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TIME AND CLOCKS.


[Illustration:

  [_Frontispiece._

NUREMBERG CLOCK. CONVERTED FROM A VERGE ESCAPEMENT TO A PENDULUM
MOVEMENT.]


TIME AND CLOCKS:

A Description of Ancient and Modern Methods of
Measuring Time.

by

H. H. CUNYNGHAME M.A. C.B. M.I.E.E.

With Many Illustrations.






London:
Archibald Constable & Co. Ltd.
16 James Street Haymarket.
1906.

Bradbury, Agnew, & Co. Ld., Printers,
London and Tonbridge.




CONTENTS.


                                                   PAGE
  INTRODUCTION                                        1

  CHAPTER   I.                                        7

  CHAPTER  II.                                       50

  CHAPTER III.                                       90

  CHAPTER  IV.                                      123

  APPENDIX ON THE SHAPE OF THE TEETH OF WHEELS      187

  INDEX                                             199




TIME AND CLOCKS.




INTRODUCTION.


When we read the works of Homer, or Virgil, or Plato, or turn to the
later productions of Dante, of Shakespeare, of Milton, and the host
of writers and poets who have done so much to instruct and amuse us,
and to make our lives good and agreeable, we are apt to look with some
disappointment upon present times. And when we turn to the field of art
and compare Greek statues and Gothic or Renaissance architecture with
our modern efforts, we must feel bound to admit our inferiority to our
ancestors. And this leads us perhaps to question whether our age is the
equal of those which have gone before, or whether the human intellect
is not on the decline.

This feeling, however, proceeds from a failure to remember that each
age of the world has its peculiar points of strength, as well as of
weakness. During one period that self-denying patriotism and zeal
for the common good will be developing, which is necessary for the
formation of society. During another, the study of the principles of
morality and religion will be in the ascendant. During another the arts
will take the lead; during another, poetry, tragedy, and lyric poetry
and prose will be cultivated; during another, music will take its turn,
and out of rude peasant songs will evolve the harmony of the opera.

To our age is reserved the glory of being easily the foremost in
scientific discovery. Future ages may despise our literature,
surpass us in poetry, complain that in philosophy we have done
nothing, and even deride and forget our music; but they will only
be able to look back with admiration on the band of scientific
thinkers who in the seventeenth century reduced to a system the
laws that govern the motions of worlds no less than those of atoms,
and who in the eighteenth and nineteenth founded the sciences of
chemistry, electricity, sound, heat, light, and who gave to mankind
the steam-engine, the telegraph, railways, the methods of making
huge structures of iron, the dynamo, the telephone, and the thousand
applications of science to the service of man.

And future students of history who shall be familiar with the
conditions of our life will, I think, be also struck with surprise at
our estimate of our own peculiar capabilities and faculties. They will
note with astonishment that a gentleman of the nineteenth century, an
age mighty in science, and by no means pre-eminent in art, literature
and philosophy, should have considered it disgraceful to be ignorant
of the accent with which a Greek or a Roman thought fit to pronounce
a word, should have been ashamed to be unable to construe a Latin
aphorism, and yet should have considered it no shame at all not to
know how a telephone was made and why it worked. They will smile when
they observe that our highest university degrees, our most lucrative
rewards, were given for the study of dead languages or archæological
investigations, and that science, our glory and that for which we have
shown real ability, should only have occupied a secondary place in our
education.

They will smile when they learn that we considered that a knowledge
of public affairs could only be acquired by a grounding in Greek
particles, or that it could ever have been thought that men could not
command an army without a study of the tactics employed at the battle
of Marathon.

But the battle between classical and scientific education is not in
reality so much a dispute regarding subjects to be taught, as between
methods of teaching. It is possible to teach classics so that they
become a mental training of the highest value. It is possible to teach
science so that it becomes a mere enslaving routine.

The one great requirement for the education of the future is firmly to
grasp the fact that a study of words is not a study of things, and that
a man cannot become a carpenter merely by learning the names of his
tools.

It was the mistake of the teachers of the Middle Ages to believe that
the first step in knowledge was to get a correct set of concepts of
all things, and then to deduce or bring out all knowledge from them.
Admirable plan if you can get your concepts! But unfortunately concepts
do not exist ready made—they must be grown; and as your knowledge
increases, so do your concepts change. A concept of a thing is not a
mere definition, it is a complete history of it. And you must build up
your edifice of scientific knowledge from the earth, brick by brick and
stone by stone. There is no magic process by which it can with a word
be conjured into existence like a palace in the Arabian Nights.

For nothing is more fatal than a juggle with words such as force,
weight, attraction, mass, time, space, capacity, or gravity. Words are
like purses, they contain only as much money as you put into them. You
may jingle your bag of pennies till they sound like sovereigns, but
when you come to pay your bills the difference is soon discovered.

This fatal practice of learning words without trying to obtain a
clear comprehension of their meaning, causes many teachers to use
mathematical formulæ not as mere steps in a logical chain, but like
magical chaldrons into which they put the premises as the witches put
herbs and babies’ thumbs into their pots, and expect the answers to
rise like apparitions by some occult process that they cannot explain.
This tendency is encouraged by foolish parents who like to see their
infant prodigies appear to understand things too hard for themselves,
and look on at their children’s lessons in mathematics like rustics
gaping at a fair. They forget that for the practical purposes of
life one thing well understood is worth a whole book-full of muddled
ill-digested formulæ. Unfortunately it is possible to cram boys up
and run them through the examination sieves with the appearance of
knowledge without its reality. If it were cricket or golf that were
being tested how soon would the fraud be discovered. No humbug would be
permitted in those interesting and absorbing subjects. And really, when
one reflects how easy it is to present the appearance of book knowledge
without the reality, one can hardly blame those who select men for
service in India and Egypt a good deal for their proficiency in sports
and games. Better a good cricketer than a silly pedant stuffed full of
learning that “lies like marl upon a barren soil encumbering what is
not in its power to fertilize.”

Another kindred error is to expect too much of science. For with all
our efforts to obtain a further knowledge of the mysteries of nature,
we are only like travellers in a forest. The deeper we penetrate it,
the darker becomes the shade. For science is “but an exchange of
ignorance for that which is another kind of ignorance”[A] and all
our analysis of incomprehensible things leads us only to things more
incomprehensible still.

    [A] _Manfred_, Act II., scene iv.

It is, therefore, by the firm resolution never to juggle with words or
ideas, or to try and persuade ourselves or others that we understand
what we do not understand, that any scientific advance can be made.




CHAPTER I.


All students of any subject are at first apt to be perplexed with the
number and complication of the new ideas presented to them.

The need of comprehending these ideas is felt, and yet they are
difficult to grasp and to define. Thus, for instance, we are all apt
to think we know what is meant when force, weight, length, capacity,
motion, rest, size, are spoken of. And yet when we come to examine
these ideas more closely, we find that we know very little about
them. Indeed, the more elementary they are, the less we are able to
understand them.

The most primordial of our ideas seem to be those of number and
quantity; we can count things, and we can measure them, or compare
them with one another. Arithmetic is the science which deals with the
numbers of things and enables us to multiply and divide them. The
estimation of quantities is made by the application of our faculty of
comparison to different subjects. The ideas of number and quantity
appear to pervade all our conceptions.

The study of natural phenomena of the world around us is called the
study of physics from the Greek word φυσίς or “inanimate nature,”
the term physics is usually confined to such part of nature as is not
alive. The study of living things is usually termed biology (from βια,
life).

In the study of natural phenomena there are, however, three ideas which
occupy a peculiar and important position, because they may be used
as the means of measuring or estimating all the rest. In this sense
they seem to be the most primitive and fundamental that we possess.
We are not entitled to say that all other ideas are formed from and
compounded of these ideas, but we are entitled to say that our correct
understanding of physics, that is of the study of nature, depends
in no slight degree upon our clear understanding of them. The three
fundamental ideas are those of space, time and mass.

Space appears to be the universal accompaniment of all our impressions
of the world around us. Try as we may, we cannot think of material
bodies except in space, and occupying space. Though we can imagine
space as empty we cannot conceive it as destroyed. And this space has
three dimensions, length, breadth measured across or at right angles to
length, and thickness measured at right angles to length and breadth.
More dimensions than this we cannot have. For some inscrutable reason
it has been arranged that space shall present these three dimensions
and no more. A fourth dimension is to us unimaginable—I will not say
inconceivable—we can conceive that a world might be with space in four
dimensions, but we cannot imagine it to ourselves or think what things
would be like in it.

With difficulty we can perhaps imagine a world with space of only two
dimensions, a “flat land,” where flat beings of different shapes,
like figures cut out of paper, slide or float about on a flat table.
They could not hop over one another, for they would only have length
and breadth; to hop up you would want to be able to move in a third
dimension, but having two dimensions only you could only slide forward
and sideways in a plane. To such beings a ring would be a box. You
would have to break the ring to get anything out of it, for if you
tried to slide out you would be met by a wall in every direction. You
could not jump out of it like a sheep would jump out of a pen over the
hurdles, for to jump would require a third dimension, which you have
not got. Beings in a world with one dimension only would be in a worse
plight still. Like beads on a string they could slide about in one
direction as far as the others would let them. They could not pass one
another. To such a being two other beings would be a box one on each
side of him, for if thus imprisoned, he could not get away. Like a
waggon on a railway, he could not walk round another waggon. That would
want power of moving in two dimensions, still less could he jump over
them, that would want three.

We have not the smallest idea why our world has been thus limited. Some
philosophers think that the limitation is in us, not in the world, and
that perhaps when our minds are free from the limitations imposed by
their sojourn in our bodies, and death has set us free, we may see not
only what is the length and breadth and height, but a great deal more
also of which we can now form no conception. But these speculations
lead us out of science into the shadowy land of metaphysics, of which
we long to know something, but are condemned to know so little. Area
is got by multiplying length by breadth. Cubic content is got by
multiplying length by breadth and by height. Of all the conceptions
respecting space, that of a line is the simplest. It has direction, and
length.

The idea of mass is more difficult to grasp than that of space. It
means quantity of matter. But what is matter? That we do not know. It
is not weight, though it is true that all matter has weight. Yet matter
would still have mass even if its property of weight were taken away.

For consider such a thing as a pound packet of tea. It has size, it
occupies space, it has length, breadth, and thickness. It has also
weight. But what gives it weight? The attraction of the earth. Suppose
you double the size of the earth. The earth being bigger would attract
the package of tea more strongly. The weight of the tea, that is, the
attraction of the earth on the package of tea, would be increased—the
tea would weigh more than before. Take the package of tea to the planet
Jupiter, which, being very large, has an attraction at the surface 2½
times that of the earth. Its size would be the same, but it would feel
to carry like a package of sand. Yet there would be the same “mass”
of tea. You could make no more cups of tea out of it in Jupiter than
on earth. Take it to the moon, and it would weigh a little over two
ounces, but still it would be a pound of tea. We are in the habit of
estimating mass by its weight, and we do so rightly, for at any place
on the earth, as London, the weights of masses are always proportioned
to the masses, and if you want to find out what mass of tea you have
got, you weigh it, and you know for certain. Hence in our minds we
confuse mass with weight. And even in our Acts of Parliament we have
done the same thing, so that it is difficult in the statutes respecting
standard weights to know what was meant by those who drew them up, and
whether a pound of tea means the _mass_ of a certain amount of tea or
the _weight_ of that mass. For accurate thinking we must, of course,
always deal with masses, not with weights. For so far as we can tell
_mass_ appears indestructible. A mass is a mass wherever it is, and
for all time, whereas its weight varies with the attractive force of
the planet upon which it happens to be, and with its distance from that
planet’s centre. A flea on this earth can skip perhaps eight inches
high; put that flea on the moon, and with the expenditure of the same
energy he could skip four feet high. Put him on the planet Jupiter and
he could only skip 3⅕ inches high. A man in a street in the moon could
jump up into a window on the first floor of a house. One pound of tea
taken to the sun would be as heavy as twenty-eight pounds of it at the
earth’s surface; and weight varies at different parts of the earth.
Hence the true measure of quantity of matter is mass, not weight.

The mass of bodies varies according to their size; if you have the same
nature of material, then for a double size you have a double mass. Some
bodies are more concentrated than others, that is to say, more dense;
it is as though they were more tightly squeezed together. Thus a ball
of lead of an inch in diameter contains forty-eight times as much mass
as a ball of cork an inch in diameter. In order to know the weight of a
certain mass of matter, we should have to multiply the mass by a figure
representing the attractive force or pull of the earth.

In physics it is usual to employ the letters of the alphabet as a sort
of shorthand to represent words. So that the letter _m_ stands for
the mass of a body. So again _g_ stands for the attractive pull of the
earth at a given place. _w_ stands for the weight of the body. Hence
then, since the weight of a body depends on its mass and also on the
attractive pull of the earth, we express this in short language by
saying, w = m × g; or _w_ is equal to _m_ multiplied by _g_; the symbol
= being used for equality, and × the sign of multiplication. In common
use × is usually omitted, and when letters are put together they are
intended to be understood as multiplied. So that this is written

                                w = mg.

Of course by this equation we do not mean that weight is mass
multiplied into the force of gravity, we only mean that the number of
units of weight is to be found by multiplying the number of units of
mass into the number of units of the earth’s force of gravity.

In the same way, if when estimating the number of waggons, _w_, that
would be wanted for an army of men, _n_, which consumed a number of
pounds, _p_, of provisions a day, we might put

                                w = np.

But this would not mean that we were multiplying soldiers into food
to produce waggons, but only that we were performing a numerical
calculation.

Time is one of the most mysterious of our elementary ideas. It seems to
exist or not to exist, according as we are thinking or not thinking.
It seems to run or stand still and to go fast or slowly. How it drags
through a wearisome lesson; how it flies during a game of cricket; how
it seems to stop in sleep. If we measured time by our own thoughts it
would be a very uncertain quantity. But other considerations seem to
show us that Nature knows no such uncertainty as regards time, that she
produces her phenomena in a uniform manner in uniform times, and that
time has an existence independent of our thoughts and wills.

The idea of a state of things in which time existed no more was quite
familiar to mediæval thinkers, and was regarded by many of them as the
condition that would exist after the Day of Judgment. In recent times
Kant propounded the theory that time was only a necessary condition of
our thoughts, and had no existence apart from thinking beings—in fact,
that it was our way of looking at things.

Scientifically, however, we are warranted in treating time as perfectly
real and capable of the most exact measurement. For example, if we
arrange a stream of sand to run out of an orifice, and observe how
much will run out while an egg is being boiled hard, we find as a fact
that if the same quantity of sand runs out, the state of the egg is
uniform. If we walk for an hour by a watch, we find that we can go
half the distance that we should if we walked two hours. It is the
correspondence of these various experiments that gives us faith in the
treatment of time as a thing existing independently of ourselves, or,
at all events, independent of our transient moods.

The ideas of time acquired by the races of men that first evolved from
a state of barbarism were no doubt derived from the observation of day
and night, the month and the year.

[Illustration: FIG. 1.]

For, suppose that a shepherd were on the plains of Chaldea, or perhaps
on those mountains of India known as the roof of the world, which
according to some archæologists was the site of the garden of Eden and
the early home of the European race, what would he see?

He would see the sun rise in the east, slowly mount the heavens till it
stood over the south at middle day, then it would sink towards the west
and disappear. In summer the rising point of the sun would be more to
the northward than in winter, and so also would be its point of setting
_A´_. In winter it would rise a little to the south of east, and set
a little to the south of west, and not rise so high in the heavens at
midday, so that the summer day would be longer than the winter day.
If the day were always divided into twelve hours, whether it were long
or short, then in summer the hours of the day would be long; in winter
they would be short. This mode of dividing the day was that used by
the Greeks. The Egyptians, on the other hand, averaged their day by
dividing the whole round of the sun into twenty-four hours, so that the
summer day contained more hours than the winter day. Hence, for the
Egyptians, sun-rise did not always take place at six o’clock. For in
winter it took place after six, and in summer before six; and this is
the system that has descended to us.

The moon also would rise at different places, varying between _A_ and
_B_, and set at places varying between _A´_ and _B´_, but these would
be independent of those at which the sun rose and set.

Moreover, the moon each day would appear to get further and further
away from the sun in the direction of the arrow, as shown in the
sketch. If the moon rose an hour after the sun on one day, the next day
it would rise more than two hours after the sun, and so on. This delay
in rising of the moon would go on day by day till at last she came
right round to the sun again, as shown at _M´_. And in her path she
would change her form from a crescent, as at _M_, up to a full moon,
when she would be half way round from the sun, that is, when she would
rise twelve hours after him, or just be rising as the sun set. This
delay and accompanying change of form would go on, till after three
weeks she would have got round to a position _A´_, when she would rise
eighteen hours after the sun, and have become a crescent with her back
to the sun; in fact, she would always turn her convex side to the sun.
At length, when twenty-eight days had passed, she would be round again
about opposite to the sun, and consequently her pale light would be
extinguished in his beams, and she would gradually reappear as a new
moon on the other side of him. This series of changes of the moon takes
place once every twenty-eight days, and is called a lunar or “moon”
month, and was used as a division of time by very early nations. The
changes of the seasons recurred with the changes in the times of rising
of the sun, and took a year to bring about. And there were nearly
thirteen moon changes in the year.

