressure
  applied to it, balanced by the _resistances_ or pressures exerted
  between it and the contiguous pieces, furnishes the means of
  determining the magnitude, position and direction of the resistances
  required at each joint in order to produce equilibrium; and the
  _conditions of stability_ are, first, that the _position_, and,
  secondly, that the _direction_, of the resistance required at each
  joint shall, under all the variations to which the load is subject, be
  such as the joint is capable of exerting--conditions which are
  fulfilled by suitably adjusting the figures and positions of the
  joints, and the _ratios_ of the gross loads of the pieces. As for the
  _magnitude_ of the resistance, it is limited by conditions, not of
  stability, but of strength and stiffness.

  § 6. _Principle of Least Resistance._--Where more than one system of
  resistances are alike capable of balancing the same system of loads
  applied to a given structure, the _smallest_ of those alternative
  systems, as was demonstrated by the Rev. Henry Moseley in his
  _Mechanics of Engineering and Architecture_, is that which will
  actually be exerted--because the resistances to displacement are the
  effect of a strained state of the pieces, which strained state is the
  effect of the load, and when the load is applied the strained state
  and the resistances produced by it increase until the resistances
  acquire just those magnitudes which are sufficient to balance the
  load, after which they increase no further.

  This principle of least resistance renders determinate many problems
  in the statics of structures which were formerly considered
  indeterminate.

  § 7. _Relations between Polygons of Loads and of Resistances._--In a
  structure in which each piece is supported at two joints only, the
  well-known laws of statics show that the directions of the gross load
  on each piece and of the two resistances by which it is supported must
  lie in one plane, must either be parallel or meet in one point, and
  must bear to each other, if not parallel, the proportions of the sides
  of a triangle respectively parallel to their directions, and, if
  parallel, such proportions that each of the three forces shall be
  proportional to the distance between the other two,--all the three
  distances being measured along one direction.

  [Illustration: FIG. 86.]

  Considering, in the first place, the case in which the load and the
  two resistances by which each piece is balanced meet in one point,
  which may be called the _centre of load_, there will be as many such
  points of intersection, or centres of load, as there are pieces in the
  structure; and the directions and positions of the resistances or
  mutual pressures exerted between the pieces will be represented by the
  sides of a polygon joining those points, as in fig. 86 where P1, P2,
  P3, P4 represent the centres of load in a structure of four pieces,
  and the sides of the _polygon of resistances_ P1 P2 P3 P4 represent
  respectively the directions and positions of the resistances exerted
  at the joints. Further, at any one of the centres of load let PL
  represent the magnitude and direction of the gross load, and Pa, Pb
  the two resistances by which the piece to which that load is applied
  is supported; then will those three lines be respectively the diagonal
  and sides of a parallelogram; or, what is the same thing, they will be
  equal to the three sides of a triangle; and they must be in the same
  plane, although the sides of the polygon of resistances may be in
  different planes.

  [Illustration: FIG. 87.]

  According to a well-known principle of statics, because the loads or
  external pressures P1L1, &c., balance each other, they must be
  proportional to the sides of a closed polygon drawn respectively
  parallel to their directions. In fig. 87 construct such a _polygon of
  loads_ by drawing the lines L1, &c., parallel and proportional to, and
  joined end to end in the order of, the gross loads on the pieces of
  the structure. Then from the proportionality and parallelism of the
  load and the two resistances applied to each piece of the structure to
  the three sides of a triangle, there results the following theorem
  (originally due to Rankine):--

  _If from the angles of the polygon of loads there be drawn lines (R1,
  R2, &c.), each of which is parallel to the resistance (as P1P2, &c.)
  exerted at the joint between the pieces to which the two loads
  represented by the contiguous sides of the polygon of loads (such as
  L1, L2, &c.) are applied; then will all those lines meet in one point
  (O), and their lengths, measured from that point to the angles of the
  polygon, will represent the magnitudes of the resistances to which
  they are respectively parallel._

  When the load on one of the pieces is parallel to the resistances
  which balance it, the polygon of resistances ceases to be closed, two
  of the sides becoming parallel to each other and to the load in
  question, and extending indefinitely. In the polygon of loads the
  direction of a load sustained by parallel resistances traverses the
  point O.[2]

  § 8. _How the Earth's Resistance is to be treated_.... When the
  pressure exerted by a structure on the earth (to which the earth's
  resistance is equal and opposite) consists either of one pressure,
  which is necessarily the resultant of the weight of the structure and
  of all the other forces applied to it, or of two or more parallel
  vertical forces, whose amount can be determined at the outset of the
  investigation, the resistance of the earth can be treated as one or
  more upward loads applied to the structure. But in other cases the
  earth is to be treated as _one of the pieces of the structure_, loaded
  with a force equal and opposite in direction and position to the
  resultant of the weight of the structure and of the other pressures
  applied to it.

  § 9. _Partial Polygons of Resistance._--In a structure in which there
  are pieces supported at more than two joints, let a polygon be
  constructed of lines connecting the centres of load of any continuous
  series of pieces. This may be called a _partial polygon of
  resistances_. In considering its properties, the load at each centre
  of load is to be held to _include_ the resistances of those joints
  which are not comprehended in the partial polygon of resistances, to
  which the theorem of § 7 will then apply in every respect. By
  constructing several partial polygons, and computing the relations
  between the loads and resistances which are determined by the
  application of that theorem to each of them, with the aid, if
  necessary, of Moseley's principle of the least resistance, the whole
  of the relations amongst the loads and resistances may be found.

  § 10. _Line of Pressures--Centres and Line of Resistance._--The line
  of pressures is a line to which the directions of all the resistances
  in one polygon are tangents. The _centre of resistance_ at any joint
  is the point where the line representing the total resistance exerted
  at that joint intersects the joint. The _line of resistance_ is a line
  traversing all the centres of resistance of a series of joints,--its
  form, in the positions intermediate between the actual joints of the
  structure, being determined by supposing the pieces and their loads to
  be subdivided by the introduction of intermediate joints _ad
  infinitum_, and finding the continuous line, curved or straight, in
  which the intermediate centres of resistance are all situated, however
  great their number. The difference between the line of resistance and
  the line of pressures was first pointed out by Moseley.

  [Illustration: FIG. 88.]

  § 11.* The principles of the two preceding sections may be illustrated
  by the consideration of a particular case of a buttress of blocks
  forming a continuous series of pieces (fig. 88), where aa, bb, cc, dd
  represent plane joints. Let the centre of pressure C at the first
  joint aa be known, and also the pressure P acting at C in direction
  and magnitude. Find R1 the resultant of this pressure, the weight of
  the block aabb acting through its centre of gravity, and any other
  external force which may be acting on the block, and produce its line
  of action to cut the joint bb in C1. C1 is then the centre of pressure
  for the joint bb, and R1 is the total force acting there. Repeating
  this process for each block in succession there will be found the
  centres of pressure C2, C3, &c., and also the resultant pressures R2,
  R3, &c., acting at these respective centres. The centres of pressure
  at the joints are also called _centres of resistance_, and the curve
  passing through these points is called a _line of resistance_. Let all
  the resultants acting at the several centres of resistance be produced
  until they cut one another in a series of points so as to form an
  unclosed polygon. This polygon is the _partial polygon of resistance_.
  A curve tangential to all the sides of the polygon is the _line of
  pressures_.

  § 12. _Stability of Position, and Stability of Friction._--The
  resistances at the several joints having been determined by the
  principles set forth in §§ 6, 7, 8, 9 and 10, not only under the
  ordinary load of the structure, but under all the variations to which
  the load is subject as to amount and distribution, the joints are now
  to be placed and shaped so that the pieces shall not suffer relative
  displacement under any of those loads. The relative displacement of
  the two pieces which abut against each other at a joint may take place
  either by turning or by sliding. Safety against displacement by
  turning is called _stability of position_; safety against displacement
  by sliding, _stability of friction_.

  § 13. _Condition of Stability of Position._--If the materials of a
  structure were infinitely stiff and strong, stability of position at
  any joint would be insured simply by making the centre of resistance
  fall within the joint under all possible variations of load. In order
  to allow for the finite stiffness and strength of materials, the least
  distance of the centre of resistance inward from the nearest edge of
  the joint is made to bear a definite proportion to the depth of the
  joint measured in the same direction, which proportion is fixed,
  sometimes empirically, sometimes by theoretical deduction from the
  laws of the strength of materials. That least distance is called by
  Moseley the _modulus of stability_. The following are some of the
  ratios of the modulus of stability to the depth of the joint which
  occur in practice:--

    Retaining walls, as designed by British engineers                1:8
    Retaining walls, as designed by French engineers                 1:5
    Rectangular piers of bridges and other buildings, and
      arch-stones                                                    1:3
    Rectangular foundations, firm ground                             1:3
    Rectangular foundations, very soft ground                        1:2
    Rectangular foundations, intermediate kinds of ground     1:3 to 1:2
    Thin, hollow towers (such as furnace chimneys exposed
      to high winds), square                                         1:6
    Thin, hollow towers, circular                                    1:4
    Frames of timber or metal, under their ordinary or
      average distribution of load                                   1:3
    Frames of timber or metal, under the greatest
      irregularities of load                                         1:3

  In the case of the towers, the _depth of the joint_ is to be
  understood to mean the _diameter of the tower_.

  [Illustration: FIG. 89.]

  § 14. _Condition of Stability of Friction._--If the resistance to be
  exerted at a joint is always perpendicular to the surfaces which abut
  at and form that joint, there is no tendency of the pieces to be
  displaced by sliding. If the resistance be oblique, let JK (fig. 89)
  be the joint, C its centre of resistance, CR a line representing the
  resistance, CN a perpendicular to the joint at the centre of
  resistance. The angle NCR is the _obliquity_ of the resistance. From R
  draw RP parallel and RQ perpendicular to the joint; then, by the
  principles of statics, the component of the resistance _normal_ to the
  joint is--

    CP = CR · cos PCR;

  and the component _tangential_ to the joint is--

    CQ = CR · sin PCR = CP · tan PCR.

  If the joint be provided either with projections and recesses, such as
  mortises and tenons, or with fastenings, such as pins or bolts, so as
  to resist displacement by sliding, the question of the utmost amount
  of the tangential resistance CQ which it is capable of exerting
  depends on the _strength_ of such projections, recesses, or
  fastenings; and belongs to the subject of strength, and not to that of
  stability. In other cases the safety of the joint against displacement
  by sliding depends on its power of exerting friction, and that power
  depends on the law, known by experiment, that the friction between two
  surfaces bears a constant ratio, depending on the nature of the
  surfaces, to the force by which they are pressed together. In order
  that the surfaces which abut at the joint JK may be pressed together,
  the resistance required by the conditions of equilibrium CR, must be a
  _thrust_ and not a _pull_; and in that case the force by which the
  surfaces are pressed together is equal and opposite to the normal
  component CP of the resistance. The condition of stability of friction
  is that the tangential component CQ of the resistance required shall
  not exceed the friction due to the normal component; that is, that

    CQ [/>] f · CP,

  where f denotes the _coefficient of friction_ for the surfaces in
  question. The angle whose tangent is the coefficient of friction is
  called _the angle of repose_, and is expressed symbolically by--

    [phi] = tan^-1 f.

    Now CQ = CP · tan PCR;

  consequently the condition of stability of friction is fulfilled if
  the angle PCR is not greater than [phi]; that is to say, if _the
  obliquity of the resistance required at the joint does not exceed the
  angle of repose_; and this condition ought to be fulfilled under all
  possible variations of the load.

  It is chiefly in masonry and earthwork that stability of friction is
  relied on.

  § 15. _Stability of Friction in Earth._--The grains of a mass of loose
  earth are to be regarded as so many separate pieces abutting against
  each other at joints in all possible positions, and depending for
  their stability on friction. To determine whether a mass of earth is
  stable at a given point, conceive that point to be traversed by planes
  in all possible positions, and determine which position gives the
  greatest obliquity to the total pressure exerted between the portions
  of the mass which abut against each other at the plane. The condition
  of stability is that this obliquity shall not exceed the angle of
  repose of the earth. The consequences of this principle are developed
  in a paper, "On the Stability of Loose Earth," already cited in § 2.

  § 16. _Parallel Projections of Figures._--If any figure be referred to
  a system of co-ordinates, rectangular or oblique, and if a second
  figure be constructed by means of a second system of co-ordinates,
  rectangular or oblique, and either agreeing with or differing from the
  first system in rectangularity or obliquity, but so related to the
  co-ordinates of the first figure that for each point in the first
  figure there shall be a corresponding point in the second figure, the
  lengths of whose co-ordinates shall bear respectively to the three
  corresponding co-ordinates of the corresponding point in the first
  figure three ratios which are the same for every pair of corresponding
  points in the two figures, these corresponding figures are called
  _parallel projections_ of each other. The properties of parallel
  projections of most importance to the subject of the present article
  are the following:--

  (1) A parallel projection of a straight line is a straight line.

  (2) A parallel projection of a plane is a plane.

  (3) A parallel projection of a straight line or a plane surface
  divided in a given ratio is a straight line or a plane surface divided
  in the same ratio.

  (4) A parallel projection of a pair of equal and parallel straight
  lines, or plain surfaces, is a pair of equal and parallel straight lines,
  or plane surfaces; whence it follows

  (5) That a parallel projection of a parallelogram is a parallelogram,
  and

  (6) That a parallel projection of a parallelepiped is a parallelepiped.

  (7) A parallel projection of a pair of solids having a given ratio
  is a pair of solids having the same ratio.

  Though not essential for the purposes of the present article, the
  following consequence will serve to illustrate the principle of
  parallel projections:--

  (8) A parallel projection of a curve, or of a surface of a given
  algebraical order, is a curve or a surface of the same order.

  For example, all ellipsoids referred to co-ordinates parallel to any
  three conjugate diameters are parallel projections of each other and
  of a sphere referred to rectangular co-ordinates.

  § 17. _Parallel Projections of Systems of Forces._--If a balanced
  system of forces be represented by a system of lines, then will every
  parallel projection of that system of lines represent a balanced
  system of forces.

  For the condition of equilibrium of forces not parallel is that they
  shall be represented in direction and magnitude by the sides and
  diagonals of certain parallelograms, and of parallel forces that they
  shall divide certain straight lines in certain ratios; and the
  parallel projection of a parallelogram is a parallelogram, and that of
  a straight line divided in a given ratio is a straight line divided in
  the same ratio.

  The resultant of a parallel projection of any system of forces is the
  projection of their resultant; and the centre of gravity of a parallel
  projection of a solid is the projection of the centre of gravity of
  the first solid.

  § 18. _Principle of the Transformation of Structures._--Here we have
  the following theorem: If a structure of a given figure have stability
  of position under a system of forces represented by a given system of
  lines, then will any structure whose figure is a parallel projection
  of that of the first structure have stability of position under a
  system of forces represented by the corresponding projection of the
  first system of lines.

  For in the second structure the weights, external pressures, and
  resistances will balance each other as in the first structure; the
  weights of the pieces and all other parallel systems of forces will
  have the same ratios as in the first structure; and the several
  centres of resistance will divide the depths of the joints in the same
  proportions as in the first structure.

  If the first structure have stability of friction, the second
  structure will have stability of friction also, so long as the effect
  of the projection is not to increase the obliquity of the resistance
  at any joint beyond the angle of repose.

  The lines representing the forces in the second figure show their
  _relative_ directions and magnitudes. To find their _absolute_
  directions and magnitudes, a vertical line is to be drawn in the first
  figure, of such a length as to represent the weight of a particular
  portion of the structure. Then will the projection of that line in the
  projected figure indicate the vertical direction, and represent the
  weight of the part of the second structure corresponding to the
  before-mentioned portion of the first structure.

  The foregoing "principle of the transformation of structures" was
  first announced, though in a somewhat less comprehensive form, to the
  Royal Society on the 6th of March 1856. It is useful in practice, by
  enabling the engineer easily to deduce the conditions of equilibrium
  and stability of structures of complex and unsymmetrical figures from
  those of structures of simple and symmetrical figures. By its aid, for
  example, the whole of the properties of elliptical arches, whether
  square or skew, whether level or sloping in their span, are at once
  deduced by projection from those of symmetrical circular arches, and
  the properties of ellipsoidal and elliptic-conoidal domes from those
  of hemispherical and circular-conoidal domes; and the figures of
  arches fitted to resist the thrust of earth, which is less
  horizontally than vertically in a certain given ratio, can be deduced
  by a projection from those of arches fitted to resist the thrust of a
  liquid, which is of equal intensity, horizontally and vertically.

  § 19. _Conditions of Stiffness and Strength._--After the arrangement
  of the pieces of a structure and the size and figure of their joints
  or surfaces of contact have been determined so as to fulfil the
  conditions of _stability_,--conditions which depend mainly on the
  position and direction of the _resultant_ or _total_ load on each
  piece, and the _relative_ magnitude of the loads on the different
  pieces--the dimensions of each piece singly have to be adjusted so as
  to fulfil the conditions of _stiffness_ and _strength_--conditions
  which depend not only on the _absolute_ magnitude of the load on each
  piece, and of the resistances by which it is balanced, but also on the
  _mode of distribution_ of the load over the piece, and of the
  resistances over the joints.

  The effect of the pressures applied to a piece, consisting of the load
  and the supporting resistances, is to force the piece into a state of
  _strain_ or disfigurement, which increases until the elasticity, or
  resistance to strain, of the material causes it to exert a _stress_,
  or effort to recover its figure, equal and opposite to the system of
  applied pressures. The condition of _stiffness_ is that the strain or
  disfigurement shall not be greater than is consistent with the
  purposes of the structure; and the condition of _strength_ is that the
  stress shall be within the limits of that which the material can bear
  with safety against breaking. The ratio in which the utmost stress
  before breaking exceeds the safe working stress is called the _factor
  of safety_, and is determined empirically. It varies from three to
  twelve for various materials and structures. (See STRENGTH OF
  MATERIALS.)


  PART II. THEORY OF MACHINES

  § 20. _Parts of a Machine: Frame and Mechanism._--The parts of a
  machine may be distinguished into two principal divisions,--the frame,
  or fixed parts, and the _mechanism_, or moving parts. The frame is a
  structure which supports the pieces of the mechanism, and to a certain
  extent determines the nature of their motions.

  The form and arrangement of the pieces of the frame depend upon the
  arrangement and the motions of the mechanism; the dimensions of the
  pieces of the frame required in order to give it stability and
  strength are determined from the pressures applied to it by means of
  the mechanism. It appears therefore that in general the mechanism is
  to be designed first and the frame afterwards, and that the designing
  of the frame is regulated by the principles of the stability of
  structures and of the strength and stiffness of materials,--care being
  taken to adapt the frame to the most severe load which can be thrown
  upon it at any period of the action of the mechanism.

  Each independent piece of the mechanism also is a structure, and its
  dimensions are to be adapted, according to the principles of the
  strength and stiffness of materials, to the most severe load to which
  it can be subjected during the action of the machine.

  § 21. _Definition and Division of the Theory of Machines._--From what
  has been said in the last section it appears that the department of
  the art of designing machines which has reference to the stability of
  the frame and to the stiffness and strength of the frame and mechanism
  is a branch of the art of construction. It is therefore to be
  separated from the _theory of machines_, properly speaking, which has
  reference to the action of machines considered as moving. In the
  action of a machine the following three things take place:--

  _Firstly_, Some natural source of energy communicates motion and force
  to a piece or pieces of the mechanism, called the _receiver of power_
  or _prime mover_.

  _Secondly_, The motion and force are transmitted from the prime mover
  through the _train of mechanism_ to the _working piece_ or _pieces_,
  and during that transmission the motion and force are modified in
  amount and direction, so as to be rendered suitable for the purpose to
  which they are to be applied.

  _Thirdly_, The working piece or pieces by their motion, or by their
  motion and force combined, produce some useful effect.

  Such are the phenomena of the action of a machine, arranged in the
  order of _causation_. But in studying or treating of the theory of
  machines, the order of _simplicity_ is the best; and in this order the
  first branch of the subject is the modification of motion and force by
  the train of mechanism; the next is the effect or purpose of the
  machine; and the last, or most complex, is the action of the prime
  mover.

  The modification of motion and the modification of force take place
  together, and are connected by certain laws; but in the study of the
  theory of machines, as well as in that of pure mechanics, much
  advantage has been gained in point of clearness and simplicity by
  first considering alone the principles of the modification of motion,
  which are founded upon what is now known as Kinematics, and afterwards
  considering the principles of the combined modification of motion and
  force, which are founded both on geometry and on the laws of dynamics.
  The separation of kinematics from dynamics is due mainly to G. Monge,
  Ampère and R. Willis.

  The theory of machines in the present article will be considered under
  the following heads:--

  I. PURE MECHANISM, or APPLIED KINEMATICS; being the theory of machines
  considered simply as modifying motion.

  II. APPLIED DYNAMICS; being the theory of machines considered as
  modifying both motion and force.


  CHAP. I. ON PURE MECHANISM

  § 22. _Division of the Subject._--Proceeding in the order of
  simplicity, the subject of Pure Mechanism, or Applied Kinematics, may
  be thus divided:--

  _Division 1._--Motion of a point.

  _Division 2._--Motion of the surface of a fluid.

  _Division 3._--Motion of a rigid solid.

  _Division 4._--Motions of a pair of connected pieces, or of an
    "elementary combination" in mechanism.

  _Division 5._--Motions of trains of pieces of mechanism.

  _Division 6._--Motions of sets of more than two connected pieces, or of
    "aggregate combinations."

  A point is the boundary of a line, which is the boundary of a surface,
  which is the boundary of a volume. Points, lines and surfaces have no
  independent existence, and consequently those divisions of this
  chapter which relate to their motions are only preliminary to the
  subsequent divisions, which relate to the motions of bodies.


  _Division 1. Motion of a Point._

  § 23. _Comparative Motion._--The comparative motion of two points is
  the relation which exists between their motions, without having regard
  to their absolute amounts. It consists of two elements,--the _velocity
  ratio_, which is the ratio of any two magnitudes bearing to each other
  the proportions of the respective velocities of the two points at a
  given instant, and the _directional relation_, which is the relation
  borne to each other by the respective directions of the motions of the
  two points at the same given instant.

  It is obvious that the motions of a pair of points may be varied in
  any manner, whether by direct or by lateral deviation, and yet that
  their _comparative motion_ may remain constant, in consequence of the
  deviations taking place in the same proportions, in the same
  directions and at the same instants for both points.

  Robert Willis (1800-1875) has the merit of having been the first to
  simplify considerably the theory of pure mechanism, by pointing out
  that that branch of mechanics relates wholly to comparative motions.

  The comparative motion of two points at a given instant is capable of
  being completely expressed by one of Sir William Hamilton's
  Quaternions,--the "tensor" expressing the velocity ratio, and the
  "versor" the directional relation.

  Graphical methods of analysis founded on this way of representing
  velocity and acceleration were developed by R. H. Smith in a paper
  communicated to the Royal Society of Edinburgh in 1885, and
  illustrations of the method will be found below.


  _Division 2. Motion of the Surface of a Fluid Mass._

  § 24. _General Principle._--A mass of fluid is used in mechanism to
  transmit motion and force between two or more movable portions (called
  _pistons_ or _plungers_) of the solid envelope or vessel in which the
  fluid is contained; and, when such transmission is the sole action, or
  the only appreciable action of the fluid mass, its volume is either
  absolutely constant, by reason of its temperature and pressure being
  maintained constant, or not sensibly varied.

  Let a represent the area of the section of a piston made by a plane
  perpendicular to its direction of motion, and v its velocity, which is
  to be considered as positive when outward, and negative when inward.
  Then the variation of the cubic contents of the vessel in a unit of
  time by reason of the motion of one piston is va. The condition that
  the volume of the fluid mass shall remain unchanged requires that
  there shall be more than one piston, and that the velocities and areas
  of the pistons shall be connected by the equation--

    [Sigma]·va = 0.  (1)

  § 25. _Comparative Motion of Two Pistons._--If there be but two
  pistons, whose areas are a1 and a2, and their velocities v1 and v2,
  their comparative motion is expressed by the equation--

    v2/v1 = -a1/a2;  (2)

  that is to say, their velocities are opposite as to inwardness and
  outwardness and inversely proportional to their areas.

  § 26. _Applications: Hydraulic Press: Pneumatic
  Power-Transmitter._--In the hydraulic press the vessel consists of two
  cylinders, viz. the pump-barrel and the press-barrel, each having its
  piston, and of a passage connecting them having a valve opening
  towards the press-barrel. The action of the enclosed water in
  transmitting motion takes place during the inward stroke of the
  pump-plunger, when the above-mentioned valve is open; and at that time
  the press-plunger moves outwards with a velocity which is less than
  the inward velocity of the pump-plunger, in the same ratio that the
  area of the pump-plunger is less than the area of the press-plunger.
  (See HYDRAULICS.)

  In the pneumatic power-transmitter the motion of one piston is
  transmitted to another at a distance by means of a mass of air
  contained in two cylinders and an intervening tube. When the pressure
  and temperature of the air can be maintained constant, this machine
  fulfils equation (2), like the hydraulic press. The amount and effect
  of the variations of pressure and temperature undergone by the air
  depend on the principles of the mechanical action of heat, or
  THERMODYNAMICS (q.v.), and are foreign to the subject of pure
  mechanism.


  _Division 3. Motion of a Rigid Solid._

  § 27. _Motions Classed._--In problems of mechanism, each solid piece
  of the machine is supposed to be so stiff and strong as not to undergo
  any sensible change of figure or dimensions by the forces applied to
  it--a supposition which is realized in practice if the machine is
  skilfully designed.

  This being the case, the various possible motions of a rigid solid
  body may all be classed under the following heads: (1) _Shifting or
  Translation_; (2) _Turning or Rotation_; (3) _Motions compounded of
  Shifting and Turning_.

  The most common forms for the paths of the points of a piece of
  mechanism, whose motion is simple shifting, are the straight line and
  the circle.

  Shifting in a straight line is regulated either by straight fixed
  guides, in contact with which the moving piece slides, or by
  combinations of link-work, called _parallel motions_, which will be
  described in the sequel. Shifting in a straight line is usually
  _reciprocating_; that is to say, the piece, after shifting through a
  certain distance, returns to its original position by reversing its
  motion.

  Circular shifting is regulated by attaching two or more points of the
  shifting piece to ends of equal and parallel rotating cranks, or by
  combinations of wheel-work to be afterwards described. As an example
  of circular shifting may be cited the motion of the coupling rod, by
  which the parallel and equal cranks upon two or more axles of a
  locomotive engine are connected and made to rotate simultaneously. The
  coupling rod remains always parallel to itself, and all its points
  describe equal and similar circles relatively to the frame of the
  engine, and move in parallel directions with equal velocities at the
  same instant.

  § 28. _Rotation about a Fixed Axis: Lever, Wheel and Axle._--The fixed
  axis of a turning body is a line fixed relatively to the body and
  relatively to the fixed space in which the body turns. In mechanism it
  is usually the central line either of a rotating shaft or axle having
  journals, gudgeons, or pivots turning in fixed bearings, or of a fixed
  spindle or dead centre round which a rotating bush turns; but it may
  sometimes be entirely beyond the limits of the turning body. For
  example, if a sliding piece moves in circular fixed guides, that piece
  rotates about an ideal fixed axis traversing the centre of those
  guides.