It was also observed that during its cycle of changes, the sun was
slowly moving round backwards among the stars in the same direction as
the moon, only it made its retrograde cycle in a year, and thus arose
the division of time into months and years. The stars turned round in
the heavens once in the complete day. The sun, therefore, appeared to
move back among them, passing successively through groups of stars, so
as to make the circuit of them all in a year. The stars through which
he passed in a year, and through which the moon travelled in a month,
were divided by the ancients into groups called constellations, and its
yearly path in the heavens was called the zodiac. There were twelve of
these constellations in the zodiac called the signs. Hence, then, the
sun passed through a sign in every month, making the tour of them all
in the year. To these signs fanciful names were given, such as “the
Ram,” “the Water-bearer,” “the Virgin,” “the Scorpion,” and so on, and
the sun and moon were then said to pass through the signs of the zodiac.

Hence, since the path of the sun marked the year, you could tell the
seasons by knowing what sign of the zodiac the sun was in. The age of
the moon was easily known by her form.

When the winter was over, then, just as the sun set the dog star would
be rising in the east, and this would show that the spring was at hand.
Then the peasants prepared to till the earth and sow the seed and lead
the oxen out to pasture, and celebrated with joyful mirth the glad
advent of the spring, corresponding to our Easter, when the sun had
run through three constellations of the zodiac. Then came the summer
heat, and with many a mystic rite they celebrated Midsummer’s Day. In
autumn, after three more signs of the zodiac have been traversed by the
sun, the sun again rises exactly in the east and sets in the west, and
the days and nights are equal. This is the autumnal equinox, and was
once celebrated by the feast which we now know as Michaelmas Day, and
the goose is the remnant of the ancient festival.

[Illustration: FIG. 2.]

And the great winter feast of the ancients is now known to us as
Christmas, and chosen to celebrate the birth of our Lord; for when
Christianity came into the world and the heathens were converted, the
old feast days were deliberately changed into Christian festivals.

To us, therefore, the whole heavens, and the fixed stars with them,
appear to turn from east to west, or from left to right, as we look
towards the south, as shown by the big arrow. But the moon and sun,
though apparently placed in the heavens, move backwards among the fixed
stars, as shown by the small arrows. The sun moves at such a rate that
he goes round the circle of the heavens in a year of three hundred and
sixty-five days. The moon goes round the circle in twenty-eight and a
half days, or a lunar month. Of course, in reality the sun is at rest,
and it is the earth that moves round the sun and spins on its axis
as it moves. But it will presently be shown that the appearance to a
person on the earth is the same whether the earth goes round the sun or
the sun round the earth.

From the works of Greek writers we know a good deal about the ideas of
the world that were entertained by the ancients. The most early notions
were, of course, connected with the worship of the gods. The sun was
considered as a huge light carried in a chariot, driven by Apollo, with
four spirited steeds. It descended to the ocean when the day declined,
and then the horses were unyoked by the nymphs of the ocean and led
round to the east, so as to be ready for the journey of the following
day. The Egyptians figured the sun as placed in a boat which sailed
over the heavens. At night the sun god descended into the infernal
regions, carrying with him the souls of those who had died during the
day. There they passed through different regions of hell, with portals
guarded by hideous monsters. Those who had well learned the ritual of
the dead knew the words of power wherewith to appease the demons. Those
unprovided with the watchwords were subjected to terrible dangers.
Then the soul appeared before Minos, and was weighed and dealt with
according to its deserts.

[Illustration: FIG. 3.]

The earth was considered as a huge island in the midst of a circular
sea. Gradually, however, astronomical ideas became subjected to
science. One of the first truths that dawned on astronomers was the
fact that the earth was a sphere. For they noticed that as people
went further and further to the north, the elevation of the sun at
midday above the horizon became smaller and smaller. This can easily
be seen from the diagram. When an observer is at _A_ the sun appears
at an altitude above the horizon equal to the angle α, but as he goes
along the curved surface of the earth to a point _B_ nearer to the
north pole, the sun appears to be lower and only to have an altitude
β. From this it was easy for men so skilled in geometry as the Greeks
to calculate how big the earth was. They did so, and it appeared to
have the enormous diameter of 8,000 miles. They only knew quite a
small portion of it. They thought that the rest was ocean. But they
had, of course, a clear idea of the “antipodes” or up-side-down side
of it, and they believed that if men were on the other side of it
that their feet must all point towards its centre. From this they
got the idea of the centre of the earth as a point of attraction for
all things that had an earth-seeking or earthy nature. Fire appeared
always to desire to go upwards, so they thought that fire had an
earth-repellent, heaven-seeking character. Water they thought partly
earth-seeking, partly heaven-seeking, for it appeared in the ocean or
floated as clouds. Air they thought to be indifferent. And out of the
four elements fire, water, earth, and air they believed the world was
made. The earth they thought must be at rest; for if it was in motion
things would fly off from it. They saw that either the sun must be
moving round the earth, or else the earth must be turning on its axis.
They chose the former hypothesis, because they argued that if the earth
were twisting round once in twenty-four hours then such a country as
Greece must be flying round like a spot on the surface of a top, at the
rate of about 18,000 miles in twenty-four hours, that is, at the rate
of about 180 yards in a second, or faster than an arrow from a bow.
But if that was the case then a bird that flew up from the earth would
be left far behind. If a ball were thrown up it would fall hundreds
of yards behind the person who threw it. They could not conceive how
it was possible for a ball thrown up by someone standing on a moving
object not to fall behind the thrower.

This decided them in their error. The mistaken astronomy of the Greeks
was also much forwarded by Aristotle, the tutor of Alexander the Great.
This great genius in politics and philosophy was only in the second
rank as a man of science, and, as I think, hardly equal to Archimedes
or Hipparchus, or even to Ptolemy. Aristotle wrote a book concerning
the heavens which bristles with the most wantonly erroneous scientific
ideas, such as, for instance, that the motion of the heavenly bodies
must be circular because the most perfect curve is a circle, and
similar assumptions as to the course of nature.

The earth, then, being fixed, they thought that the moon, the sun,
and the seven planets were carried round it, fixed each of them in
an enormous crystal spherical shell. These spheres, like coats of
an onion, slid round one upon another, each carrying his celestial
luminary. The moon was the nearest, then Mercury, then Venus, then the
sun, then Mars, Jupiter and Saturn. Outside them was the sphere of the
stars, and outside all the “_primum mobile_,” or great Prime Mover of
the universe. When one of the celestial bodies, such as the moon, got
in front of another, such as the sun, there was an eclipse. They soon
observed that the moon derived its light from the sun. As they knew
the size of the earth, by comparison they got some vague idea of the
huge distances that the heavenly bodies must be from us. In fact, they
measured the distance of the moon with approximate accuracy, making it
240,000 miles, or about thirty times the earth’s diameter.

This, of course, gave them the moon’s diameter, for they were easily
able to calculate how big an object must be, that looks as big as the
moon and is 240,000 miles away.

This large size of the moon gave them some idea of the distance of the
sun, but they failed to realise how big and far away he really is.

Several ancient nations used weeks as means of measuring time. They
made four weeks to the lunar month. The order of the days was rather
curiously arranged. For, assuming that the earth is the centre of the
planetary system, put the planets in a column, putting the nearest (the
moon) at the bottom and the furthest off at the top—

  Saturn,
  Jupiter,
  Mars,
  The Sun,
  Venus,
  Mercury,
  The Moon.

Then divide the day into three watches of eight hours each, and let
each watch be presided over by one of the planet-gods: begin with
Saturn. We then have Saturn as the first god ruling Saturday, and
Jupiter and Mars, the two other gods, for that day. The first watch for
Sunday will be the sun; Venus and Mercury will preside over the next
two watches of that day. The planet that will preside over the first
watch of the next day will be the moon, and the day will, therefore, be
called Monday; Saturn and Jupiter will be the other gods for Monday.
The first watch of the next day will be presided over by Mars, and the
day will, therefore, be called Mars-day or Mardi, or, in the Teutonic
languages, Tuesday, after Tuesco, a Scandinavian god of war. Mercury
will give a name to Mercredi, or to Wednesday, or Wodin’s-day. Jupiter
to Jeudi, or “Thurs” day. Venus to Vendredi, or in the Scandinavian,
Friday, the day of the Scandinavian goddess Freya, the goddess of love
and beauty, who corresponds to Venus, and thus the week is completed.

[Illustration: FIG. 4.]

This weekly scheme came probably from the Chaldean astronomers. It
appears probable that the great tower of Babel, the ruins of which
exist to this day, consisted of seven stages, one over the other, the
top one painted white, or perhaps purple, to represent the Moon, the
next lower blue for Mercury, then green for Venus, yellow for the Sun,
red for Mars, orange for Jupiter, and black for Saturn. Unfortunately,
of the colours no trace now remains.

But nightly on the long terraces the Babylonian priests observed
eclipses and other celestial phenomena. Their records were afterwards
taken to Alexandria and kept in the great library that was subsequently
burned by the Turks. In that library they were seen by the astronomer
Ptolemy, who used them in the writing of his work on astronomy called
the “Great Syntaxis” or “Collection.” The original work perished, but
it had been translated into Arabic by the Arab astronomers, who called
it “Al Magest,” the Great Book. It was translated from Arabic into
Latin, and remained the textbook for astronomers in Europe quite down
to the time of Queen Elizabeth, when a better system took its place.

For the use of men engaged in practical astronomy, it is very
convenient to consider the sun, moon, stars, and planets as going
round the earth at rest. For instance, seamen use the heavenly bodies
as in a way hands of a huge clock from which they can know the time and
their position on the earth. “The Nautical Almanac,” which is printed
yearly, gives the true position of these heavenly bodies for every
hour, minute, and second of the year, and I will presently show how
useful this is to sailors.

We will deal with the sun first. From the motions of the sun we can
observe the time. This is done in every garden by means of sun-dials,
and I will now describe how they are constructed. If a light, such as
the light of a candle, be moved round in a circle at a uniform pace so
as to go round once in some given period, such as twenty-four hours, it
is obvious that it would serve to measure time. Thus, for example, if a
sheet of white paper be placed upon the table, and a pencil be stuck on
to it upright with some sealing wax, or a pen propped up in an ink-pot,
then a candle held by anyone will cast the shadow of the pen on the
paper.

[Illustration: FIG. 5.]

If the person holding the candle walk round the table at a uniform
speed, the shadow will go round like the hand of a clock, and might be
made to mark the time. If the candle took twenty-four hours to go round
the table, as the sun takes twenty-four hours to go round the earth,
then marks placed on the paper would serve to measure the hours, and
the paper and pen would serve as a sort of sun-dial.

But the sun does not go round the earth as the candle round the
table. Its path is an inclined one, like that shown by the dotted
line. Sometimes it is above the level of the table, sometimes below
it. And, moreover, its winter path is different from its summer path.
Whence then it follows that the hour-marks on the paper cannot be
put equidistant like the hours on the dial of a clock, and that some
arrangement must be made so that the line as shown by the summer sun
shall correspond with the time as shown by the winter sun.

[Illustration: FIG. 6.]

Let us suppose that _N O S_ is the axis of the heavens, and the lines
_N A S_, _N B S_, _N C S_, are meridian lines drawn from one of the
poles _N_ of the heavens round on the surface of a celestial sphere
whose centre is at _O_. Let _A B C_ be a circle also on this sphere,
passing through _O_, the centre of the sphere, in a plane at right
angles to _N S_, the axis. Then _A B C_ is called the equatorial. It
is a circle in the heavens corresponding to the equator on the earth.
At the vernal and autumnal equinox, namely on March 25 and September
25, the sun is in the equatorial. In midsummer and midwinter it is on
opposite sides of the equatorial. In midsummer it is nearer to _N_, as
at _V_; in midwinter it is nearer to _S_, as at _W_. Suppose we were
on an island in the midst of a surrounding ocean, we should only have
a limited range of view. If the highest point on the island were 100
feet, then from that altitude we should be able to see about thirteen
miles to the horizon. More than that could not be seen on account of
the rotundity of the earth.

Let us suppose then such an island surrounded for thirteen miles
distant on every side by an ocean, and let us consider what would be
the apparent motions of the sun when seen from such an island. At the
vernal and autumnal equinoxes, when the sun is on the equatorial, it
would appear to rise out of the ocean at a point _E_, due east; it
traverses half the equatorial and sets in the ocean at a point _W_, due
west. The day is twelve hours long, from 6 a.m. to 6 p.m.

[Illustration: FIG. 7.]

In summer the sun is higher, and nearer to the pole _N_, say at a point
_s_. It rises at a point _a_ in the ocean more to the north than _E_,
the eastern point, and sets at a point _b_, also more north than _W_,
the western point, and traverses the path _a s b_. But to traverse
this path it takes longer than twelve hours, for _a s b_ is more than
half the circle _a s b_. Hence then it rises say at 4.30 a.m. and
sets at 7.30 p.m. The night, during which the sun moves round the path
from _b_ to _a_, is correspondingly short, being only nine hours in
length, from 7.30 p.m. till 4.30 a.m. So you have a long summer day and
a short summer night. But in winter, when the sun gets nearer to the
south pole of the heavens, it rises at a point _C_ in the ocean at 7.30
a.m., and traverses the arc _c t d_, and sets at the point _d_ at 4.30
p.m. So that the winter day is only nine hours long. But the winter
night lasts from 4.30 p.m. till 7.30 a.m., and is therefore fifteen
hours long, the sun going round the path _d r c_ in the interval. It is
therefore the obliquity of the poles _N S_, coupled with the fact that
the sun’s position is nearer to one pole, _N_, in summer, and nearer to
the other pole, _S_, in winter, that produces the inequality of days
and nights in our latitudes. Suppose our island were on the equator.
The north pole and the south pole would appear to be on the horizon,
and then whether the sun moved in the circle _a s b_ in the summer, or
_E S W_ at the vernal or autumnal equinoxes, or _c t d_ in the winter,
in each of these cases, though the places of rising and setting in the
ocean might vary in summer from _a_ and _b_ to _c_ and _d_ in winter,
yet in each of these cases the path from _a_ to _b_, _A_ to _B_, and
_c_ to _d_ would still always be a half-circle and occupy twelve hours.
Hence at the equator the days and nights never vary in length, but the
sun always rises at six and sets at six. And, besides, it always rises
straight up from the ocean and plunges down vertically into it, so that
there is but little twilight and dawn.

[Illustration: FIG. 8.]

But now let us suppose we were living at the north pole. In this case
the north pole would be directly overhead, the south pole directly
under our feet. At the vernal and autumnal equinoxes the sun would
appear with half its disc above the ocean, and go round the ocean
horizon, always appearing with half its disc above the sea. In summer
it would appear at a point _s_ nearer to the pole _N_. It would go
round in the heavens, always appearing above the horizon, and would
never set at all. As the summer waned the sun would become lower and
lower, still, however, going round and round without setting till at
the autumn equinox it reached the horizon. So that for six months it
would never have set. But when it did set, there would then be six
months without a sun at all.

[Illustration: FIG. 9.]

Thus then all over the world the period of darkness and light is
equivalent. At the tropics the days and nights are always equal. At
the poles light for six months is followed by darkness for six months.
In the intermediate temperate regions nights of varying lengths follow
days of varying lengths, a short night following a long day and _vice
versâ_.

[Illustration: FIG. 10.]

It is evident that for a person living on the north pole a sun-dial
would be an easy thing to make. All that would be needful would be to
put a post vertically in the ground, and observe its shadow as the sun
went round (Fig. 10).

[Illustration: FIG. 11.]

In latitudes such as that of England, where the pole of the earth is
inclined at an angle to the horizon, it is necessary that the rod, or
“style” as it is called, of the sun-dial should be inclined to the
horizontal. For if we used an upright “style,” as _O A_, then when the
sun was in the south, at midday, the shadow would lie along the same
direction, _O B_, whether the sun were high in summer, as at _S_, or
low in winter, as at _s_. But at other hours, such as nine o’clock in
the morning, the shadow of the “style” _O A_ would, when the sun was
at its summer position _T_, lie along _O D_, whereas when the sun was
at its winter position _t_ the shadow would lie along _O C_. Thus the
time would appear different in summer and in winter; and the dial
would lead to errors. But if the “style” is inclined in the direction
of the poles, then, however, the sun moves from or towards the pole.
As its position varies in winter and summer, the shadow still remains
unchanged for any particular hour, and it is only the circular motion
of the sun round in its daily path that affects the position of the
shadows.

[Illustration: FIG. 12.]

Therefore the first condition of making a sun-dial is that the “style”
which casts the shadow should be parallel to the earth’s axis, that
is to say should point to the polar star. This is the case whether
the sun-dial is horizontal or is vertical, and whether it stands on a
pillar in the garden or is attached to the wall of a house.

To divide the dial, we have only to imagine it surrounded by a sort
of cage formed of twenty-four arcs drawn from the north pole to the
south pole, and equidistant from one another. In its course the sun
would cross one of them every hour. Hence the points to which the
shadows _o a_, _o b_, _o c_, _o d_, of the inclined “style” _O N_ would
point are the points where these arcs meet the horizontal circle. This
consideration leads to a simple method of constructing a sun-dial,
which is given at the end of this chapter in an appendix.