  Let the angular velocity of the rotation be denoted by [alpha] =
  d[theta]/dt, then the linear velocity of any point A at the distance r
  from the axis is [alpha]r; and the path of that point is a circle of
  the radius r described about the axis.

  This is the principle of the modification of motion by the lever,
  which consists of a rigid body turning about a fixed axis called a
  fulcrum, and having two points at the same or different distances from
  that axis, and in the same or different directions, one of which
  receives motion and the other transmits motion, modified in direction
  and velocity according to the above law.

  In the wheel and axle, motion is received and transmitted by two
  cylindrical surfaces of different radii described about their common
  fixed axis of turning, their velocity-ratio being that of their radii.

  [Illustration: FIG. 90.]

  § 29. _Velocity Ratio of Components of Motion._--As the distance
  between any two points in a rigid body is invariable, the projections
  of their velocities upon the line joining them must be equal. Hence it
  follows that, if A in fig. 90 be a point in a rigid body CD, rotating
  round the fixed axis F, the component of the velocity of A in any
  direction AP parallel to the plane of rotation is equal to the total
  velocity of the point m, found by letting fall Fm perpendicular to AP;
  that is to say, is equal to

    [alpha]·Fm.

  Hence also the ratio of the components of the velocities of two points
  A and B in the directions AP and BW respectively, both in the plane of
  rotation, is equal to the ratio of the perpendiculars Fm and Fn.

  § 30. _Instantaneous Axis of a Cylinder rolling on a Cylinder._--Let a
  cylinder bbb, whose axis of figure is B and angular velocity [gamma],
  roll on a fixed cylinder [alpha][alpha][alpha], whose axis of figure
  is A, either outside (as in fig. 91), when the rolling will be towards
  the same hand as the rotation, or inside (as in fig. 92), when the
  rolling will be towards the opposite hand; and at a given instant let
  T be the line of contact of the two cylindrical surfaces, which is at
  their common intersection with the plane AB traversing the two axes of
  figure.

  The line T on the surface bbb has for the instant no velocity in a
  direction perpendicular to AB; because for the instant it touches,
  without sliding, the line T on the fixed surface aaa.

  The line T on the surface bbb has also for the instant no velocity in
  the plane AB; for it has just ceased to move towards the fixed surface
  aaa, and is just about to begin to move away from that surface.

  The line of contact T, therefore, on the surface of the cylinder bbb,
  is _for the instant_ at rest, and is the "instantaneous axis" about
  which the cylinder bbb turns, together with any body rigidly attached
  to that cylinder.

  [Illustration: FIG. 91.]

  [Illustration: FIG. 92.]

  To find, then, the direction and velocity at the given instant of any
  point P, either in or rigidly attached to the rolling cylinder T, draw
  the plane PT; the direction of motion of P will be perpendicular to
  that plane, and towards the right or left hand according to the
  direction of the rotation of bbb; and the velocity of P will be

    v_P = [gamma]·PT,  (3)

  PT denoting the perpendicular distance of P from T. The path of P is a
  curve of the kind called _epitrochoids_. If P is in the circumference
  of bbb, that path becomes an _epicycloid_.

  The velocity of any point in the axis of figure B is

    v_B = [gamma]·TB;  (4)

  and the path of such a point is a circle described about A with the
  radius AB, being for outside rolling the sum, and for inside rolling
  the difference, of the radii of the cylinders.

  Let [alpha] denote the angular velocity with which the _plane of axes_
  AB rotates about the fixed axis A. Then it is evident that

    v_B = [alpha]·AB,  (5)

  and consequently that

    [alpha] = [gamma]·TB/AB.  (6)

  For internal rolling, as in fig. 92, AB is to be treated as negative,
  which will give a negative value to [alpha], indicating that in this
  case the rotation of AB round A is contrary to that of the cylinder
  bbb.

  The angular velocity of the rolling cylinder, _relatively to the plane
  of axes_ AB, is obviously given by the equation--

    [beta] = [gamma] - [alpha]     \
                                    >,  (7)
    whence [beta] = [gamma]·TA/AB  /

  care being taken to attend to the sign of [alpha], so that when that
  is negative the arithmetical values of [gamma] and [alpha] are to be
  added in order to give that of [beta].

  The whole of the foregoing reasonings are applicable, not merely when
  aaa and bbb are actual cylinders, but also when they are the
  osculating cylinders of a pair of cylindroidal surfaces of varying
  curvature, A and B being the axes of curvature of the parts of those
  surfaces which are in contact for the instant under consideration.

  [Illustration: FIG. 93.]

  § 31. _Instantaneous Axis of a Cone rolling on a Cone._--Let Oaa (fig.
  93) be a fixed cone, OA its axis, Obb a cone rolling on it, OB the
  axis of the rolling cone, OT the line of contact of the two cones at
  the instant under consideration. By reasoning similar to that of § 30,
  it appears that OT is the instantaneous axis of rotation of the
  rolling cone.

  Let [gamma] denote the total angular velocity of the rotation of the
  cone B about the instantaneous axis, [beta] its angular velocity about
  the axis OB _relatively_ to the plane AOB, and [alpha] the angular
  velocity with which the plane AOB turns round the axis OA. It is
  required to find the ratios of those angular velocities.

  _Solution._--In OT take any point E, from which draw EC parallel to
  OA, and ED parallel to OB, so as to construct the parallelogram OCED.
  Then

    OD : OC : OE :: [alpha] : [beta] : [gamma].  (8)

  Or because of the proportionality of the sides of triangles to the
  sines of the opposite angles,

    sin TOB : sin TOA : sin AOB :: [alpha] : [beta] : [gamma],  (8 A)

  that is to say, the angular velocity about each axis is proportional
  to the sine of the angle between the other two.

  _Demonstration._--From C draw CF perpendicular to OA, and CG
  perpendicular to OE

                  area ECO
    Then CF = 2 × --------,
                     CE

                  area ECO
    and CG = 2 ×  --------;
                     OE

    :. CG : CF :: CE = OD : OE.

  Let v_c denote the linear velocity of the point C. Then

    v_c = [alpha] · CF = [gamma]·CG
    :. [gamma] : [alpha] :: CF : CG :: OE : OD,

  which is one part of the solution above stated. From E draw EH
  perpendicular to OB, and EK to OA. Then it can be shown as before that

    EK : EH :: OC : OD.

  Let v_E be the linear velocity of the point E _fixed in the plane of
  axes_ AOB. Then

    v_K = [alpha] · EK.

  Now, as the line of contact OT is for the instant at rest on the
  rolling cone as well as on the fixed cone, the linear velocity of the
  point E fixed to the plane AOB relatively to the rolling cone is the
  same with its velocity relatively to the fixed cone. That is to say,

    [beta]·EH = v_E = [alpha]·EK;

  therefore

    [alpha] : [beta] :: EH : EK :: OD : OC,

  which is the remainder of the solution.

  The path of a point P in or attached to the rolling cone is a
  spherical epitrochoid traced on the surface of a sphere of the radius
  OP. From P draw PQ perpendicular to the instantaneous axis. Then the
  motion of P is perpendicular to the plane OPQ, and its velocity is

    v_P = [gamma]·PQ.  (9)

  The whole of the foregoing reasonings are applicable, not merely when
  A and B are actual regular cones, but also when they are the
  osculating regular cones of a pair of irregular conical surfaces,
  having a common apex at O.

  § 32. _Screw-like or Helical Motion._--Since any displacement in a
  plane can be represented in general by a rotation, it follows that the
  only combination of translation and rotation, in which a complex
  movement which is not a mere rotation is produced, occurs when there
  is a translation _perpendicular to the plane and parallel to the axis_
  of rotation.

  [Illustration: FIG. 94.]

  Such a complex motion is called _screw-like_ or _helical_ motion; for
  each point in the body describes a _helix_ or _screw_ round the axis
  of rotation, fixed or instantaneous as the case may be. To cause a
  body to move in this manner it is usually made of a helical or
  screw-like figure, and moves in a guide of a corresponding figure.
  Helical motion and screws adapted to it are said to be right- or
  left-handed according to the appearance presented by the rotation to
  an observer looking towards the direction of the translation. Thus the
  screw G in fig. 94 is right-handed.

  The translation of a body in helical motion is called its _advance_.
  Let v_x denote the velocity of advance at a given instant, which of
  course is common to all the particles of the body; [alpha] the angular
  velocity of the rotation at the same instant; 2[pi] = 6.2832 nearly,
  the circumference of a circle of the radius unity. Then

    T = 2[pi]/[alpha]  (10)

  is the time of one turn at the rate [alpha]; and

    p = v_x T = 2[pi]v_x/[alpha]  (11)

  is the _pitch_ or _advance per turn_--a length which expresses the
  _comparative motion_ of the translation and the rotation.

  The pitch of a screw is the distance, measured parallel to its axis,
  between two successive turns of the same _thread_ or helical
  projection.

  Let r denote the perpendicular distance of a point in a body moving
  helically from the axis. Then

    v_r = [alpha]r  (12)

  is the component of the velocity of that point in a plane
  perpendicular to the axis, and its total velocity is

    v = [root](v_x² + v_r²).  (13)

  The ratio of the two components of that velocity is

    v_x/v_r = p/2[pi]r = tan [theta].  (14)

  where [theta] denotes the angle made by the helical path of the point
  with a plane perpendicular to the axis.


  _Division 4. Elementary Combinations in Mechanism_

  § 33. _Definitions._--An _elementary combination_ in mechanism
  consists of two pieces whose kinds of motion are determined by their
  connexion with the frame, and their comparative motion by their
  connexion with each other--that connexion being effected either by
  direct contact of the pieces, or by a connecting piece, which is not
  connected with the frame, and whose motion depends entirely on the
  motions of the pieces which it connects.

  The piece whose motion is the cause is called the _driver_; the piece
  whose motion is the effect, the _follower_.

  The connexion of each of those two pieces with the frame is in general
  such as to determine the path of every point in it. In the
  investigation, therefore, of the comparative motion of the driver and
  follower, in an elementary combination, it is unnecessary to consider
  relations of angular direction, which are already fixed by the
  connexion of each piece with the frame; so that the inquiry is
  confined to the determination of the velocity ratio, and of the
  directional relation, so far only as it expresses the connexion
  between _forward_ and _backward_ movements of the driver and follower.
  When a continuous motion of the driver produces a continuous motion of
  the follower, forward or backward, and a reciprocating motion a motion
  reciprocating at the same instant, the directional relation is said to
  be _constant_. When a continuous motion produces a reciprocating
  motion, or vice versa, or when a reciprocating motion produces a
  motion not reciprocating at the same instant, the directional relation
  is said to be _variable_.

  The _line of action_ or _of connexion_ of the driver and follower is a
  line traversing a pair of points in the driver and follower
  respectively, which are so connected that the component of their
  velocity relatively to each other, resolved along the line of
  connexion, is null. There may be several or an indefinite number of
  lines of connexion, or there may be but one; and a line of connexion
  may connect either the same pair of points or a succession of
  different pairs.

  § 34. _General Principle._--From the definition of a line of connexion
  it follows that _the components of the velocities of a pair of
  connected points along their line of connexion are equal_. And from
  this, and from the property of a rigid body, already stated in § 29,
  it follows, that _the components along a line of connexion of all the
  points traversed by that line, whether in the driver or in the
  follower, are equal_; and consequently, _that the velocities of any
  pair of points traversed by a line of connexion are to each other
  inversely as the cosines, or directly as the secants, of the angles
  made by the paths of those points with the line of connexion_.

  The general principle stated above in different forms serves to solve
  every problem in which--the mode of connexion of a pair of pieces
  being given--it is required to find their comparative motion at a
  given instant, or vice versa.

  [Illustration: FIG. 95.]

  § 35. _Application to a Pair of Shifting Pieces._--In fig. 95, let
  P1P2 be the line of connexion of a pair of pieces, each of which has a
  motion of translation or shifting. Through any point T in that line
  draw TV1, TV2, respectively parallel to the simultaneous direction of
  motion of the pieces; through any other point A in the line of
  connexion draw a plane perpendicular to that line, cutting TV1, TV2 in
  V1, V2; then, velocity of piece 1 : velocity of piece 2 :: TV1 : TV2.
  Also TA represents the equal components of the velocities of the
  pieces parallel to their line of connexion, and the line V1V2
  represents their velocity relatively to each other.

  § 36. _Application to a Pair of Turning Pieces._--Let [alpha]1,
  [alpha]2 be the angular velocities of a pair of turning pieces;
  [theta]1, [theta]2 the angles which their line of connexion makes with
  their respective planes of rotation; r1, r2 the common perpendiculars
  let fall from the line of connexion upon the respective axes of
  rotation of the pieces. Then the equal components, along the line of
  connexion, of the velocities of the points where those perpendiculars
  meet that line are--

    [alpha]1r1 cos [theta]1 = [alpha]2r2 cos [theta]2;

  consequently, the comparative motion of the pieces is given by the
  equation

    [alpha]2   r1 cos [theta]1
    -------- = ---------------.  (15)
    [alpha]1   r2 cos [theta]2

  § 37. _Application to a Shifting Piece and a Turning Piece._--Let a
  shifting piece be connected with a turning piece, and at a given
  instant let [alpha]1 be the angular velocity of the turning piece, r1
  the common perpendicular of its axis of rotation and the line of
  connexion, [theta]1 the angle made by the line of connexion with the
  plane of rotation, [theta]2 the angle made by the line of connexion
  with the direction of motion of the shifting piece, v2 the linear
  velocity of that piece. Then

    [alpha]1r1 cos [theta]1 = v2 cos [theta]2;  (16)

  which equation expresses the comparative motion of the two pieces.

  § 38. _Classification of Elementary Combinations in Mechanism._--The
  first systematic classification of elementary combinations in
  mechanism was that founded by Monge, and fully developed by Lanz and
  Bétancourt, which has been generally received, and has been adopted in
  most treatises on applied mechanics. But that classification is
  founded on the absolute instead of the comparative motions of the
  pieces, and is, for that reason, defective, as Willis pointed out in
  his admirable treatise _On the Principles of Mechanism_.

  Willis's classification is founded, in the first place, on comparative
  motion, as expressed by velocity ratio and directional relation, and
  in the second place, on the mode of connexion of the driver and
  follower. He divides the elementary combinations in mechanism into
  three classes, of which the characters are as follows:--

  Class A: Directional relation constant; velocity ratio constant.

  Class B: Directional relation constant; velocity ratio varying.

  Class C: Directional relation changing periodically; velocity ratio
  constant or varying.

  Each of those classes is subdivided by Willis into five divisions, of
  which the characters are as follows:--

    Division A: Connexion by rolling contact.
        "    B:     "      "  sliding contact.
        "    C:     "      "  wrapping connectors.
        "    D:     "      "  link-work.
        "    E:     "      "  reduplication.

  In the Reuleaux system of analysis of mechanisms the principle of
  comparative motion is generalized, and mechanisms apparently very
  diverse in character are shown to be founded on the same sequence of
  elementary combinations forming a kinematic chain. A short description
  of this system is given in § 80, but in the present article the
  principle of Willis's classification is followed mainly. The
  arrangement is, however, modified by taking the _mode of connexion_ as
  the basis of the primary classification, and by removing the subject
  of connexion by reduplication to the section of aggregate
  combinations. This modified arrangement is adopted as being better
  suited than the original arrangement to the limits of an article in an
  encyclopaedia; but it is not disputed that the original arrangement
  may be the best for a separate treatise.

  § 39. _Rolling Contact: Smooth Wheels and Racks._--In order that two
  pieces may move in rolling contact, it is necessary that each pair of
  points in the two pieces which touch each other should at the instant
  of contact be moving in the same direction with the same velocity. In
  the case of two _shifting_ pieces this would involve equal and
  parallel velocities for all the points of each piece, so that there
  could be no rolling, and, in fact, the two pieces would move like one;
  hence, in the case of rolling contact, either one or both of the
  pieces must rotate.

  The direction of motion of a point in a turning piece being
  perpendicular to a plane passing through its axis, the condition that
  each pair of points in contact with each other must move in the same
  direction leads to the following consequences:--

  I. That, when both pieces rotate, their axes, and all their points of
  contact, lie in the same plane.

  II. That, when one piece rotates, and the other shifts, the axis of
  the rotating piece, and all the points of contact, lie in a plane
  perpendicular to the direction of motion of the shifting piece.

  The condition that the velocity of each pair of points of contact must
  be equal leads to the following consequences:--

  III. That the angular velocities of a pair of turning pieces in
  rolling contact must be inversely as the perpendicular distances of
  any pair of points of contact from the respective axes.

  IV. That the linear velocity of a shifting piece in rolling contact
  with a turning piece is equal to the product of the angular velocity
  of the turning piece by the perpendicular distance from its axis to a
  pair of points of contact.

  The _line of contact_ is that line in which the points of contact are
  all situated. Respecting this line, the above Principles III. and IV.
  lead to the following conclusions:--

  V. That for a pair of turning pieces with parallel axes, and for a
  turning piece and a shifting piece, the line of contact is straight,
  and parallel to the axes or axis; and hence that the rolling surfaces
  are either plane or cylindrical (the term "cylindrical" including all
  surfaces generated by the motion of a straight line parallel to
  itself).

  VI. That for a pair of turning pieces with intersecting axes the line
  of contact is also straight, and traverses the point of intersection
  of the axes; and hence that the rolling surfaces are conical, with a
  common apex (the term "conical" including all surfaces generated by
  the motion of a straight line which traverses a fixed point).

  Turning pieces in rolling contact are called _smooth_ or _toothless
  wheels_. Shifting pieces in rolling contact with turning pieces may be
  called _smooth_ or _toothless racks_.

  VII. In a pair of pieces in rolling contact every straight line
  traversing the line of contact is a line of connexion.

  § 40. _Cylindrical Wheels and Smooth Racks._--In designing cylindrical
  wheels and smooth racks, and determining their comparative motion, it
  is sufficient to consider a section of the pair of pieces made by a
  plane perpendicular to the axis or axes.

  The points where axes intersect the plane of section are called
  _centres_; the point where the line of contact intersects it, the
  _point of contact_, or _pitch-point_; and the wheels are described as
  _circular_, _elliptical_, &c., according to the forms of their
  sections made by that plane.

  When the point of contact of two wheels lies between their centres,
  they are said to be in _outside gearing_; when beyond their centres,
  in _inside gearing_, because the rolling surface of the larger wheel
  must in this case be turned inward or towards its centre.

  From Principle III. of § 39 it appears that the angular velocity-ratio
  of a pair of wheels is the inverse ratio of the distances of the point
  of contact from the centres respectively.

  [Illustration: FIG. 96.]

  For outside gearing that ratio is _negative_, because the wheels turn
  contrary ways; for inside gearing it is _positive_, because they turn
  the same way.

  If the velocity ratio is to be constant, as in Willis's Class A, the
  wheels must be circular; and this is the most common form for wheels.

  If the velocity ratio is to be variable, as in Willis's Class B, the
  figures of the wheels are a pair of _rolling curves_, subject to the
  condition that the distance between their _poles_ (which are the
  centres of rotation) shall be constant.

  The following is the geometrical relation which must exist between
  such a pair of curves:--

  Let C1, C2 (fig. 96) be the poles of a pair of rolling curves; T1, T2
  any pair of points of contact; U1, U2 any other pair of points of
  contact. Then, for every possible pair of points of contact, the two
  following equations must be simultaneously fulfilled:--

    Sum of radii, C1U1 + C2U2 = C1T1 + C2T2 = constant;
    arc, T2U2 = T1U1.  (17)

  A condition equivalent to the above, and necessarily connected with
  it, is, that at each pair of points of contact the inclinations of the
  curves to their radii-vectores shall be equal and contrary; or,
  denoting by r1, r2 the radii-vectores at any given pair of points of
  contact, and s the length of the equal arcs measured from a certain
  fixed pair of points of contact--

    dr2/ds = -dr1/ds;  (18)

  which is the differential equation of a pair of rolling curves whose
  poles are at a constant distance apart.

  For full details as to rolling curves, see Willis's work, already
  mentioned, and Clerk Maxwell's paper on Rolling Curves, _Trans. Roy.
  Soc. Edin._, 1849.

  A rack, to work with a circular wheel, must be straight. To work with
  a wheel of any other figure, its section must be a rolling curve,
  subject to the condition that the perpendicular distance from the pole
  or centre of the wheel to a straight line parallel to the direction of
  the motion of the rack shall be constant. Let r1 be the radius-vector
  of a point of contact on the wheel, x2 the ordinate from the straight
  line before mentioned to the corresponding point of contact on the
  rack. Then

    dx2/ds = -dr1/ds  (19)

  is the differential equation of the pair of rolling curves.

  To illustrate this subject, it may be mentioned that an ellipse
  rotating about one focus rolls completely round in outside gearing
  with an equal and similar ellipse also rotating about one focus, the
  distance between the axes of rotation being equal to the major axis of
  the ellipses, and the velocity ratio varying from (1 +
  eccentricity)/(1 - eccentricity) to (1 - eccentricity)/(1 +
  eccentricity); an hyperbola rotating about its further focus rolls in
  inside gearing, through a limited arc, with an equal and similar
  hyperbola rotating about its nearer focus, the distance between the
  axes of rotation being equal to the axis of the hyperbolas, and the
  velocity ratio varying between (eccentricity + 1)/(eccentricity - 1)
  and unity; and a parabola rotating about its focus rolls with an equal
  and similar parabola, shifting parallel to its directrix.

  [Illustration: FIG. 97.]

  § 41. _Conical or Bevel and Disk Wheels._--From Principles III. and
  VI. of § 39 it appears that the angular velocities of a pair of wheels
  whose axes meet in a point are to each other inversely as the sines of
  the angles which the axes of the wheels make with the line of contact.
  Hence we have the following construction (figs. 97 and 98).--Let O be
  the apex or point of intersection of the two axes OC1, OC2. The
  angular velocity ratio being given, it is required to find the line of
  contact. On OC1, OC2 take lengths OA1, OA2, respectively proportional
  to the angular velocities of the pieces on whose axes they are taken.
  Complete the parallelogram OA1EA2; the diagonal OET will be the line
  of contact required.

  When the velocity ratio is variable, the line of contact will shift
  its position in the plane C1OC2, and the wheels will be cones, with
  eccentric or irregular bases. In every case which occurs in practice,
  however, the velocity ratio is constant; the line of contact is
  constant in position, and the rolling surfaces of the wheels are
  regular circular cones (when they are called _bevel wheels_); or one
  of a pair of wheels may have a flat disk for its rolling surface, as
  W2 in fig. 98, in which case it is a _disk wheel_. The rolling
  surfaces of actual wheels consist of frusta or zones of the complete
  cones or disks, as shown by W1, W2 in figs. 97 and 98.

  [Illustration: FIG. 98.]

  § 42. _Sliding Contact (lateral): Skew-Bevel Wheels._--An hyperboloid
  of revolution is a surface resembling a sheaf or a dice box, generated
  by the rotation of a straight line round an axis from which it is at a
  constant distance, and to which it is inclined at a constant angle. If
  two such hyperboloids E, F, equal or unequal, be placed in the closest
  possible contact, as in fig. 99, they will touch each other along one
  of the generating straight lines of each, which will form their line
  of contact, and will be inclined to the axes AG, BH in opposite
  directions. The axes will not be parallel, nor will they intersect
  each other.

  [Illustration: FIG. 99.]

  The motion of two such hyperboloids, turning in contact with each
  other, has hitherto been classed amongst cases of rolling contact; but
  that classification is not strictly correct, for, although the
  component velocities of a pair of points of contact in a direction at
  right angles to the line of contact are equal, still, as the axes are
  parallel neither to each other nor to the line of contact, the
  velocities of a pair of points of contact have components along the
  line of contact which are unequal, and their difference constitutes a
  _lateral sliding_.

  The directions and positions of the axes being given, and the required
  angular velocity ratio, the following construction serves to determine
  the line of contact, by whose rotation round the two axes respectively
  the hyperboloids are generated:--

  [Illustration: FIG. 100.]

  In fig. 100, let B1C1, B2C2 be the two axes; B1B2 their common
  perpendicular. Through any point O in this common perpendicular draw
  OA1 parallel to B1C1 and OA2 parallel to B2C2; make those lines
  proportional to the angular velocities about the axes to which they
  are respectively parallel; complete the parallelogram OA1EA2, and draw
  the diagonal OE; divide B1B2 in D into two parts, _inversely_
  proportional to the angular velocities about the axes which they
  respectively adjoin; through D parallel to OE draw DT. This will be
  the line of contact.

  A pair of thin frusta of a pair of hyperboloids are used in practice
  to communicate motion between a pair of axes neither parallel nor
  intersecting, and are called _skew-bevel wheels_.

  In skew-bevel wheels the properties of a line of connexion are not
  possessed by every line traversing the line of contact, but only by
  every line traversing the line of contact at right angles.

  If the velocity ratio to be communicated were variable, the point D
  would alter its position, and the line DT its direction, at different
  periods of the motion, and the wheels would be hyperboloids of an
  eccentric or irregular cross-section; but forms of this kind are not
  used in practice.

  § 43. _Sliding Contact (circular): Grooved Wheels._--As the adhesion
  or friction between a pair of smooth wheels is seldom sufficient to
  prevent their slipping on each other, contrivances are used to
  increase their mutual hold. One of those consists in forming the rim
  of each wheel into a series of alternate ridges and grooves parallel
  to the plane of rotation; it is applicable to cylindrical and bevel
  wheels, but not to skew-bevel wheels. The comparative motion of a pair
  of wheels so ridged and grooved is the same as that of a pair of
  smooth wheels in rolling contact, whose cylindrical or conical
  surfaces lie midway between the tops of the ridges and bottoms of the
  grooves, and those ideal smooth surfaces are called the _pitch
  surfaces_ of the wheels.

  The relative motion of the faces of contact of the ridges and grooves
  is a _rotatory sliding_ or _grinding_ motion, about the line of
  contact of the pitch-surfaces as an instantaneous axis.

  Grooved wheels have hitherto been but little used.

  § 44. _Sliding Contact (direct): Teeth of Wheels, their Number and
  Pitch._--The ordinary method of connecting a pair of wheels, or a
  wheel and a rack, and the only method which ensures the exact
  maintenance of a given numerical velocity ratio, is by means of a
  series of alternate ridges and hollows parallel or nearly parallel to
  the successive lines of contact of the ideal smooth wheels whose
  velocity ratio would be the same with that of the toothed wheels. The
  ridges are called _teeth_; the hollows, _spaces_. The teeth of the
  driver push those of the follower before them, and in so doing
  sliding takes place between them in a direction across their lines of
  contact.

  The _pitch-surfaces_ of a pair of toothed wheels are the ideal smooth
  surfaces which would have the same comparative motion by rolling
  contact that the actual wheels have by the sliding contact of their
  teeth. The _pitch-circles_ of a pair of circular toothed wheels are
  sections of their pitch-surfaces, made for _spur-wheels_ (that is, for
  wheels whose axes are parallel) by a plane at right angles to the
  axes, and for bevel wheels by a sphere described about the common
  apex. For a pair of skew-bevel wheels the pitch-circles are a pair of
  contiguous rectangular sections of the pitch-surfaces. The
  _pitch-point_ is the point of contact of the pitch-circles.

  The pitch-surface of a wheel lies intermediate between the points of
  the teeth and the bottoms of the hollows between them. That part of
  the acting surface of a tooth which projects beyond the pitch-surface
  is called the _face_; that part which lies within the pitch-surface,
  the _flank_.

  Teeth, when not otherwise specified, are understood to be made in one
  piece with the wheel, the material being generally cast-iron, brass or
  bronze. Separate teeth, fixed into mortises in the rim of the wheel,
  are called _cogs_. A _pinion_ is a small toothed wheel; a _trundle_ is
  a pinion with cylindrical _staves_ for teeth.

  The radius of the pitch-circle of a wheel is called the _geometrical
  radius_; a circle touching the ends of the teeth is called the
  _addendum circle_, and its radius the _real radius_; the difference
  between these radii, being the projection of the teeth beyond the
  pitch-surface, is called the _addendum_.

  The distance, measured along the pitch-circle, from the face of one
  tooth to the face of the next, is called the _pitch_. The pitch and
  the number of teeth in wheels are regulated by the following
  principles:--

  I. In wheels which rotate continuously for one revolution or more, it
  is obviously necessary _that the pitch should be an aliquot part of
  the circumference_.

  In wheels which reciprocate without performing a complete revolution
  this condition is not necessary. Such wheels are called _sectors_.

  II. In order that a pair of wheels, or a wheel and a rack, may work
  correctly together, it is in all cases essential _that the pitch
  should be the same in each_.

  III. Hence, in any pair of circular wheels which work together, the
  numbers of teeth in a complete circumference are directly as the radii
  and inversely as the angular velocities.

  IV. Hence also, in any pair of circular wheels which rotate
  continuously for one revolution or more, the ratio of the numbers of
  teeth and its reciprocal the angular velocity ratio must be
  expressible in whole numbers.

  From this principle arise problems of a kind which will be referred to
  in treating of _Trains of Mechanism_.

  V. Let n, N be the respective numbers of teeth in a pair of wheels, N
  being the greater. Let t, T be a pair of teeth in the smaller and
  larger wheel respectively, which at a particular instant work
  together. It is required to find, first, how many pairs of teeth must
  pass the line of contact of the pitch-surfaces before t and T work
  together again (let this number be called a); and, secondly, with how
  many different teeth of the larger wheel the tooth t will work at
  different times (let this number be called b); thirdly, with how many
  different teeth of the smaller wheel the tooth T will work at
  different times (let this be called c).

  CASE 1. If n is a divisor of N,

    a = N; b = N/n; c = 1.  (20)

  CASE 2. If the greatest common divisor of N and n be d, a number less
  than n, so that n = md, N = Md; then

    a = mN = Mn = Mmd; b = M; c = m.  (21)

  CASE 3. If N and n be prime to each other,

    a = nN; b = N; c = n.  (22)

  It is considered desirable by millwrights, with a view to the
  preservation of the uniformity of shape of the teeth of a pair of
  wheels, that each given tooth in one wheel should work with as many
  different teeth in the other wheel as possible. They therefore study
  that the numbers of teeth in each pair of wheels which work together
  shall either be prime to each other, or shall have their greatest
  common divisor as small as is consistent with a velocity ratio suited
  for the purposes of the machine.

  § 45. _Sliding Contact: Forms of the Teeth of Spur-wheels and
  Racks._--A line of connexion of two pieces in sliding contact is a
  line perpendicular to their surfaces at a point where they touch.
  Bearing this in mind, the principle of the comparative motion of a
  pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel
  and a rack, is found by applying the principles stated generally in §§
  36 and 37 to the case of parallel axes for a pair of spur-wheels, and
  to the case of an axis perpendicular to the direction of shifting for
  a wheel and a rack.

  In fig. 101, let C1, C2 be the centres of a pair of spur-wheels;
  B1IB1´, B2IB2´ portions of their pitch-circles, touching at I, the
  pitch-point. Let the wheel 1 be the driver, and the wheel 2 the
  follower.

  [Illustration: FIG. 101.]

  Let D1TB1A1, D2TB2A2 be the positions, at a given instant, of the
  acting surfaces of a pair of teeth in the driver and follower
  respectively, touching each other at T; the line of connexion of those
  teeth is P1P2, perpendicular to their surfaces at T. Let C1P1, C2P2 be
  perpendiculars let fall from the centres of the wheels on the line of
  contact. Then, by § 36, the angular velocity-ratio is

    [alpha]2/[alpha]1 = C1P1/C2P2.  (23)

  The following principles regulate the forms of the teeth and their
  relative motions:--

  I. The angular velocity ratio due to the sliding contact of the teeth
  will be the same with that due to the rolling contact of the
  pitch-circles, if the line of connexion of the teeth cuts the line of
  centres at the pitch-point.

  For, let P1P2 cut the line of centres at I; then, by similar
  triangles,

    [alpha]1 : [alpha]2 :: C2P2 : C1P1 :: IC2 :: IC1;  (24)

  which is also the angular velocity ratio due to the rolling contact of
  the circles B1IB1´, B2IB2´.

  This principle determines the _forms_ of all teeth of spur-wheels. It
  also determines the forms of the teeth of straight racks, if one of
  the centres be removed, and a straight line EIE´, parallel to the
  direction of motion of the rack, and perpendicular to C1IC2, be
  substituted for a pitch-circle.

  II. The component of the velocity of the point of contact of the teeth
  T along the line of connexion is

    [alpha]1·C1P1 = [alpha]2·C2P2.  (25)

  III. The relative velocity perpendicular to P1P2 of the teeth at their
  point of contact--that is, their _velocity of sliding_ on each
  other--is found by supposing one of the wheels, such as 1, to be
  fixed, the line of centres C1C2 to rotate backwards round C1 with the
  angular velocity [alpha]1, and the wheel 2 to rotate round C2 as
  before, with the angular velocity [alpha]2 relatively to the line of
  centres C1C2, so as to have the same motion as if its pitch-circle
  _rolled_ on the pitch-circle of the first wheel. Thus the _relative_
  motion of the wheels is unchanged; but 1 is considered as fixed, and 2
  has the total motion, that is, a rotation about the instantaneous axis
  I, with the angular velocity [alpha]1 + [alpha]2. Hence the _velocity
  of sliding_ is that due to this rotation about I, with the radius IT;
  that is to say, its value is

    ([alpha]1 + [alpha]2)·IT;  (26)

  so that it is greater the farther the point of contact is from the
  line of centres; and at the instant when that point passes the line of
  centres, and coincides with the _pitch-point_, the velocity of sliding
  is null, and the action of the teeth is, for the instant, that of
  rolling contact.

  IV. The _path of contact_ is the line traversing the various positions
  of the point T. If the line of connexion preserves always the same
  position, the path of contact coincides with it, and is straight; in
  other cases the path of contact is curved.

  It is divided by the pitch-point I into two parts--the _arc_ or _line
  of approach_ described by T in approaching the line of centres, and
  the _arc_ or _line of recess_ described by T after having passed the
  line of centres.

  During the _approach_, the _flank_ D1B1 of the driving tooth drives
  the face D2B2 of the following tooth, and the teeth are sliding
  _towards_ each other. During the _recess_ (in which the position of
  the teeth is exemplified in the figure by curves marked with accented
  letters), the _face_ B1´A1´ of the driving tooth drives the _flank_
  B2´A2´ of the following tooth, and the teeth are sliding _from_ each
  other.

  The path of contact is bounded where the approach commences by the
  addendum-circle of the follower, and where the recess terminates by
  the addendum-circle of the driver. The length of the path of contact
  should be such that there shall always be at least one pair of teeth
  in contact; and it is better still to make it so long that there shall
  always be at least two pairs of teeth in contact.

  V. The _obliquity_ of the action of the teeth is the angle EIT = IC1,
  P1 = IC2P2.

  In practice it is found desirable that the mean value of the obliquity
  of action during the contact of teeth should not exceed 15°, nor the
  maximum value 30°.

  It is unnecessary to give separate figures and demonstrations for
  inside gearing. The only modification required in the formulae is,
  that in equation (26) the _difference_ of the angular velocities
  should be substituted for their sum.

  § 46. _Involute Teeth._--The simplest form of tooth which fulfils the
  conditions of § 45 is obtained in the following manner (see fig. 102).
  Let C1, C2 be the centres of two wheels, B1IB1´, B2IB2´ their
  pitch-circles, I the pitch-point; let the obliquity of action of the
  teeth be constant, so that the same straight line P1IP2 shall
  represent at once the constant line of connexion of teeth and the path
  of contact. Draw C1P1, C2P2 perpendicular to P1IP2, and with those
  lines as radii describe about the centres of the wheels the circles
  D1D1´, D2D2´, called _base-circles_. It is evident that the radii of
  the base-circles bear to each other the same proportions as the radii
  of the pitch-circles, and also that

    C1P1 = IC1 · cos obliquity  \  (27)
    C2P2 = IC2 · cos obliquity  /

  (The obliquity which is found to answer best in practice is about
  14½°; its cosine is about 31/22, and its sine about ¼. These values
  though not absolutely exact, are near enough to the truth for
  practical purposes.)

  [Illustration: FIG. 102.]

  Suppose the base-circles to be a pair of circular pulleys connected by
  means of a cord whose course from pulley to pulley is P1IP2. As the
  line of connexion of those pulleys is the same as that of the proposed
  teeth, they will rotate with the required velocity ratio. Now, suppose
  a tracing point T to be fixed to the cord, so as to be carried along
  the path of contact P1IP2, that point will trace on a plane rotating
  along with the wheel 1 part of the involute of the base-circle D1D1´,
  and on a plane rotating along with the wheel 2 part of the involute of
  the base-circle D2D2´; and the two curves so traced will always touch
  each other in the required point of contact T, and will therefore
  fulfil the condition required by Principle I. of § 45.

  Consequently, one of the forms suitable for the teeth of wheels is the
  involute of a circle; and the obliquity of the action of such teeth is
  the angle whose cosine is the ratio of the radius of their base-circle
  to that of the pitch-circle of the wheel.

  All involute teeth of the same pitch work smoothly together.

  To find the length of the path of contact on either side of the
  pitch-point I, it is to be observed that the distance between the
  fronts of two successive teeth, as measured along P1IP2, is less than
  the pitch in the ratio of cos obliquity : I; and consequently that, if
  distances equal to the pitch be marked off either way from I towards
  P1 and P2 respectively, as the extremities of the path of contact, and
  if, according to Principle IV. of § 45, the addendum-circles be
  described through the points so found, there will always be at least
  two pairs of teeth in action at once. In practice it is usual to make
  the path of contact somewhat longer, viz. about 2.4 times the pitch;
  and with this length of path, and the obliquity already mentioned of
  14½°, the addendum is about 3.1 of the pitch.

  The teeth of a _rack_, to work correctly with wheels having involute
  teeth, should have plane surfaces perpendicular to the line of
  connexion, and consequently making with the direction of motion of the
  rack angles equal to the complement of the obliquity of action.

  § 47. _Teeth for a given Path of Contact: Sang's Method._--In the
  preceding section the form of the teeth is found by assuming a figure
  for the path of contact, viz. the straight line. Any other convenient
  figure may be assumed for the path of contact, and the corresponding
  forms of the teeth found by determining what curves a point T, moving
  along the assumed path of contact, will trace on two disks rotating
  round the centres of the wheels with angular velocities bearing that
  relation to the component velocity of T along TI, which is given by
  Principle II. of § 45, and by equation (25). This method of finding
  the forms of the teeth of wheels forms the subject of an elaborate and
  most interesting treatise by Edward Sang.

  All wheels having teeth of the same pitch, traced from the same path
  of contact, work correctly together, and are said to belong to the
  same set.

  [Illustration: FIG. 103.]

  § 48. _Teeth traced by Rolling Curves._--If any curve R (fig. 103) be
  rolled on the inside of the pitch-circle BB of a wheel, it appears,
  from § 30, that the instantaneous axis of the rolling curve at any
  instant will be at the point I, where it touches the pitch-circle for
  the moment, and that consequently the line AT, traced by a
  tracing-point T, fixed to the rolling curve upon the plane of the
  wheel, will be everywhere perpendicular to the straight line TI; so
  that the traced curve AT will be suitable for the flank of a tooth, in
  which T is the point of contact corresponding to the position I of the
  pitch-point. If the same rolling curve R, with the same tracing-point
  T, be rolled on the _outside_ of any other pitch-circle, it will have
  the _face_ of a tooth suitable to work with the _flank_ AT.

  In like manner, if either the same or any other rolling curve R´ be
  rolled the opposite way, on the _outside_ of the pitch-circle BB, so
  that the tracing point T´ shall start from A, it will trace the face
  AT´ of a tooth suitable to work with a _flank_ traced by rolling the
  same curve R´ with the same tracing-point T´ _inside_ any other
  pitch-circle.

  The figure of the _path of contact_ is that traced on a fixed plane by
  the tracing-point, when the rolling curve is rotated in such a manner
  as always to touch a fixed straight line EIE (or E´I´E´, as the case
  may be) at a fixed point I (or I´).

  If the same rolling curve and tracing-point be used to trace both the
  faces and the flanks of the teeth of a number of wheels of different
  sizes but of the same pitch, all those wheels will work correctly
  together, and will form a _set_. The teeth of a _rack_, of the same
  set, are traced by rolling the rolling curve on both sides of a
  straight line.

  The teeth of wheels of any figure, as well as of circular wheels, may
  be traced by rolling curves on their pitch-surfaces; and all teeth of
  the same pitch, traced by the same rolling curve with the same
  tracing-point, will work together correctly if their pitch-surfaces
  are in rolling contact.

  [Illustration: FIG. 104.]

  § 49. _Epicycloidal Teeth._--The most convenient rolling curve is the
  circle. The path of contact which it traces is identical with itself;
  and the flanks of the teeth are internal and their faces external
  epicycloids for wheels, and both flanks and faces are cycloids for a
  rack.

  For a pitch-circle of twice the radius of the rolling or _describing_
  circle (as it is called) the internal epicycloid is a straight line,
  being, in fact, a diameter of the pitch-circle, so that the flanks of
  the teeth for such a pitch-circle are planes radiating from the axis.
  For a smaller pitch-circle the flanks would be convex and _in-curved_
  or _under-cut_, which would be inconvenient; therefore the smallest
  wheel of a set should have its pitch-circle of twice the radius of the
  describing circle, so that the flanks may be either straight or
  concave.

  In fig. 104 let BB´ be part of the pitch-circle of a wheel with
  epicycloidal teeth; CIC´ the line of centres; I the pitch-point; EIE´
  a straight tangent to the pitch-circle at that point; R the internal
  and R´ the equal external describing circles, so placed as to touch
  the pitch-circle and each other at I. Let DID´ be the path of contact,
  consisting of the arc of approach DI and the arc of recess ID´. In
  order that there may always be at least two pairs of teeth in action,
  each of those arcs should be equal to the pitch.

  The obliquity of the action in passing the line of centres is nothing;
  the maximum obliquity is the angle EID = E´ID; and the mean obliquity
  is one-half of that angle.

  It appears from experience that the mean obliquity should not exceed
  15°; therefore the maximum obliquity should be about 30°; therefore
  the equal arcs DI and ID´ should each be one-sixth of a circumference;
  therefore the circumference of the describing circle should be _six
  times the pitch_.

  It follows that the smallest pinion of a set in which pinion the
  flanks are straight should have twelve teeth.

  § 50. _Nearly Epicycloidal Teeth: Willis's Method._--To facilitate the
  drawing of epicycloidal teeth in practice, Willis showed how to
  approximate to their figure by means of two circular arcs--one
  concave, for the flank, and the other convex, for the face--and each
  having for its radius the _mean_ radius of curvature of the
  epicycloidal arc. Willis's formulae are founded on the following
  properties of epicycloids:--

  Let R be the radius of the pitch-circle; r that of the describing
  circle; [theta] the angle made by the normal TI to the epicycloid at a
  given point T, with a tangent to the circle at I--that is, the
  obliquity of the action at T.

  Then the radius of curvature of the epicycloid at T is--

                                                      R - r    \
    For an internal epicycloid, [rho] = 4r sin [theta]------   |
                                                      R - 2r   |
                                                                >  (28)
                                                       R + r   |
    For an external epicycloid, [rho]´ = 4r sin [theta]------  |
                                                       R + 2r  /

  Also, to find the position of the centres of curvature relatively to
  the pitch-circle, we have, denoting the chord of the describing circle
  TI by c, c = 2r sin [theta]; and therefore

                                               R      \
    For the flank, [rho] - c = 2r sin [theta]------   |
                                             R - 2r   |
                                                       >  (29)
                                                R     |
    For the face, [rho]´ - c = 2r sin [theta]------   |
                                             R + 2r   /


  For the proportions approved of by Willis, sin [theta] = ¼ nearly; r =
  p (the pitch) nearly; c = ½p nearly; and, if N be the number of teeth
  in the wheel, r/R = 6/N nearly; therefore, approximately,

    [rho] - c = p/2 · N/N - 12   \  (30)
    [rho]´ - c = p/2 · N/N + 12  /

  [Illustration: FIG. 105.]

  Hence the following construction (fig. 105). Let BB be part of the
  pitch-circle, and a the point where a tooth is to cross it. Set off ab
  = ac - ½p. Draw radii bd, ce; draw fb, cg, making angles of 75½° with
  those radii. Make bf = p´ - c, cg = p - c. From f, with the radius fa,
  draw the circular arc ah; from g, with the radius ga, draw the
  circular arc ak. Then ah is the face and ak the flank of the tooth
  required.

  To facilitate the application of this rule, Willis published tables of
  [rho] - c and [rho]´ - c, and invented an instrument called the
  "odontograph."

  § 51. _Trundles and Pin-Wheels._--If a wheel or trundle have
  cylindrical pins or staves for teeth, the faces of the teeth of a
  wheel suitable for driving it are described by first tracing external
  epicycloids, by rolling the pitch-circle of the pin-wheel or trundle
  on the pitch-circle of the driving-wheel, with the centre of a stave
  for a tracing-point, and then drawing curves parallel to, and within
  the epicycloids, at a distance from them equal to the radius of a
  stave. Trundles having only six staves will work with large wheels.

  § 52. _Backs of Teeth and Spaces._--Toothed wheels being in general
  intended to rotate either way, the _backs_ of the teeth are made
  similar to the fronts. The _space_ between two teeth, measured on the
  pitch-circle, is made about (1/6)th part wider than the thickness of
  the tooth on the pitch-circle--that is to say,

    Thickness of tooth = 5/11 pitch;
    Width of space = 6/11 pitch.

  The difference of 1/11 of the pitch is called the _back-lash_. The
  clearance allowed between the points of teeth and the bottoms of the
  spaces between the teeth of the other wheel is about one-tenth of the
  pitch.

  § 53. _Stepped and Helical Teeth._--R. J. Hooke invented the making of
  the fronts of teeth in a series of steps with a view to increase the
  smoothness of action. A wheel thus formed resembles in shape a series
  of equal and similar toothed disks placed side by side, with the teeth
  of each a little behind those of the preceding disk. He also invented,
  with the same object, teeth whose fronts, instead of being parallel to
  the line of contact of the pitch-circles, cross it obliquely, so as to
  be of a screw-like or helical form. In wheel-work of this kind the
  contact of each pair of teeth commences at the foremost end of the
  helical front, and terminates at the aftermost end; and the helix is
  of such a pitch that the contact of one pair of teeth shall not
  terminate until that of the next pair has commenced.

  Stepped and helical teeth have the desired effect of increasing the
  smoothness of motion, but they require more difficult and expensive
  workmanship than common teeth; and helical teeth are, besides, open to
  the objection that they exert a laterally oblique pressure, which
  tends to increase resistance, and unduly strain the machinery.

  § 54. _Teeth of Bevel-Wheels._--The acting surfaces of the teeth of
  bevel-wheels are of the conical kind, generated by the motion of a
  line passing through the common apex of the pitch-cones, while its
  extremity is carried round the outlines of the cross section of the
  teeth made by a sphere described about that apex.

  [Illustration: FIG. 106.]

  The operations of describing the exact figures of the teeth of
  bevel-wheels, whether by involutes or by rolling curves, are in every
  respect analogous to those for describing the figures of the teeth of
  spur-wheels, except that in the case of bevel-wheels all those
  operations are to be performed on the surface of a sphere described
  about the apex instead of on a plane, substituting _poles_ for
  _centres_, and _great circles_ for _straight lines_.

  In consideration of the practical difficulty, especially in the case
  of large wheels, of obtaining an accurate spherical surface, and of
  drawing upon it when obtained, the following approximate method,
  proposed originally by Tredgold, is generally used:--

  Let O (fig. 106) be the common apex of a pair of bevel-wheels; OB1I,
  OB2I their pitch cones; OC1, OC2 their axes; OI their line of contact.
  Perpendicular to OI draw A1IA2, cutting the axes in A1, A2; make the
  outer rims of the patterns and of the wheels portions of the cones
  A1B1I, A2B2I, of which the narrow zones occupied by the teeth will be
  sufficiently near to a spherical surface described about O for
  practical purposes. To find the figures of the teeth, draw on a flat
  surface circular arcs ID1, ID2, with the radii A1I, A2I; those arcs
  will be the _developments_ of arcs of the pitch-circles B1I, B2I, when
  the conical surfaces A1B1I, A2B2I are spread out flat. Describe the
  figures of teeth for the developed arcs as for a pair of spur-wheels;
  then wrap the developed arcs on the cones, so as to make them coincide
  with the pitch-circles, and trace the teeth on the conical surfaces.

  § 55. _Teeth of Skew-Bevel Wheels._--The crests of the teeth of a
  skew-bevel wheel are parallel to the generating straight line of the
  hyperboloidal pitch-surface; and the transverse sections of the teeth
  at a given pitch-circle are similar to those of the teeth of a
  bevel-wheel whose pitch surface is a cone touching the hyperboloidal
  surface at the given circle.

  § 56. _Cams._--A _cam_ is a single tooth, either rotating continuously
  or oscillating, and driving a sliding or turning piece either
  constantly or at intervals. All the principles which have been stated
  in § 45 as being applicable to teeth are applicable to cams; but in
  designing cams it is not usual to determine or take into consideration
  the form of the ideal pitch-surface, which would give the same
  comparative motion by rolling contact that the cam gives by sliding
  contact.

  § 57. _Screws._--The figure of a screw is that of a convex or concave
  cylinder, with one or more helical projections, called _threads_,
  winding round it. Convex and concave screws are distinguished
  technically by the respective names of _male_ and _female_; a short
  concave screw is called a _nut_; and when a _screw_ is spoken of
  without qualification a _convex_ screw is usually understood.

  The relation between the _advance_ and the _rotation_, which compose
  the motion of a screw working in contact with a fixed screw or helical
  guide, has already been demonstrated in § 32; and the same relation
  exists between the magnitudes of the rotation of a screw about a fixed
  axis and the advance of a shifting nut in which it rotates. The
  advance of the nut takes place in the opposite direction to that of
  the advance of the screw in the case in which the nut is fixed. The
  _pitch_ or _axial pitch_ of a screw has the meaning assigned to it in
  that section, viz. the distance, measured parallel to the axis,
  between the corresponding points in two successive turns of the _same
  thread_. If, therefore, the screw has several equidistant threads, the
  true pitch is equal to the _divided axial pitch_, as measured between
  two adjacent threads, multiplied by the number of threads.

  If a helix be described round the screw, crossing each turn of the
  thread at right angles, the distance between two corresponding points
  on two successive turns of the same thread, measured along this
  _normal helix_, may be called the _normal pitch_; and when the screw
  has more than one thread the normal pitch from thread to thread may be
  called the _normal divided pitch_.

  The distance from thread to thread, measured on a circle described
  about the axis of the screw, called the pitch-circle, may be called
  the _circumferential pitch_; for a screw of one thread it is one
  circumference; for a screw of n threads, (one circumference)/n.

  Let r denote the radius of the pitch circle;
    n the number of threads;
    [theta] the obliquity of the threads to the pitch circle, and of the
       normal helix to the axis;

          P_a \             / pitch
    P_a        > the axial <
    --- = p_a |             |
     n        /             \ divided pitch;

           P_n \              / pitch
    P_n         > the normal <
    ---  = p_n |              |
     n         /              \ divided pitch;

    P_c the circumferential pitch;

  then

                                              2[pi]r               \
    p_c = p_a cot [theta] = p_n cos [theta] = ------,              |
                                                 n                 |
                                                                   |
                                              2[pi]r tan [theta]   |
    p_a = p_n sec [theta] = p_c tan [theta] = ------------------,   >  (31)
                                                      n            |
                                                                   |
                                              2[pi]r sin [theta]   |
    p_n = p_c sin [theta] = p_a cos [theta] = ------------------,  |
                                                      n            /

  If a screw rotates, the number of threads which pass a fixed point in
  one revolution is the number of threads in the screw.

  A pair of convex screws, each rotating about its axis, are used as an
  elementary combination to transmit motion by the sliding contact of
  their threads. Such screws are commonly called _endless screws_. At
  the point of contact of the screws their threads must be parallel; and
  their line of connexion is the common perpendicular to the acting
  surfaces of the threads at their point of contact. Hence the following
  principles:--

  I. If the screws are both right-handed or both left-handed, the angle
  between the directions of their axes is the sum of their obliquities;
  if one is right-handed and the other left-handed, that angle is the
  difference of their obliquities.

  II. The normal pitch for a screw of one thread, and the normal divided
  pitch for a screw of more than one thread, must be the same in each
  screw.

  III. The angular velocities of the screws are inversely as their
  numbers of threads.

  Hooke's wheels with oblique or helical teeth are in fact screws of
  many threads, and of large diameters as compared with their lengths.

  The ordinary position of a pair of endless screws is with their axes
  at right angles to each other. When one is of considerably greater
  diameter than the other, the larger is commonly called in practice a
  _wheel_, the name _screw_ being applied to the smaller only; but they
  are nevertheless both screws in fact.

  To make the teeth of a pair of endless screws fit correctly and work
  smoothly, a hardened steel screw is made of the figure of the smaller
  screw, with its thread or threads notched so as to form a cutting
  tool; the larger screw, or "wheel," is cast approximately of the
  required figure; the larger screw and the steel screw are fitted up in
  their proper relative position, and made to rotate in contact with
  each other by turning the steel screw, which cuts the threads of the
  larger screw to their true figure.

  [Illustration: FIG. 107.]

  § 58. _Coupling of Parallel Axes--Oldham's Coupling._--A _coupling_ is
  a mode of connecting a pair of shafts so that they shall rotate in the
  same direction with the same mean angular velocity. If the axes of the
  shafts are in the same straight line, the coupling consists in so
  connecting their contiguous ends that they shall rotate as one piece;
  but if the axes are not in the same straight line combinations of
  mechanism are required. A coupling for parallel shafts which acts by
  _sliding contact_ was invented by Oldham, and is represented in fig.
  107. C1, C2 are the axes of the two parallel shafts; D1, D2 two disks
  facing each other, fixed on the ends of the two shafts respectively;
  E1E1 a bar sliding in a diametral groove in the face of D1; E2E2 a bar
  sliding in a diametral groove in the face of D2: those bars are fixed
  together at A, so as to form a rigid cross. The angular velocities of
  the two disks and of the cross are all equal at every instant; the
  middle point of the cross, at A, revolves in the dotted circle
  described upon the line of centres C1C2 as a diameter twice for each
  turn of the disks and cross; the instantaneous axis of rotation of the
  cross at any instant is at I, the point in the circle C1C2
  diametrically opposite to A.

  Oldham's coupling may be used with advantage where the axes of the
  shafts are intended to be as nearly in the same straight line as is
  possible, but where there is some doubt as to the practibility or
  permanency of their exact continuity.