Sun-dials were largely in use in ancient times. It is thought that the
circular rows of stones used by the Druids were used to mark the sun’s
path, and indicate the times and seasons. Obelisks are also supposed
to have been used to cast sun-shadows. The Greeks were perfectly
acquainted with the method of making sun-dials with inclined “styles,”
or “gnomons.”

[Illustration: FIG. 13.]

Small portable sun-dials were once much used before the introduction of
watches, and were provided with compasses by which they could be turned
round, so that the “style” pointed to the north.

Sun-dials were only available during the hours of the day when the
sun was shining. The desire to mark the hours of the night led to
the adoption of water clocks, which measured time by the amount of
water which escaped from a small hole in a level of water. Some care,
however, is required to secure correct registration. For suppose that
we have a vessel with a small pipe leading out near the bottom, then
the amount of water which will run out of the pipe in a given time
depends upon the pressure of the water at the pipe, and this depends
in its turn upon _P Q_, the head of water in the vessel. Whence it
follows that the division _Q R_, due to say an hour’s run of the clock
at _Q R_, will be more than _q r_, the division corresponding to an
hour, at _q_, a point lower down between _P_ and _Q_, and hence the
divisions marked on the vessel to show the hours by means of the level
of the water would be uneven, becoming smaller and smaller as the water
fell in the vessel.

To avoid the inconvenience of unequal divisions, the water to be
measured was allowed to escape into an empty vessel from a vessel in
which its surface was always kept at a constant level. Inasmuch as the
pressure on the pipe or orifice in the vessel in which the water was
always kept at a constant level was always constant, it followed that
equal volumes of water indicated equal times, and the vessel into which
the water fell needed only to be equally divided.

As a measure of hours of the day in countries such as Egypt, where the
hours were always equal, and thus where the longer days contained more
hours, the water clock was very suitable; but in Greece and Rome, where
the day, whatever its length, was always divided into twelve hours, the
simple water clock was as unsuitable as a modern clock would be, for it
always divided the hours equally, and took no account of the fact that
by such a system the hours in summer were longer than in winter.

In order, therefore, to make the water clock available in Greece and
Italy, it became necessary to make the hours unequal, and to arrange
them to correspond with unequal hours of the Greek day. This plan was
accomplished as follows. Upon the water which was poured into the
vessel that measured the hours was placed a float; and on the float
stood a figure made of thin copper, with a wand in its hand. This wand
pointed to an unequally divided scale. A separate scale was provided
for every day in the year, and these scales were mounted on a drum
which revolved so as to turn round once in the year. Thus as the figure
rose each day by means of a cogwheel it moved the drum round one
division, or one three hundred and sixty-fifth part of a revolution.
By this means the scale corresponding to any particular day of winter
or summer was brought opposite the wand of the figure, and thus the
scale of hours was kept true. In fact, the water clock, which kept
true time, was made by artificial means to keep untrue time, in order
to correspond with the unequal hours of the Greek days. In the picture
_A_ is the receiving water vessel, _P_ the pipe through which the
water flows; _B_ is the figure, _C_ the rod; _D_ is the drum, made to
revolve by the cogwheel _E_, containing 365 teeth, of which one tooth
was driven forward at the close of each day. A syphon _G_ was fixed in
the vessel _A_, so that when the figure had risen to the top and pushed
forward the lever _F_, the syphon suddenly emptied the vessel through
the pipe _H_, and the figure fell to the bottom of the vessel _A_ and
became ready to rise and register another day. The divisions on the
drum are, of course, uneven. On one side, corresponding to the summer,
the day hours would reckon about seventy minutes each, the night hours
would be only about fifty minutes each, so that the day divisions on
the scale would be long, and the night divisions short. The reverse
would be the case in winter. And, therefore, the lines round the drum
would go in an uneven wavy form.

[Illustration: FIG. 14.]

Such water clocks as these were used by the ancient Romans.

Sand was also used to measure time. As soon as the art of blowing glass
had been perfected by the people of Byzantium, from whom the art passed
to the Venetians, sand-glasses were made. These glasses were used for
all sorts of purposes, for speeches and for cooking, but their most
important use was at sea. For it was very important in the early days
of navigation to know the speed at which the vessel was proceeding in
order that one’s place at sea might be calculated. The earliest method
was to throw over a heavy piece of wood of a shape that resisted being
dragged through the water, and with a string tied to it. The block of
wood was called the log, and the string had knots in it. The knots
were so arranged that when one of them ran through one’s fingers in
a half-minute measured by a sand-glass it indicated that the vessel
was going at the speed of one nautical mile in an hour. The nautical
mile was taken so that sixty of them constituted one degree, that is
one three hundred and sixtieth part of a great circle of the earth.
Each nautical mile has, therefore, 6,080 feet. This is bigger than an
ordinary mile on land, which has only 5,280 feet. The knots, therefore,
have to be arranged so that when the ship is going one nautical
mile—that is to say, 6,080 feet—in an hour, a knot shall run out during
the half-minute run of the minute glass. This is attained by putting
the knots 1/120 × 6,080 = 50 feet 7 inches apart. As one sailor heaved
the log over he gave a stamp on the deck and allowed the cord to run
out through his fingers. Another sailor then turned the sand-glass.
When the sand had all run out, showing that half a minute had passed,
the man who was letting the cord run through his fingers gripped it
fast, and observed how many knots or parts of knots of string had run
out, and thus was able to tell how many “knots” per half-minute the
vessel was going, that is to say, how many nautical miles an hour.

The modern plan of observing the speed of vessels is different. Now we
use a patent log, consisting of a miniature screw propeller tied to a
string and dragged through the water after the vessel. As it is pulled
through the water it revolves, and the number of revolutions it makes
shows how much water it has passed through, and thus what distance
it has gone. The number of revolutions is measured by a counting
mechanism, and can be read off when the log is pulled in. Or sometimes
the screw is attached to a stiff wire, and the counting mechanism is
kept on board the ship.

We use the expression “knots an hour” quite incorrectly. It should be
“knots per half-minute,” or “nautical miles an hour.”

It is easy to use the flow of sand for all sorts of purposes to measure
time. Thus, if sand be allowed to flow from a hopper through a fine
hole into a bucket, the bucket may be arranged so that when a given
time has elapsed, and a given weight of sand has therefore fallen, the
bucket shall tip over, and release a catch, which shall then allow
a weight to fall and any mechanical operation to be done that is
required. Thus, for example, we might put an egg in a small holder tied
to a string and lower it into a saucepan of boiling water. The string
might have a counter-weight attached to it, acting over a pulley and
thus always trying to pull it up out of the water. But this might be
prevented by a pin passing through a loop in the string and preventing
it moving. A hopper or funnel might be filled with sand which was
allowed gradually to escape into a small tip-waggon or other similar
device, so that when a given amount of sand had entered the tip-waggon
would tip over, lurch the pin out of the loop, and thus release the
weight, which in its turn would pull the egg up out of the water in
three minutes or any desired time after it had been put in, or a hole
could be made in the saucepan, furnished with a little tap, and the
water that ran out might be made to fall into a tip-waggon and tip it
over, and thus when it had run out to put an extinguisher on to the
spirit lamp that was heating the saucepan, and at the same time make
a contact and ring an electric bell. By this means the egg would be
always exactly cooked to the right amount, would be kept warm after it
was cooked, and a signal given when it was ready.

[Illustration: FIG. 15.]

The sketch shows such an arrangement. The saucepan is about three
inches in diameter and two inches high. When filled with water it will
hold an egg comfortably. The extinguisher _E_, mounted on a hinge _Q_,
is turned back, and the spirit lamp _L_ is lit. As soon as the water
boils, the tap _T_ is turned, and the water gradually trickles away
into the tip-waggon. As soon as it is full it tips over and strikes the
arm _X_ of the extinguisher, and turns the lamp out. The little hot
water left in the saucepan will keep the egg warm for some time. The
waggon _W_ must have a weight _P_ at one end of it, and the fulcrum
must be nearer to that end, so that when empty it rests with the end
_P_ down, but when full it tips over on the fulcrum, when the waggon
has received the right quantity of water. I leave to the ingenious
reader the task of working out the details of such a machine, which, if
made properly, will act very well and may be made for a number of eggs
and worked with very little trouble.

[Illustration: FIG. 16.]

Mercury has been used also as an hour-glass. The orifice must be
exceedingly fine. Or a bubble of mercury may be put into a tube which
contains air, and made gradually as it falls to drive the air out
through a minute hole. The difficulty is to get the hole fine enough.
All that can be done is to draw out a fine tube in the blow-lamp, break
it off, and put the broken point in the blow-lamp until it is almost
completely closed up. A tube may thus be made about twelve inches long
that will take twelve hours for a bubble of mercury to descend in it.
But the trouble of making so small a hole is considerable.

[Illustration: FIG. 17.]

King Alfred is said to have used candles made of wax to mark the time.
As they blew about with the draughts, he put them in lanterns of horn.
They had no glass windows in those days, but only openings closed
with heavy wooden shutters. These large shutters were for use in fine
weather. Smaller shutters were made in them, so as to let a little
light in in rainy weather without letting in too much wind and rain.

Rooms must then have been very draughty, so that people required to
wear caps and gowns, and beds had thick curtains drawn round them.
When glass was first invented it was only used by kings and princes,
and glass casements were carried about with them to be fixed into
the windows of the houses to which they came, and removed at their
departure.

Oil lamps were also used to mark the time. Some of them certainly as
early as the fifteenth century were made like bird-bottles; that is
to say, they consisted of a reservoir closed at the top with a pipe
leading out of the bottom. When full, the pressure of the external
atmosphere keeps the oil in the bottle, and the oil stands in the neck
and feeds the wick. As the oil is consumed bubbles of air pass back
along the neck and rise up to the top of the oil, the level of which
gradually sinks. Of course the time shown by the lamp varies with the
rate of burning of the oil, and hence with the size of the wick, so
that the method of measuring time is a very rough one.


APPENDIX.

To make a sun-dial, procure a circular piece of zinc, about ⅛ inch
thick, and say twelve inches in diameter. Have a “style” or “gnomon”
cast such that the angle of its edge equals the latitude of the place
where the sun-dial is to be set up. This for London will be equal to
51° 30´´. A pattern may be made for this in wood; it should then be
cast in gun-metal, which is much better for out-of-door exposure than
brass. On a sheet of paper draw a circle _A B C_ with centre _O_. Make
the angle _B O D_ equal to the latitude of the place for London = 51°
30´´. From _A_ draw _A E_ parallel to _O B_ to meet _O D_ in _E_, and
with radius _O E_ describe another circle about _O_. Divide the inner
circle _A B C_ into twenty-four parts, and draw radii through them
from _O_ to meet the larger circle. Through any divisions (say that
corresponding to two o’clock) draw lines parallel to _O B_, _O C_,
respectively to meet in _a_. Then the line _O a_ is the shadow line
of the gnomon at two o’clock. The lines thus drawn on paper may be
transferred to the dial and engraved on it, or else eaten in with acid
in the manner in which etchings are done.

[Illustration: FIG. 18.]

The centre _O_ need not be in the centre of the zinc disc, but may
be on one side of it, so as to give better room for the hours, etc.
A motto may be etched upon the dial, such as “Horas non numero nisi
serenas,” or “Qual ’hom senza Dio, son senza sol io,” or any suitable
inscription, and the dial is ready for use. It is best put up by
turning it till the hour is shown truly as compared with a correctly
timed watch. It must be levelled with a spirit level. It must be
remembered that the sun does not move quite uniformly in his yearly
path among the fixed stars. This is because he moves not in a circle,
but in an ellipse of which the earth is in one of the foci. Hence the
hours shown on the dial are slightly irregular, the sun being sometimes
in advance of the clock, sometimes behind it. The difference is never
more than a quarter of an hour. There is no difference at midsummer and
midwinter.

[Illustration: FIG. 19.]

Civil time is solar time averaged, so as to make the hours and days all
equal. The difference between civil time and apparent solar time is
called the equation of time, and is the amount by which the sun-dial is
in advance of or in retard of the clock. In setting a dial by means of
a watch, of course allowance must be made for the equation of time.




CHAPTER II.


In the last chapter a short description has been given of the ideas of
the ancients as to the nature of the earth and heavens. Before we pass
to the changes introduced by modern science, it will be well to devote
a short space to an examination of ancient scientific ideas.

All science is really based upon a combination of two methods,
called respectively inductive and deductive reasoning. The first of
these consists in gathering together the results of observation and
experiment, and, having put them all together, in the formulation of
universal laws. Having, for example, long observed that all heavy
things tended to go towards the centre of the earth, we might conclude
that, since the stars remain up in the sky, they can have no weight.
The conclusion would be wrong in this case, not because the method
is wrong, but because it is wrongly applied. It is true that all
heavy things _tend_ to go to the centre of the earth, but if they are
being whirled round like a stone in a sling the centrifugal force
will counteract this tendency. The first part of the reasoning would
be inductive, the second deductive. All this reasoning consists,
therefore, in forming as complete an idea as possible respecting the
nature of a thing, and then concluding from that idea what the thing
will do or what its other properties will be. In fact, you form correct
ideas, or “concepts,” as they are called, and reason from them.

But the danger arises when you begin to reason before you are sure of
the nature of your concepts, and this has been the great source of
error, and it was this error that all men of science so commonly fell
into all through ancient and modern times up to the seventeenth century.

Of course, if it were possible by mere observation to derive a
complete knowledge of any objects, it would be the simplest method.
All that would be necessary to do would be to reason correctly from
this knowledge. Unfortunately, however, it is not possible to obtain
knowledge of this kind in any branch of science.

The ancient method resembled the action of one who should contend that
by observing and talking to a man you could acquire such a knowledge of
his character as would infallibly enable you to understand and predict
all his actions, and to take little trouble to see whether what he did
verified your predictions.

The only difference between the old methods and the new is that in
modern times men have learned to give far more care to the formation
of correct ideas to start with, are much more cautious in arguing
from them, and keep testing them again and again on every possible
opportunity.

The constant insistence on the formation of clear ideas and the
practice of, as Lord Bacon called it, “putting nature to the torture,”
is the main cause of the advance of physical science in modern times,
and the want of application of these principles explains why so little
progress is being made in the so-called “humanitarian” studies, such as
philosophy, ethics, and politics.

The works of Aristotle are full of the fallacious method of the old
system. In his work on the heavens he repeatedly argues that the
heavenly bodies must move in circles, because the circle is the most
perfect figure. He affects a perplexity as to how a circle can at the
same time be convex and also its opposite, concave, and repeatedly
entangles his readers in similar mere word confusion.

Regarded as a man of science, he must be placed, I think, in spite
of his great genius, below Archimedes, Hipparchus, and several other
ancient astronomers and physicists.

His errors lived after him and dominated the thought of the middle
ages, and for a long time delayed the progress of science.

The other great writer on astronomy of ancient times was Ptolemy of
Alexandria.

His work was called the “Great Collection,” and was what we should
now term a compendium of astronomy. Although based on a fundamental
error, it is a thoroughly scientific work. There is none of the false
philosophy in it that so much disfigures the work of Aristotle. The
reasons for believing that the earth is at rest are interesting.
Ptolemy argues that if the earth were moving round on its axis once in
twenty-four hours a bird that flew up from it would be left behind.
At first sight this argument seems very convincing, for it appears
impossible to conceive a body spinning at the rate at which the earth
is alleged to move, and yet not leaving behind any bodies that become
detached from it.

On the other hand, the system which taught that the sun and planets
moved round the earth, and which had been adopted largely on account
of its supposed simplicity, proved, on further examination, to be
exceedingly complicated. Each planet, instead of moving simply and
uniformly round the earth in a circle, had to be supposed to move
uniformly in a circle round another point that moved round the earth in
a circle. This secondary circle, in which the planet moved, was called
an epicycle. And even this more complicated view failed to explain the
facts.

A system which, like that of Aristotle and Ptolemy, was based on
deductions from concepts, and which consisted rather of drawing
conclusions than of examining premises, was very well adapted to
mediæval thought, and formed the foundation of astronomy and geography
as taught by the schoolmen.

[Illustration: FIG. 20.]

The poem of Dante accurately represents the best scientific knowledge
of his day. According to his views, the centre of the earth was a fixed
point, such that all things of a heavy nature tended towards it. Thus
the earth and water collected round it in the form of a ball. He had no
idea of the attraction of one particle of matter for another particle.
The only conception he had of gravity was of a force drawing all heavy
things to a certain point, which thus became the point round which the
world was formed. The habitable part of the earth was an island, with
Jerusalem in the middle of it _J_. Round this island was an ocean _O_.
Under the island, in the form of a hollow cone, was hell, with its
seven circles of torment, each circle becoming smaller and smaller,
till it got down into the centre _C_. Heaven was at the opposite side
_H_ of the earth to Jerusalem, and was beyond the circles of the
planets, in the _primum mobile_. When Lucifer was expelled from heaven
after his rebellion against God, having become of a nature to be
attracted to the centre of the earth, and no longer drawn heavenwards,
he fell from heaven, and impinged upon the earth just at the antipodes
of Jerusalem, with such violence that he plunged right through it to
the centre, throwing up behind him a hill. On the summit of this hill
was the Garden of Eden, where our first parents lived, and down the
sides of the hill was a spiral winding way which constituted purgatory.
Dante, having descended into hell, and passed the centre, found his
head immediately turned round so as to point the other way up, and,
having ascended a tortuous path, came out upon the hill of Purgatory.
Having seen this, he was conducted to the various spheres of the
planets, and in each sphere he became put into spiritual communion with
the spirits of the blessed who were of the character represented by
that sphere, and he supposes that he was thus allowed to proceed from
sphere to sphere until he was permitted to come into the presence of
the Almighty, who in the _primum mobile_ presided over the celestial
hosts.