  § 59. _Wrapping Connectors--Belts, Cords and Chains._--Flat belts of
  leather or of gutta percha, round cords of catgut, hemp or other
  material, and metal chains are used as wrapping connectors to transmit
  rotatory motion between pairs of pulleys and drums.

  _Belts_ (the most frequently used of all wrapping connectors) require
  nearly cylindrical pulleys. A belt tends to move towards that part of
  a pulley whose radius is greatest; pulleys for belts, therefore, are
  slightly swelled in the middle, in order that the belt may remain on
  the pulley, unless forcibly shifted. A belt when in motion is shifted
  off a pulley, or from one pulley on to another of equal size alongside
  of it, by pressing against that part of the belt which is moving
  _towards_ the pulley.

  _Cords_ require either cylindrical drums with ledges or grooved
  pulleys.

  _Chains_ require pulleys or drums, grooved, notched and toothed, so as
  to fit the links of the chain.

  Wrapping connectors for communicating continuous motion are endless.

  Wrapping connectors for communicating reciprocating motion have
  usually their ends made fast to the pulleys or drums which they
  connect, and which in this case may be sectors.

  [Illustration: FIG. 108.]

  The line of connexion of two pieces connected by a wrapping connector
  is the centre line of the belt, cord or chain; and the comparative
  motions of the pieces are determined by the principles of § 36 if both
  pieces turn, and of § 37 if one turns and the other shifts, in which
  latter case the motion must be reciprocating.

  The _pitch-line_ of a pulley or drum is a curve to which the line of
  connexion is always a tangent--that is to say, it is a curve parallel
  to the acting surface of the pulley or drum, and distant from it by
  half the thickness of the wrapping connector.

  Pulleys and drums for communicating a constant velocity ratio are
  circular. The _effective radius_, or radius of the pitch-circle of a
  circular pulley or drum, is equal to the real radius added to half the
  thickness of the connector. The angular velocities of a pair of
  connected circular pulleys or drums are inversely as the effective
  radii.

  A _crossed_ belt, as in fig. 108, A, reverses the direction of the
  rotation communicated; an _uncrossed_ belt, as in fig. 108, B,
  preserves that direction.

  The _length_ L of an endless belt connecting a pair of pulleys whose
  effective radii are r1, r2, with parallel axes whose distance apart is
  c, is given by the following formulae, in each of which the first
  term, containing the radical, expresses the length of the straight
  parts of the belt, and the remainder of the formula the length of the
  curved parts.

  For a crossed belt:--

                                             /                r1 + r2 \
    L = 2[root][c² - (r1 + r2)²] + (r1 + r2)( [pi] - 2 sin^-1 -------  );  (32 A)
                                             \                   c    /
  and for an uncrossed belt:--

                                                                    r1 - r2
    L = 2[root][c² - (r1 - r2)²] + [pi](r1 + r2 + 2(r1 - r2) sin^-1 -------;  (32 B)
                                                                       c
  in which r1 is the greater radius, and r2 the less.

  When the axes of a pair of pulleys are not parallel, the pulleys
  should be so placed that the part of the belt which is _approaching_
  each pulley shall be in the plane of the pulley.

  § 60. _Speed-Cones._--A pair of speed-cones (fig. 109) is a
  contrivance for varying and adjusting the velocity ratio communicated
  between a pair of parallel shafts by means of a belt. The speed-cones
  are either continuous cones or conoids, as A, B, whose velocity ratio
  can be varied gradually while they are in motion by shifting the belt,
  or sets of pulleys whose radii vary by steps, as C, D, in which case
  the velocity ratio can be changed by shifting the belt from one pair
  of pulleys to another.

  [Illustration: FIG. 109.]

  In order that the belt may fit accurately in every possible position
  on a pair of speed-cones, the quantity L must be constant, in
  equations (32 A) or (32 B), according as the belt is crossed or
  uncrossed.

  For a _crossed_ belt, as in A and C, fig. 109, L depends solely on c
  and on r1 + r2. Now c is constant because the axes are parallel;
  therefore the _sum of the radii_ of the pitch-circles connected in
  every position of the belt is to be constant. That condition is
  fulfilled by a pair of continuous cones generated by the revolution of
  two straight lines inclined opposite ways to their respective axes at
  equal angles.

  For an uncrossed belt, the quantity L in equation (32 B) is to be made
  constant. The exact fulfilment of this condition requires the solution
  of a transcendental equation; but it may be fulfilled with accuracy
  sufficient for practical purposes by using, instead of (32 B) the
  following _approximate_ equation:--

    L nearly = 2c + [pi](r1 + r2) + (r1 - r2)²/c.  (33)

  The following is the most convenient practical rule for the
  application of this equation:--

  Let the speed-cones be equal and similar conoids, as in B, fig. 109,
  but with their large and small ends turned opposite ways. Let r1 be
  the radius of the large end of each, r2 that of the small end, r0 that
  of the middle; and let v be the _sagitta_, measured perpendicular to
  the axes, of the arc by whose revolution each of the conoids is
  generated, or, in other words, the _bulging_ of the conoids in the
  middle of their length. Then

    v = r0 - (r1 + r2)/2 = (r1 - r2)²/2[pi]c.  (34)

  2[pi] = 6.2832; but 6 may be used in most practical cases without
  sensible error.

  The radii at the middle and end being thus determined, make the
  generating curve an arc either of a circle or of a parabola.

  § 61. _Linkwork in General._--The pieces which are connected by
  linkwork, if they rotate or oscillate, are usually called _cranks_,
  _beams_ and levers. The _link_ by which they are connected is a rigid
  rod or bar, which may be straight or of any other figure; the straight
  figure being the most favourable to strength, is always used when
  there is no special reason to the contrary. The link is known by
  various names in various circumstances, such as _coupling-rod_,
  _connecting-rod_, _crank-rod_, _eccentric-rod_, &c. It is attached to
  the pieces which it connects by two pins, about which it is free to
  turn. The effect of the link is to maintain the distance between the
  axes of those pins invariable; hence the common perpendicular of the
  axes of the pins is _the line of connexion_, and its extremities may
  be called the _connected points_. In a turning piece, the
  perpendicular let fall from its connected point upon its axis of
  rotation is the _arm_ or _crank-arm_.

  The axes of rotation of a pair of turning pieces connected by a link
  are almost always parallel, and perpendicular to the line of connexion
  in which case the angular velocity ratio at any instant is the
  reciprocal of the ratio of the common perpendiculars let fall from the
  line of connexion upon the respective axes of rotation.

  If at any instant the direction of one of the crank-arms coincides
  with the line of connexion, the common perpendicular of the line of
  connexion and the axis of that crank-arm vanishes, and the directional
  relation of the motions becomes indeterminate. The position of the
  connected point of the crank-arm in question at such an instant is
  called a _dead-point_. The velocity of the other connected point at
  such an instant is null, unless it also reaches a dead-point at the
  same instant, so that the line of connexion is in the plane of the two
  axes of rotation, in which case the velocity ratio is indeterminate.
  Examples of dead-points, and of the means of preventing the
  inconvenience which they tend to occasion, will appear in the sequel.

  § 62. _Coupling of Parallel Axes._--Two or more parallel shafts (such
  as those of a locomotive engine, with two or more pairs of driving
  wheels) are made to rotate with constantly equal angular velocities by
  having equal cranks, which are maintained parallel by a coupling-rod
  of such a length that the line of connexion is equal to the distance
  between the axes. The cranks pass their dead-points simultaneously. To
  obviate the unsteadiness of motion which this tends to cause, the
  shafts are provided with a second set of cranks at right angles to the
  first, connected by means of a similar coupling-rod, so that one set
  of cranks pass their dead points at the instant when the other set are
  farthest from theirs.

  § 63. _Comparative Motion of Connected Points._--As the link is a
  rigid body, it is obvious that its action in communicating motion may
  be determined by finding the comparative motion of the connected
  points, and this is often the most convenient method of proceeding.

  If a connected point belongs to a turning piece, the direction of its
  motion at a given instant is perpendicular to the plane containing the
  axis and crank-arm of the piece. If a connected point belongs to a
  shifting piece, the direction of its motion at any instant is given,
  and a plane can be drawn perpendicular to that direction.

  The line of intersection of the planes perpendicular to the paths of
  the two connected points at a given instant is the _instantaneous axis
  of the link_ at that instant; and the _velocities of the connected
  points are directly as their distances from that axis_.

  [Illustration: FIG. 110.]

  In drawing on a plane surface, the two planes perpendicular to the
  paths of the connected points are represented by two lines (being
  their sections by a plane normal to them), and the instantaneous axis
  by a point (fig. 110); and, should the length of the two lines render
  it impracticable to produce them until they actually intersect, the
  velocity ratio of the connected points may be found by the principle
  that it is equal to the ratio of the segments which a line parallel to
  the line of connexion cuts off from any two lines drawn from a given
  point, perpendicular respectively to the paths of the connected
  points.

  To illustrate this by one example. Let C1 be the axis, and T1 the
  connected point of the beam of a steam-engine; T1T2 the connecting or
  crank-rod; T2 the other connected point, and the centre of the
  crank-pin; C2 the axis of the crank and its shaft. Let v1 denote the
  velocity of T1 at any given instant; v2 that of T2. To find the ratio
  of these velocities, produce C1T1, C2T2 till they intersect in K; K is
  the instantaneous axis of the connecting rod, and the velocity ratio
  is

    v1 : v2 :: KT1 : KT2.  (35)

  Should K be inconveniently far off, draw any triangle with its sides
  respectively parallel to C1T1, C2T2 and T1T2; the ratio of the two
  sides first mentioned will be the velocity ratio required. For
  example, draw C2A parallel to C1T1, cutting T1T2 in A; then

    v1 : v2 :: C2A : C2T2.  (36)

  § 64. _Eccentric._--An eccentric circular disk fixed on a shaft, and
  used to give a reciprocating motion to a rod, is in effect a crank-pin
  of sufficiently large diameter to surround the shaft, and so to avoid
  the weakening of the shaft which would arise from bending it so as to
  form an ordinary crank. The centre of the eccentric is its connected
  point; and its eccentricity, or the distance from that centre to the
  axis of the shaft, is its crank-arm.

  An eccentric may be made capable of having its eccentricity altered by
  means of an adjusting screw, so as to vary the extent of the
  reciprocating motion which it communicates.

  § 65. _Reciprocating Pieces--Stroke--Dead-Points._--The distance
  between the extremities of the path of the connected point in a
  reciprocating piece (such as the piston of a steam-engine) is called
  the _stroke_ or _length of stroke_ of that piece. When it is connected
  with a continuously turning piece (such as the crank of a
  steam-engine) the ends of the stroke of the reciprocating piece
  correspond to the _dead-points_ of the path of the connected point of
  the turning piece, where the line of connexion is continuous with or
  coincides with the crank-arm.

  Let S be the length of stroke of the reciprocating piece, L the length
  of the line of connexion, and R the crank-arm of the continuously
  turning piece. Then, if the two ends of the stroke be in one straight
  line with the axis of the crank,

    S = 2R;  (37)

  and if these ends be not in one straight line with that axis, then S,
  L - R, and L + R, are the three sides of a triangle, having the angle
  opposite S at that axis; so that, if [theta] be the supplement of the
  arc between the dead-points,

    S² = 2(L² + R²) - 2(L² - R²) cos [theta], \
                                              |
                  2L² + 2R² - S²               >  (38)
    cos [theta] = --------------              |
                    2(L² - R²)                /

  [Illustration: FIG. 111.]

  § 66. _Coupling of Intersecting Axes--Hooke's Universal
  Joint._--Intersecting axes are coupled by a contrivance of Hooke's,
  known as the "universal joint," which belongs to the class of linkwork
  (see fig. 111). Let O be the point of intersection of the axes OC1,
  OC2, and [theta] their angle of inclination to each other. The pair of
  shafts C1, C2 terminate in a pair of forks F1, F2 in bearings at the
  extremities of which turn the gudgeons at the ends of the arms of a
  rectangular cross, having its centre at O. This cross is the link; the
  connected points are the centres of the bearings F1, F2. At each
  instant each of those points moves at right angles to the central
  plane of its shaft and fork, therefore the line of intersection of the
  central planes of the two forks at any instant is the instantaneous
  axis of the cross, and the _velocity ratio_ of the points F1, F2
  (which, as the forks are equal, is also the _angular velocity ratio_
  of the shafts) is equal to the ratio of the distances of those points
  from that instantaneous axis. The _mean_ value of that velocity ratio
  is that of equality, for each successive _quarter-turn_ is made by
  both shafts in the same time; but its actual value fluctuates between
  the limits:--

    [alpha]2        1                                      \
    -------- = -----------  when F1 is the plane of OC1C2  |
    [alpha]1   cos [theta]                                 |
                                                            > (39)
        [alpha]2                                           |
    and --------  = cos [theta] when F2 is in that plane.  |
        [alpha]1                                           /

  Its value at intermediate instants is given by the following
  equations: let [phi]1, [phi]2 be the angles respectively made by the
  central planes of the forks and shafts with the plane OC1C2 at a given
  instant; then

    cos [theta] = tan [phi]1 tan [phi]2,             \
                                                     |
    [alpha]2      d[phi]2   tan [phi]1 + cot [phi]1   > (40)
    --------- = - ------- = -----------------------. |
    [alpha]1      d[phi]1   tan [phi]2 + cot [phi]2  /

  § 67. _Intermittent Linkwork--Click and Ratchet._--A click acting upon
  a ratchet-wheel or rack, which it pushes or pulls through a certain
  arc at each forward stroke and leaves at rest at each backward stroke,
  is an example of intermittent linkwork. During the forward stroke the
  action of the click is governed by the principles of linkwork; during
  the backward stroke that action ceases. A _catch_ or _pall_, turning
  on a fixed axis, prevents the ratchet-wheel or rack from reversing its
  motion.


  _Division 5.--Trains of Mechanism._

  § 68. _General Principles.--A train of mechanism_ consists of a series
  of pieces each of which is follower to that which drives it and driver
  to that which follows it.

  The comparative motion of the first driver and last follower is
  obtained by combining the proportions expressing by their terms the
  velocity ratios and by their signs the directional relations of the
  several elementary combinations of which the train consists.

  § 69. _Trains of Wheelwork._--Let A1, A2, A3, &c., A_(m-1), A_m denote
  a series of axes, and [alpha]1, [alpha]2, [alpha]3, &c.,
  [alpha]_(m-1), [alpha]_m their angular velocities. Let the axis A1
  carry a wheel of N1 teeth, driving a wheel of n2 teeth on the axis A2,
  which carries also a wheel of N2 teeth, driving a wheel of n3 teeth on
  the axis A3, and so on; the numbers of teeth in drivers being denoted
  by N´s, and in followers by n's, and the axes to which the wheels are
  fixed being denoted by numbers. Then the resulting velocity ratio is
  denoted by

    [alpha]_m   [alpha]2   [alpha]3            [alpha]_m      N1 · N2 ... &c. ... N_(m-1)
    --------- = -------- · -------- · &c. ... ------------- = ---------------------------;  (41)
    [alpha]1    [alpha]1   [alpha]2           [alpha]_(m-1)     n2 · n3 ... &c. ... n_m

  that is to say, the velocity ratio of the last and first axes is the
  ratio of the product of the numbers of teeth in the drivers to the
  product of the numbers of teeth in the followers.

  Supposing all the wheels to be in outside gearing, then, as each
  elementary combination reverses the direction of rotation, and as the
  number of elementary combinations m - 1 is one less than the number
  of axes m, it is evident that if m is odd the direction of rotation is
  preserved, and if even reversed.

  It is often a question of importance to determine the number of teeth
  in a train of wheels best suited for giving a determinate velocity
  ratio to two axes. It was shown by Young that, to do this with the
  _least total number of teeth_, the velocity ratio of each elementary
  combination should approximate as nearly as possible to 3.59. This
  would in many cases give too many axes; and, as a useful practical
  rule, it may be laid down that from 3 to 6 ought to be the limit of
  the velocity ratio of an elementary combination in wheel-work. The
  smallest number of teeth in a pinion for epicycloidal teeth ought to
  be _twelve_ (see § 49)--but it is better, for smoothness of motion,
  not to go below _fifteen_; and for involute teeth the smallest number
  is about _twenty-four_.

  Let B/C be the velocity ratio required, reduced to its least terms,
  and let B be greater than C. If B/C is not greater than 6, and C lies
  between the prescribed minimum number of teeth (which may be called t)
  and its double 2t, then one pair of wheels will answer the purpose,
  and B and C will themselves be the numbers required. Should B and C be
  inconveniently large, they are, if possible, to be resolved into
  factors, and those factors (or if they are too small, multiples of
  them) used for the number of teeth. Should B or C, or both, be at once
  inconveniently large and prime, then, instead of the exact ratio B/C
  some ratio approximating to that ratio, and capable of resolution into
  convenient factors, is to be found by the method of continued
  fractions.

  Should B/C be greater than 6, the best number of elementary
  combinations m - 1 will lie between

    (log B - log C)     log B - log C
    --------------- and -------------.
         log 6              log 3

  Then, if possible, B and C themselves are to be resolved each into m -
  1 factors (counting 1 as a factor), which factors, or multiples of
  them, shall be not less than t nor greater than 6t; or if B and C
  contain inconveniently large prime factors, an approximate velocity
  ratio, found by the method of continued fractions, is to be
  substituted for B/C as before.

  So far as the resultant velocity ratio is concerned, the _order_ of
  the drivers N and of the followers n is immaterial: but to secure
  equable wear of the teeth, as explained in § 44, the wheels ought to
  be so arranged that, for each elementary combination, the greatest
  common divisor of N and n shall be either 1, or as small as possible.

  § 70. _Double Hooke's Coupling._--It has been shown in § 66 that the
  velocity ratio of a pair of shafts coupled by a universal joint
  fluctuates between the limits cos [theta] and 1/cos [theta]. Hence one
  or both of the shafts must have a vibratory and unsteady motion,
  injurious to the mechanism and framework. To obviate this evil a short
  intermediate shaft is introduced, making equal angles with the first
  and last shaft, coupled with each of them by a Hooke's joint, and
  having its own two forks in the same plane. Let [alpha]1, [alpha]2,
  [alpha]3 be the angular velocities of the first, intermediate, and
  last shaft in this _train of two Hooke's couplings_. Then, from the
  principles of § 60 it is evident that at each instant
  [alpha]2/[alpha]1 = [alpha]2/[alpha]3, and consequently that [alpha]3
  = [alpha]1; so that the fluctuations of angular velocity ratio caused
  by the first coupling are exactly neutralized by the second, and the
  first and last shafts have equal angular velocities at each instant.

  § 71. _Converging and Diverging Trains of Mechanism._--Two or more
  trains of mechanism may converge into one--as when the two pistons of
  a pair of steam-engines, each through its own connecting-rod, act upon
  one crank-shaft. One train of mechanism may _diverge_ into two or
  more--as when a single shaft, driven by a prime mover, carries several
  pulleys, each of which drives a different machine. The principles of
  comparative motion in such converging and diverging trains are the
  same as in simple trains.


  _Division 6.--Aggregate Combinations._

  § 72. _General Principles._--Willis designated as "aggregate
  combinations" those assemblages of pieces of mechanism in which the
  motion of one follower is the _resultant_ of component motions
  impressed on it by more than one driver. Two classes of aggregate
  combinations may be distinguished which, though not different in their
  actual nature, differ in the _data_ which they present to the
  designer, and in the method of solution to be followed in questions
  respecting them.

  Class I. comprises those cases in which a piece A is not carried
  directly by the frame C, but by another piece B, _relatively_ to which
  the motion of A is given--the motion of the piece B relatively to the
  frame C being also given. Then the motion of A relatively to the frame
  C is the _resultant_ of the motion of A relatively to B and of B
  relatively to C; and that resultant is to be found by the principles
  already explained in Division 3 of this Chapter §§ 27-32.

  Class II. comprises those cases in which the motions of three points
  in one follower are determined by their connexions with two or with
  three different drivers.

  This classification is founded on the kinds of problems arising from
  the combinations. Willis adopts another classification founded on the
  _objects_ of the combinations, which objects he divides into two
  classes, viz. (1) to produce _aggregate velocity_, or a velocity which
  is the resultant of two or more components in the same path, and (2)
  to produce _an aggregate path_--that is, to make a given point in a
  rigid body move in an assigned path by communicating certain motions
  to other points in that body.

  It is seldom that one of these effects is produced without at the same
  time producing the other; but the classification of Willis depends
  upon which of those two effects, even supposing them to occur
  together, is the practical object of the mechanism.

  [Illustration: FIG. 112.]

  § 73. _Differential Windlass._--The axis C (fig. 112) carries a larger
  barrel AE and a smaller barrel DB, rotating as one piece with the
  angular velocity [alpha]1 in the direction AE. The pulley or _sheave_
  FG has a weight W hung to its centre. A cord has one end made fast to
  and wrapped round the barrel AE; it passes from A under the sheave FG,
  and has the other end wrapped round and made fast to the barrel BD.
  Required the relation between the velocity of translation v2 of W and
  the angular velocity [alpha]1 of the _differential barrel_.

  In this case v2 is an _aggregate velocity_, produced by the joint
  action of the two drivers AE and BD, transmitted by wrapping
  connectors to FG, and combined by that sheave so as to act on the
  follower W, whose motion is the same with that of the centre of FG.

  The velocity of the point F is [alpha]1·AC, _upward_ motion being
  considered positive. The velocity of the point G is -[alpha]1·CB,
  _downward_ motion being negative. Hence the instantaneous axis of the
  sheave FG is in the diameter FG, at the distance

    FG     AC - BC
    --- ·  -------
     2     AC + BC

  from the centre towards G; the angular velocity of the sheave is

                          AC + BC
    [alpha]2 = [alpha]1 · -------;
                             FG

  and, consequently, the velocity of its centre is

                    FG    AC - BC   [alpha]1(AC - BC)
    v2 = [alpha]2 · --- · ------- = -----------------,  (42)
                     2    AC + BC           2

  or the _mean between the velocities of the two vertical parts of the
  cord_.

  If the cord be fixed to the framework at the point B, instead of being
  wound on a barrel, the velocity of W is half that of AF.

  A case containing several sheaves is called a _block_. A _fall-block_
  is attached to a fixed point; a _running-block_ is movable to and from
  a fall-block, with which it is connected by two or more plies of a
  rope. The whole combination constitutes a _tackle_ or _purchase_. (See
  PULLEYS for practical applications of these principles.)

  § 74. _Differential Screw._--On the same axis let there be two screws
  of the respective pitches p1 and p2, made in one piece, and rotating
  with the angular velocity [alpha]. Let this piece be called B. Let the
  first screw turn in a fixed nut C, and the second in a sliding nut A.
  The velocity of advance of B relatively to C is (according to § 32)
  [alpha]p1, and of A relatively to B (according to § 57) -[alpha]p2;
  hence the velocity of A relatively to C is

    [alpha](p1 - p2),  (46)

  being the same with the velocity of advance of a screw of the pitch p1
  - p2. This combination, called _Hunter's_ or the _differential screw_,
  combines the strength of a large thread with the slowness of motion
  due to a small one.

  § 75. _Epicyclic Trains._--The term _epicyclic train_ is used by
  Willis to denote a train of wheels carried by an arm, and having
  certain rotations relatively to that arm, which itself rotates. The
  arm may either be driven by the wheels or assist in driving them. The
  comparative motions of the wheels and of the arm, and the _aggregate
  paths_ traced by points in the wheels, are determined by the
  principles of the composition of rotations, and of the description of
  rolling curves, explained in §§ 30, 31.

  § 76. _Link Motion._--A slide valve operated by a link motion receives
  an aggregate motion from the mechanism driving it. (See STEAM-ENGINE
  for a description of this and other types of mechanism of this class.)

  [Illustration: FIG. 113.]

  § 77. _Parallel Motions._--A _parallel motion_ is a combination of
  turning pieces in mechanism designed to guide the motion of a
  reciprocating piece either exactly or approximately in a straight
  line, so as to avoid the friction which arises from the use of
  straight guides for that purpose.

  Fig. 113 represents an exact parallel motion, first proposed, it is
  believed, by Scott Russell. The arm CD turns on the axis C, and is
  jointed at D to the middle of the bar ADB, whose length is double of
  that of CD, and one of whose ends B is jointed to a slider, sliding in
  straight guides along the line CB. Draw BE perpendicular to CB,
  cutting CD produced in E, then E is the instantaneous axis of the bar
  ADB; and the direction of motion of A is at every instant
  perpendicular to EA--that is, along the straight line ACa. While the
  stroke of A is ACa, extending to equal distances on either side of C,
  and equal to twice the chord of the arc Dd, the stroke of B is only
  equal to twice the sagitta; and thus A is guided through a
  comparatively long stroke by the sliding of B through a comparatively
  short stroke, and by rotatory motions at the joints C, D, B.

  [Illustration: FIG. 114.]

  [Illustration: FIG. 115.]

  § 78.* An example of an approximate straight-line motion composed of
  three bars fixed to a frame is shown in fig. 114. It is due to P. L.
  Tchebichev of St Petersburg. The links AB and CD are equal in length
  and are centred respectively at A and C. The ends D and B are joined
  by a link DB. If the respective lengths are made in the proportions AC
  : CD : DB = 1 : 1.3 : 0.4 the middle point P of DB will describe an
  approximately straight line parallel to AC within limits of length
  about equal to AC. C. N. Peaucellier, a French engineer officer, was
  the first, in 1864, to invent a linkwork with which an exact straight
  line could be drawn. The linkwork is shown in fig. 115, from which it
  will be seen that it consists of a rhombus of four equal bars ABCD,
  jointed at opposite corners with two equal bars BE and DE. The seventh
  link AF is equal in length to halt the distance EA when the mechanism
  is in its central position. The points E and F are fixed. It can be
  proved that the point C always moves in a straight line at right
  angles to the line EF. The more general property of the mechanism
  corresponding to proportions between the lengths FA and EF other than
  that of equality is that the curve described by the point C is the
  inverse of the curve described by A. There are other arrangements of
  bars giving straight-line motions, and these arrangements together
  with the general properties of mechanisms of this kind are discussed
  in _How to Draw a Straight Line_ by A. B. Kempe (London, 1877).

  [Illustration: FIG. 116.]

  [Illustration: FIG. 117.]

  § 79.* _The Pantograph._--If a parallelogram of links (fig. 116), be
  fixed at any one point a in any one of the links produced in either
  direction, and if any straight line be drawn from this point to cut
  the links in the points b and c, then the points a, b, c will be in a
  straight line for all positions of the mechanism, and if the point b
  be guided in any curve whatever, the point c will trace a similar
  curve to a scale enlarged in the ratio ab : ac. This property of the
  parallelogram is utilized in the construction of the pantograph, an
  instrument used for obtaining a copy of a map or drawing on a
  different scale. Professor J. J. Sylvester discovered that this
  property of the parallelogram is not confined to points lying in one
  line with the fixed point. Thus if b (fig. 117) be any point on the
  link CD, and if a point c be taken on the link DE such that the
  triangles CbD and DcE are similar and similarly situated with regard
  to their respective links, then the ratio of the distances ab and ac
  is constant, and the angle bac is constant for all positions of the
  mechanism; so that, if b is guided in any curve, the point c will
  describe a similar curve turned through an angle bac, the scales of
  the curves being in the ratio ab to ac. Sylvester called an instrument
  based on this property a plagiograph or a skew pantograph.

  The combination of the parallelogram with a straight-line motion, for
  guiding one of the points in a straight line, is illustrated in Watt's
  parallel motion for steam-engines. (See STEAM-ENGINE.)

  § 80.* _The Reuleaux System of Analysis._--If two pieces, A and B,
  (fig. 118) are jointed together by a pin, the pin being fixed, say, to
  A, the only relative motion possible between the pieces is one of
  turning about the axis of the pin. Whatever motion the pair of pieces
  may have as a whole each separate piece shares in common, and this
  common motion in no way affects the relative motion of A and B. The
  motion of one piece is said to be completely constrained relatively to
  the other piece. Again, the pieces A and B (fig. 119) are paired
  together as a slide, and the only relative motion possible between
  them now is that of sliding, and therefore the motion of one
  relatively to the other is completely constrained. The pieces may be
  paired together as a screw and nut, in which case the relative motion
  is compounded of turning with sliding.

  [Illustration: FIG. 118.]

  [Illustration: FIG. 119.]

  These combinations of pieces are known individually as _kinematic
  pairs of elements_, or briefly _kinematic pairs_. The three pairs
  mentioned above have each the peculiarity that contact between the two
  pieces forming the pair is distributed over a surface. Kinematic pairs
  which have surface contact are classified as _lower pairs_. Kinematic
  pairs in which contact takes place along a line only are classified as
  _higher pairs_. A pair of spur wheels in gear is an example of a
  higher pair, because the wheels have contact between their teeth along
  lines only.

  A _kinematic link_ of the simplest form is made by joining up the
  halves of two kinematic pairs by means of a rigid link. Thus if A1B1
  represent a turning pair, and A2B2 a second turning pair, the rigid
  link formed by joining B1 to B2 is a kinematic link. Four links of
  this kind are shown in fig. 120 joined up to form a _closed kinematic
  chain_.

  [Illustration: FIG. 120.]

  In order that a kinematic chain may be made the basis of a mechanism,
  every point in any link of it must be completely constrained with
  regard to every other link. Thus in fig. 120 the motion of a point a
  in the link A1A2 is completely constrained with regard to the link
  B1B4 by the turning pair A1B1, and it can be proved that the motion of
  a relatively to the non-adjacent link A3A4 is completely constrained,
  and therefore the four-bar chain, as it is called, can be and is used
  as the basis of many mechanisms. Another way of considering the
  question of constraint is to imagine any one link of the chain fixed;
  then, however the chain be moved, the path of a point, as a, will
  always remain the same. In a five-bar chain, if a is a point in a link
  non-adjacent to a fixed link, its path is indeterminate. Still another
  way of stating the matter is to say that, if any one link in the chain
  be fixed, any point in the chain must have only one degree of freedom.
  In a five-bar chain a point, as a, in a link non-adjacent to the fixed
  link has two degrees of freedom and the chain cannot therefore be used
  for a mechanism. These principles may be applied to examine any
  possible combination of links forming a kinematic chain in order to
  test its suitability for use as a mechanism. Compound chains are
  formed by the superposition of two or more simple chains, and in these
  more complex chains links will be found carrying three, or even more,
  halves of kinematic pairs. The Joy valve gear mechanism is a good
  example of a compound kinematic chain.

  [Illustration: FIG. 121.]

  A chain built up of three turning pairs and one sliding pair, and
  known as the _slider crank chain_, is shown in fig. 121. It will be
  seen that the piece A1 can only slide relatively to the piece B1, and
  these two pieces therefore form the sliding pair. The piece A1 carries
  the pin B4, which is one half of the turning pair A4 B4. The piece A1
  together with the pin B4 therefore form a kinematic link A1B4. The
  other links of the chain are, B1A2, B2B3, A3A4. In order to convert a
  chain into a mechanism it is necessary to fix one link in it. Any one
  of the links may be fixed. It follows therefore that there are as many
  possible mechanisms as there are links in the chain. For example,
  there is a well-known mechanism corresponding to the fixing of three
  of the four links of the slider crank chain (fig. 121). If the link d
  is fixed the chain at once becomes the mechanism of the ordinary steam
  engine; if the link e is fixed the mechanism obtained is that of the
  oscillating cylinder steam engine; if the link c is fixed the
  mechanism becomes either the Whitworth quick-return motion or the
  slot-bar motion, depending upon the proportion between the lengths of
  the links c and e. These different mechanisms are called _inversions_
  of the slider crank chain. What was the fixed framework of the
  mechanism in one case becomes a moving link in an inversion.

  The Reuleaux system, therefore, consists essentially of the analysis
  of every mechanism into a kinematic chain, and since each link of the
  chain may be the fixed frame of a mechanism quite diverse mechanisms
  are found to be merely inversions of the same kinematic chain. Franz
  Reuleaux's _Kinematics of Machinery_, translated by Sir A. B. W.
  Kennedy (London, 1876), is the book in which the system is set forth
  in all its completeness. In _Mechanics of Machinery_, by Sir A. B. W.
  Kennedy (London, 1886), the system was used for the first time in an
  English textbook, and now it has found its way into most modern
  textbooks relating to the subject of mechanism.

  § 81.* _Centrodes, Instantaneous Centres, Velocity Image, Velocity
  Diagram._--Problems concerning the relative motion of the several
  parts of a kinematic chain may be considered in two ways, in addition
  to the way hitherto used in this article and based on the principle of
  § 34. The first is by the method of instantaneous centres, already
  exemplified in § 63, and rolling centroids, developed by Reuleaux in
  connexion with his method of analysis. The second is by means of
  Professor R. H. Smith's method already referred to in § 23.

  _Method 1._--By reference to § 30 it will be seen that the motion of a
  cylinder rolling on a fixed cylinder is one of rotation about an
  instantaneous axis T, and that the velocity both as regards direction
  and magnitude is the same as if the rolling piece B were for the
  instant turning about a fixed axis coincident with the instantaneous
  axis. If the rolling cylinder B and its path A now be assumed to
  receive a common plane motion, what was before the velocity of the
  point P becomes the velocity of P relatively to the cylinder A, since
  the motion of B relatively to A still takes place about the
  instantaneous axis T. If B stops rolling, then the two cylinders
  continue to move as though they were parts of a rigid body. Notice
  that the shape of either rolling curve (fig. 91 or 92) may be found by
  considering each fixed in turn and then tracing out the locus of the
  instantaneous axis. These rolling cylinders are sometimes called
  axodes, and a section of an axode in a plane parallel to the plane of
  motion is called a centrode. The axode is hence the locus of the
  instantaneous axis, whilst the centrode is the locus of the
  instantaneous centre in any plane parallel to the plane of motion.
  There is no restriction on the shape of these rolling axodes; they may
  have any shape consistent with rolling (that is, no slipping is
  permitted), and the relative velocity of a point P is still found by
  considering it with regard to the instantaneous centre.

  Reuleaux has shown that the relative motion of any pair of
  non-adjacent links of a kinematic chain is determined by the rolling
  together of two ideal cylindrical surfaces (cylindrical being used
  here in the general sense), each of which may be assumed to be formed
  by the extension of the material of the link to which it corresponds.
  These surfaces have contact at the instantaneous axis, which is now
  called the instantaneous axis of the two links concerned. To find the
  form of these surfaces corresponding to a particular pair of
  non-adjacent links, consider each link of the pair fixed in turn, then
  the locus of the instantaneous axis is the axode corresponding to the
  fixed link, or, considering a plane of motion only, the locus of the
  instantaneous centre is the centrode corresponding to the fixed link.

  To find the instantaneous centre for a particular link corresponding
  to any given configuration of the kinematic chain, it is only
  necessary to know the direction of motion of any two points in the
  link, since lines through these points respectively at right angles to
  their directions of motion intersect in the instantaneous centre.

  [Illustration: FIG. 122.]

  To illustrate this principle, consider the four-bar chain shown in
  fig. 122 made up of the four links, a, b, c, d. Let a be the fixed
  link, and consider the link c. Its extremities are moving respectively
  in directions at right angles to the links b and d; hence produce the
  links b and d to meet in the point O_(ac). This point is the
  instantaneous centre of the motion of the link c relatively to the
  fixed link a, a fact indicated by the suffix ac placed after the
  letter O. The process being repeated for different values of the angle
  [theta] the curve through the several points Oac is the centroid which
  may be imagined as formed by an extension of the material of the link
  a. To find the corresponding centroid for the link c, fix c and repeat
  the process. Again, imagine d fixed, then the instantaneous centre
  O_(bd) of b with regard to d is found by producing the links c and a
  to intersect in O_(bd), and the shapes of the centroids belonging
  respectively to the links b and d can be found as before. The axis
  about which a pair of adjacent links turn is a permanent axis, and is
  of course the axis of the pin which forms the point. Adding the
  centres corresponding to these several axes to the figure, it will be
  seen that there are six centres in connexion with the four-bar chain
  of which four are permanent and two are instantaneous or virtual
  centres; and, further, that whatever be the configuration of the chain
  these centres group themselves into three sets of three, each set
  lying on a straight line. This peculiarity is not an accident or a
  special property of the four-bar chain, but is an illustration of a
  general law regarding the subject discovered by Aronhold and Sir A. B.
  W. Kennedy independently, which may be thus stated: If any three
  bodies, a, b, c, have plane motion their three virtual centres,
  O_(ab), O_(bc), O_(ac), are three points on one straight line. A proof
  of this will be found in _The Mechanics of Machinery_ quoted above.
  Having obtained the set of instantaneous centres for a chain, suppose
  a is the fixed link of the chain and c any other link; then O_(ac) is
  the instantaneous centre of the two links and may be considered for
  the instant as the trace of an axis fixed to an extension of the link
  a about which c is turning, and thus problems of instantaneous
  velocity concerning the link c are solved as though the link c were
  merely rotating for the instant about a fixed axis coincident with the
  instantaneous axis.

  [Illustration: FIG. 123.]

  [Illustration: FIG. 124.]

  _Method 2._--The second method is based upon the vector representation
  of velocity, and may be illustrated by applying it to the four-bar
  chain. Let AD (fig. 123) be the fixed link. Consider the link BC, and
  let it be required to find the velocity of the point B having given
  the velocity of the point C. The principle upon which the solution is
  based is that the only motion which B can have relatively to an axis
  through C fixed to the link CD is one of turning about C. Choose any
  pole O (fig. 124). From this pole set out Oc to represent the velocity
  of the point C. The direction of this must be at right angles to the
  line CD, because this is the only direction possible to the point C.
  If the link BC moves without turning, Oc will also represent the
  velocity of the point B; but, if the link is turning, B can only move
  about the axis C, and its direction of motion is therefore at right
  angles to the line CB. Hence set out the possible direction of B´s
  motion in the velocity diagram, namely cb1, at right angles to CB. But
  the point B must also move at right angles to AB in the case under
  consideration. Hence draw a line through O in the velocity diagram at
  right angles to AB to cut cb1 in b. Then Ob is the velocity of the
  point b in magnitude and direction, and cb is the tangential velocity
  of B relatively to C. Moreover, whatever be the actual magnitudes of
  the velocities, the instantaneous velocity ratio of the points C and B
  is given by the ratio Oc/Ob.

  A most important property of the diagram (figs. 123 and 124) is the
  following: If points X and x are taken dividing the link BC and the
  tangential velocity cb, so that cx:xb = CX:XB, then Ox represents the
  velocity of the point X in magnitude and direction. The line cb has
  been called the _velocity image_ of the rod, since it may be looked
  upon as a scale drawing of the rod turned through 90° from the actual
  rod. Or, put in another way, if the link CB is drawn to scale on the
  new length cb in the velocity diagram (fig. 124), then a vector drawn
  from O to any point on the new drawing of the rod will represent the
  velocity of that point of the actual rod in magnitude and direction.
  It will be understood that there is a new velocity diagram for every
  new configuration of the mechanism, and that in each new diagram the
  image of the rod will be different in scale. Following the method
  indicated above for a kinematic chain in general, there will be
  obtained a velocity diagram similar to that of fig. 124 for each
  configuration of the mechanism, a diagram in which the velocity of the
  several points in the chain utilized for drawing the diagram will
  appear to the same scale, all radiating from the pole O. The lines
  joining the ends of these several velocities are the several
  tangential velocities, each being the velocity image of a link in the
  chain. These several images are not to the same scale, so that
  although the images may be considered to form collectively an image of
  the chain itself, the several members of this chain-image are to
  different scales in any one velocity diagram, and thus the chain-image
  is distorted from the actual proportions of the mechanism which it
  represents.

  [Illustration: FIG. 125.]

  § 82.* _Acceleration Diagram. Acceleration Image._--Although it is
  possible to obtain the acceleration of points in a kinematic chain
  with one link fixed by methods which utilize the instantaneous centres
  of the chain, the vector method more readily lends itself to this
  purpose. It should be understood that the instantaneous centre
  considered in the preceding paragraphs is available only for
  estimating relative velocities; it cannot be used in a similar manner
  for questions regarding acceleration. That is to say, although the
  instantaneous centre is a centre of no velocity for the instant, it is
  not a centre of no acceleration, and in fact the centre of no
  acceleration is in general a quite different point. The general
  principle on which the method of drawing an acceleration diagram
  depends is that if a link CB (fig. 125) have plane motion and the
  acceleration of any point C be given in magnitude and direction, the
  acceleration of any other point B is the vector sum of the
  acceleration of C, the radial acceleration of B about C and the
  tangential acceleration of B about C. Let A be any origin, and let Ac
  represent the acceleration of the point C, ct the radial acceleration
  of B about C which must be in a direction parallel to BC, and tb the
  tangential acceleration of B about C, which must of course be at right
  angles to ct; then the vector sum of these three magnitudes is Ab, and
  this vector represents the acceleration of the point B. The directions
  of the radial and tangential accelerations of the point B are always
  known when the position of the link is assigned, since these are to be
  drawn respectively parallel to and at right angles to the link itself.
  The magnitude of the radial acceleration is given by the expression
  v²/BC, v being the velocity of the point B about the point C. This
  velocity can always be found from the velocity diagram of the chain of
  which the link forms a part. If dw/dt is the angular acceleration of
  the link, dw/dt × CB is the tangential acceleration of the point B
  about the point C. Generally this tangential acceleration is unknown
  in magnitude, and it becomes part of the problem to find it. An
  important property of the diagram is that if points X and x are taken
  dividing the link CB and the whole acceleration of B about C, namely,
  cb in the same ratio, then Ax represents the acceleration of the point
  X in magnitude and direction; cb is called the acceleration image of
  the rod. In applying this principle to the drawing of an acceleration
  diagram for a mechanism, the velocity diagram of the mechanism must be
  first drawn in order to afford the means of calculating the several
  radial accelerations of the links. Then assuming that the acceleration
  of one point of a particular link of the mechanism is known together
  with the corresponding configuration of the mechanism, the two vectors
  Ac and ct can be drawn. The direction of tb, the third vector in the
  diagram, is also known, so that the problem is reduced to the
  condition that b is somewhere on the line tb. Then other conditions
  consequent upon the fact that the link forms part of a kinematic chain
  operate to enable b to be fixed. These methods are set forth and
  exemplified in _Graphics_, by R. H. Smith (London, 1889). Examples,
  completely worked out, of velocity and acceleration diagrams for the
  slider crank chain, the four-bar chain, and the mechanism of the Joy
  valve gear will be found in ch. ix. of _Valves and Valve Gear
  Mechanism_, by W. E. Dalby (London, 1906).


  CHAPTER II. ON APPLIED DYNAMICS.

  § 83. _Laws of Motion._--The action of a machine in transmitting
  _force_ and _motion_ simultaneously, or performing _work_, is
  governed, in common with the phenomena of moving bodies in general, by
  two "laws of motion."


  _Division 1. Balanced Forces in Machines of Uniform Velocity._

  § 84. _Application of Force to Mechanism._--Forces are applied in
  units of weight; and the unit most commonly employed in Britain is the
  _pound avoirdupois_. The action of a force applied to a body is always
  in reality distributed over some definite space, either a volume of
  three dimensions or a surface of two. An example of a force
  distributed throughout a volume is the _weight_ of the body itself,
  which acts on every particle, however small. The _pressure_ exerted
  between two bodies at their surface of contact, or between the two
  parts of one body on either side of an ideal surface of separation, is
  an example of a force distributed over a surface. The mode of
  distribution of a force applied to a solid body requires to be
  considered when its stiffness and strength are treated of; but, in
  questions respecting the action of a force upon a rigid body
  considered as a whole, the _resultant_ of the distributed force,
  determined according to the principles of statics, and considered as
  acting in a _single line_ and applied at a _single point_, may, for
  the occasion, be substituted for the force as really distributed.
  Thus, the weight of each separate piece in a machine is treated as
  acting wholly at its _centre of gravity_, and each pressure applied to
  it as acting at a point called the _centre of pressure_ of the surface
  to which the pressure is really applied.

  § 85. _Forces applied to Mechanism Classed._--If [theta] be the
  _obliquity_ of a force F applied to a piece of a machine--that is, the
  angle made by the direction of the force with the direction of motion
  of its point of application--then by the principles of statics, F may
  be resolved into two rectangular components, viz.:--

    Along the direction of motion, P = F cos [theta]   \  (49)
    Across the direction of motion, Q = F sin [theta]  /

  If the component along the direction of motion acts with the motion,
  it is called an _effort_; if _against_ the motion, a _resistance_. The
  component _across_ the direction of motion is a _lateral pressure_;
  the unbalanced lateral pressure on any piece, or part of a piece, is
  _deflecting force_. A lateral pressure may increase resistance by
  causing friction; the friction so caused acts against the motion, and
  is a resistance, but the lateral pressure causing it is not a
  resistance. Resistances are distinguished into _useful_ and
  _prejudicial_, according as they arise from the useful effect produced
  by the machine or from other causes.

  § 86. _Work._--_Work_ consists in moving against resistance. The work
  is said to be _performed_, and the resistance _overcome_. Work is
  measured by the product of the resistance into the distance through
  which its point of application is moved. The _unit of work_ commonly
  used in Britain is a resistance of one pound overcome through a
  distance of one foot, and is called a _foot-pound_.

  Work is distinguished into _useful work_ and _prejudicial_ or _lost
  work_, according as it is performed in producing the useful effect of
  the machine, or in overcoming prejudicial resistance.

  § 87. _Energy: Potential Energy._--_Energy_ means _capacity for
  performing work_. The _energy of an effort_, or _potential energy_, is
  measured by the product of the effort into the distance through which
  its point of application is _capable_ of being moved. The unit of
  energy is the same with the unit of work.

  When the point of application of an effort _has been moved_ through a
  given distance, energy is said to have been _exerted_ to an amount
  expressed by the product of the effort into the distance through which
  its point of application has been moved.

  § 88. _Variable Effort and Resistance._--If an effort has different
  magnitudes during different portions of the motion of its point of
  application through a given distance, let each different magnitude of
  the effort P be multiplied by the length [Delta]s of the corresponding
  portion of the path of the point of application; the sum

    [Sigma] · P[Delta]s  (50)

  is the whole energy exerted. If the effort varies by insensible
  gradations, the energy exerted is the integral or limit towards which
  that sum approaches continually as the divisions of the path are made
  smaller and more numerous, and is expressed by

    [int]P ds.  (51)

  Similar processes are applicable to the finding of the work performed
  in overcoming a varying resistance.

  The work done by a machine can be actually measured by means of a
  dynamometer (q.v.).

  § 89. _Principle of the Equality of Energy and Work._--From the first
  law of motion it follows that in a machine whose pieces move with
  uniform velocities the efforts and resistances must balance each
  other. Now from the laws of statics it is known that, in order that a
  system of forces applied to a system of connected points may be in
  equilibrium, it is necessary that the sum formed by putting together
  the products of the forces by the respective distances through which
  their points of application are capable of moving simultaneously, each
  along the direction of the force applied to it, shall be
  zero,--products being considered positive or negative according as the
  direction of the forces and the possible motions of their points of
  application are the same or opposite.

  In other words, the sum of the negative products is equal to the sum
  of the positive products. This principle, applied to a machine whose
  parts move with uniform velocities, is equivalent to saying that in
  any given interval of time _the energy exerted is equal to the work
  performed_.

  The symbolical expression of this law is as follows: let efforts be
  applied to one or any number of points of a machine; let any one of
  these efforts be represented by P, and the distance traversed by its
  point of application in a given interval of time by ds; let
  resistances be overcome at one or any number of points of the same
  machine; let any one of these resistances be denoted by R, and the
  distance traversed by its point of application in the given interval
  of time by ds´; then

    [Sigma] · P ds = [Sigma] · R ds´.  (52)

  The lengths ds, ds´ are proportional to the velocities of the points
  to whose paths they belong, and the proportions of those velocities to
  each other are deducible from the construction of the machine by the
  principles of pure mechanism explained in Chapter I.

  § 90. _Static Equilibrium of Mechanisms._--The principle stated in the
  preceding section, namely, that the energy exerted is equal to the
  work performed, enables the ratio of the components of the forces
  acting in the respective directions of motion at two points of a
  mechanism, one being the point of application of the effort, and the
  other the point of application of the resistance, to be readily found.
  Removing the summation signs in equation (52) in order to restrict its
  application to two points and dividing by the common time interval
  during which the respective small displacements ds and ds´ were made,
  it becomes P ds/dt = R ds´/dt, that is, Pv = Rv´, which shows that the
  force ratio is the inverse of the velocity ratio. It follows at once
  that any method which may be available for the determination of the
  velocity ratio is equally available for the determination of the force
  ratio, it being clearly understood that the forces involved are the
  components of the actual forces resolved in the direction of motion
  of the points. The relation between the effort and the resistance may
  be found by means of this principle for all kinds of mechanisms, when
  the friction produced by the components of the forces across the
  direction of motion of the two points is neglected. Consider the
  following example:--

  [Illustration: FIG. 126.]

  A four-bar chain having the configuration shown in fig. 126 supports a
  load P at the point x. What load is required at the point y to
  maintain the configuration shown, both loads being supposed to act
  vertically? Find the instantaneous centre O_(bd), and resolve each
  load in the respective directions of motion of the points x and y;
  thus there are obtained the components P cos [theta] and R cos [phi].
  Let the mechanism have a small motion; then, for the instant, the link
  b is turning about its instantaneous centre O_(bd), and, if [omega] is
  its instantaneous angular velocity, the velocity of the point x is
  [omega]r, and the velocity of the point y is [omega]s. Hence, by the
  principle just stated, P cos [theta] × [omega]r = R cos [phi] ×
  [omega]s. But, p and q being respectively the perpendiculars to the
  lines of action of the forces, this equation reduces to P_p = R_q,
  which shows that the ratio of the two forces may be found by taking
  moments about the instantaneous centre of the link on which they act.

  The forces P and R may, however, act on different links. The general
  problem may then be thus stated: Given a mechanism of which r is the
  fixed link, and s and t any other two links, given also a force f_s,
  acting on the link s, to find the force f_t acting in a given
  direction on the link t, which will keep the mechanism in static
  equilibrium. The graphic solution of this problem may be effected
  thus:--

    (1) Find the three virtual centres O_(rs), O_(rt), O_(st), which
    must be three points in a line.

    (2) Resolve f_s into two components, one of which, namely, f_q,
    passes through O_(rs) and may be neglected, and the other f_p passes
    through O_(st).

    (3) Find the point M, where f_p joins the given direction of f_t,
    and resolve f_p into two components, of which one is in the
    direction MO_(rt), and may be neglected because it passes through
    O_(rt), and the other is in the given direction of f_t and is
    therefore the force required.

  [Illustration: FIG. 127.]

  This statement of the problem and the solution is due to Sir A. B. W.
  Kennedy, and is given in ch. 8 of his _Mechanics of Machinery_.
  Another general solution of the problem is given in the _Proc. Lond.
  Math. Soc._ (1878-1879), by the same author. An example of the method
  of solution stated above, and taken from the _Mechanics of Machinery_,
  is illustrated by the mechanism fig. 127, which is an epicyclic train
  of three wheels with the first wheel r fixed. Let it be required to
  find the vertical force which must act at the pitch radius of the last
  wheel t to balance exactly a force f_s acting vertically downwards on
  the arm at the point indicated in the figure. The two links concerned
  are the last wheel t and the arm s, the wheel r being the fixed link
  of the mechanism. The virtual centres O_(rs), O_(st) are at the
  respective axes of the wheels r and t, and the centre O_(rt) divides
  the line through these two points externally in the ratio of the train
  of wheels. The figure sufficiently indicates the various steps of the
  solution.

  The relation between the effort and the resistance in a machine to
  include the effect of friction at the joints has been investigated in
  a paper by Professor Fleeming Jenkin, "On the application of graphic
  methods to the determination of the efficiency of machinery" (_Trans.
  Roy. Soc. Ed._, vol. 28). It is shown that a machine may at any
  instant be represented by a frame of links the stresses in which are
  identical with the pressures at the joints of the mechanism. This
  self-strained frame is called the _dynamic frame_ of the machine. The
  driving and resisting efforts are represented by elastic links in the
  dynamic frame, and when the frame with its elastic links is drawn the
  stresses in the several members of it may be determined by means of
  reciprocal figures. Incidentally the method gives the pressures at
  every joint of the mechanism.

  § 91. _Efficiency._--The _efficiency_ of a machine is the ratio of the
  _useful_ work to the _total_ work--that is, to the energy exerted--and
  is represented by

    [Sigma]·R_u ds´             [Sigma]·R_u ds´           [Sigma]·R_u ds´    U
    --------------- = --------------------------------- = --------------- = ---.  (53)
    [Sigma]·R ds´     [Sigma]·R_u ds´ + [Sigma]·R_p ds´    [Sigma]·P ds      E

  R_u being taken to represent useful and R_p prejudicial resistances.
  The more nearly the efficiency of a machine approaches to unity the
  better is the machine.

  § 92. _Power and Effect._--The _power_ of a machine is the energy
  exerted, and the _effect_ the useful work performed, in some interval
  of time of definite length, such as a second, an hour, or a day.

  The unit of power, called conventionally a horse-power, is 550
  foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000
  foot-pounds per hour.

  § 93. _Modulus of a Machine._--In the investigation of the properties
  of a machine, the useful resistances to be overcome and the useful
  work to be performed are usually given. The prejudicial resistances
  arc generally functions of the useful resistances of the weights of
  the pieces of the mechanism, and of their form and arrangement; and,
  having been determined, they serve for the computation of the _lost_
  work, which, being added to the useful work, gives the expenditure of
  energy required. The result of this investigation, expressed in the
  form of an equation between this energy and the useful work, is called
  by Moseley the _modulus_ of the machine. The general form of the
  modulus may be expressed thus--

    E = U + [phi](U, A) + [psi](A),  (54)

  where A denotes some quantity or set of quantities depending on the
  form, arrangement, weight and other properties of the mechanism.
  Moseley, however, has pointed out that in most cases this equation
  takes the much more simple form of

    E = (1 + A)U + B,  (55)

  where A and B are _constants_, depending on the form, arrangement and
  weight of the mechanism. The efficiency corresponding to the last
  equation is

     U        1
    --- = -----------.  (56)
     E    1 + A + B/U

  § 94. _Trains of Mechanism._--In applying the preceding principles to
  a train of mechanism, it may either be treated as a whole, or it may
  be considered in sections consisting of single pieces, or of any
  convenient portion of the train--each section being treated as a
  machine, driven by the effort applied to it and energy exerted upon it
  through its line of connexion with the preceding section, performing
  useful work by driving the following section, and losing work by
  overcoming its own prejudicial resistances. It is evident that _the
  efficiency of the whole train is the product of the efficiencies of
  its sections_.