The astronomical descriptions given by Dante of the rising and setting
of the sun and moon and planets are quite accurate, according to the
system of the world as conceived by him, and show not only that he was
a competent astronomer, but that he probably possessed an astrolabe and
some tables of the motions of the heavenly bodies.

Our own poet Chaucer may also be credited with accurate knowledge of
the astronomy of his day. His poems often mention the constellations,
and one of them is devoted to a description of the astrolabe, an
instrument somewhat like the celestial globe which used to be employed
in schools.

But with the revival of learning in Europe and the rise of freedom of
thought, the old theories were questioned in more than one quarter.

It occurred to Copernicus, an ecclesiastic who lived in the sixteenth
century, to re-examine the theory that had been started in ancient
times, and to consider what explanation of the appearance of the
heavenly bodies could be given on the hypothesis put forward by
Pythagoras, that the earth moved round on its own axis, and also round
the sun.

It may appear rather curious that two theories so different, one that
the sun goes round the earth and the other that the earth goes round
the sun, should each be capable of explaining the observed appearances
of those bodies. But it must be remembered that motion is relative. If
in a waltz the gentleman goes round the lady, the lady also goes round
the gentleman. If you take away the room in which they are turning,
and consider them as spinning round like two insects in space, who is
to say which of them is at rest and which in motion? For motion is
relative. I can consider motion in a train from London to York. As I
leave London I get nearer to York, and I move with respect to London
and York. But if both London and York were annihilated how should I
know that I was in motion at all? Or, again, if, while I was at rest
in the train at a station on the way, instead of the train moving the
whole earth began to move in a southward direction, and the train in
some way were left stationary, then, though the earth was moving, and
the train was at rest, yet, so far as I was concerned, the train would
appear to have started again on its journey to York, at which place it
would appear to arrive in due time. The trees and hedges would fly by
at the proper rate, and who was to say whether the train was in motion
or the earth?

The theory of Copernicus, however, remained but a theory. It was
opposed to the evidence of the senses, which certainly leads us to
think that the earth is at rest, and it was opposed also to the ideas
of some among the theologians who thought that the Bible taught us that
the earth was so fast that it could not be moved. Therefore the theory
found but little favour. It was in fact necessary before the question
could be properly considered on its merits that more should be known
about the laws of motion, and this was the principal work of Galileo.

The merit of Galileo is not only to have placed on a firm basis the
study of mechanics, but to have set himself definitely and consciously
to reverse the ancient methods of learning.

He discarded authority, basing all knowledge upon reason, and protested
against the theory that the study of words could be any substitute for
the study of things.

Alluding to the mathematicians of his day, “This sort of men,” says
Galileo in a letter to the astronomer Kepler, “fancied that philosophy
was to be studied like the ‘Æneid’ or ‘Odyssey,’ and that the true
reading of nature was to be detected by the collating of texts.” And
most of his life was spent in fighting against preconceived ideas. It
was maintained that there could only be seven planets, because God
had ordered all things in nature by sevens (“Dianoia Astronomica,”
1610); and even the discoveries of the spots on the sun and the
mountains in the moon were discredited on the ground that celestial
bodies could have no blemishes. “How great and common an error,”
writes Galileo, “appears to me the mistake of those who persist in
making their knowledge and apprehension the measure of the knowledge
and apprehension of God, as if that alone were perfect which they
understand to be so. But ... nature has other scales of perfection,
which we, being unable to comprehend, class among imperfections.

“If one of our most celebrated architects had had to distribute the
vast multitude of fixed stars over the great vault of heaven, I believe
he would have disposed them with beautiful arrangements of squares,
hexagons, and octagons; he would have dispersed the larger ones among
the middle-sized or lesser, so as to correspond exactly with each
other; and then he would think he had contrived admirable proportions;
but God, on the contrary, has shaken them out from His hand as if by
chance, and we, forsooth, must think that He has scattered them up
yonder without any regularity, symmetry, or elegance.”

In one of Galileo’s “Dialogues” Simplicio says, “That the cause that
the parts of the earth move downwards is notorious, and everyone knows
that it is gravity.” Salviati replies, “You are out, Master Simplicio:
you should say that everyone knows that _it is called_ gravity; I do
not ask you for the name, but for the nature, of the thing of which
nature neither you nor I know anything.”

Too often are we still inclined to put the name for the thing, and to
think when we use big words such as art, empire, liberty, and the
rights of man, that we explain matters instead of obscuring them. Not
one man in a thousand who uses them knows what he means; no two men
agree as to their signification.

The relativity of motion mentioned above was very elegantly illustrated
by Galileo. He called attention to the fact that if an artist were
making a drawing with a pen while in a ship that was in rapid passage
through the water, the true line drawn by the pen with regard to the
surface of the earth would be a long straight line with some small
dents or variations in it. Yet the very same line traced by the pen
upon a paper carried along in the ship made up a drawing. Whether you
saw a long uneven line or a drawing in the path that the pen had traced
depended altogether on the point of view with which you regarded its
motion.

[Illustration: FIG. 21.]

But the first great step in science which Galileo made when quite a
young professor at Pisa was the refutation of Aristotle’s opinion that
heavy bodies fell to the earth faster than light ones. In the presence
of a number of professors he dropped two balls, a large and a small
one, from the parapet of the leaning tower of Pisa. They fell to the
ground almost exactly in the same time. This experiment is quite an
easy one to try. One of the simplest ways is as follows: Into any beam
(the lintel of a door will do), and about four inches apart, drive
three smooth pins so as to project each about a quarter of an inch;
they must not have any heads. Take two unequal weights, say of 1 lb.
and 3 lbs. Anything will do, say a boot for one and pocket-knife for
the other; fasten loops of fine string to them, put the loops over the
centre peg of the three, and pass the strings one over each of the side
pegs. Now of course if you hitch the loops off the centre peg _P_ the
objects will be released together. This can be done by making a loop
at the end of another piece of string, _A_, and putting it on to the
centre peg behind the other loops. If the string be pulled of course
the loop on it pulls the other two loops off the central peg, and
allows the boot and the knife to drop. The boot and the knife should be
hung so as to be at the same height. They will then fall to the ground
together. The same experiment can be tried by dropping two objects from
an upper window, holding one in each hand, and taking care to let them
go together.

[Illustration: FIG. 22.]

This result is very puzzling; one does not understand it. It appears as
though two unequal forces produced the same effect. It is as though a
strong horse could run no faster than a weaker one.

The professors were so irritated at the result of this experiment, and
indeed at the general character of young Professor Galileo’s attacks on
the time-honoured ideas of Aristotle, that they never rested till they
worried him out of his very poorly paid chair at Pisa. He then took a
professorship at Padua.

Let us now examine this result and see why it is that the ideas we
should at first naturally form are wrong, and that the heavy body will
fall in exactly the same time as the light one.

We may reason the matter in this way. The heavy body has more force
pulling on it; that is true, but then, on the other hand there is more
matter which has got to be moved. If a crowd of persons are rushing out
of a building, the total force of the crowd will be greater than the
force of one man, but the speed at which they can get out will not be
greater than the speed of one man; in fact, each man in the crowd has
only force enough to move his own mass. And so it is with the weights:
each part of the body is occupied in moving itself. If you add more to
the body you only add another part which has itself to move. A hundred
men by taking hands cannot run faster than one man.

But, you will say, cannot a man run faster than a child? Yes, because
his impelling power is greater in proportion to his weight than that of
a child.

If it were the fact that the attraction of gravity due to the earth
acted on some bodies with forces greater in proportion to their
masses than the forces that acted on other bodies, then it is true
that those different bodies would fall in unequal time. But it is
an experimental fact that the attractive force of gravity is always
exactly proportional to the mass of a body, and the resistance to
motion is also proportional to mass, hence the force with which a
body is moved by the earth’s attraction is always proportional to the
difficulty of moving the body. This would not be the case with other
methods of setting a body in motion. If I kick a small ball with all
my might, I shall send it further than a kick of equal strength would
send a heavier ball. Why? Because the impulse is the same in each case,
but the masses are different. But if those balls are pulled by gravity,
then, by the very nature of the earth’s attraction (the reason of which
we cannot explain), the small ball receives a little pull, and the big
ball receives a big pull, the earth exactly apportioning its pull in
each case to the mass of the body on which it has to act. It is to this
fact, that the earth pulls bodies with a strength always in each case
exactly proportional to their masses, that is due the result that they
fall in equal times, each body having a pull given to it proportional
to its needs.

The error of the view of Aristotle was not only demonstrated by
Galileo by experiment, but was also demonstrated by argument. In this
argument Galileo imitated the abstract methods of the Aristotelians,
and turned those methods against themselves. For he said, “You” (the
Aristotelians) “say that a lighter body will fall more slowly than a
heavy one. Well, then, if you bind a light body on to a heavy one by
means of a string, and let them fall together, the light body ought
to hang behind, and impede the heavy body, and thus the two bodies
together ought to fall more slowly than the heavy body alone; this
follows from your view: but see the contradiction. For the two bodies
tied together constitute a heavier body than the heavy body alone, and
thus, on your own theory, ought to fall more quickly than the heavy
body alone. Your theory, therefore, contradicts itself.”

The truth is that each body is occupied in moving itself without
troubling about moving its neighbour, so that if you put any number of
marbles into a bag and let them drop they all go down individually, as
it were, and all in the time which a single marble would take to fall.
For any other result would be a contradiction. If you cut a piece of
bread in two, and put the two halves together, and tie them together
with a thread, will the mere fact that they are two pieces make each of
them fall more slowly than if they were one? Yet that is what you would
be bound to assert on the Aristotelian theory. Hold an egg in your
open hand and jump down from a chair. The egg is not left behind; it
falls with you. Yet you are the heavier of the two, and on Aristotelian
principles you ought to leave the egg behind you. It is true that when
you jump down a bank your straw hat will often come off, but that is
because the air offers more resistance to it than the air offers to
your body. It is the downward rush through the air that causes your hat
to be left behind, just as wind will blow your hat off without blowing
you away. For since motion is relative, it is all one whether you jump
down through the air, or the air rushes past you, as in a wind. If
there were no air, the hat would fall as fast as your body.

This is easy to see if we have an airpump and are thus enabled to
pump out almost all the air from a glass vessel. In that vessel so
exhausted, a feather and a coin will fall in equal times. If we have
not an airpump, we can try the experiment in a more simple way. For
let us put a feather into a metal egg-cup and drop them together. The
cup will keep the air from the feather, and the feather will not come
out of the cup. Both will fall to the ground together. But if the
lighter body fall more slowly, the feather ought to be left behind. If,
however, you tie some strings across a napkin ring so as to make a sort
of rough sieve, and put a feather in it, and then drop the ring, then
as the ring falls the air can get through the bottom of the ring and
act on the feather, which will be left floating as the ring falls.

Let us now go on to examine the second fallacy that was derived from
the Aristotelians, and that so long impeded the advance of science,
namely, that the earth must be at rest.

The principal reason given for this was that if bodies were thrown
up from the earth they ought, if the earth were in motion, to remain
behind. Now, if this were so, then it would follow that if a person
in a train which was moving rapidly threw a ball vertically, that is
perpendicularly, up into the air, the ball, instead of coming back into
his hand, ought to hit the side of the carriage behind him. The next
time any of my readers travel by train he can easily satisfy himself
that this is not so. But there are other ways of proving it. For
instance, if a little waggon running on rails has a spring gun fixed in
it in a perpendicular position, so arranged that when the waggon comes
to a particular point on the rails a catch releases the trigger and
shoots a ball perpendicularly upwards, it will be found that the ball,
instead of going upwards in a vertical line, is carried along over the
waggon, and the ball as it ascends and descends keeps always above the
waggon, just as a hawk might hover over a running mouse, and finally
falls not behind the waggon, but into it.

So, again, if an article is dropped out of the window of a train, it
will not simply be left behind as it falls, but while it falls it will
also partake of the motion of the train, and touch the ground, not
behind the point from which it was dropped, but just underneath it.

The reason is, that when the ball is dropped or thrown it acquires
not only the motion given to it by the throw, or by gravity, but it
takes also the motion of the train from which it is thrown. If a ball
is thrown from the hand, it derives its motion from the motion of the
hand, and if at the time of throwing the person who does so is moving
rapidly along in a train, his hand has not only the outward motion
of the throw, but also the onward motion of the train, and the ball
therefore acquires both motions simultaneously. Hence then it is not
correct reasoning to say, because a ball thrown up vertically falls
vertically back to the spot from which it was thrown, that therefore
the earth must be at rest; the same result will happen whether the
earth is at rest or in motion. You can no more tell whether the earth
is at rest or in motion from the behaviour of falling bodies than you
can tell whether a ship on the ocean is at rest or in motion from the
behaviour of bodies on it.

But you will say. Then why do we feel sea-sick on a ship? The answer
is, that that is because the motion of the ship is not uniform. If the
earth, instead of turning round uniformly, were to rock to and fro,
everything on it would be flung about in the wildest fashion. For as
soon as the earth had communicated its motion to a body which then
moved with the earth, if the earth’s motion were reversed, the body
would go on like a passenger in a train on which the break is quickly
applied, and he would be shot up against the side of the room. Nay,
more, the houses would be shaken off their foundations. Changes of
motion are perceptible _so long as the change is going on_. We are
therefore justified in inferring from the behaviour of bodies on the
earth, not that the earth is at rest, but that it is either at rest, or
else, if it is in motion, that its motion is uniform and not in jerks
or variable.

[Illustration: FIG. 23.]

For if it were not so, consider what would be happening around us. The
earth is about 8,000 miles in diameter, and a parallel of latitude
through London is therefore about 19,000 miles long, and this space
is travelled in twenty-four hours. So that London is spinning through
space at the rate of over 1,000 feet a second, due to the earth’s
rotary motion alone, not to speak of the motion due to the earth’s
path round the sun. If a boy jumped up two and a half feet into the
air, he would take about half a second to go up and come down, but if
in jumping he did not partake of the earth’s motion, he would land
more than 500 feet to the westward of the point from which he jumped
up, and if he did it in a room, he would be dashed against the wall
with a force greater than he would experience from a drop down from
the top of Mont Blanc. He would be not only killed, but dashed into an
indistinguishable mass. If the earth suddenly stood still, everything
on it would be shaken to pieces. It is bad enough to have the
concussion of a train going thirty miles an hour when dashed against
some obstacle. But the concussion due to the earth’s stoppage would
be as of a train going about 800 miles an hour, which would smash up
everything and everybody.

Thus, then, the first effect of the new ideas formulated by Galileo was
to show that the Copernican theory that the earth moved round on its
axis, and round the sun, was in agreement with the laws of motion. In
fact, he introduced quite new ideas of force, and these ideas I must
now endeavour to explain.

Let us consider what is meant by the word “force.” If I press my
hand against the table, I exert force. The harder I press, the more
force there is. If I put a weight on a stand, the weight presses the
stand down with a force. If I squeeze a spring, the spring tries to
recover itself and exerts a certain force. In all these cases force
is considered as a pressure. And I can measure the force by seeing
how much it will press things. If I take a spring, and press it in an
inch, it takes perhaps a force of 1 lb. It will take a force of 2 lbs.
to press it in another inch. Or again, if I pull it out an inch, it
takes a force of 1 lb. If I pull it out another inch, it takes a force
of 2 lbs. We thus always get into the habit of conceiving forces as
producing pressures and being measured by pressures.

[Illustration: FIG. 24.]

This is a perfectly legitimate way of looking at the matter, just as
the cook’s method of employing a spring balance to weigh masses of meat
is a perfectly legitimate way of estimating the forces acting upon
bodies at rest. But when you come to consider the laws of the pendulums
of clocks, to which all that I am saying is a preparation, then you
have to deal with bodies in motion. And for this purpose a new idea of
force altogether is requisite. We shall no longer speak of forces as
producing _pressures_. We shall treat them quite independently of their
pressing power. The sun exerts a force of attraction on the earth, but
it does not press upon it. It exerts its force at a distance. Hence
then we want a new idea of “force.” This idea is to be the following.
We will consider that when a force acts upon a body it endeavours to
cause it to move; in fact, it tries to impart motion to the body. We
may treat this motion as a sort of thing or property. The longer the
force acts on the body, the more motion it imparts to it, provided the
body is free to receive that motion. So that we may say that the test
of the strength of the force is how much motion it can give to a body
of a given mass in a given time. It does not matter how the force acts.
It may act by means of a string and pull it; it may act by means of a
stick and push it; it may act by attraction and draw it; it may act
by repulsion and repel it; it may act as a sort of little spirit and
fly away with it. In all these cases it _acts_. The more it acts, the
more effect it has. In double the time it produces double the motion.
If the mass is big, it takes more force to make the mass move; if the
mass of the body is small, it is moved more easily. Therefore when we
want to measure a force in this way we do not press it against springs
to see how much it will press them in. What we do is to cause it to act
on bodies that are free to move and see what motions it will produce
in them. Of course we can only do this with things that are free to
move. You cannot treat force in this way if you have only a pair of
scales; in that case you would have to be content with simply measuring
pressures. It is important clearly to grasp this idea. If a body has
a certain mass, then the force acting on it is measured by the amount
of motion that will in a given time be imparted to that mass, provided
that the mass is free to move. This is to be our definition of force.