  § 95. _Rotating Pieces: Couples of Forces._--It is often convenient to
  express the energy exerted upon and the work performed by a turning
  piece in a machine in terms of the _moment_ of the _couples of forces_
  acting on it, and of the angular velocity. The ordinary British unit
  of moment is a _foot-pound_; but it is to be remembered that this is a
  foot-pound of a different sort from the unit of energy and work.

  If a force be applied to a turning piece in a line not passing through
  its axis, the axis will press against its bearings with an equal and
  parallel force, and the equal and opposite reaction of the bearings
  will constitute, together with the first-mentioned force, a couple
  whose arm is the perpendicular distance from the axis to the line of
  action of the first force.

  A couple is said to be _right_ or _left handed_ with reference to the
  observer, according to the direction in which it tends to turn the
  body, and is a _driving_ couple or a _resisting_ couple according as
  its tendency is with or against that of the actual rotation.

  Let dt be an interval of time, [alpha] the angular velocity of the
  piece; then [alpha]dt is the angle through which it turns in the
  interval dt, and ds = vdt = r[alpha]dt is the distance through which
  the point of application of the force moves. Let P represent an
  effort, so that Pr is a driving couple, then

    P ds = Pv dt = Pr[alpha] dt = M[alpha] dt  (57)

  is the energy exerted by the couple M in the interval dt; and a
  similar equation gives the work performed in overcoming a resisting
  couple. When several couples act on one piece, the resultant of their
  moments is to be multiplied by the common angular velocity of the
  whole piece.

  § 96. _Reduction of Forces to a given Point, and of Couples to the
  Axis of a given Piece._--In computations respecting machines it is
  often convenient to substitute for a force applied to a given point,
  or a couple applied to a given piece, the _equivalent_ force or couple
  applied to some other point or piece; that is to say, the force or
  couple, which, if applied to the other point or piece, would exert
  equal energy or employ equal work. The principles of this reduction
  are that the ratio of the given to the equivalent force is the
  reciprocal of the ratio of the velocities of their points of
  application, and the ratio of the given to the equivalent couple is
  the reciprocal of the ratio of the angular velocities of the pieces to
  which they are applied.

  These velocity ratios are known by the construction of the mechanism,
  and are independent of the absolute speed.

  § 97. _Balanced Lateral Pressure of Guides and Bearings._--The most
  important part of the lateral pressure on a piece of mechanism is the
  reaction of its guides, if it is a sliding piece, or of the bearings
  of its axis, if it is a turning piece; and the balanced portion of
  this reaction is equal and opposite to the resultant of all the other
  forces applied to the piece, its own weight included. There may be or
  may not be an unbalanced component in this pressure, due to the
  deviated motion. Its laws will be considered in the sequel.

  § 98. _Friction. Unguents._--The most important kind of resistance in
  machines is the _friction_ or _rubbing resistance_ of surfaces which
  slide over each other. The _direction_ of the resistance of friction
  is opposite to that in which the sliding takes place. Its _magnitude_
  is the product of the _normal pressure_ or force which presses the
  rubbing surfaces together in a direction perpendicular to themselves
  into a specific constant already mentioned in § 14, as the
  _coefficient of friction_, which depends on the nature and condition
  of the surfaces of the unguent, if any, with which they are covered.
  The _total pressure_ exerted between the rubbing surfaces is the
  resultant of the normal pressure and of the friction, and its
  _obliquity_, or inclination to the common perpendicular of the
  surfaces, is the _angle of repose_ formerly mentioned in § 14, whose
  tangent is the coefficient of friction. Thus, let N be the normal
  pressure, R the friction, T the total pressure, f the coefficient of
  friction, and [phi] the angle of repose; then

    f = tan [phi]                      \  (58)
    R = fN = N tan [phi] = T sin [phi] /

  Experiments on friction have been made by Coulomb, Samuel Vince, John
  Rennie, James Wood, D. Rankine and others. The most complete and
  elaborate experiments are those of Morin, published in his _Notions
  fondamentales de mécanique_, and republished in Britain in the works
  of Moseley and Gordon.

  The experiments of Beauchamp Tower ("Report of Friction Experiments,"
  _Proc. Inst. Mech. Eng._, 1883) showed that when oil is supplied to a
  journal by means of an oil bath the coefficient of friction varies
  nearly inversely as the load on the bearing, thus making the product
  of the load on the bearing and the coefficient of friction a constant.
  Mr Tower's experiments were carried out at nearly constant
  temperature. The more recent experiments of Lasche (_Zeitsch, Verein
  Deutsche Ingen._, 1902, 46, 1881) show that the product of the
  coefficient of friction, the load on the bearing, and the temperature
  is approximately constant. For further information on this point and
  on Osborne Reynolds's theory of lubrication see BEARINGS and
  LUBRICATION.

  § 99. _Work of Friction. Moment of Friction._--The work performed in a
  unit of time in overcoming the friction of a pair of surfaces is the
  product of the friction by the velocity of sliding of the surfaces
  over each other, if that is the same throughout the whole extent of
  the rubbing surfaces. If that velocity is different for different
  portions of the rubbing surfaces, the velocity of each portion is to
  be multiplied by the friction of that portion, and the results summed
  or integrated.

  When the relative motion of the rubbing surfaces is one of rotation,
  the work of friction in a unit of time, for a portion of the rubbing
  surfaces at a given distance from the axis of rotation, may be found
  by multiplying together the friction of that portion, its distance
  from the axis, and the angular velocity. The product of the force of
  friction by the distance at which it acts from the axis of rotation is
  called the _moment of friction_. The total moment of friction of a
  pair of rotating rubbing surfaces is the sum or integral of the
  moments of friction of their several portions.

  To express this symbolically, let du represent the area of a portion
  of a pair of rubbing surfaces at a distance r from the axis of their
  relative rotation; p the intensity of the normal pressure at du per
  unit of area; and f the coefficient of friction. Then the moment of
  friction of du is fprdu;

    the total moment of friction is f [integral] pr·du;              \
    and the work performed in a unit cf time in overcoming friction,  >  (59)
    when the angular velocity is [alpha], is [alpha]f [int] pr·du.   /

  It is evident that the moment of friction, and the work lost by being
  performed in overcoming friction, are less in a rotating piece as the
  bearings are of smaller radius. But a limit is put to the diminution
  of the radii of journals and pivots by the conditions of durability
  and of proper lubrication, and also by conditions of strength and
  stiffness.

  § 100. _Total Pressure between Journal and Bearing._--A single piece
  rotating with a uniform velocity has four mutually balanced forces
  applied to it: (l) the effort exerted on it by the piece which drives
  it; (2) the resistance of the piece which follows it--which may be
  considered for the purposes of the present question as useful
  resistance; (3) its weight; and (4) the reaction of its own
  cylindrical bearings. There are given the following data:--

    The direction of the effort.
    The direction of the useful resistance.
    The weight of the piece and the direction in which it acts.
    The magnitude of the useful resistance.
    The radius of the bearing r.
    The angle of repose [phi], corresponding to the friction of the
      journal on the bearing.

  And there are required the following:--

    The direction of the reaction of the bearing.
    The magnitude of that reaction.
    The magnitude of the effort.

  Let the useful resistance and the weight of the piece be compounded by
  the principles of statics into one force, and let this be called _the
  given force_.

  [Illustration: FIG. 128.]

  The directions of the effort and of the given force are either
  parallel or meet in a point. If they are parallel, the direction of
  the reaction of the bearing is also parallel to them; if they meet in
  a point, the direction of the reaction traverses the same point.

  Also, let AAA, fig. 128, be a section of the bearing, and C its axis;
  then the direction of the reaction, at the point where it intersects
  the circle AAA, must make the angle [phi] with the radius of that
  circle; that is to say, it must be a line such as PT touching the
  smaller circle BB, whose radius is r · sin [phi]. The side on which it
  touches that circle is determined by the fact that the obliquity of
  the reaction is such as to oppose the rotation.

  Thus is determined the direction of the reaction of the bearing; and
  the magnitude of that reaction and of the effort are then found by the
  principles of the equilibrium of three forces already stated in § 7.

  The work lost in overcoming the friction of the bearing is the same as
  that which would be performed in overcoming at the circumference of
  the small circle BB a resistance equal to the whole pressure between
  the journal and bearing.

  In order to diminish that pressure to the smallest possible amount,
  the effort, and the resultant of the useful resistance, and the weight
  of the piece (called above the "given force") ought to be opposed to
  each other as directly as is practicable consistently with the
  purposes of the machine.

  An investigation of the forces acting on a bearing and journal
  lubricated by an oil bath will be found in a paper by Osborne Reynolds
  in the _Phil. Trans._ pt. i. (1886). (See also BEARINGS.)

  § 101. _Friction of Pivots and Collars._--When a shaft is acted upon
  by a force tending to shift it lengthways, that force must be balanced
  by the reaction of a bearing against a _pivot_ at the end of the
  shaft; or, if that be impossible, against one or more _collars_, or
  rings _projecting_ from the body of the shaft. The bearing of the
  pivot is called a _step_ or _footstep_. Pivots require great hardness,
  and are usually made of steel. The _flat_ pivot is a cylinder of steel
  having a plane circular end as a rubbing surface. Let N be the total
  pressure sustained by a flat pivot of the radius r; if that pressure
  be uniformly distributed, which is the case when the rubbing surfaces
  of the pivot and its step are both true planes, the _intensity_ of the
  pressure is

    p = N/[pi]r²;  (60)

  and, introducing this value into equation 59, the _moment of friction
  of the flat pivot_ is found to be

    (2/3)fNr  (61)

  or two-thirds of that of a cylindrical journal of the same radius
  under the same normal pressure.

  The friction of a _conical_ pivot exceeds that of a flat pivot of the
  same radius, and under the same pressure, in the proportion of the
  side of the cone to the radius of its base.

  The moment of friction of a _collar_ is given by the formula--

            r³ - r´³
    (2/3)fN --------,  (62)
            r² - r´²

  where r is the external and r´ the internal radius.

  [Illustration: FIG. 129.]

  In the _cup and ball_ pivot the end of the shaft and the step present
  two recesses facing each other, into which art fitted two shallow cups
  of steel or hard bronze. Between the concave spherical surfaces of
  those cups is placed a steel ball, being either a complete sphere or a
  lens having convex surfaces of a somewhat less radius than the concave
  surfaces of the cups. The moment of friction of this pivot is at first
  almost inappreciable from the extreme smallness of the radius of the
  circles of contact of the ball and cups, but, as they wear, that
  radius and the moment of friction increase.

  It appears that the rapidity with which a rubbing surface wears away
  is proportional to the friction and to the velocity jointly, or nearly
  so. Hence the pivots already mentioned wear unequally at different
  points, and tend to alter their figures. Schiele has invented a pivot
  which preserves its original figure by wearing equally at all points
  in a direction parallel to its axis. The following are the principles
  on which this equality of wear depends:--

  The rapidity of wear of a surface measured in an _oblique_ direction
  is to the rapidity of wear measured normally as the secant of the
  obliquity is to unity. Let OX (fig. 129) be the axis of a pivot, and
  let RPC be a portion of a curve such that at any point P the secant of
  the obliquity to the normal of the curve of a line parallel to the
  axis is inversely proportional to the ordinate PY, to which the
  velocity of P is proportional. The rotation of that curve round OX
  will generate the form of pivot required. Now let PT be a tangent to
  the curve at P, cutting OX in T; PT = PY × _secant obliquity_, and
  this is to be a constant quantity; hence the curve is that known as
  the _tractory_ of the straight line OX, in which PT = OR = constant.
  This curve is described by having a fixed straight edge parallel to
  OX, along which slides a slider carrying a pin whose centre is T. On
  that pin turns an arm, carrying at a point P a tracing-point, pencil
  or pen. Should the pen have a nib of two jaws, like those of an
  ordinary drawing-pen, the plane of the jaws must pass through PT.
  Then, while T is slid along the axis from O towards X, P will be drawn
  after it from R towards C along the tractory. This curve, being an
  asymptote to its axis, is capable of being indefinitely prolonged
  towards X; but in designing pivots it should stop before the angle PTY
  becomes less than the angle of repose of the rubbing surfaces,
  otherwise the pivot will be liable to stick in its bearing. The moment
  of friction of "Schiele's anti-friction pivot," as it is called, is
  equal to that of a cylindrical journal of the radius OR = PT the
  constant tangent, under the same pressure.

  Records of experiments on the friction of a pivot bearing will be
  found in the _Proc. Inst. Mech. Eng._ (1891), and on the friction of a
  collar bearing ib. May 1888.

  § 102. _Friction of Teeth._--Let N be the normal pressure exerted
  between a pair of teeth of a pair of wheels; s the total distance
  through which they slide upon each other; n the number of pairs of
  teeth which pass the plane of axis in a unit of time; then

    nfNs  (63)

  is the work lost in unity of time by the friction of the teeth. The
  sliding s is composed of two parts, which take place during the
  approach and recess respectively. Let those be denoted by s1 and s2,
  so that s = s1 + s2. In § 45 the _velocity_ of sliding at any instant
  has been given, viz. u = c ([alpha]1 + [alpha]2), where u is that
  velocity, c the distance T1 at any instant from the point of contact
  of the teeth to the pitch-point, and [alpha]1, [alpha]2 the respective
  angular velocities of the wheels.

  Let v be the common velocity of the two pitch-circles, r1, r2, their
  radii; then the above equation becomes

            / 1     1 \
    u = cv ( --- + --- ).
            \r1    r2 /

  To apply this to involute teeth, let c1 be the length of the approach,
  c2 that of the recess, u1, the _mean_ volocity of sliding during the
  approach, u2 that during the recess; then

         c1v  / 1     1 \        c2v  / 1     1 \
    u1 = --- ( --- + --- ); u2 = --- ( --- + --- )
          2   \r1    r2 /         2   \r1    r2 /

  also, let [theta] be the obliquity of the action; then the times
  occupied by the approach and recess are respectively

         c1              c2
    -------------,  -------------;
    v cos [theta]   v cos [theta]

  giving, finally, for the length of sliding between each pair of teeth,

                    c1² + c2²    / 1     1 \
    s = s1 + s2 = ------------- ( --- + --- )  (64)
                  2 cos [theta]  \r1    r2 /

  which, substituted in equation (63), gives the work lost in a unit of
  time by the friction of involute teeth. This result, which is exact
  for involute teeth, is approximately true for teeth of any figure.

  For inside gearing, if r1 be the less radius and r2 the greater, 1/r1
  - 1/r2 is to be substituted for 1/r1 + 1/r2.

  § 103. _Friction of Cords and Belts._--A flexible band, such as a
  cord, rope, belt or strap, may be used either to exert an effort or a
  resistance upon a pulley round which it wraps. In either case the
  tangential force, whether effort or resistance, exerted between the
  band and the pulley is their mutual friction, caused by and
  proportional to the normal pressure between them.

  Let T1 be the tension of the free part of the band at that side
  _towards_ which it tends to draw the pulley, or _from_ which the
  pulley tends to draw it; T2 the tension of the free part at the other
  side; T the tension of the band at any intermediate point of its arc
  of contact with the pulley; [theta] the ratio of the length of that
  arc to the radius of the pulley; d[theta] the ratio of an indefinitely
  small element of that arc to the radius; F = T1 - T2 the total
  friction between the band and the pulley; dF the elementary portion of
  that friction due to the elementary arc d[theta]; f the coefficient of
  friction between the materials of the band and pulley.

  Then, according to a well-known principle in statics, the normal
  pressure at the elementary arc d[theta] is Td[theta], T being the mean
  tension of the band at that elementary arc; consequently the friction
  on that arc is dF = fTd[theta]. Now that friction is also the
  difference between the tensions of the band at the two ends of the
  elementary arc, or dT = dF = fTd[theta]; which equation, being
  integrated throughout the entire arc of contact, gives the following
  formulae:--

             T1                                              \
    hyp log. -- = f^[theta]                                  |
             T2                                              |
                                                             |
    T1                                                        >  (65)
    -- = ef^[theta]                                          |
    T2                                                       |
                                                             |
    F = T1 - T2 = T1(1 - e - f^[theta]) = T2(ef^[theta] - 1) /

  When a belt connecting a pair of pulleys has the tensions of its two
  sides originally equal, the pulleys being at rest, and when the
  pulleys are next set in motion, so that one of them drives the other
  by means of the belt, it is found that the advancing side of the belt
  is exactly as much tightened as the returning side is slackened, so
  that the _mean_ tension remains unchanged. Its value is given by this
  formula--

    T1 + T2    ef^[theta] + 1
    ------- = -----------------  (66)
       2      2(ef^[theta] - 1)

  which is useful in determining the original tension required to enable
  a belt to transmit a given force between two pulleys.

  The equations 65 and 66 are applicable to a kind of _brake_ called a
  _friction-strap_, used to stop or moderate the velocity of machines by
  being tightened round a pulley. The strap is usually of iron, and the
  pulley of hard wood.

  Let [alpha] denote the arc of contact expressed in _turns and
  fractions of a turn_; then

    [theta] = 6.2832a                                       \  (67)
    ef^[theta] = number whose common logarithm is 2.7288fa  /

  See also DYNAMOMETER for illustrations of the use of what are
  essentially friction-straps of different forms for the measurement of
  the brake horse-power of an engine or motor.

  § 104. _Stiffness of Ropes._--Ropes offer a resistance to being bent,
  and, when bent, to being straightened again, which arises from the
  mutual friction of their fibres. It increases with the sectional area
  of the rope, and is inversely proportional to the radius of the curve
  into which it is bent.

  The _work lost_ in pulling a given length of rope over a pulley is
  found by multiplying the length of the rope in feet by its stiffness
  in pounds, that stiffness being the excess of the tension at the
  leading side of the rope above that at the following side, which is
  necessary to bend it into a curve fitting the pulley, and then to
  straighten it again.

  The following empirical formulae for the stiffness of hempen ropes
  have been deduced by Morin from the experiments of Coulomb:--

  Let F be the stiffness in pounds avoirdupois; d the diameter of the
  rope in inches, n = 48d² for white ropes and 35d² for tarred ropes; r
  the _effective_ radius of the pulley in inches; T the tension in
  pounds. Then

                          n                                  \
    For white ropes, F = --- (0.0012 + 0.001026n + 0.0012T)  |
                          r                                  |
                                                              >  (68)
                           n                                 |
    For tarred ropes, F = --- (0.006 + 0.001392n + 0.00168T) |
                           r                                 /

  § 105. _Friction-Couplings._--Friction is useful as a means of
  communicating motion where sudden changes either of force or velocity
  take place, because, being limited in amount, it may be so adjusted as
  to limit the forces which strain the pieces of the mechanism within
  the bounds of safety. Amongst contrivances for effecting this object
  are _friction-cones_. A rotating shaft carries upon a cylindrical
  portion of its figure a wheel or pulley turning loosely on it, and
  consequently capable of remaining at rest when the shaft is in motion.
  This pulley has fixed to one side, and concentric with it, a short
  frustum of a hollow cone. At a small distance from the pulley the
  shaft carries a short frustum of a solid cone accurately turned to fit
  the hollow cone. This frustum is made always to turn along with the
  shaft by being fitted on a square portion of it, or by means of a rib
  and groove, or otherwise, but is capable of a slight longitudinal
  motion, so as to be pressed into, or withdrawn from, the hollow cone
  by means of a lever. When the cones are pressed together or engaged,
  their friction causes the pulley to rotate along with the shaft; when
  they are disengaged, the pulley is free to stand still. The angle made
  by the sides of the cones with the axis should not be less than the
  angle of repose. In the _friction-clutch_, a pulley loose on a shaft
  has a hoop or gland made to embrace it more or less tightly by means
  of a screw; this hoop has short projecting arms or ears. A fork or
  _clutch_ rotates along with the shaft, and is capable of being moved
  longitudinally by a handle. When the clutch is moved towards the hoop,
  its arms catch those of the hoop, and cause the hoop to rotate and to
  communicate its rotation to the pulley by friction. There are many
  other contrivances of the same class, but the two just mentioned may
  serve for examples.

  § 106. _Heat of Friction: Unguents._--The work lost in friction is
  employed in producing heat. This fact is very obvious, and has been
  known from a remote period; but the _exact_ determination of the
  proportion of the work lost to the heat produced, and the experimental
  proof that that proportion is the same under all circumstances and
  with all materials, solid, liquid and gaseous, are comparatively
  recent achievements of J. P. Joule. The quantity of work which
  produces a British unit of heat (or so much heat as elevates the
  temperature of one pound of pure water, at or near ordinary
  atmospheric temperatures, by 1° F.) is 772 foot-pounds. This constant,
  now designated as "Joule's equivalent," is the principal experimental
  datum of the science of thermodynamics.

  A more recent determination (_Phil. Trans._, 1897), by Osborne
  Reynolds and W. M. Moorby, gives 778 as the mean value of Joule's
  equivalent through the range of 32° to 212° F. See also the papers of
  Rowland in the _Proc. Amer. Acad._ (1879), and Griffiths, _Phil.
  Trans._ (1893).

  The heat produced by friction, when moderate in amount, is useful in
  softening and liquefying thick unguents; but when excessive it is
  prejudicial, by decomposing the unguents, and sometimes even by
  softening the metal of the bearings, and raising their temperature so
  high as to set fire to neighbouring combustible matters.

  Excessive heating is prevented by a constant and copious supply of a
  good unguent. The elevation of temperature produced by the friction of
  a journal is sometimes used as an experimental test of the quality of
  unguents. For modern methods of forced lubrication see BEARINGS.

  § 107. _Rolling Resistance._--By the rolling of two surfaces over each
  other without sliding a resistance is caused which is called sometimes
  "rolling friction," but more correctly _rolling resistance_. It is of
  the nature of a _couple_, resisting rotation. Its _moment_ is found by
  multiplying the normal pressure between the rolling surfaces by an
  _arm_, whose length depends on the nature of the rolling surfaces, and
  the work lost in a unit of time in overcoming it is the product of its
  moment by the _angular velocity_ of the rolling surfaces relatively to
  each other. The following are approximate values of the arm in
  decimals of a foot:--

    Oak upon oak             0.006 (Coulomb).
    Lignum vitae on oak      0.004     "
    Cast iron on cast iron   0.002 (Tredgold).

  § 108. _Reciprocating Forces: Stored and Restored Energy._--When a
  force acts on a machine alternately as an effort and as a resistance,
  it may be called a _reciprocating force_. Of this kind is the weight
  of any piece in the mechanism whose centre of gravity alternately
  rises and falls; for during the rise of the centre of gravity that
  weight acts as a resistance, and energy is employed in lifting it to
  an amount expressed by the product of the weight into the vertical
  height of its rise; and during the fall of the centre of gravity the
  weight acts as an effort, and exerts in assisting to perform the work
  of the machine an amount of energy exactly equal to that which had
  previously been employed in lifting it. Thus that amount of energy is
  not lost, but has its operation deferred; and it is said to be
  _stored_ when the weight is lifted, and _restored_ when it falls.

  In a machine of which each piece is to move with a uniform velocity,
  if the effort and the resistance be constant, the weight of each piece
  must be balanced on its axis, so that it may produce lateral pressure
  only, and not act as a reciprocating force. But if the effort and the
  resistance be alternately in excess, the uniformity of speed may still
  be preserved by so adjusting some moving weight in the mechanism that
  when the effort is in excess it may be lifted, and so balance and
  employ the excess of effort, and that when the resistance is in excess
  it may fall, and so balance and overcome the excess of
  resistance--thus _storing_ the periodical excess of energy and
  _restoring_ that energy to perform the periodical excess of work.

  Other forces besides gravity may be used as reciprocating forces for
  storing and restoring energy--for example, the elasticity of a spring
  or of a mass of air.

  In most of the delusive machines commonly called "perpetual motions,"
  of which so many are patented in each year, and which are expected by
  their inventors to perform work without receiving energy, the
  fundamental fallacy consists in an expectation that some reciprocating
  force shall restore more energy than it has been the means of storing.


  _Division 2. Deflecting Forces._

  § 109. _Deflecting Force for Translation in a Curved Path._--In
  machinery, deflecting force is supplied by the tenacity of some piece,
  such as a crank, which guides the deflected body in its curved path,
  and is _unbalanced_, being employed in producing deflexion, and not in
  balancing another force.

  § 110. _Centrifugal Force of a Rotating Body._--_The centrifugal force
  exerted by a rotating body on its axis of rotation is the same in
  magnitude as if the mass of the body were concentrated at its centre
  of gravity, and acts in a plane passing through the axis of rotation
  and the centre of gravity of the body._

  The particles of a rotating body exert centrifugal forces on each
  other, which strain the body, and tend to tear it asunder, but these
  forces balance each other, and do not affect the resultant centrifugal
  force exerted on the axis of rotation.[3]

  _If the axis of rotation traverses the centre of gravity of the body,
  the centrifugal force exerted on that axis is nothing._

  Hence, unless there be some reason to the contrary, each piece of a
  machine should be balanced on its axis of rotation; otherwise the
  centrifugal force will cause strains, vibration and increased
  friction, and a tendency of the shafts to jump out of their bearings.

  § 111. _Centrifugal Couples of a Rotating Body._--Besides the tendency
  (if any) of the combined centrifugal forces of the particles of a
  rotating body to _shift_ the axis of rotation, they may also tend to
  _turn_ it out of its original direction. The latter tendency is called
  _a centrifugal couple_, and vanishes for rotation about a principal
  axis.

  It is essential to the steady motion of every rapidly rotating piece
  in a machine that its axis of rotation should not merely traverse its
  centre of gravity, but should be a permanent axis; for otherwise the
  centrifugal couples will increase friction, produce oscillation of the
  shaft and tend to make it leave its bearings.

  The principles of this and the preceding section are those which
  regulate the adjustment of the weight and position of the
  counterpoises which are placed between the spokes of the
  driving-wheels of locomotive engines.

  [Illustration: (From _Balancing of Engines_, by permission of Edward
  Arnold.)

  FIG. 130.]

  § 112.* _Method of computing the position and magnitudes of balance
  weights which must be added to a given system of arbitrarily chosen
  rotating masses in order to make the common axis of rotation a
  permanent axis._--The method here briefly explained is taken from a
  paper by W. E. Dalby, "The Balancing of Engines with special reference
  to Marine Work," _Trans. Inst. Nav. Arch._ (1899). Let the weight
  (fig. 130), attached to a truly turned disk, be rotated by the shaft
  OX, and conceive that the shaft is held in a bearing at one point, O.
  The force required to constrain the weight to move in a circle, that
  is the deviating force, produces an equal and opposite reaction on the
  shaft, whose amount F is equal to the centrifugal force Wa²r/g lb.,
  where r is the radius of the mass centre of the weight, and a is its
  angular velocity in radians per second. Transferring this force to the
  point O, it is equivalent to, (1) a force at O equal and parallel to
  F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX
  may be a permanent axis it is necessary that there should be a
  sufficient number of weights attached to the shaft and so distributed
  that when each is referred to the point O

    (1) [Sigma]F  = 0  \  (a)
    (2) [Sigma]Fa = 0  /

  The plane through O to which the shaft is perpendicular is called the
  _reference plane_, because all the transferred forces act in that
  plane at the point O. The plane through the radius of the weight
  containing the axis OX is called the _axial plane_ because it contains
  the forces forming the couple due to the transference of F to the
  reference plane. Substituting the values of F in (a) the two
  conditions become

                                   a²
    (1) (W1r1 + W2r2 + W3r3 + ...)--- = 0
                                   g
                                 a²         (b)
    (2) (W1a1r1 + W2a2r2 + ... )--- = 0
                                 g

  In order that these conditions may obtain, the quantities in the
  brackets must be zero, since the factor a²/g is not zero. Hence
  finally the conditions which must be satisfied by the system of
  weights in order that the axis of rotation may be a permanent axis is

    (1) (W1r1 + W2r2 + W3r3) = 0
    (2) (W1a1r1 + W2a2r2 + W3a3r3) = 0  (c)

  It must be remembered that these are all directed quantities, and that
  their respective sums are to be taken by drawing vector polygons. In
  drawing these polygons the magnitude of the vector of the type Wr is
  the product Wr, and the direction of the vector is from the shaft
  outwards towards the weight W, parallel to the radius r. For the
  vector representing a couple of the type War, if the masses are all on
  the same side of the reference plane, the direction of drawing is from
  the axis outwards; if the masses are some on one side of the reference
  plane and some on the other side, the direction of drawing is from the
  axis outwards towards the weight for all masses on the one side, and
  from the mass inwards towards the axis for all weights on the other
  side, drawing always parallel to the direction defined by the radius
  r. The magnitude of the vector is the product War. The conditions (c)
  may thus be expressed: first, that the sum of the vectors Wr must form
  a closed polygon, and, second, that the sum of the vectors War must
  form a closed polygon. The general problem in practice is, given a
  system of weights attached to a shaft, to find the respective weights
  and positions of two balance weights or counterpoises which must be
  added to the system in order to make the shaft a permanent axis, the
  planes in which the balance weights are to revolve also being given.
  To solve this the reference plane must be chosen so that it coincides
  with the plane of revolution of one of the as yet unknown balance
  weights. The balance weight in this plane has therefore no couple
  corresponding to it. Hence by drawing a couple polygon for the given
  weights the vector which is required to close the polygon is at once
  found and from it the magnitude and position of the balance weight
  which must be added to the system to balance the couples follow at
  once. Then, transferring the product Wr corresponding with this
  balance weight to the reference plane, proceed to draw the force
  polygon. The vector required to close it will determine the second
  balance weight, the work may be checked by taking the reference plane
  to coincide with the plane of revolution of the second balance weight
  and then re-determining them, or by taking a reference plane anywhere
  and including the two balance weights trying if condition (c) is
  satisfied.

  When a weight is reciprocated, the equal and opposite force required
  for its acceleration at any instant appears as an unbalanced force on
  the frame of the machine to which the weight belongs. In the
  particular case, where the motion is of the kind known as "simple
  harmonic" the disturbing force on the frame due to the reciprocation
  of the weight is equal to the component of the centrifugal force in
  the line of stroke due to a weight equal to the reciprocated weight
  supposed concentrated at the crank pin. Using this principle the
  method of finding the balance weights to be added to a given system of
  reciprocating weights in order to produce a system of forces on the
  frame continuously in equilibrium is exactly the same as that just
  explained for a system of revolving weights, because for the purpose
  of finding the balance weights each reciprocating weight may be
  supposed attached to the crank pin which operates it, thus forming an
  equivalent revolving system. The balance weights found as part of the
  equivalent revolving system when reciprocated by their respective
  crank pins form the balance weights for the given reciprocating
  system. These conditions may be exactly realized by a system of
  weights reciprocated by slotted bars, the crank shaft driving the
  slotted bars rotating uniformly. In practice reciprocation is usually
  effected through a connecting rod, as in the case of steam engines. In
  balancing the mechanism of a steam engine it is often sufficiently
  accurate to consider the motion of the pistons as simple harmonic, and
  the effect on the framework of the acceleration of the connecting rod
  may be approximately allowed for by distributing the weight of the rod
  between the crank pin and the piston inversely as the centre of
  gravity of the rod divides the distance between the centre of the
  cross head pin and the centre of the crank pin. The moving parts of
  the engine are then divided into two complete and independent systems,
  namely, one system of revolving weights consisting of crank pins,
  crank arms, &c., attached to and revolving with the crank shaft, and a
  second system of reciprocating weights consisting of the pistons,
  cross-heads, &c., supposed to be moving each in its line of stroke
  with simple harmonic motion. The balance weights are to be separately
  calculated for each system, the one set being added to the crank shaft
  as revolving weights, and the second set being included with the
  reciprocating weights and operated by a properly placed crank on the
  crank shaft. Balance weights added in this way to a set of
  reciprocating weights are sometimes called bob-weights. In the case of
  locomotives the balance weights required to balance the pistons are
  added as revolving weights to the crank shaft system, and in fact are
  generally combined with the weights required to balance the revolving
  system so as to form one weight, the counterpoise referred to in the
  preceding section, which is seen between the spokes of the wheels of a
  locomotive. Although this method balances the pistons in the
  horizontal plane, and thus allows the pull of the engine on the train
  to be exerted without the variation due to the reciprocation of the
  pistons, yet the force balanced horizontally is introduced vertically
  and appears as a variation of pressure on the rail. In practice about
  two-thirds of the reciprocating weight is balanced in order to keep
  this variation of rail pressure within safe limits. The assumption
  that the pistons of an engine move with simple harmonic motion is
  increasingly erroneous as the ratio of the length of the crank r, to
  the length of the connecting rod l increases. A more accurate though
  still approximate expression for the force on the frame due to the
  acceleration of the piston whose weight is W is given by

     W              /               r              \
    --- [omega]² r ( cos [theta] + --- cos 2[theta] )
     g              \               l              /

  The conditions regulating the balancing of a system of weights
  reciprocating under the action of accelerating forces given by the
  above expression are investigated in a paper by Otto Schlick, "On
  Balancing of Steam Engines," _Trans, Inst. Nav. Arch._ (1900), and in
  a paper by W. E. Dalby, "On the Balancing of the Reciprocating Parts
  of Engines, including the Effect of the Connecting Rod" (ibid., 1901).
  A still more accurate expression than the above is obtained by
  expansion in a Fourier series, regarding which and its bearing on
  balancing engines see a paper by J. H. Macalpine, "A Solution of the
  Vibration Problem" (ibid., 1901). The whole subject is dealt with in a
  treatise, _The Balancing of Engines_, by W. E. Dalby (London, 1906).
  Most of the original papers on this subject of engine balancing are to
  be found in the _Transactions_ of the Institution of Naval Architects.

  § 113.* _Centrifugal Whirling of Shafts._--When a system of revolving
  masses is balanced so that the conditions of the preceding section are
  fulfilled, the centre of gravity of the system lies on the axis of
  revolution. If there is the slightest displacement of the centre of
  gravity of the system from the axis of revolution a force acts on the
  shaft tending to deflect it, and varies as the deflexion and as the
  square of the speed. If the shaft is therefore to revolve stably, this
  force must be balanced at any instant by the elastic resistance of the
  shaft to deflexion. To take a simple case, suppose a shaft, supported
  on two bearings to carry a disk of weight W at its centre, and let the
  centre of gravity of the disk be at a distance e from the axis of
  rotation, this small distance being due to imperfections of material
  or faulty construction. Neglecting the mass of the shaft itself, when
  the shaft rotates with an angular velocity a, the centrifugal force
  Wa²e/g will act upon the shaft and cause its axis to deflect from the
  axis of rotation a distance, y say. The elastic resistance evoked by
  this deflexion is proportional to the deflexion, so that if c is a
  constant depending upon the form, material and method of support of
  the shaft, the following equality must hold if the shaft is to rotate
  stably at the stated speed--

     W
    ---(y + e)a² = cy,
     g

  from which y = Wa²e/(gc - Wa²).

  This expression shows that as a increases y increases until when Wa² =
  gc, y becomes infinitely large. The corresponding value of a, namely
  [root]gc/W, is called the _critical velocity_ of the shaft, and is the
  speed at which the shaft ceases to rotate stably and at which
  centrifugal whirling begins. The general problem is to find the value
  of a corresponding to all kinds of loadings on shafts supported in any
  manner. The question was investigated by Rankine in an article in the
  _Engineer_ (April 9, 1869). Professor A. G. Greenhill treated the
  problem of the centrifugal whirling of an unloaded shaft with
  different supporting conditions in a paper "On the Strength of
  Shafting exposed both to torsion and to end thrust," _Proc. Inst.
  Mech. Eng._ (1883). Professor S. Dunkerley ("On the Whirling and
  Vibration of Shafts," _Phil. Trans._, 1894) investigated the question
  for the cases of loaded and unloaded shafts, and, owing to the
  complication arising from the application of the general theory to the
  cases of loaded shafts, devised empirical formulae for the critical
  speeds of shafts loaded with heavy pulleys, based generally upon the
  following assumption, which is stated for the case of a shaft carrying
  one pulley: If N1, N2 be the separate speeds of whirl of the shaft and
  pulley on the assumption that the effect of one is neglected when that
  of the other is under consideration, then the resulting speed of whirl
  due to both causes combined may be taken to be of the form N1N2
  [root][(N²1 + N1²)] where N means revolutions per minute. This form is
  extended to include the cases of several pulleys on the same shaft.
  The interesting and important part of the investigation is that a
  number of experiments were made on small shafts arranged in different
  ways and loaded in different ways, and the speed at which whirling
  actually occurred was compared with the speed calculated from formulae
  of the general type indicated above. The agreement between the
  observed and calculated values of the critical speeds was in most
  cases quite remarkable. In a paper by Dr C. Chree, "The Whirling and
  Transverse Vibrations of Rotating Shafts," _Proc. Phys. Soc. Lon._,
  vol. 19 (1904); also _Phil. Mag._, vol. 7 (1904), the question is
  investigated from a new mathematical point of view, and expressions
  for the whirling of loaded shafts are obtained without the necessity
  of any assumption of the kind stated above. An elementary presentation
  of the problem from a practical point of view will be found in _Steam
  Turbines_, by Dr A. Stodola (London, 1905).

  [Illustration: FIG. 131.]

  § 114. _Revolving Pendulum. Governors._--In fig. 131 AO represents an
  upright axis or spindle; B a weight called a _bob_, suspended by rod
  OB from a horizontal axis at O, carried by the vertical axis. When the
  spindle is at rest the bob hangs close to it; when the spindle
  rotates, the bob, being made to revolve round it, diverges until the
  resultant of the centrifugal force and the weight of the bob is a
  force acting at O in the direction OB, and then it revolves steadily
  in a circle. This combination is called a _revolving_, _centrifugal_,
  or _conical pendulum_. Revolving pendulums are usually constructed
  with _pairs_ of rods and bobs, as OB, Ob, hung at opposite sides of
  the spindle, that the centrifugal forces exerted at the point O may
  balance each other.

  In finding the position in which the bob will revolve with a given
  angular velocity, a, for most practical cases connected with machinery
  the mass of the rod may be considered as insensible compared with that
  of the bob. Let the bob be a sphere, and from the centre of that
  sphere draw BH = y perpendicular to OA. Let OH = z; let W be the
  weight of the bob, F its centrifugal force. Then the condition of its
  steady revolution is W : F :: z : y; that is to say, y/z = F/W =
  ya²/g; consequently

    z = g/[alpha]² (69)

  Or, if n = [alpha] 2[pi] = [alpha]/6.2832 be the number of turns or
  fractions of a turn in a second,

           g       0.8165 ft.   9.79771 in.  \
    z = -------- = ---------- = -----------   >  (70)
        4[pi]²n²       n²            n²      /

  z is called the _altitude of the pendulum_.

  [Illustration: FIG. 132.]

  If the rod of a revolving pendulum be jointed, as in fig. 132, not to
  a point in the vertical axis, but to the end of a projecting arm C,
  the position in which the bob will revolve will be the same as if the
  rod were jointed to the point O, where its prolongation cuts the
  vertical axis.

  A revolving pendulum is an essential part of most of the contrivances
  called _governors_, for regulating the speed of prime movers, for
  further particulars of which see STEAM ENGINE.


  _Division 3. Working of Machines of Varying Velocity._

  § 115. _General Principles._--In order that the velocity of every
  piece of a machine may be uniform, it is necessary that the forces
  acting on each piece should be always exactly balanced. Also, in order
  that the forces acting on each piece of a machine may be always
  exactly balanced, it is necessary that the velocity of that piece
  should be uniform.

  An excess of the effort exerted on any piece, above that which is
  necessary to balance the resistance, is accompanied with acceleration;
  a deficiency of the effort, with retardation.

  When a machine is being started from a state of rest, and brought by
  degrees up to its proper speed, the effort must be in excess; when it
  is being retarded for the purpose of stopping it, the resistance must
  be in excess.

  An excess of effort above resistance involves an excess of energy
  exerted above work performed; that excess of energy is employed in
  producing acceleration.

  An excess of resistance above effort involves an excess of work
  performed above energy expended; that excess of work is performed by
  means of the retardation of the machinery.

  When a machine undergoes alternate acceleration and retardation, so
  that at certain instants of time, occurring at the end of intervals
  called _periods_ or _cycles_, it returns to its original speed, then
  in each of those periods or cycles the alternate excesses of energy
  and of work neutralize each other; and at the end of each cycle the
  principle of the equality of energy and work stated in § 87, with all
  its consequences, is verified exactly as in the case of machines of
  uniform speed.

  At intermediate instants, however, other principles have also to be
  taken into account, which are deduced from the second law of motion,
  as applied to _direct deviation_, or acceleration and retardation.

  § 116. _Energy of Acceleration and Work of Retardation for a Shifting
  Body._--Let w be the weight of a body which has a motion of
  translation in any path, and in the course of the interval of time
  [Delta]t let its velocity be increased at a uniform rate of
  acceleration from v1 to v2. The rate of acceleration will be

    dv/dt = const. = (v2 - v1)[Delta]t;

  and to produce this acceleration a uniform effort will be required,
  expressed by

    P = w(v2 - v1)g[Delta]t  (71)

  (The product wv/g of the mass of a body by its velocity is called its
  _momentum_; so that the effort required is found by dividing the
  increase of momentum by the time in which it is produced.)

  To find the _energy_ which has to be exerted to produce the
  acceleration from v1 to v2, it is to be observed that the _distance_
  through which the effort P acts during the acceleration is

    [Delta]s = (v2 + v1)[Delta]t/2;

  consequently, the _energy of acceleration_ is

    P[Delta]s = w(v2 - v1) (v2 + v1)/2g = w(v2² - v1²)2g,  (72)

  being proportional to the increase in the square of the velocity, and
  _independent of the time_.

  In order to produce a _retardation_ from the greater velocity v2 to
  the less velocity v1, it is necessary to apply to the body a
  _resistance_ connected with the retardation and the time by an
  equation identical in every respect with equation (71), except by the
  substitution of a resistance for an effort; and in overcoming that
  resistance the body _performs work_ to an amount determined by
  equation (72), putting Rds for Pas.

  § 117. _Energy Stored and Restored by Deviations of Velocity._--Thus a
  body alternately accelerated and retarded, so as to be brought back to
  its original speed, performs work during its retardation exactly equal
  in amount to the energy exerted upon it during its acceleration; so
  that that energy may be considered as _stored_ during the
  acceleration, and _restored_ during the retardation, in a manner
  analogous to the operation of a reciprocating force (§ 108).

  Let there be given the mean velocity V = ½(v2 + v1) of a body whose
  weight is w, and let it be required to determine the fluctuation of
  velocity v2 - v1, and the extreme velocities v1, v2, which that body
  must have, in order alternately to store and restore an amount of
  energy E. By equation (72) we have

    E = w(v2² - v1²)´2g

  which, being divided by V = ½(v2 + v1), gives

    E/V = w(v2 - v1)/g;

  and consequently

    v2 - v1 = gE/Vw  (73)

  The ratio of this fluctuation to the mean velocity, sometimes called
  the unsteadiness of the motion of the body, is

    (v2 - v1)V = gE/V²w.  (74)

  § 118. _Actual Energy of a Shifting Body._--The energy which must be
  exerted on a body of the weight w, to accelerate it from a state of
  rest up to a given velocity of translation v, and the equal amount of
  work which that body is capable of performing by overcoming resistance
  while being retarded from the same velocity of translation v to a
  state of rest, is

    wv²/2g.  (75)

  This is called the _actual energy_ of the motion of the body, and is
  half the quantity which in some treatises is called vis viva.

  The energy stored or restored, as the case may be, by the deviations
  of velocity of a body or a system of bodies, is the amount by which
  the actual energy is increased or diminished.

  § 119. _Principle of the Conservation of Energy in Machines._--The
  following principle, expressing the general law of the action of
  machines with a velocity uniform or varying, includes the law of the
  equality of energy and work stated in § 89 for machines of uniform
  speed.

  _In any given interval during the working of a machine, the energy
  exerted added to the energy restored is equal to the energy stored
  added to the work performed._

  § 120. _Actual Energy of Circular Translation--Moment of
  Inertia._--Let a small body of the weight w undergo translation in a
  circular path of the radius [rho], with the angular velocity of
  deflexion [alpha], so that the common linear velocity of all its
  particles is v = [alpha][rho]. Then the actual energy of that body is

    wv²/2g = w[alpha]²p²/2g.  (76)

  By comparing this with the expression for the centrifugal force
  (w[alpha]²p/g), it appears that the actual energy of a revolving body
  is equal to the potential energy Fp/2 due to the action of the
  deflecting force along one-half of the radius of curvature of the path
  of the body.

  The product wp²/g, by which the half-square of the angular velocity is
  multiplied, is called the _moment of inertia_ of the revolving body.

  § 121. _Flywheels._--A flywheel is a rotating piece in a machine,
  generally shaped like a wheel (that is to say, consisting of a rim
  with spokes), and suited to store and restore energy by the periodical
  variations in its angular velocity.

  The principles according to which variations of angular velocity store
  and restore energy are the same as those of § 117, only substituting
  _moment of inertia_ for _mass_, and _angular_ for _linear_ velocity.

  Let W be the weight of a flywheel, R its radius of gyration, a2 its
  maximum, a1 its minimum, and A = ½([alpha]2 + [alpha]1) its mean
  angular velocity. Let

    I/S  = ([alpha]2 - [alpha]2)/A

  denote the _unsteadiness_ of the motion of the flywheel; the
  denominator S of this fraction is called the _steadiness_. Let e
  denote the quantity by which the energy exerted in each cycle of the
  working of the machine alternately exceeds and falls short of the work
  performed, and which has consequently to be alternately stored by
  acceleration and restored by retardation of the flywheel. The value of
  this _periodical excess_ is--

    e = R²W ([alpha]2² - [alpha]1²), 2g,  (77)

  from which, dividing both sides by A², we obtain the following
  equations:--

    e/A² = R²W/gS     \
                       >.   (78)
    R²WA²/2g = Se/2   /

  The latter of these equations may be thus expressed in words: _The
  actual energy due to the rotation of the fly, with its mean angular
  velocity, is equal to one-half of the periodical excess of energy
  multiplied by the steadiness._

  In ordinary machinery S = about 32; in machinery for fine purposes S =
  from 50 to 60; and when great steadiness is required S = from 100 to
  150.

  The periodical excess e may arise either from variations in the effort
  exerted by the prime mover, or from variations in the resistance of
  the work, or from both these causes combined. When but one flywheel is
  used, it should be placed in as direct connexion as possible with that
  part of the mechanism where the greatest amount of the periodical
  excess originates; but when it originates at two or more points, it is
  best to have a flywheel in connexion with each of these points. For
  example, in a machine-work, the steam-engine, which is the prime mover
  of the various tools, has a flywheel on the crank-shaft to store and
  restore the periodical excess of energy arising from the variations in
  the effort exerted by the connecting-rod upon the crank; and each of
  the slotting machines, punching machines, riveting machines, and other
  tools has a flywheel of its own to store and restore energy, so as to
  enable the very different resistances opposed to those tools at
  different times to be overcome without too great unsteadiness of
  motion. For tools performing useful work at intervals, and having only
  their own friction to overcome during the intermediate intervals, e
  should be assumed equal to the whole work performed at each separate
  operation.

  § 122. _Brakes._--A brake is an apparatus for stopping and diminishing
  the velocity of a machine by friction, such as the friction-strap
  already referred to in § 103. To find the distance s through which a
  brake, exerting the friction F, must rub in order to stop a machine
  having the total actual energy E at the moment when the brake begins
  to act, reduce, by the principles of § 96, the various efforts and
  other resistances of the machine which act at the same time with the
  friction of the brake to the rubbing surface of the brake, and let R
  be their resultant--positive if _resistance_, _negative_ if effort
  preponderates. Then

    s = E/(F + R).  (79)

  § 123. _Energy distributed between two Bodies: Projection and
  Propulsion._--Hitherto the effort by which a machine is moved has been
  treated as a force exerted between a movable body and a fixed body, so
  that the whole energy exerted by it is employed upon the movable body,
  and none upon the fixed body. This conception is sensibly realized in
  practice when one of the two bodies between which the effort acts is
  either so heavy as compared with the other, or has so great a
  resistance opposed to its motion, that it may, without sensible error,
  be treated as fixed. But there are cases in which the motions of both
  bodies are appreciable, and must be taken into account--such as the
  projection of projectiles, where the velocity of the _recoil_ or
  backward motion of the gun bears an appreciable proportion to the
  forward motion of the projectile; and such as the propulsion of
  vessels, where the velocity of the water thrown backward by the
  paddle, screw or other propeller bears a very considerable proportion
  to the velocity of the water moved forwards and sideways by the ship.
  In cases of this kind the energy exerted by the effort is
  _distributed_ between the two bodies between which the effort is
  exerted in shares proportional to the velocities of the two bodies
  during the action of the effort; and those velocities are to each
  other directly as the portions of the effort unbalanced by resistance
  on the respective bodies, and inversely as the weights of the bodies.

  To express this symbolically, let W1, W2 be the weights of the bodies;
  P the effort exerted between them; S the distance through which it
  acts; R1, R2 the resistances opposed to the effort overcome by W1, W2
  respectively; E1, E2 the shares of the whole energy E exerted upon W1,
  W2 respectively. Then

                         E     :   E1    :   E2     \
       W2(P - R1) + W1(P - R2)   P - R1    P - R2   |
    :: ----------------------- : ------  : ------    >.  (80)
                 W1W2              W1        W2     /

  If R1 = R2, which is the case when the resistance, as well as the
  effort, arises from the mutual actions of the two bodies, the above
  becomes,

             E : E1 : E2  \
    :: W1 + W2 : W2 : W1  /,  (81)

  that is to say, the energy is exerted on the bodies in shares
  inversely proportional to their weights; and they receive
  accelerations inversely proportional to their weights, according to
  the principle of dynamics, already quoted in a note to § 110, that the
  mutual actions of a system of bodies do not affect the motion of their
  common centre of gravity.

  For example, if the weight of a gun be 160 times that of its ball
  160/161 of the energy exerted by the powder in exploding will be
  employed in propelling the ball, and 1/161 in producing the recoil of
  the gun, provided the gun up to the instant of the ball's quitting the
  muzzle meets with no resistance to its recoil except the friction of
  the ball.

  § 124. _Centre of Percussion._--It is obviously desirable that the
  deviations or changes of motion of oscillating pieces in machinery
  should, as far as possible, be effected by forces applied at their
  centres of percussion.

  If the deviation be a _translation_--that is, an equal change of
  motion of all the particles of the body--the centre of percussion is
  obviously the centre of gravity itself; and, according to the second
  law of motion, if dv be the deviation of velocity to be produced in
  the interval dt, and W the weight of the body, then

         W    dv
    P = --- · --  (82)
         g    dt

  is the unbalanced effort required.

  If the deviation be a rotation about an axis traversing the centre of
  gravity, there is no centre of percussion; for such a deviation can
  only be produced by a _couple_ of forces, and not by any single force.
  Let d[alpha] be the deviation of angular velocity to be produced in
  the interval dt, and I the moment of the inertia of the body about an
  axis through its centre of gravity; then ½Id([alpha]^2) = I[alpha]
  d[alpha] is the variation of the body's actual energy. Let M be the
  moment of the unbalanced couple required to produce the deviation;
  then by equation 57, § 104, the energy exerted by this couple in the
  interval dt is M[alpha] dt, which, being equated to the variation of
  energy, gives

         d[alpha]   R²W   d[alpha]
    M = I-------- = --- · --------.  (83)
            dt       g       dt

  R is called the radius of gyration of the body with regard to an axis
  through its centre of gravity.

  [Illustration: FIG. 133.]

  Now (fig. 133) let the required deviation be a rotation of the body BB
  about an axis O, not traversing the centre of gravity G, d[alpha]
  being, as before, the deviation of angular velocity to be produced in
  the interval dt. A rotation with the angular velocity [alpha] about an
  axis O may be considered as compounded of a rotation with the same
  angular velocity about an axis drawn through G parallel to O and a
  translation with the velocity [alpha]. OG, OG being the perpendicular
  distance between the two axes. Hence the required deviation may be
  regarded as compounded of a deviation of translation dv = OG·d[alpha],
  to produce which there would be required, according to equation (82),
  a force applied at G perpendicular to the plane OG--

         W         d[alpha]
    P = --- · OG · --------  (84)
         g            dt

  and a deviation d[alpha] of rotation about an axis drawn through G
  parallel to O, to produce which there would be required a couple of
  the moment M given by equation (83). According to the principles of
  statics, the resultant of the force P, applied at G perpendicular to
  the plane OG, and the couple M is a force equal and parallel to P, but
  applied at a distance GC from G, in the prolongation of the
  perpendicular OG, whose value is

    GC = M/P = R²/OG.  (85)

  Thus is determined the position of the centre of percussion C,
  corresponding to the axis of rotation O. It is obvious from this
  equation that, for an axis of rotation parallel to O traversing C, the
  centre of percussion is at the point where the perpendicular OG meets
  O.

  § 125.* _To find the moment of inertia of a body about an axis through
  its centre of gravity experimentally._--Suspend the body from any
  conveniently selected axis O (fig. 48) and hang near it a small plumb
  bob. Adjust the length of the plumb-line until it and the body
  oscillate together in unison. The length of the plumb-line, measured
  from its point of suspension to the centre of the bob, is for all
  practical purposes equal to the length OC, C being therefore the
  centre of percussion corresponding to the selected axis O. From
  equation (85)

    R^2 = CG × OG = (OC - OG)OG.

  The position of G can be found experimentally; hence OG is known, and
  the quantity R² can be calculated, from which and the ascertained
  weight W of the body the moment of inertia about an axis through G,
  namely, W/g × R², can be computed.

  [Illustration: FIG. 134.]