Therefore, by the action of an attraction or any other force on a body
free to move; motion is continually being imparted to the body. Motion
is, as it were, poured into it, and therefore the body continually
moves faster and faster.

Here is a ball flying through the air. Let us suppose that forces are
acting on it. How can we measure them? We cannot feel what pressures
are being exerted on it. The only thing we can do is to watch its
motions, and see how it flies. If it goes more and more quickly, we
say, “There is propelling force acting on it”; if it begins to stop,
we say again, “There is retarding force acting on it.” So long as it
does not change its speed or direction, we say, “There is no force
acting on it.” By this method, therefore, we tell whether a body is
being acted on by force, simply by observing its speed or its change of
speed. Merely to say a body is _moving_ does not tell us that force is
acting on it. All we know is that, if it is moving, force _has_ acted
on it. It is only when we see it changing its speed or direction, that
is changing its motion, that we say _force_ is acting. Every change of
motion, either in direction or speed, must be the result of force, and
must be proportional to that force. This is what we mean when we say
motion is the test and measure of force.

This most interesting way of looking at the matter lies at the root
of the whole theory of mechanics. It is the foundation of the system
which the stupendous genius of Newton conceived in order to explain the
motion of the sun, moon, and stars.

Forces were treated by him as proportional to the motions, and the
motions proportional to the forces, and with this idea he solved a part
of the riddle of the universe. Galileo had partly seen the same thing,
but he never saw it so clearly as Newton. Great discoveries are only
made by seeing things clearly. What required the force of a genius in
one age to see in the next may be understood by a child.

Hence then we say a force is that which in a given time produces a
given motion in a given mass which is free to move.

You must have time for a force to act in; for however great the force,
in no time there can be no motion. You must have mass for a force to
act on; no mass, no effect. You must have free space for the mass to
move in; no freedom to move, no movement.

But what is this “mass”? We do not know; it is a mystery. We call it
“quantity of matter.” In uniform substances it varies with size. Double
the volume, double the mass. Cut a cake in half, each half has the
same “mass.” But then is mass “weight”? No, it is not. _Weight_ is the
action of the earth’s attraction on matter. No earth to attract, and
you would have no weight, but you would still have “mass.” What then
is matter? Of that we have no idea. The greatest minds are now at work
upon it. But _mass_ is quantity of matter. Knock a brick against your
head, and you will know what mass is. It is not the weight of the brick
that gives you a bump; it is the mass. Try to throw a ball of lead, and
you will know what mass is. Try to push a heavy waggon, and you will
know what mass is. _Weights_, that is earth attractions on masses, are
proportional to the masses at the same place. This, as we have seen, is
known by experiment.

Therefore, when a force acts for a certain time on a mass that is free
to move, however small the force and however small the time, that body
will move. When a baby in a temper stamps upon the earth it makes the
earth move—not much, it is true, but still it moves; nay, more, in
theory, not a fly can jump into the air without moving the earth and
the whole solar system. Only, as you may imagine they do not show it
appreciably. Still, in theory the motion is there.

Hence then there are two different ways of considering and estimating
forces, one suitable for observations on bodies at rest, the other
suitable for observations of bodies that are free to move. The force
of course always tends to produce motion. If, however, motion is
impossible, then it develops pressures which we can measure, and
calculate, and observe. If the body is free to move, then the force
produces motions which we can also measure, calculate, and observe.
And we can compare these two sets of effects. We can say, “A force
which, acting on a ball of a mass of one pound, would produce such and
such motions, would if it acted on a certain spring produce so much
compression.”

The attraction of the earth on masses of matter that are not free to
move gives rise to forces which are called weights. Thus the attraction
of gravitation on a mass of one pound produces a pressure equal to a
weight of one pound. Unfortunately the same word “pound” is used to
express both the mass and the weight, and has come down to us from days
when the nature of mass was not very well appreciated. But great care
must be taken not to confuse these two meanings.

But the earth’s attractions and all other forces acting upon matter
which is free to move give rise to changes of motion. The word used for
a change of motion is “acceleration” or a quickening. “He accelerated
his pace,” we say. That is, he quickened it; he added to his motion. So
that _force_, acting on _mass_ during a _time_, produces acceleration.

From this, then, it follows that if a _force_ continues to act on a
body the body keeps moving quicker and quicker. When the force stops
acting, the motion already acquired goes on, but the acceleration
stops. That is to say, the body goes on moving in a straight line
uniformly at the pace it had when the force stopped.

If, then, a body is exposed to the action of a force, and held tight,
what will happen? It will, of course, remain fixed. Now let it go—it
will then, being a free body, begin to move. As long as the force
acts, the force keeps putting more and more motion into the body, like
pouring water into a jug, the longer you pour the faster the motion
becomes. The body keeps all the motion it had, and keeps adding all the
motion it gains. It is like a boy saving up his weekly pocket-money:
he has what he had, and he keeps adding to that. So if in one second
a motion is imparted of one foot a second, then in another second a
motion of one foot a second more will be added, making together a
motion of two feet a second; in another second of force action the
motion will have been increased or “accelerated” by another foot per
second, and so on. The speed will thus be always proportional to
the force and the time. If we write the letter V to represent the
motion, or speed, or velocity; F to represent the acceleration or gain
of motion; and T to represent the time, then V = FT. Here V is the
velocity the body will have acquired at the end of the time T, if free
to move and submitted to a force capable of producing an acceleration
of F feet per second in a unit of time.

V is the final velocity. The average velocity will be 1/2 V, for it
began with no velocity and increased uniformly. How far will the body
have fallen in the interval? Manifestly we get that by multiplying the
time by the average velocity, that is S = 1/2 VT, where V, as I said,
is the final velocity, but we found that V = FT. Hence by substitution
S = 1/2 FT × T = 1/2 FT².

It is to be carefully borne in mind that these letters V, S, and T
do not represent velocities, spaces, and times, but merely represent
arithmetical numbers of units of velocities, spaces, and times. Thus
V represents V feet per second, S represents S feet, and T represents
T seconds. And when we use the equation V = FT we do not mean that
by multiplying a force by a time you can produce a velocity. If, for
instance, it be true that you can obtain the number of inhabitants (H)
in London by multiplying the average number of persons (P) who live
in a house by the number of houses (N), this may be expressed by the
equation H = PN. But this does not mean that by multiplying people into
houses you can produce inhabitants. H, P, and N are numbers of units,
and they are _numbers only_.

Therefore when a body is being acted on by an accelerating force
it tends to go faster and faster as it proceeds, and therefore its
velocity increases with the time. But the space passed through
increases faster still, for as the time runs on not only does the
space passed through increase, but the rate of passing also gets
bigger. It goes on increasing at an increasing rate. It is like a man
who has an increasing income and always goes on saving it. His total
mounts up not merely in proportion to the time, but the very rate of
increase also increases with the time, so that the total increase is
in proportion to the time multiplied into the time, in other words to
the square of the time. So then, if I let a body drop from rest under
the action of any force capable of producing an acceleration, the space
passed through will be as the square of the time.

Now let us see what the speed will be if the force is gravity, that is
the attraction of the earth.

Turning back to what was said about Galileo, it will be remembered that
he showed that all bodies, big and small, light and heavy, fell to the
earth at the same speeds. What is that speed? Let us denominate by G
the number of feet per second of increase of motion produced in a body
by the earth’s action during one second. Then the velocity at the end
of that second will be V = GT. The space fallen through will be S = 1/2
GT².

What I want to know then is this: how far will a body under the action
of gravity fall in a second of time?

This, of course, is a matter for measurement. If we can get a machine
to measure seconds, we shall be able to do it; but inasmuch as falling
bodies begin by falling sixteen feet in the first second and afterwards
go on falling quicker and quicker, the measurements are difficult.
Galileo wanted to see if he could make it easier to observe. He said
to himself, “If I can only water down the force of gravity and make
it weaker, so that the body will move very slowly under its action,
then the time of falling will be easier to observe.” But how to do it?
This is one of those things the discovery of which at once marks the
inventor.

[Illustration: FIG. 25.]

The idea of Galileo was, instead of letting the body drop vertically,
to make it roll slowly down an incline, for a body put upon an incline
is not urged down the incline with the same force which tends to make
it fall vertically.

Can any law be discovered tending to show what the force is with which
gravity tends to drag a mass down an incline?

There is a simple one, and before Galileo’s time it had been
discovered by Stevinus, an engineer. Stevinus’ solution was as follows.
Suppose that _A B C_ is a wedge-shaped block of wood. Let a loop of
heavy chain be hung over it, and suppose that there is a little pulley
at _C_ and no friction anywhere. Then the chain will hang at rest. But
the lower part, from _A_ to _B_, is symmetrical; that is to say, it
is even in shape on both sides. Hence, so far as any pull it exerts
is concerned, the half from _A_ to _D_ will balance the other half
from _B_ to _D_. Therefore, like weights in a scale, you may remove
both, and then the force of gravity acting down the plane on the part
_A C_ will balance the force of gravity acting vertically on the part
_C B_. Now the weight of any part of the chain, since it is uniform,
is proportional to its length. Hence, then, the gravitational force
down the plane of a piece whose weight equals _C A_ is equal to the
gravitational force vertically of a piece whose weight equals _C B_. In
other words, the force of gravity acting down a plane is diminished in
the ratio of _C B_ to _C A_.

But when a body falls vertically, then, as we have seen, S = 1/2 GT²,
where S is the space it will fall through, G the number of feet per
second of velocity that gravity, acting vertically on a body, will
produce in it in a second, and T the number of seconds of time. If
then, instead of falling vertically, the body is to fall obliquely down
a plane, instead of G we must put as the accelerating force

  G × (vertical height of the end of the plane)/(length of the plane).

To try the experiment, he took a beam of wood thirty-six feet long with
a groove in it. He inclined it so that one end was one foot higher than
the other. Hence the acceleration down the plane was 1/36 G, where G is
the vertical acceleration due to gravity which he wanted to discover.
Then he measured the time a brass ball took to run down the plane
thirty-six feet long, and found it to be nine seconds. Whence from
the equation given above 36 feet = 1/2 acceleration of gravity down
the plane × (9 seconds)². Whence it follows that the acceleration of
gravity down the plane is (36 × 2)/(9)² feet per second.

But the slope of the plane is one thirty-sixth to the vertical.
Therefore the vertical acceleration of gravity, _i.e._, the velocity
which gravity would induce in a vertical direction in a second, is
equal to thirty-six times that which it exercises down the plane,
_i.e._,

36 × (36 × 2)/(9)²; and this equals 32 feet per second.

Though this method is ingenious, it possesses two defects. One is the
error produced by friction, the other from failure to observe that
the force of gravity on the ball is not only exerted in getting it
down the plane, but also in rotating it, and for this no allowance has
been made. The allowance to be made for rotation is complicated, and
involves more knowledge than Galileo possessed. Still the result is
approximately true.

[Illustration: FIG. 26.]

The next attempt to measure G, that is the velocity that gravity will
produce on a body in a second of time, was made by Attwood, a Cambridge
professor. His idea was to weaken the force of gravity and thus make
the action slow, not by making it act obliquely, but by allowing it to
act, not on the whole, but only on a portion of the mass to be moved.
For this purpose he hung two equal weights over a very delicately
constructed pulley. Gravity, of course, could not act on these, for
any effect it produced on one would be negatived by its effect on
the other. The weights would therefore remain at rest. If, however,
a small weight _W_, equal say to a hundredth of the combined weight
of the weights _A_ and _B_ and _W_, were suddenly put on _A_, then it
would descend under an accelerating force equal to a hundredth part of
ordinary gravity. We should then have

    S (the space moved through by the weights) = 1/2 × G/100 × t².

With such a system, he found that in 7½ seconds the weights moved
through 9 feet. Whence he got

                        9 = 1/2 G/100 × (7½)².

From which

         G = (2 × 9 × 100)/(7½)² = 32 feet per second nearly.

Thus by letting gravity only act on a hundredth part of the total
weight moved, namely _A_, _B_, and _W_, he weakened its action 100
times, and thus made the time of falling and the space fallen through
sufficiently large to be capable of measurement. To sum up, when a body
free to move is acted upon by the force of gravity, its speed will
increase in proportion to the time it has been acted upon, and the
space it will pass through from rest is proportional to the square of
the time during which the accelerating force has acted on it.

Gravity is, of course, not the only accelerating force with which
we are acquainted. If a spring be suddenly allowed to act on a body
and pull it, the body begins to move, and its action is gradually
accelerated, just as though it were attracted, and the acceleration
of its motion will be proportional to the time during which the
accelerating force acts. Similarly, if gunpowder be exploded in a
gun-barrel, and the force thus produced be allowed to act on a bullet,
the motion of the bullet is accelerated so long as it is in the barrel.
When the bullet leaves the barrel it goes on with a uniform pace in a
straight line, which, however, by the earth’s attraction is at once
deflected into a curve, and altered by the resistance of the air.

[Illustration: FIG. 27.]

It has been already stated that motions may be considered independently
one of another, so that if a body be exposed to two different forces
the action of these forces can be considered and calculated each
independently of the other. Let us take an example of this law. We have
seen if a body is propelled forwards, and then the force acting on it
ceases, that it proceeds on with uniform unchanging velocity, and if
nothing impeded it, or influenced it, it would go on in a straight line
at a uniform speed.

We have also seen that if a body is exposed to the action of an
accelerating force such as gravity it constantly keeps being
accelerated, it constantly keeps gaining motion, and its speed becomes
quicker and quicker.

[Illustration: FIG. 28.]

Let us suppose a body exposed to both of these forces at the same time.
Shoot it out of a cannon, and let an accelerating force act on it, not
in the direction it is going, but in some other direction, say at right
angles. What will happen? In the direction in which it is going, its
speed will remain uniform. In the direction in which the accelerating
force is acting, it will move faster and faster. Thus along _A B_ it
will proceed uniformly. If it proceeded uniformly also along _A C_ (as
it would do if a simple force acted on it and then ceased to act), then
as a result it would go in the oblique line _A D_, the obliquity being
determined by the relative magnitude of the forces acting on it. But
how if it went uniformly along _A B_, but at an accelerated pace along
_A C_? Then while in equal times the distances along _A B_ would be
uniform the distances in the same times along _A C_ would be getting
bigger and bigger. It _would not describe a straight line; it would go
in a curve_. This is very interesting. Let us take an example of it.
Suppose we give a ball a blow horizontally; as soon as it quits the bat
it would of course go on horizontally in a straight line at a uniform
speed; but now if I at the same instant expose it to the accelerating
force of gravity, then, of course, while its horizontal movement will
go on uniformly, its downward drop will keep increasing at a speed
varying as the time. And while the total distances horizontally will
be uniform in equal times, the total downward drop from _A B_ will
be as the squares of the times. Here, then, you have a point moving
uniformly in a horizontal direction, but as the squares of the times in
a vertical direction. It describes a curve. What curve? Why, one whose
distances go uniformly one way, but increase as the squares the other
way.

[Illustration: FIG. 29.]

This interesting curve is called a parabola. With a ball simply hit by
a bat, the motion is so very fast that we cannot see it well. Cannot
we make it go slowly? Let us remember what Galileo did. He used an
inclined plane to water down his force of gravity. Let us do the same.
Let us take an inclined plane and throw on it a ball horizontally.
It will go in a curve. Its speed is uniform horizontally, but is
accelerated downwards. If we desire to trace the curve it is easy to
do. We coat the ball with cloth and then dip it in the inkpot. It will
then describe a visible parabola. If I tilt up the plane and make the
force of gravity big, the parabola is long and thin; if I weaken down
the force of gravity by making the plane nearly horizontal, then it is
wide and flat.

One can also show this by a stream of peas or shot. The little bullets
go each with a uniform velocity horizontally, and an accelerated force
downwards.

Instead of peas we can use water. A stream of it rushing horizontally
out of an orifice will soon bend down into a parabola.

Thus then I have tried to show what force is and how it is measured. I
repeat again, when a body is free to move, then, if no further force
acts on it, it will go on in a straight line at a uniform speed, but
if a force continues to act on it in any direction, then that force
produces in each unit of time a unit of acceleration in the direction
in which the force acts, and the result is that the body goes on moving
towards the direction of acceleration at a constantly increasing speed,
and hence passing over spaces that are greater and greater as the speed
increases. This is the notion of a “force.” In all that has been said
above it has been assumed that the attraction of gravity on a body
does not increase as that body gets nearer to the earth. This is not
strictly true; in reality the attractive force of gravity increases as
the earth’s centre is approached. But small distances through which
the weights in Attwood’s machine fall make no appreciable difference,
being as nothing compared to the radius of earth. For practical
purposes, therefore, the force may be considered uniform on bodies that
are being moved within a few feet of the earth’s surface. It is only
when we have to consider the motions of the planets that considerations
of the change of attractive force due to distance have to be considered.

I am glad to say that the most tiresome, or rather the most difficult,
part of our inquiry is now over. With the help of the notions already
acquired, we are now ready to get to the pendulum, and to show how it
came about that a boy who once in church amused himself by watching the
swinging of the great lamps instead of attending to the service laid
the foundation of our modern methods of measuring time.




CHAPTER III.


We have examined the action of a body under the accelerating or
speed-quickening force due to gravity, the attractive force of which on
any body is always proportional to the mass of that body. Let us now
consider another form of acceleration.

[Illustration: FIG. 30.]

Take the case of a strip of indiarubber. If pulled it resists and tends
to spring back. The more I pull it out the harder is the pull I have
to exert. This is true of all springs. It is true of spiral springs,
whether they are pulled out or pushed in, and in each case the amount
by which the spring is pulled out or pushed in is proportional to the
pressure. This law is called Hooke’s law. It was expressed by him in
Latin, “Ut tensio, sic vis”: “As the extension, so the force.” It is
true of all elastic bodies, and it is true whether they are pulled out
or pushed in or bent aside. The common spring balance is devised on
this principle. The body to be weighed is hung on a hook suspended from
a spring. The amount by which the spring is pulled out is a measure
of the weight of the body. If you take a fishing rod and put the butt
end of it on a table and secure it by putting something heavy on the
end, then the tip will bend down on account of its own weight. Mark the
point to which it goes. Now, if you hang a weight on the tip, the tip
will bend down a little further. If you put double the weight the tip
will go down double the distance, and so on until the fishing rod is
considerably bent, so that its form is altered and a new law of flexure
comes into play. Suppose I use a spring as an accelerating force. For
example, suppose I suspend a heavy ball by a string and then attach a
spiral spring to it and pull the spring aside. The ball will be drawn
after the spring. If then I let the ball go, it will begin to move. The
force of the spring will act upon it as an accelerating force, and the
ball will go on moving quicker and quicker. But the acceleration will
not be like that of gravity. There will be two differences. The pull of
the spring will in no way depend on the mass of the ball, and the pull
of the spring, instead of being constant, like the pull of gravity,
will become weaker and weaker as the ball yields to it. Consequently
the equations above given which determine the relations between this
space passed through, the velocity, and the time which were determined
in the case of gravity are no longer true, and a different set of
relations has to be determined. This can be easily done by mathematics.
But I do not propose to go into it. I prefer to offer a rough and ready
explanation, which, though it does not amount to a proof, yet enables
us to accept the truth that can be established both by experiment and
by calculation.

[Illustration: FIG. 31.]

Let a heavy ball (_A_) be suspended by a long string, so that the
action of gravity sideways on the ball is very small and may be
neglected, and to each side attach an indiarubber thread fastened
at _B_ and _C_. Then when the ball is pulled aside a little, say
to a position _D_, it will tend to fly back to _A_ with a force
proportioned to the distance _A D_. What will be the time it will take
to do this? If the distance _A D_ is small, the ball has only a small
distance to go, but then, on the other hand, it has only small forces
acting on it. If the distance _A D_ is bigger, then it has a longer
distance to go, but larger forces to urge it. These counteract one
another, so that the time in each case will be the same.

[Illustration: FIG. 32.]

The question is this:—Will you go a long distance with a powerful
horse, or a small distance with a weak horse? If the distance in each
case is proportioned to the power of the horse, then the amount of the
distance does not matter. The powerful horse goes the long distance in
the same time that the weak horse goes the short distance. And so it
is here. However far you pull out the spring, the accelerative pull on
the ball is proportioned to the distance. But the time of pulling the
ball in depends on the distance. So that each neutralises the other.
Whence then we have this most important fact, that springs are all
isochronous; that is to say, any body attached to any spring whatever,
whether it is big or small, straight or curly, long or short, has a
time of vibration quite independent of the bigness of the vibration.
The experiment is easy to try with a ball mounted on a long arm that
can swing horizontally. It is attached on each side to an elastic
thread. If pulled aside, it vibrates, but observe, the vibration is
exactly the same whether the bigness of the vibration is great or
small. If the pull aside is big, the force of restitution is big; if
the pull is small, the force of restitution is small. In one case the
ball has a longer distance to go, but then at all points of its path
it has a proportionally stronger force to pull it; if the ball has a
smaller distance to go, then at all the corresponding points of its
path it has a proportionally weaker force to pull it. Thus the time
remains the same whether you have the powerful horse for the long
journey or the weaker horse for the smaller journey.

[Illustration: FIG. 33.]

Next take a short, stiff spring of steel. One of the kind known as
tuning forks may be employed.

The reader is probably aware that sounds are produced by very rapid
pulsations of the air. Any series of taps becomes a continuous sound
if it is only rapid enough. For example, if I tap a card at the rate
of 264 times in a second, I should get a continuous sound such as that
given by the middle C note of the piano. That, in fact, is the rate
at which the piano string is vibrating when C is struck, and that
vibration it is that gives the taps to the air by which the note is
produced.

This can be very easily proved. For if you lift up the end of a bicycle
and cause the driving wheel to spin pretty rapidly by turning the pedal
with the hand, then the wheel will rotate perhaps about three times in
a second. If a visiting card be held so as to be flipped by the spokes
as they fly by, since there are about thirty-six of them, we should
get a series of taps at the rate of about 108 a second. This on trial
will be found to nearly correspond to the note A, the lowest space on
the bass clef of music. As the speed of rotation is lowered, the tone
of the note becomes lower; if the speed is made greater, the pitch of
the note becomes higher, and the note more shrill. However far or near
the card is held from the centre of the wheel makes no difference, for
the number of taps per second remains the same. So, again, if a bit of
watch-spring be rapidly drawn over a file, you hear a musical note. The
finer the file, and the more rapid the action, the higher the note. The
action of a tuning fork and of a vibrating string in producing a note
depends simply on the beating of the air. The hum of insects is also
similarly produced by the rapid flapping of their wings.

It is an experimental fact that when a piano note is struck, as the
vibration gradually ceases the sound dies away, but the pitch of the
note remains unchanged. A tune played softly, so that the strings
vibrate but little, remains the same tune still, and with the same
pitch for the notes.

A “siren” is an ingenious apparatus for producing a series of very
rapid puffs of air. It consists of a small wheel with oblique holes in
it, mounted so as to revolve in close proximity to a fixed wheel with
similar holes in it. If air be forced through the wheels, by reason
of the obliquity of the orifices in the movable wheel it is caused to
rotate. As it does so, the air is alternately interrupted and allowed
to pass, so that a series of very rapid puffs is produced. As the
air is forced in, the wheel turns faster and faster. The rapidity of
succession of the puffs increases so that the note produced by them
gradually increases in pitch till it rises to a sort of scream. For
steamers these “sirens” are worked by steam, and make a very loud noise.

It is, however, impossible to make a tuning fork or a stretched piano
spring alter the pitch of its note without altering the elastic force
of the spring by altering its tension, or without putting weights on
the arms of the tuning fork to make it go more slowly. And this is
because the tuning fork and the piano spring, being elastic, obey
Hooke’s law, “As the deflection, so the force”; and therefore the time
of back spring is in each case invariable, and the pitch of the note
produced therefore remains invariable, whatever the amplitude of the
vibration may be.

Upon this law depends the correct going of both clocks and watches.

Wonderful nature, that causes the uniformity of sounds of a piano, or a
violin, to depend on the same laws that govern the uniform going of a
watch! Nay, more, all creation is vibrating. The surge of the sea upon
the coast that swishes in at regular intervals, the colours of light,
which consist of ripples made in an elastic ether, which springs back
with a restitutional force proportioned to its displacement, all depend
upon the same law. This grand law by which so many phenomena of nature
are governed has a very beautiful name, which I hope you will remember.
It is called “harmonic motion,” by which is meant that when the atoms
of nature vibrate they vibrate, like piano strings, according to the
laws of harmony. The ancient Pythagorean philosophers thought that all
nature moved to music, and that dying souls could begin to hear the
tones to which the stars moved in their orbits. They called it, as you
know, the music of the spheres. But could they have seen what science
has revealed to man’s patient efforts, they would have seen a vision
of harmony in which not a ray of light, not a string of a musical
instrument, not a pipe of an organ, not an undulation of all-pervading
electricity, not a wing of a fly, but vibrates according to the law
of harmony, the simple easy law of which a boy’s catapult is the type,
and which, as we have seen, teaches us that when an elastic body is
displaced the force of restitution, in other words, the force tending
to restore it to its old position, is proportional to the displacement,
and the time of vibration is uniform. The last is the important thing
for us; we seem to get a gleam of a notion of how the clock and watch
problem is going to be solved.

But before we get to that we have yet to go back a little.

About the year 1580 an inattentive youth (it was our friend Galileo
again) watched the swing of one of the great chandeliers in the
cathedral church at Pisa. The chandeliers have been renewed since his
day, it was one of the old lamps that he watched. It had been lit, and
allowed to swing through a considerable space. He expected that as it
gradually came to rest it would swing in a quicker and quicker time,
but it seemed to be uniform. This was curious. He wanted to measure the
time of its swing. For this purpose he counted his pulse-beats. So far
as he could judge, there were exactly the same number in each pendulum
swing.

This greatly interested him, and at home he began to try some
experiments. As he got older his attention was repeatedly turned to
that subject, and he finally established in a satisfactory way the law
that, if a weight is hung to the end of a string and caused to vibrate,
it is isochronous, or equal-timed, no matter what the extent of the arc
of vibration.

The first use of this that he made was to make a little machine with
a string of which you could vary the length, for use by doctors. For
the doctors of that day had no gold watch to pull out while with
solemn face they watched the ticks. They were delighted with the new
invention, and for years doctors used to take out the little string and
weight, and put one hand on the patient’s pulse while they adjusted the
string till the pendulum beat in unison with the pulse. By observing
the length of the string, they were then able to tell how many beats
the pulse made in a minute. But Galileo did not stop there. He
proceeded to examine the laws which govern the pendulum.

We will follow these investigations, which will largely depend on what
we have already learned.

Before, however, it is possible to understand the laws which govern
the pendulum, there are one or two simple matters connected with the
balance and operation of forces which have to be grasped.

Suppose that we have a flat piece of wood of any shape like Fig. 34,
and that we put a screw through any spot _A_ in it, no matter where,
and screw it to a wall, so that it can turn round the screw as round a
pivot.

[Illustration: FIG. 34.]

[Illustration: FIG. 35.]

Next we will knock a tintack into any point _B_, and tie a string on
to _B_. Then if I pull at the string in any direction _B C_ the board
tends to twist round the screw at _A_. What will the strength of the
twisting force be? It will depend on the strength of the pull, and
on the “leverage,” or distance of the line _C B_ from _A_. We might
imagine the string, instead of being attached at _B_, to be attached
at _D_; then, if I put _P_ as the strength of the pull, the twisting
power would be represented by _P_ × _A D_. This is called the “moment”
of the force _P_ round the centre _A_. It would be the same as if I
had simply an arm _A D_, and pulled upon it with the force _P_. It
is an experimental truth, known to the old Greek philosophers, that
moments, or twisting powers, are equal when in each case the result of
multiplying the arm by the power acting at right angles to it is equal.

Now suppose _A B_ is a pendulum, with a bob _B_ of 10 lbs. weight, and
suppose it has been drawn aside out of the vertical so that the bob is
in the position _B_. Then the weight of the bob will act vertically
downwards along the line _B C_. The moment, or twisting power, of the
weight will be equal to 10 lbs. multiplied by _A D_, _A D_ being a
line perpendicular to _B C_.

[Illustration: FIG. 36.]

Now suppose that another string were tied to the bob _B_, and pulled
in a direction at right angles to _A B_, with a force _P_ just enough
to hold the bob back in the position _B_. The pull along _D B_ × _A B_
would be the moment of that pull round the point _A_. But, because this
moment just holds the pendulum up, it follows that the moment of the
weight of the pendulum round _A_ is equal to the moment of the pull of
the string _B D_ round _A_.

                 Whence P × A B = 10 lbs. × A D.

                       Whence P = 10 lbs. × (A D)/(A B).

But _A B_ is always the same, whatever the side deflection or
displacement of the pendulum may be. Whence then we see that when
a pendulum is pulled aside a distance _E B_ (which is always equal
to _A D_), then the force tending to bring it back to _E_ is always
proportional to _E B_. But if the pendulum be fairly long, say 39-1/7
inches, and the displacement _E B_ be small,—in other words, if we do
not drag it much out of the vertical,—then we may say that the force
tending to bring it back to _F_, its position of rest, is not very
different from the force tending to bring it back to _E_. But _F B_
is the “displacement” of the pendulum, and, therefore, we find that
when a pendulum is displaced, or deflected, or pulled aside a little,
the amount of the deflection is always very nearly proportional to the
force which was used to produce the deflection. This important law
can be verified by experiment. If _C_ is a small pulley, and _B C_ a
string attached to a pendulum _A B_ whose bob is _B_. Then if a weight
_D_ be tied to the string and passed over a pulley _C_, the amount _F
B_ by which the weight _D_ will deflect the bob _B_ is almost exactly
proportional to _D_, so long as we only make the deflection _E B_
small, that is two or three inches, where say 39-1/7 inches is the
length _A B_ of the pendulum.

If _F B_ is made too big, then the line _B F_ can no longer be
considered nearly equal to the arc of deflection _E B_, and the
proposition is no longer true.

Hence then, both by experiment and on theory, we find that for small
distances the displacement of a pendulum bob is approximately equal to
the force by which that displacement is produced.

But if so, then from what has gone before, we have an example of
harmonic motion. The weight of the bob, tending to pull the bob back to
_E_, acts just as an elastic band would act, that is to say pulls more
strongly in proportion as the distance _F B_ is bigger. In fact, if we
could remove the force of gravity still leaving the mass _B_ of the
pendulum bob, the force of an elastic band acting so as to tend to pull
the bob back to rest might be used to replace it. It would be all one
whether the bob were brought back to rest by the downward force of its
own gravity, or by the horizontal force of a properly arranged elastic
band of suitable length.

[Illustration: FIG. 37.]

But the motion of the bob, under the influence of the pull of
an elastic band where the strain was always proportional to the
displacement, would, as we have seen, be harmonic motion, and performed
in equal times whatever the extent of the swing. Whence then we
conclude that if the swings of a pendulum are not too big, say not
exceeding two and a half inches each way, the motion may be considered
harmonic motion, and the swings will be made in equal times whether
they are large or small ones. In other words, a clock with a 39-1/7
inch pendulum and side swing on each side if not over two inches will
keep time, whatever the arc of swing may be.

This may be verified experimentally. Take a pendulum of wood 39-1/7
inches long, and affix to its end a bob of 10 lbs. weight. The pendulum
will swing once in each second. To pull it aside two inches we should
want a weight such that its moment about the point of support was equal
to the moment of the force of gravity acting on the bob, about the
point of support. In other words, the weight required × 39-1/7 inches =
10 lbs. × 2 inches. Whence the weight required = 1/2 lb. (nearly).

Now fix a similar pendulum _A B_ 39-1/7 inches long, horizontally,
with a weight _B_ of 10 lbs. on it. Fasten it to a vertical shaft _C
D_, with a tie rod of wire or string _A B_ so as to keep it up, and
attach to each side of the rod _A B_ elastic threads _E F_ and _E G_.
Let these threads be tied on at such a point that when _B_ is pulled
aside two inches the force tending to bring it back to rest is half a
pound. Then if set vibrating the rod will swing backwards and forwards
in equal times, no matter how big, the arc of vibration (provided the
arc is kept small), and the time of oscillation will be that of a
pendulum, namely, one swing in a second. In fact, whether you put _A B_
vertically and let it swing on the pivots _C_ and _D_ by the force of
gravity, or put it horizontally, and thus prevent gravity acting on it,
but make it swing under the accelerating influence of a pair of elastic
bands so arranged as to be equivalent to gravity, in each case it will
swing in seconds.

[Illustration: FIG. 38.]

It is this curious property of the circle that makes the vertical
force of gravity on a pendulum pull it as though it were a
horizontally acting elastic band; that is the reason why a pendulum is
equal-time-swinging, or, as it is called, isochronous, from two Greek
words that mean “the same” and “time.”

But it must be remembered that this equal swinging is only approximate,
and only true when the arc of vibration is small.

Here then we have a proof which shows us that the pendulum of a clock
and the balance wheel of a watch depend on exactly the same principles.
They are each an example of harmonic motion.

The next question that arises is whether the weight of the pendulum has
any influence upon the time of its vibration.

A little reflection will soon convince us that it has none. For we know
that the time that bodies take to fall to the ground under the action
of gravity is independent of the weight. A falling 2 lb. weight is only
equivalent to two pound-weights falling side by side.

In the same way and by the same reasoning we might take two pendulums
of equal length, and each with a bob weighing 1 lb. They would, if put
side by side close together swing in equal times. But the time would be
the same if they were fastened together, and made into one pendulum.