  § 126.* _To find the force competent to produce the instantaneous
  acceleration of any link of a mechanism._--In many practical problems
  it is necessary to know the magnitude and position of the forces
  acting to produce the accelerations of the several links of a
  mechanism. For a given link, this force is the resultant of all the
  accelerating forces distributed through the substance of the material
  of the link required to produce the requisite acceleration of each
  particle, and the determination of this force depends upon the
  principles of the two preceding sections. The investigation of the
  distribution of the forces through the material and the stress
  consequently produced belongs to the subject of the STRENGTH OF
  MATERIALS (q.v.). Let BK (fig. 134) be any link moving in any manner
  in a plane, and let G be its centre of gravity. Then its motion may be
  analysed into (1) a translation of its centre of gravity; and (2) a
  rotation about an axis through its centre of gravity perpendicular to
  its plane of motion. Let [alpha] be the acceleration of the centre of
  gravity and let A be the angular acceleration about the axis through
  the centre of gravity; then the force required to produce the
  translation of the centre of gravity is F = W[alpha]/g, and the couple
  required to produce the angular acceleration about the centre of
  gravity is M = IA/g, W and I being respectively the weight and the
  moment of inertia of the link about the axis through the centre of
  gravity. The couple M may be produced by shifting the force F parallel
  to itself through a distance x. such that Fx = M. When the link forms
  part of a mechanism the respective accelerations of two points in the
  link can be determined by means of the velocity and acceleration
  diagrams described in § 82, it being understood that the motion of one
  link in the mechanism is prescribed, for instance, in the
  steam-engine's mechanism that the crank shall revolve uniformly. Let
  the acceleration of the two points B and K therefore be supposed
  known. The problem is now to find the acceleration [alpha] and A. Take
  any pole O (fig. 49), and set out Ob equal to the acceleration of B
  and Ok equal to the acceleration of K. Join bk and take the point g so
  that KG: GB = kg : gb. Og is then the acceleration of the centre of
  gravity and the force F can therefore be immediately calculated. To
  find the angular acceleration A, draw kt, bt respectively parallel to
  and at right angles to the link KB. Then tb represents the angular
  acceleration of the point B relatively to the point K and hence tb/KB
  is the value of A, the angular acceleration of the link. Its moment of
  inertia about G can be found experimentally by the method explained in
  § 125, and then the value of the couple M can be computed. The value
  of x is found immediately from the quotient M/F. Hence the magnitude F
  and the position of F relatively to the centre of gravity of the link,
  necessary to give rise to the couple M, are known, and this force is
  therefore the resultant force required.

  [Illustration: FIG. 135.]

  § 127.* _Alternative construction for finding the position of F
  relatively to the centre of gravity of the link._--Let B and K be any
  two points in the link which for greater generality are taken in fig.
  135, so that the centre of gravity G is not in the line joining them.
  First find the value of R experimentally. Then produce the given
  directions of acceleration of B and K to meet in O; draw a circle
  through the three points B, K and O; produce the line joining O and G
  to cut the circle in Y; and take a point Z on the line OY so that YG ×
  GZ = R². Then Z is a point in the line of action of the force F. This
  useful theorem is due to G. T. Bennett, of Emmanuel College,
  Cambridge. A proof of it and three corollaries are given in appendix 4
  of the second edition of Dalby's _Balancing of Engines_ (London,
  1906). It is to be noticed that only the directions of the
  accelerations of two points are required to find the point Z.

  For an example of the application of the principles of the two
  preceding sections to a practical problem see _Valve and Valve Gear
  Mechanisms_, by W. E. Dalby (London, 1906), where the inertia stresses
  brought upon the several links of a Joy valve gear, belonging to an
  express passenger engine of the Lancashire & Yorkshire railway, are
  investigated for an engine-speed of 68 m. an hour.

  [Illustration: FIG. 136.]

  § 128.* _The Connecting Rod Problem._--A particular problem of
  practical importance is the determination of the force producing the
  motion of the connecting rod of a steam-engine mechanism of the usual
  type. The methods of the two preceding sections may be used when the
  acceleration of two points in the rod are known. In this problem it is
  usually assumed that the crank pin K (fig. 136) moves with uniform
  velocity, so that if [alpha] is its angular velocity and r its radius,
  the acceleration is [alpha]²r in a direction along the crank arm from
  the crank pin to the centre of the shaft. Thus the acceleration of one
  point K is known completely. The acceleration of a second point,
  usually taken at the centre of the crosshead pin, can be found by the
  principles of § 82, but several special geometrical constructions have
  been devised for this purpose, notably the construction of Klein,[4]
  discovered also independently by Kirsch.[5] But probably the most
  convenient is the construction due to G. T. Bennett[6] which is as
  follows: Let OK be the crank and KB the connecting rod. On the
  connecting rod take a point L such that KL × KB = KO². Then, the crank
  standing at any angle with the line of stroke, draw LP at right angles
  to the connecting rod, PN at right angles to the line of stroke OB and
  NA at right angles to the connecting rod; then AO is the acceleration
  of the point B to the scale on which KO represents the acceleration of
  the point K. The proof of this construction is given in _The Balancing
  of Engines_.

  The finding of F may be continued thus: join AK, then AK is the
  acceleration image of the rod, OKA being the acceleration diagram.
  Through G, the centre of gravity of the rod, draw Gg parallel to the
  line of stroke, thus dividing the image at g in the proportion that
  the connecting rod is divided by G. Hence Og represents the
  acceleration of the centre of gravity and, the weight of the
  connecting rod being ascertained, F can be immediately calculated. To
  find a point in its line of action, take a point Q on the rod such
  that KG × GQ = R², R having been determined experimentally by the
  method of § 125; join G with O and through Q draw a line parallel to
  BO to cut GO in Z. Z is a point in the line of action of the resultant
  force F; hence through Z draw a line parallel to Og. The force F acts
  in this line, and thus the problem is completely solved. The above
  construction for Z is a corollary of the general theorem given in §
  127.

  § 129. _Impact._ Impact or collision is a pressure of short duration
  exerted between two bodies.

  The effects of impact are sometimes an alteration of the distribution
  of actual energy between the two bodies, and always a loss of a
  portion of that energy, depending on the imperfection of the
  elasticity of the bodies, in permanently altering their figures, and
  producing heat. The determination of the distribution of the actual
  energy after collision and of the loss of energy is effected by means
  of the following principles:--

  I. The motion of the common centre of gravity of the two bodies is
  unchanged by the collision.

  II. The loss of energy consists of a certain proportion of that part
  of the actual energy of the bodies which is due to their motion
  relatively to their common centre of gravity.

  Unless there is some special reason for using impact in machines, it
  ought to be avoided, on account not only of the waste of energy which
  it causes, but from the damage which it occasions to the frame and
  mechanism.     (W. J. M. R.; W. E. D.)


FOOTNOTES:

  [1] In view of the great authority of the author, the late Professor
    Macquorn Rankine, it has been thought desirable to retain the greater
    part of this article as it appeared in the 9th edition of the
    _Encyclopaedia Britannica_. Considerable additions, however, have
    been introduced in order to indicate subsequent developments of the
    subject; the new sections are numbered continuously with the old, but
    are distinguished by an asterisk. Also, two short chapters which
    concluded the original article have been omitted--ch. iii., "On
    Purposes and Effects of Machines," which was really a classification
    of machines, because the classification of Franz Reuleaux is now
    usually followed, and ch. iv., "Applied Energetics, or Theory of
    Prime Movers," because its subject matter is now treated in various
    special articles, e.g. Hydraulics, Steam Engine, Gas Engine, Oil
    Engine, and fully developed in Rankine's The Steam Engine and Other
    Prime Movers (London, 1902). (Ed. _E.B._)

  [2] Since the relation discussed in § 7 was enunciated by Rankine, an
    enormous development has taken place in the subject of Graphic
    Statics, the first comprehensive textbook on the subject being _Die
    Graphische Statik_ by K. Culmann, published at Zürich in 1866. Many
    of the graphical methods therein given have now passed into the
    textbooks usually studied by engineers. One of the most beautiful
    graphical constructions regularly used by engineers and known as "the
    method of reciprocal figures" is that for finding the loads supported
    by the several members of a braced structure, having given a system
    of external loads. The method was discovered by Clerk Maxwell, and
    the complete theory is discussed and exemplified in a paper "On
    Reciprocal Figures, Frames and Diagrams of Forces," _Trans. Roy. Soc.
    Ed._, vol. xxvi. (1870). Professor M. W. Crofton read a paper on
    "Stress-Diagrams in Warren and Lattice Girders" at the meeting of the
    Mathematical Society (April 13, 1871), and Professor O. Henrici
    illustrated the subject by a simple and ingenious notation. The
    application of the method of reciprocal figures was facilitated by a
    system of notation published in _Economics of Construction in
    relation to framed Structures_, by Robert H. Bow (London, 1873). A
    notable work on the general subject is that of Luigi Cremona,
    translated from the Italian by Professor T. H. Beare (Oxford, 1890),
    and a discussion of the subject of reciprocal figures from the
    special point of view of the engineering student is given in _Vectors
    and Rotors_ by Henrici and Turner (London, 1903). See also above
    under "_Theoretical Mechanics_," Part 1. § 5.

  [3] This is a particular case of a more general principle, that _the
    motion of the centre of gravity of a body is not affected by the
    mutual actions of its parts_.

  [4] J. F. Klein, "New Constructions of the Force of Inertia of
    Connecting Rods and Couplers and Constructions of the Pressures on
    their Pins," _Journ. Franklin Inst._, vol. 132 (Sept. and Oct.,
    1891).

  [5] Prof. Kirsch, "Über die graphische Bestimmung der
    Kolbenbeschleunigung," _Zeitsch. Verein deutsche Ingen_. (1890), p.
    1320.

  [6] Dalby, _The Balancing of Engines_ (London, 1906), app. 1.




MECHANICVILLE, a village of Saratoga county, New York, U.S.A., on the
west bank of the Hudson River, about 20 m. N. of Albany; on the Delaware
& Hudson and Boston & Maine railways. Pop. (1900), 4695 (702
foreign-born); (1905, state census), 5877; (1910) 6,634. It lies partly
within Stillwater and partly within Half-Moon townships, in the
bottom-lands at the mouth of the Anthony Kill, about 1-1/2 m. S. of the
mouth of the Hoosick River. On the north and south are hills reaching a
maximum height of 200 ft. There is ample water power, and there are
manufactures of paper, sash and blinds, fibre, &c. From a dam here power
is derived for the General Electric Company at Schenectady. The first
settlement in this vicinity was made in what is now Half-Moon township
about 1680. Mechanicville (originally called Burrow) was chartered by
the county court in 1859, and incorporated as a village in 1870. It was
the birthplace of Colonel Ephraim Elmer Ellsworth (1837-1861), the first
Federal officer to lose his life in the Civil War.




MECHITHARISTS, a congregation of Armenian monks in communion with the
Church of Rome. The founder, Mechithar, was born at Sebaste in Armenia,
1676. He entered a monastery, but under the influence of Western
missionaries he became possessed with the idea of propagating Western
ideas and culture in Armenia, and of converting the Armenian Church from
its monophysitism and uniting it to the Latin Church. Mechithar set out
for Rome in 1695 to make his ecclesiastical studies there, but he was
compelled by illness to abandon the journey and return to Armenia. In
1696 he was ordained priest and for four years worked among his people.
In 1700 he went to Constantinople and began to gather disciples around
him. Mechithar formally joined the Latin Church, and in 1701, with
sixteen companions, he formed a definitely religious institute of which
he became the superior. Their Uniat propaganda encountered the
opposition of the Armenians and they were compelled to move to the
Morea, at that time Venetian territory, and there built a monastery,
1706. On the outbreak of hostilities between the Turks and Venetians
they migrated to Venice, and the island of St Lazzaro was bestowed on
them, 1717. This has since been the headquarters of the congregation,
and here Mechithar died in 1749, leaving his institute firmly
established. The rule followed at first was that attributed to St
Anthony; but when they settled in the West modifications from the
Benedictine rule were introduced, and the Mechitharists are numbered
among the lesser orders affiliated to the Benedictines. They have ever
been faithful to their founder's programme. Their work has been
fourfold: (1) they have brought out editions of important patristic
works, some Armenian, others translated into Armenian from Greek and
Syriac originals no longer extant; (2) they print and circulate Armenian
literature among the Armenians, and thereby exercise a powerful
educational influence; (3) they carry on schools both in Europe and
Asia, in which Uniat Armenian boys receive a good secondary education;
(4) they work as Uniat missioners in Armenia. The congregation is
divided into two branches, the head houses being at St Lazzaro and
Vienna. They have fifteen establishments in various places in Asia Minor
and Europe. There are some 150 monks, all Armenians; they use the
Armenian language and rite in the liturgy.

  See _Vita del servo di Dio Mechitar_ (Venice, 1901); E. Boré,
  _Saint-Lazare_ (1835); Max Heimbucher, _Orden u. Kongregationen_
  (1907) I. § 37; and the articles in Wetzer u. Welte, _Kirchenlexicon_
  (ed. 2) and Herzog, _Realencyklopädie_ (ed. 3), also articles by
  Sargisean, a Mechitharist, in _Rivista storica benedettina_ (1906),
  "La Congregazione Mechitarista."     (E. C. B.)




MECKLENBURG, a territory in northern Germany, on the Baltic Sea,
extending from 53° 4´ to 54° 22´ N. and from 10° 35´ to 13° 57´ E.,
unequally divided into the two grand duchies of Mecklenburg-Schwerin and
Mecklenburg-Strelitz.

MECKLENBURG-SCHWERIN is bounded N. by the Baltic Sea, W. by the
principality of Ratzeburg and Schleswig-Holstein, S. by Brandenburg and
Hanover, and E. by Pomerania and Mecklenburg-Strelitz. It embraces the
duchies of Schwerin and Güstrow, the district of Rostock, the
principality of Schwerin, and the barony of Wismar, besides several
small enclaves (Ahrensberg, Rosson, Tretzeband, &c.) in the adjacent
territories. Its area is 5080 sq. m. Pop. (1905), 625,045.

MECKLENBURG-STRELITZ consists of two detached parts, the duchy of
Strelitz on the E. of Mecklenburg-Schwerin, and the principality of
Ratzeburg on the W. The first is bounded by Mecklenburg-Schwerin,
Pomerania and Brandenburg, the second by Mecklenburg-Schwerin,
Lauenburg, and the territory of the free town of Lübeck. Their joint
area is 1130 sq. m. Pop. (1905), 103,451.

  Mecklenburg lies wholly within the great North-European plain, and its
  flat surface is interrupted only by one range of low hills,
  intersecting the country from south-east to north-west, and forming
  the watershed between the Baltic Sea and the Elbe. Its highest point,
  the Helpter Berg, is 587 ft. above sea-level. The coast-line runs for
  65 m. along the Baltic (without including indentations), for the most
  part in flat sandy stretches covered with dunes. The chief inlets are
  Wismar Bay, the Salzhaff, and the roads of Warnemünde. The rivers are
  numerous though small; most of them are affluents of the Elbe, which
  traverses a small portion of Mecklenburg. Several are navigable, and
  the facilities for inland water traffic are increased by canals. Lakes
  are numerous; about four hundred, covering an area of 500 sq. m., are
  reckoned in the two duchies. The largest is Lake Müritz, 52 sq. m. in
  extent. The climate resembles that of Great Britain, but the winters
  are generally more severe; the mean annual temperature is 48° F., and
  the annual rainfall is about 28 in. Although there are long stretches
  of marshy moorland along the coast, the soil is on the whole
  productive. About 57% of the total area of Mecklenburg-Schwerin
  consists of cultivated land, 18% of forest, and 13% of heath and
  pasture. In Mecklenburg-Strelitz the corresponding figures are 47, 21
  and 10%. Agriculture is by far the most important industry in both
  duchies. The chief crops are rye, oats, wheat, potatoes and hay.
  Smaller areas are devoted to maize, buckwheat, pease, rape, hemp,
  flax, hops and tobacco. The extensive pastures support large herds of
  sheep and cattle, including a noteworthy breed of merino sheep. The
  horses of Mecklenburg are of a fine sturdy quality and highly
  esteemed. Red deer, wild swine and various other game are found in the
  forests. The industrial establishments include a few iron-foundries,
  wool-spinning mills, carriage and machine factories, dyeworks,
  tanneries, brick-fields, soap-works, breweries, distilleries, numerous
  limekilns and tar-boiling works, tobacco and cigar factories, and
  numerous mills of various kinds. Mining is insignificant, though a
  fair variety of minerals is represented in the district. Amber is
  found on and near the Baltic coast. Rostock, Warnemünde and Wismar are
  the principal commercial centres. The chief exports are grain and
  other agricultural produce, live stock, spirits, wood and wool; the
  chief imports are colonial produce, iron, coal, salt, wine, beer and
  tobacco. The horse and wool markets of Mecklenburg are largely
  attended by buyers from various parts of Germany. Fishing is carried
  on extensively in the numerous inland lakes.

  In 1907 the grand dukes of both duchies promised a constitution to
  their subjects. The duchies had always been under a government of
  feudal character, the grand dukes having the executive entirely in
  their hands (though acting through ministers), while the duchies
  shared a diet (_Landtag_), meeting for a short session each year, and
  at other times represented by a committee, and consisting of the
  proprietors of knights' estates (_Rittergüter_), known as the
  _Ritterschaft_, and the _Landschaft_ or burgomasters of certain towns.
  Mecklenburg-Schwerin returns six members to the Reichstag and
  Mecklenburg-Strelitz one member.

  In Mecklenburg-Schwerin the chief towns are Rostock (with a
  university), Schwerin, and Wismar the capital. The capital of
  Mecklenburg-Strelitz is Neu-Strelitz. The peasantry of Mecklenburg
  retain traces of their Slavonic origin, especially in speech, but
  their peculiarities have been much modified by amalgamation with
  German colonists. The townspeople and nobility are almost wholly of
  Saxon strain. The slowness of the increase in population is chiefly
  accounted for by emigration.

_History._--The Teutonic peoples, who in the time of Tacitus occupied
the region now known as Mecklenburg, were succeeded in the 6th century
by some Slavonic tribes, one of these being the Obotrites, whose chief
fortress was Michilenburg, the modern Mecklenburg, near Wismar; hence
the name of the country. Though partly subdued by Charlemagne towards
the close of the 8th century, they soon regained their independence, and
until the 10th century no serious effort was made by their Christian
neighbours to subject them. Then the German king, Henry the Fowler,
reduced the Slavs of Mecklenburg to obedience and introduced
Christianity among them. During the period of weakness through which the
German kingdom passed under the later Ottos, however, they wrenched
themselves free from this bondage; the 11th and the early part of the
12th century saw the ebb and flow of the tide of conquest, and then came
the effective subjugation of Mecklenburg by Henry the Lion, duke of
Saxony. The Obotrite prince Niklot was killed in battle in 1160 whilst
resisting the Saxons, but his son Pribislaus (d. 1178) submitted to
Henry the Lion, married his daughter to the son of the duke, embraced
Christianity, and was permitted to retain his office. His descendants
and successors, the present grand dukes of Mecklenburg, are the only
ruling princes of Slavonic origin in Germany. Henry the Lion introduced
German settlers and restored the bishoprics of Ratzeburg and Schwerin;
in 1170 the emperor Frederick I. made Pribislaus a prince of the empire.
From 1214 to 1227 Mecklenburg was under the supremacy of Denmark; then,
in 1229, after it had been regained by the Germans, there took place the
first of the many divisions of territory which with subsequent reunions
constitute much of its complicated history. At this time the country was
divided between four princes, grandsons of duke Henry Borwin, who had
died two years previously. But in less than a century the families of
two of these princes became extinct, and after dividing into three
branches a third family suffered the same fate in 1436. There then
remained only the line ruling in Mecklenburg proper, and the princes of
this family, in addition to inheriting the lands of their dead kinsmen,
made many additions to their territory, including the counties of
Schwerin and of Strelitz. In 1352 the two princes of this family made a
division of their lands, Stargard being separated from the rest of the
country to form a principality for John (d. 1393), but on the extinction
of his line in 1471 the whole of Mecklenburg was again united under a
single ruler. One member of this family, Albert (c. 1338-1412), was king
of Sweden from 1364 to 1389. In 1348 the emperor Charles IV. had raised
Mecklenburg to the rank of a duchy, and in 1418 the university of
Rostock was founded.

The troubles which arose from the rivalry and jealousy of two or more
joint rulers incited the prelates, the nobles and the burghers to form a
union among themselves, and the results of this are still visible in the
existence of the _Landesunion_ for the whole country which was
established in 1523. About the same time the teaching of Luther and the
reformers was welcomed in Mecklenburg, although Duke Albert (d. 1547)
soon reverted to the Catholic faith; in 1549 Lutheranism was recognized
as the state religion; a little later the churches and schools were
reformed and most of the monasteries were suppressed. A division of the
land which took place in 1555 was of short duration, but a more
important one was effected in 1611, although Duke John Albert I. (d.
1576) had introduced the principle of primogeniture and had forbidden
all further divisions of territory. By this partition John Albert's
grandson Adolphus Frederick I. (d. 1658) received Schwerin, and another
grandson John Albert II. (d. 1636) received Güstrow. The town of
Rostock "with its university and high court of justice" was declared to
be common property, while the Diet or _Landtag_ also retained its joint
character, its meetings being held alternately at Sternberg and at
Malchin.

During the early part of the Thirty Years' War the dukes of
Mecklenburg-Schwerin and Mecklenburg-Güstrow were on the Protestant
side, but about 1627 they submitted to the emperor Ferdinand II. This
did not prevent Ferdinand from promising their land to Wallenstein, who,
having driven out the dukes, was invested with the duchies in 1629 and
ruled them until 1631. In this year the former rulers were restored by
Gustavus Adolphus of Sweden, and in 1635 they came to terms with the
emperor and signed the peace of Prague, but their land continued to be
ravaged by both sides until the conclusion of the war. In 1648 by the
Treaty of Westphalia, Wismar and some other parts of Mecklenburg were
surrendered to Sweden, the recompense assigned to the duchies including
the secularized bishoprics of Schwerin and of Ratzeburg. The sufferings
of the peasants in Mecklenburg during the Thirty Years' War were not
exceeded by those of their class in any other part of Germany; most of
them were reduced to a state of serfdom and in some cases whole villages
vanished. Christian Louis who ruled Mecklenburg-Schwerin from 1658 until
his death in 1692 was, like his father Adolphus Frederick, frequently at
variance with the estates of the land and with members of his family. He
was a Roman Catholic and a supporter of Louis XIV., and his country
suffered severely during the wars waged by France and her allies in
Germany.

In June 1692 when Christian Louis died in exile and without sons, a
dispute arose about the succession to his duchy between his brother
Adolphus Frederick and his nephew Frederick William. The emperor and the
rulers of Sweden and of Brandenburg took part in this struggle which was
intensified when, three years later, on the death of Duke Gustavus
Adolphus, the family ruling over Mecklenburg-Güstrow became extinct. At
length the partition Treaty of Hamburg was signed on the 8th of March
1701, and a new division of the country was made. Mecklenburg was
divided between the two claimants, the shares given to each being
represented by the existing duchies of Mecklenburg-Schwerin, the part
which fell to Frederick William, and Mecklenburg-Strelitz, the share of
Adolphus Frederick. At the same time the principle of primogeniture was
again asserted, and the right of summoning the joint _Landtag_ was
reserved to the ruler of Mecklenburg-Schwerin.

Mecklenburg-Schwerin began its existence by a series of constitutional
struggles between the duke and the nobles. The heavy debt incurred by
Duke Charles Leopold (d. 1747), who had joined Russia in a war against
Sweden, brought matters to a crisis; the emperor Charles VI. interfered
and in 1728 the imperial court of justice declared the duke incapable of
governing and his brother Christian Louis was appointed administrator of
the duchy. Under this prince, who became ruler _de jure_ in 1747, there
was signed in April 1755 the convention of Rostock by which a new
constitution was framed for the duchy. By this instrument all power was
in the hands of the duke, the nobles and the upper classes generally,
the lower classes being entirely unrepresented. During the Seven Years'
War Duke Frederick (d. 1785) took up a hostile attitude towards
Frederick the Great, and in consequence Mecklenburg was occupied by
Prussian troops, but in other ways his rule was beneficial to the
country. In the early years of the French revolutionary wars Duke
Frederick Francis I. (1756-1837) remained neutral, and in 1803 he
regained Wismar from Sweden, but in 1806 his land was overrun by the
French and in 1808 he joined the Confederation of the Rhine. He was the
first member of the confederation to abandon Napoleon, to whose armies
he had sent a contingent, and in 1813-1814 he fought against France. In
1815 he joined the Germanic Confederation (Bund) and took the title of
grand duke. In 1819 serfdom was abolished in his dominions. During the
movement of 1848 the duchy witnessed a considerable agitation in favour
of a more liberal constitution, but in the subsequent reaction all the
concessions which had been made to the democracy were withdrawn and
further restrictive measures were introduced in 1851 and 1852.

Mecklenburg-Strelitz adopted the constitution of the sister duchy by an
act of September 1755. In 1806 it was spared the infliction of a French
occupation through the good offices of the king of Bavaria; in 1808 its
duke, Charles (d. 1816), joined the confederation of the Rhine, but in
1813 he withdrew therefrom. Having been a member of the alliance against
Napoleon he joined the Germanic confederation in 1815 and assumed the
title of grand duke.

In 1866 both the grand dukes of Mecklenburg joined the North German
confederation and the _Zollverein_, and began to pass more and more
under the influence of Prussia, who in the war with Austria had been
aided by the soldiers of Mecklenburg-Schwerin. In the Franco-German War
also Prussia received valuable assistance from Mecklenburg, Duke
Frederick Francis II. (1823-1883), an ardent advocate of German unity,
holding a high command in her armies. In 1871 the two grand duchies
became states of the German Empire. There was now a renewal of the
agitation for a more democratic constitution, and the German Reichstag
gave some countenance to this movement. In 1897 Frederick Francis IV.
(b. 1882) succeeded his father Frederick Francis III. (1851-1897) as
grand duke of Mecklenburg-Schwerin, and in 1904 Adolphus Frederick (b.
1848) a son of the grand duke Frederick William (1819-1904) and his wife
Augusta Carolina, daughter of Adolphus Frederick, duke of Cambridge,
became grand duke of Mecklenburg-Strelitz. The grand dukes still style
themselves princes of the Wends.

  See F. A. Rudloff, _Pragmatisches Handbuch der mecklenburgischen
  Geschichte_ (Schwerin, 1780-1822); C. C. F. von Lützow, _Versuch einer
  pragmatischen Geschichte von Mecklenburg_ (Berlin, 1827-1835);
  _Mecklenburgische Geschichte in Einzeldarstellungen_, edited by R.
  Beltz, C. Beyer, W. P. Graff and others; C. Hegel, _Geschichte der
  mecklenburgischen Landstände bis 1555_ (Rostock, 1856); A. Mayer,
  _Geschichte des Grossherzogtums Mecklenburg-Strelitz 1816-1890_ (New
  Strelitz, 1890); Tolzien, _Die Grossherzöge von Mecklenburg-Schwerin_
  (Wismar, 1904); Lehsten, _Der Adel Mecklenburgs seit dem
  landesgrundgesetslichen Erbvergleich_ (Rostock, 1864); the
  _Mecklenburgisches Urkundenbuch_ in 21 vols. (Schwerin, 1873-1903);
  the _Jahrbücher des Vereins für mecklenburgische Geschichte und
  Altertumskunde_ (Schwerin, 1836 fol.); and W. Raabe, _Mecklenburgische
  Vaterlandskunde_ (Wismar, 1894-1896); von Hirschfeld, _Friedrich Franz
  II., Grossherzog von Mecklenburg-Schwerin und seine Vorgänger_
  (Leipzig, 1891); Volz, _Friedrich Franz II._ (Wismar, 1893); C.
  Schröder, _Friedrich Franz III._ (Schwerin, 1898); Bartold, _Friedrich
  Wilhelm, Grossherzog von Mecklenburg-Strelitz und Augusta Carolina_
  (New Strelitz, 1893); and H. Sachsse, _Mecklenburgische Urkunden und
  Daten_ (Rostock, 1900).