For inasmuch as the fall of a pendulum is due to gravity, and the
action of gravity upon a body is proportional to its mass, it follows
that in a pendulum the part of the gravitational force that acts upon
each part of the mass is occupied in moving that mass, and the whole
pendulum may be considered as a bundle of pendulums tied together and
vibrating together.

The same would be the case with a pendulum vibrating under the
influence of a spring. If you have two bobs and two springs, they will
vibrate in the same time as one bob accelerated by one spring. In
this case, however, the force of the one spring must be equal to the
combined force of the two springs. In other words, the springs must be
made proportional in strength to the masses.

Hence, then, you cannot increase the speed of the vibration of a
pendulum by adding weight to the bob.

On the other hand, if you have a bob vibrating under the influence of a
spring, like the balance wheel of a watch, then if you increase the bob
without increasing the spring, since the mass to be moved has increased
without a corresponding increase in the accelerating force acting on
it, the time of swing will alter accordingly.

But in the case of gravity, by altering the mass, you thereby
proportionally alter the attraction on it, and therefore the time of
swing is unaltered.

[Illustration: FIG. 39.]

The explanation which has been given above of the reasons why a
pendulum swings backwards and forwards in a given time independently
of the length of the arc through which it swings, that is to say of
the amount by which it sways from side to side, is only approximate,
because in the proof we assumed that the arc of swing and the line _F
B_ were equal, which is not really and exactly true. Galileo never got
at the real solution, though he tried hard. It was reserved for another
than he to find the true path of an isochronous pendulum and completely
to determine its laws. Huygens, a Dutch mathematician, found that the
true path in which a pendulum ought to swing if it is to be really
isochronous is a curve called a cycloid, that is to say the curve which
is traced out by a pencil fixed on the rim of a hoop when the hoop is
rolled along a straight ruler. It is the curve which a nail sticking
out of the rim of a waggon wheel would scratch upon a wall. I will
not go into the mathematical proof of this. Clocks are not made with
cycloidal pendulums, because when the arc of a pendulum is small the
swing is so very near a cycloid as to make no appreciable difference in
time-keeping.

I am now glad to be able to say that I have dealt with all the
mathematics that is necessary to enable the mechanism of a clock to be
understood. It all leads up to this:—

(1) A harmonic motion is one in which the accelerating force increases
with the distance of the body from some fixed point.

(2) Bodies moving harmonically make their swings about this point in
equal times.

(3) A spring of any sort or shape always has a restitutional force
proportional to the displacement.

(4) And therefore masses attached to springs vibrate in equal times
however large the vibration may be.

(5) The bob of a pendulum, oscillating backwards and forwards, acts
like a weight under the influence of a spring, and is therefore
isochronous.

(6) The time of vibration of a pendulum is uninfluenced by changes in
the weight of the bob, but is influenced by changes in the length of
the pendulum rod. The time of vibration of a mass attached to a spring
is influenced by changes in the mass.

We have now to deal with the application of these principles to clocks
and watches.

Clocks had been known before the time of Galileo, and before the
invention of the pendulum. They had what is known as balance, or verge
escapements. Strictly in order of time I ought to explain them here.
But I will not do so. I will go on to describe the pendulum clock, and
then I will go back and explain the verge escapement, which, we shall
see, is really a sort of huge watch of a very imperfect character.

As soon as Galileo had discovered that pendulums were isochronous,
that is, equi-time-swinging, he set to work to see whether he could
not contrive to make a timepiece by means of them. This would be easy
if only he could keep a pendulum swinging. When a pendulum is set
swinging, it soon comes to rest. What brings it to rest? The resistance
of the air and the friction of the pivots. Therefore what is obviously
wanted is something to give it a kick now and then, but the thing must
kick with discretion. If it kicked at the wrong time, it might actually
stop the pendulum instead of keeping it going. You want something that,
just as the pendulum is at one end and has begun to move, will give it
a further push. Suppose that I have a swing and that I put a boy in it,
and I swing him to and fro. I time my pushes. As he comes back against
my hand I let him push it back, and then just as the swing turns I give
it a further push. But I cannot stand doing that all day. I must make a
machine to do it. Now what sort of a machine?

First, the machine must have a reservoir of force. I can’t get a
machine to do work unless I wind it up, nor a man to do work unless I
feed him, which is his way of being wound up. But then what do I want
him to do? I want him, when I give him a push, to push me back harder.
I want a reservoir of force such that when a pendulum comes back and
touches it, the touch, like the pressure of the trigger of a gun, shall
allow some pent-up power to escape and to drive the pendulum forward.

This is the case in a swing. Each time that the swing returns to my
hands I give it a push, which serves to sustain the motion that would
otherwise be destroyed by friction and the resistance of the air.

Such an arrangement, if it can be contrived mechanically, is called an
“escapement.”

An arrangement of this kind was contrived by Galileo. He provided a
wheel, as is here shown, with a number of pins round it. The pendulum
_A B_ has an arm _A H_ attached to it, and there is a ratchet _C D_
which engages with the pins. The ratchet has a projecting arm _E F_.

[Illustration: FIG. 40.]

When the pendulum comes back towards the end of its beat, the arm _A H_
strikes the arm _E F_, and raises the ratchet _C D_. This releases the
wheel, which has a weight wound up upon it, and therefore at once tries
to go round. The consequence is, that the pin _G_ strikes upon the arm
_A H_, and thus on its return stroke gives an impetus to the pendulum.
As the pin _G_ moves forward it slides on the arm _A H_ till it slips
over the point _H_. The wheel now being free, would fly round were it
not that when the pendulum returned, and the arm _A H_ was lowered,
the ratchet had got into position again and its point _D_ was ready to
meet and stop the next pin that was coming on against it. At each blow
of the pins against the pendulum a “tick” is made, at each blow of a
pin against the ratchet a “tock” is sounded, so that as it moves the
pendulum makes the “tick-tock” sound with which we are all familiar.

Hence then a clock consists of a wheel, or train of wheels, urged by
a weight or spring, which strives continually to spin round, but its
rotation is controlled by an escapement and pendulum, so contrived as
only to allow it to go a step forward at regular equal intervals of
time.

But this would make only a poor sort of escapement. For the mode of
driving the pendulum adds a complication to the swing of the pendulum.
Instead of the pendulum being simply under the accelerative force of
gravity, it is also subjected to the acceleration of the pin _G_. This
acceleration is not of the “harmonic” order. Hence so far as it goes it
does not tend to assist in giving a harmonic motion to the pendulum,
but, on the contrary, disturbs that harmonic motion. Besides this, the
impulse of the pin is in practice not always uniform. For if the wheel
is at the end of a train of wheels driven by a weight, though the force
acting on it is constant, yet, as that force is transmitted through a
train of wheels, it is much affected by the friction of the oil. And
on colder days the oil becomes more coagulated, and offers greater
resistance. Moreover, as will be explained more in detail afterwards,
the fact that the impulse is administered by _G_ at the end of the
stroke of the pendulum is disadvantageous, as it interferes with the
free play of the pendulum.

From all these causes the above escapement is imperfect in character,
and would not do where precision was required.

[Illustration: FIG. 41.]

It is now time to return to the old-fashioned escapements which were
in use before the time of Galileo. These consisted of a wheel called
a crown wheel, with triangular teeth. On one side of this wheel a
vertical axis was fitted, with projecting “pallets” _e f_. Across
the axis a verge or rod _e f_ was placed, fitted with a ball at each
end. When the crown wheel attempted to move on, one of its teeth came
in contact with a pallet. This urged the pallet forward, and thereby
caused an impulse to be given to the axis, on which was mounted the
verge, carrying the balls. These of course began to move under the
acceleration of the force thus impressed upon the pallet. Meantime,
however, the other pallet was moving in the opposite direction, and by
the time the first pallet had been pushed so far that it escaped or
slid past the tooth of the crown wheel, which was pressing upon it, the
other pallet had come into contact with the tooth on the other side
of the crown wheel. This tended to arrest the motion of the verge, to
bring the balls to a standstill, and ultimately to impart a motion in
a contrary direction to them.

Thus then the arrangement was that of a pendulum not acted on by
gravity, for the balls neutralised one another. The pendulum was,
however, not subjected to a harmonic acceleration, but alternately
to a nearly uniform acceleration from _A_ to _B_ and _B_ to _A_. As
a result, therefore, the time of oscillation was not independent of
the arc of swing, but varied according to it, as also according to
the driving power of the crown wheel. At each stroke there was a
considerable “recoil.” For as each tooth of the wheel came into play it
was unable at first to overcome and drive back the pallet against which
it was pressing, but, on the contrary, was for a time itself driven
back by the pallet.

[Illustration: FIG. 42.]

Of course, so long as the motions of the wheel and verge were exactly
uniform, fair time was kept. But the least inequality of manufacture
produced differences.

Nevertheless it was on this principle that clocks were made during the
thirteenth, fourteenth and fifteenth centuries. They were mostly made
for cathedrals and monasteries. One was put up at Westminster, erected
out of money paid as a fine upon one of the few English judges who have
been convicted of taking bribes.

The time of swing of these clocks depended entirely upon the ratio of
the mass of the balls at the end of the verge as compared with the
strength of the driving force by which the acceleration on the pallets
was produced. They were very commonly driven by a spring instead of a
weight. The spring consisted of a long strip of rather poor quality
steel coiled up on a drum. As it unwound it became weaker, and thus the
acceleration on the verge became weaker, and the clock went slower.

In order, therefore, to keep the time true, it became necessary to
devise some arrangement by which the driving force on the crown wheel
should be kept more constant.

This gave rise to the invention of the fusee. The spring was put inside
a drum or cylindrical box. One end of the spring was fastened to an
axis, which was kept fixed while the clock was going; the other was
fastened to the inside of the drum. Round the drum a cord was wound,
which, as the drum was moved by the spring, tended to be wound up on
the surface of the drum. Owing to the unequal pull of the spring, this
cord was pulled by the drum strongly at first, and afterwards more
feebly. To compensate its action a conical wheel was provided, with a
spiral path cut in it in such a way and of such a size and proportion
that as the wheel was turned round by the pull of the drum the cord was
on different parts of it, so that the leverage or turning power on it
varied, becoming greater as the pull of the cord became weaker, and in
such a ratio that one just compensated the other, and the turning power
of the axle was kept uniform.

In this manner small table clocks were made which kept very tolerable
time.

[Illustration: FIG. 43.]

Huygens converted these clocks into pendulum clocks in a very simple
manner. He removed one of the balls, lengthened the verge, and slightly
increased the weight of the other ball. By this means, while the crown
wheel still continued to drive the verge and remaining ball, the
acceleration on that ball now no longer depended entirely on the force
of the crown wheel. The acceleration and retardation were now almost
entirely governed by the force of gravity on the remaining ball, and
this acceleration was harmonic.

The clock, therefore, was immensely improved as a time-keeper. Still,
however, the acceleration remained partly due to the driving power, and
this was partly non-harmonic and introduced errors.

Most of the old clocks were converted shortly after the time of
Huygens. As there was in general no room for the pendulum inside the
clock-case, they usually brought the axle on which the pallets were
mounted outside the clock and made it vibrate in front of the face.

Many old clocks exist, of which the engraving in the frontispiece is
an example, that have been thus converted. A true old verge escapement
clock is now a rarity.

The type of escapement invented by Galileo never came into vogue for
clocks, on account of its imperfections, except till after a long
interval, when, with certain modifications, it became the basis of a
new improvement at the hands of Sir George Airey.

The crown wheel fell into disuse and was replaced by the anchor
escapement, which was employed in that popular and excellent timepiece
used throughout the eighteenth and the early part of the nineteenth
century, and is now known as “The Grandfather’s Clock.” It was after
all the crown wheel in another shape. The wheel, however, was flattened
out, the teeth being put in the same plane. This made it much easier to
construct. The pallets were fixed on an axis, and were a little altered
so as to suit the changed arrangement of the teeth. The pendulum was
no longer hung on the axis which carried the pallets. A cause of a good
deal of friction and loss of power was thus removed. The pendulum was
hung from a strip of thin steel spring, which allowed it to oscillate,
and which supported it without friction. This excellent manner of
suspending pendulums is now universal. It enabled the pendulum to be
made very heavy. The bob was usually some eight or nine pounds weight.
By this means the acceleration on the pendulum was due almost entirely
to gravity acting on the bob, and thus the motion of the pendulum
became almost wholly harmonic. Whence it followed that variations in
the pendulum swing became of secondary importance, and did not greatly
alter the going of the clock.

[Illustration: FIG. 44.]

Therefore when the wheels became worn, and the pivots choked with old
oil and dust, the old clock still went on. If it showed a tendency to
stop for want of power, a little more was added to the driving weight,
and the clock kept as good time as ever.

The swing of the pendulum was by this escapement enabled to be made
small, so that the arc of swing of the bob differed but little from a
cycloid.

The secret of the time-keeping qualities of these old “Grandfather”
clocks is the length of pendulum. This renders it possible to have
but a small arc of oscillation, and therefore the motion is kept very
nearly harmonic. For practical purposes nothing will even now beat
these old clocks, of which one should be in every house. At present
the tendency is to abolish them and to substitute American clocks with
very short pendulums, which never can keep good time. They are made
of stamped metal. When they get out of order no one thinks of having
them mended. They are thrown into the ash-pit and a new one bought. In
reality this is not economy.

Good “Grandfather” clocks are not now often made. The last place I
remember to have seen them being manufactured is at Morez, in the
district of the Jura. An excellent clock, enclosed in a dust-tight iron
case, with a tall painted case of quaint old design, can be bought for
about 55_s._ The wheels are well cut, and the internal mechanism very
good.

I visited the town of Morez in the year 1893. The clock industry was
declining. The farmers of France seemed to prefer small clocks of
hideous appearance, made in Germany and in America, to the excellent
work of their own country. Probably by now the old clockmaking industry
is extinct. One I purchased at that time has gone well ever since.




CHAPTER IV.


It is now time to give a description of the various parts of an
ordinary pendulum clock. We will take the “Grandfather” clock as an
example. We shall want an hour hand and a minute hand in the centre of
the face, and a seconds hand to show seconds a little above them. There
will be a seconds pendulum 39·14 inches long, and the centre of the
face of the clock will be about seven feet above the ground, so as to
give practically about five feet of fall for the weight.

[Illustration: FIG. 45.]

In the first place, we have to consider the axle which carries the
minute hand, and which turns round once in each hour. This is usually
made of a piece of steel about one-sixth of an inch in diameter.
Clockmakers usually call an axle an “arbor,” or “tree,” whence our word
axletree.

This “arbor” is turned in the lathe, so as to have pivots on each end,
fitted into holes in the clock plates, that is to say, the flat pieces
of brass that serve as the body of the clock. The adjoining diagram
shows _S T_ the clock faces, and _C_, the arbor of the minute hand.

Inasmuch as the seconds hand is to turn round sixty times while the
minute hand turns round once, it is obvious that the arbor of the
minute hand must be connected to the arbor of the seconds hand by a
train of cogwheels so arranged as to multiply by sixty. This of course
involves us in having large and small cogwheels.

[Illustration: FIG. 46.]

The small cogwheels usually have eight teeth, and are for convenience
of manufacture, as also to stand prolonged wear, cut out of the solid
steel of the arbor. They are nicely polished.

The easiest pair of wheels to use will be two pinions of eight teeth,
or “leaves,” as they are called, and two cogwheels, one of sixty-four
teeth, the other of sixty teeth.

It is then clear that if the arbor _A_ turns round once in an hour,
the arbor _B_ will turn round eight times in an hour, and _C_ will turn
round (60 × 64)/(8 × 8) = 60 times in an hour, or once in each minute.

By having 480 teeth on the cogwheel on _A_, you could, of course, make
_C_ go round once in a minute without the use of any intermediate arbor
such as _B_.

[Illustration: FIG. 47.]

But this would not be a very convenient plan. For as the wheel on _A_
is usually about two and a quarter inches in diameter, to cut 480 teeth
on so small a wheel would involve us in cutting about sixty teeth to
the inch. The teeth would thus be microscopically small, and would
have to be set so fine that the least dirt would clog them. Moreover,
the pinion of eight leaves would have to be microscopic. For these
reasons, therefore, it is usual in clocks not to use wheels with teeth
more than sixty or sixty-four in number, and to diminish the motion
gradually by means, where needful, of intermediate arbors. We have next
to consider how the weight is to be arranged so as to turn the arbor
_A_ once round in an hour. We know that we have five feet of space for
the weights to fall in. If we arrange to have what is called a double
fall, as shown in the sketch, then, allowing room for pulley wheels, we
shall find that our string may be practically about nine feet in length.

[Illustration: FIG. 48.]

The clock will be wanted to go for a week without winding, and as
people may forget to wind it at the proper hour of the day, we will
give it a day extra, and make an “eight-day” clock of it. Hence then,
while nine feet of cord is being pulled out by a weight which falls
four and a half feet, the minute hand is to be turned round as many
times as there are hours in eight days, viz., 192 times. This could be
accomplished, of course, by winding the cord round the arbor of the
minute hand. But this would require 192 turns. If our cord is to be
ordinary whipcord, or catgut, say one-twelfth of an inch in diameter,
in order that the cord could be wound upon it, the arbor would have
to be 192/12 inches long = 14⅓ inches long. This would make the clock
case unnecessarily deep. We must therefore again have recourse to an
intermediate wheel.

[Illustration: FIG. 49.]

If we put a pinion of eight leaves on the minute hand arbor _c_, and
engage it with a wheel of sixty-four teeth on another arbor _b_, then
_b_ will obviously turn round once in eight hours, that is to say,
twenty-four times in the period of eight days. And, if we fix on _b_
a “drum” or cylinder two inches long, the twenty-four turns of our
cord will just fit upon it, since, as has been said, our cord is to be
one-twelfth of an inch in diameter. The diameter of the drum must be
such that a cord nine feet long can be wound twenty-four times round
it. That is to say, each lap must take (9 × 12)/24 = 4½ inches of cord.
From this it is easy to calculate that the diameter of the drum must be
rather less than one and a half inches. From this then it results that
we want for a “Grandfather’s” clock a drum two inches long and one and
a half inches diameter, on this a cogwheel of sixty-four teeth working
into a minute hand arbor, with a pinion wheel with eight leaves, and
a cogwheel of sixty-four teeth, an intermediate or idle wheel with
an eight-leaved pinion, and a cogwheel of sixty teeth, engaging with
a seconds hand arbor with a pinion of eight leaves. This is called
the “train of wheels.” With it a weight such as can be arranged in an
ordinary “Grandfather’s” clock case will cause by its fall during eight
days the second hand arbor to turn round once in each minute during the
whole time, and the minute hand arbor to turn round once in each hour.

[Illustration: FIG. 50.]

We must next provide an arrangement for winding the clock up. It
is obvious that we cannot do so by twisting the hands back. It is
true that this could be done, but it would take about five minutes
to do each time and be wearisome. In order to save this trouble, an
arrangement called a ratchet wheel and pall must be provided. A ratchet
wheel consists of a wheel with a series of notches cut in it, as
shown in the figure _A_. A pall is a piece of metal, mounted on a pin,
and kept pressed up against the ratchet wheel by a spring _C_. It is
obvious that if I turn the wheel _A_ round, and thus wind up a weight,
fastened to a cord wound round the drum _D_, that the pall _B_ will go
click-click-click as the ratchet wheel goes round, but that the pall
will hold it from slipping back again. When, however, I take my hands
away, and let the ratchet wheel alone, then the weight _E_ will pull on
the drum _D_, and try and turn the ratchet wheel back the opposite way
to that in which I twisted it at first. If the pall _B_ is held fast,
it is impossible to move it, but if the pall is fixed to a cogwheel
_F_, which rides loose on the arbor of the drum _D_, then the pull of
the weight _E_ will tend to twist the cogwheel _F_ round, and this, if
engaged with a pinion wheel on the minute hand arbor, will therefore
drive the clock. As the clock arbors move, of course the weight _E_
gradually runs down, and, at last all the string is unwound from the
drum _D_. The clock is said then to have “run down,” but if I take a
clock key, and by means of it wind the string up upon the drum _D_,
then the pall lets the drum and ratchet slip; the clock hands are not
affected. When I have given twenty-four turns to the arbor, the nine
feet of cord will then be wound upon the drum again, and the clock will
be ready to go for eight more days, and will begin to move as soon as I
cease to press upon the clock key.

[Illustration: FIG. 51.]

I have thus described the winding mechanism. It now remains to describe
the escapement.

It is of course obvious that, if the weight and train of wheels were
simply let go, the weight would rush down, and the seconds-hand
wheel would fly round at a tremendous pace; but we want it to be so
restrained as only to be allowed to go one-sixtieth part of its
journey round in each second. In fact, we need an “escapement” and a
pendulum.

The escapement usually employed in “Grandfather” clocks is the anchor
escapement above described. It is not by any means the best sort of
escapement, but it is the easiest to make; and hence its popularity in
the days sometimes called the “dear, good old days,” when people had to
file everything out by hand, and had to take a day to do badly what can
now be done well in five minutes.

The escape wheel of an anchor escapement has thirty sharp angular teeth
on its rim. The wheel is made as light as possible, so that the shock
of stoppage at each tick of the clock may be as slight as possible, for
a heavy blow of course wastes power and gradually wears out the clock.
The anchor consists of two arms of the shape shown in the illustration
(Fig. 44). As the escape wheel goes round in the direction of the
arrow, the anchor, mounted on its arbor, rocks to and fro. The wheel
cannot run away, because the act of pushing one arm or “pallet,” as it
is called, outwards, and thus freeing the tooth pulls the other pallet
in, and this stops the motion of the tooth opposite to it, but when the
anchor rocks back again, so as to disengage the pallet from the tooth
that holds it, then the opposite tooth is free to fly forward against
the other pallet. This tends to rock the anchor the other way, but
at that instant the pallet just engages the next tooth of the wheel,
and so the action goes on. The anchor rocks from side to side; the
pallets alternately engage the teeth of the wheel, making at each rock
of the anchor the tick-tock sound with which we are so familiar. If
the anchor were free to rock at any speed it could, the ticking of the
clock would be very quick; so, to restrain the vivacity of the anchor,
we have a pendulum. The pendulum might be simply hung on to the anchor.
But the disadvantage of doing this would be that the heavy bob of the
pendulum would cause such a pressure on the arbor of the anchor that
there would be great friction, and the arbor would soon be worn out,
and the accurate going of the clock disturbed. The pendulum therefore
is hung on a piece of steel spring on a separate hook, which lets it
go backwards and forwards and carries the weight easily, while a rod
projecting from the anchor has a pin, which works in a slot on the
pendulum. The pendulum is therefore able to control and regulate the
movements of the escapement, and thus the time of the clock.

Of course it is clear that the heavier the driving weight put on the
drum of the clock, and the better the cut and finish of the wheels, and
the greater the cleanliness and oil, the more will be the pressure
tending to drive round the escape wheel, and the harder the pressure
on the pallets, and hence the bigger the impulses on the pendulum, and
therefore the larger the amplitude of its swing.

If the amplitude of the pendulum’s swing affected the time of its
swing, then the time kept by the clock would vary with the weight, and
the dirt and friction, and the drying up of the oil. But here precisely
is where the value of the beautiful law governing the harmonic motion
of the pendulum comes in. The time of the pendulum is (for small arcs)
independent of the length of swing, and therefore of the driving force
of the clock, and hence within limits the clock, even though roughly
made and foul with the dirt of years, continues to keep good time.
But the anchor escapement has imperfections. The only way in which
a pendulum can be relied on to keep accurate time is by leaving it
unimpeded. But the pressure of the teeth on the pallets in an anchor
escapement constantly interferes with this.

[Illustration: FIG. 52.]

A little consideration will easily show that there are some times
during the swing of a pendulum at which interference is far more
fatal to its time-keeping than at others. Thus the bob of a pendulum
may be regarded as a weight shot outwards from its position of rest
against the influence of a retarding force varying as its distance
from rest—in fact, shot out against a spring. The time of going out
and coming in again will be quite independent of the force exerted to
throw it out, quite independent of its original velocity. Therefore
a variation in the impulse given to the bob is of no consequence,
provided that impulse is given when the bob is near the position of
rest. This follows from the nature of the motion. If a ball be attached
to a piece of elastic thread, and thrown from the hand, so that it
flies out, and then stops and is brought back by the elastic force of
the thread, the time of the outward motion and the return is the same
whatever be the force of the throw. And so if a pendulum be impelled
outwards from a position of rest, the time of the swing out and back
is the same, however big (within limits) is the impelling force and
the consequent length of the swing. The use of a pendulum as a measure
of time is to impel it outwards, and then let it fly _freely_ out and
back. But if its motion is not free, if forces other than gravity act
upon it while on its path, then its time of swing will be disturbed.
It does not matter with what force you originally impel it, but what
does matter is, that when it once starts it should be allowed to travel
unimpeded and uninfluenced. Now that is what an anchor escapement does
not do. The impulse is given the whole way out on one of the pallets,
and then when it is at its extreme of swing, and ought to be left
tranquil, the other pallet fastens on it. But a perfect escapement
ought to give its impulse at the middle point of the swing, when the
pendulum is at the lowest, and then cease, and allow the pendulum to
adapt its swing to the impulse it has received, and thus therefore to
keep its time constant. This is done by an escapement called the dead
beat escapement, which, though in an imperfect way, realises these
conditions.

The alteration is made in the shape of the pallets of the anchor. The
wheel is much the same. Each pallet consists of two faces: a driving
face _a b_ and a sliding face _b c_.

[Illustration]

When the tooth _b_ has done its work by pressing on the driving face,
and thus driving the anchor over, say, to the left, then the tooth on
the opposite side falls on the sliding face of the other pallet. This
being an arc of a circle, has no effect in driving the anchor one way
or the other; hence the pendulum is free to swing to the left as far as
it likes and return when it feels inclined, always with the exception
of a little friction of the tooth on the faces of the pallets, but
when it returns and begins to move towards the right, the tooth slides
back along the face of the pallet till the pendulum is almost at the
middle of its swing; then an impulse is given by the pressure of the
tooth upon the inclined plane _a´ b´_. As soon, however, as the tooth
leaves _b´_, another tooth on the other side at once engages the
sliding face _b c_ of the other pallet, and so the motion goes on.

This beautiful escapement is at present used for astronomical clocks;
the pallets are made of agate or sapphire, and therefore do not grind
away the teeth of the wheel perceptibly, and the loss by friction on
the sliding surfaces is exceedingly small.

There are several other ways even better than this for securing a free
pendulum movement. We have now to return to our clock.

The centre arbor moves round once in an hour, and carries the minute
hand. In order to provide an hour hand, which shall turn round once in
twelve hours, we fasten a cogwheel and tube _N_ on to the minute hand
arbor by means of a small spring, which keeps it rather tight, but
allows it to slip if turned round hard (see Fig. 45). This spring is
a little bent plate slipped in behind the cogwheel on which its ends
rest; its centre presses on a shoulder on the minute hand arbor; it is
a sort of small carriage spring. The cogwheel _n_ has thirty teeth.
This cogwheel engages another cogwheel _o_ with thirty teeth, on a
separate arbor, which carries a third cogwheel, _p_, with six teeth,
and this again engages a fourth cogwheel, _q_, with seventy-two teeth,
mounted on a tube which slips over the tube to which the cogwheel _a_
is attached. It is now easy to see that for each turn of the minute
hand arbor the arbor _p_ makes one turn, and for each turn of the
arbor _p_ the cogwheel _d_, makes one-twelfth of a turn. From which
it follows that for each turn of the minute hand arbor the cogwheel
_d_ with its tube, or, as it is sometimes called, its “slieve,” makes
one-twelfth of a turn, and thus makes a hand fastened to it show one
hour for every complete turn of the minute hand.

The minute hand is attached to the tube or slieve which carries the
cogwheel _N_. The hour hand is attached to the tube or slieve which
carries the cogwheel _Q_, and one goes twelve times as slowly as the
other.

But if you want to set the clock it is easy to do so by reason of the
fact that the minute hand is not fixed to the arbor, but only to the
slieve on the cogwheel that fits on the arbor, and is held somewhat
tight to the arbor by means of the spring. The hands can thus be
turned, but they are a little stiff. A washer on the minute hand arbor
keeps the slieve on the cogwheel pressed tight against the spring,
being secured in its turn by a very small lynch-pin driven through a
hole in the minute hand arbor.

It remains to explain a few subsidiary arrangements, not always found
upon all clocks, but which are useful.

In order to prevent the overwinding of the clock (see Fig. 43), which
would cause the cord to overrun the drum, an arm is provided, fitted
with a spring. As the weight is wound up the free part of the cord
travels along the drum or the fusee; and the cord, when it is near the
end of the winding, comes up against the arm and pushes it a little
aside. This causes the end of the arm to be pushed against a stop on
the axis of the fusee, and thus prevents the clock being further wound
up. The stop, being ratchet-shaped, does not prevent the weight from
pulling the ratchet wheel round the other way, and thus driving the
clock; it only prevents the rotation of that wheel when the string is
near it, and the winding is finished.

Another arrangement is the “maintaining spring.”

It will be remembered that during the process of winding the clock the
hand twisting the key takes the pressure of the ratchet wheel off the
pall, so that during that operation no force is at work to drive the
clock. In consequence the pendulum receives no impulse, but swings
simply by virtue of its former motion. If the process of winding were
done slowly enough the clock might even stop. To avoid this, a very
ingenious arrangement is made to keep the cogwheel mounted on the
winding shaft going during the winding-up process. This is called a
maintaining spring.

The arrangement shown in Fig. 53 will explain it.

[Illustration: FIG. 53.]

The cogwheel _a_ and the ratchet wheel are both mounted loosely on the
arbor carrying the drum. _a_ is linked to _b_ by a spring _c_. The
ratchet wheel _b_ is engaged by a pall fixed to some convenient place
on the body of the clock frame. When the weight pulls on the drum the
pull is communicated to the ratchet wheel _b_, and this acts on the
spring _c_ and pulls it out a little. As soon as the spring _c_ is
pulled out as far as its elasticity permits, a pull is communicated
to the cogwheel _a_, and the clock is driven round. When the clock is
wound the pressure of the weight is removed, and therefore the ratchet
wheel _e_ no longer presses on the pall, and thus no pressure is
communicated to the ratchet wheel _b_, or through it to the clock. But
here the spring _c_ comes into play. For since the ratchet wheel _b_ is
held fast by the pall _d_, the spring _c_ pulls at the wheel _a_, and
thus for a minute or so will continue to drive the clock. This driving
force, it is true, is less than that caused by the weight, but it is
just enough to keep the pendulum going for a short time, so that the
going of the clock is not interfered with.

If the reader can get possession of a clock, preferably one that does
not strike, and, with the aid of a small pair of pincers and one or two
screwdrivers, will take it to pieces and put it together again, the
mechanism above described will soon become familiar to him. Not every
clock is provided with maintaining spring and overwinding preventer.

The cause of stoppage of a clock generally is dirt. Where possible,
clocks should always be put under glass cases. “Grandfather” clocks
will go much better if brown paper covers are fitted over the works
under the cases. In this way a quantity of dust may be avoided. To get
a good oil is very important. It will be noticed that pivot-holes in
clocks are usually provided with little cup-like depressions. This is
to aid in keeping in the oil. The best clock oil is that which does not
easily solidify or evaporate. Ordinary machine oil, such as used for
sewing machines, is good as a lubricant, but rapidly evaporates. Olive
oil corrodes the brass.

It is best to procure a little clock oil, or else the oil used for gun
locks, sold by the gunsmiths. The holes should be cleaned out with the
end of a wooden lucifer match, cut to a tapering point. The pivots
should be well rubbed with a rag dipped in spirits of wine. If the
pivots are worn they should be repolished in the lathe. If the cogs of
the wheels are worn, there is no remedy but to get new ones. Old clocks
sometimes want a little addition to the driving weight to make them go.

The weight necessary to drive the clock depends on its goodness of
construction, and on the weight of the pendulum. If the clock is driven
for eight days with a cord of nine feet in length with a double fall,
then during each beat of the pendulum that weight will descend by an
amount =

             9/(2 × 24 × 60 × 60 × 8) feet or 1/12800th inch.

Whence, if the clock weight is 10 lbs., the impulse received by the
clock at each beat is equivalent to a weight of 10 lbs. falling through
1/12800th of an inch, or to the fall of six grains through an inch.

The power thus expended goes in friction of the wheels and hands, and
in maintaining the pendulum in spite of the friction of the air.

The work therefore that is put into the clock by the operation of
winding is gradually expended during the week in movement against
friction. The work is indestructible. The friction of the parts of the
clock develops heat, which is dissipated over the room and gradually
absorbed in nature. But this heat is only another form of work. Amounts
of work are estimated in pressures acting through distances. Thus, if
I draw up a weight of 1 lb. against the accelerative force of gravity
through a distance of one foot, I am said to do a foot-pound of work.

One pound of coal consumed in a perfect engine would do eight millions
foot-pounds of work. Hence, if the energy in a pound of coal could be
utilized, it would keep about 100,000 grandfather’s clocks going for a
week. As it is consumed in an ordinary steam engine it will do about
half a million foot-pounds of work. One pound of bread contains about
three million foot-pounds of energy. A man can eat about three pounds
of bread in a day, and, as he is a very good engine, he can turn this
into about three-quarters of a million foot-pounds of work. The rest of
the work contained in the bread goes off in the form of heat.

[Illustration: FIG. 54.]

As has been previously said, the power of the action of gravity in
drawing back a pendulum that has been pushed aside from its position of
rest becomes less in proportion as the pendulum is longer, and hence
as the pendulum is longer the time of vibrations increases. In the
appendix to this chapter a short proof will be given showing that the
length of a pendulum varies as the square of the time of its vibration.
A pendulum which is 39·14 inches in length vibrates at London once in
each second. Of course at other parts of the earth, where the force of
gravity is slightly different, the time of vibration will be different,
but, since the earth is nearly a globe in shape, the force of gravity
at different parts of it does not vary much, and therefore the time of
vibration of the same pendulum in different parts of the earth does not
vary very much. The length of a pendulum is measured from its point of
suspension down to a point in the bob or weight. At first sight one
would be inclined to think that the centre of gravity of the pendulum
would be the point to which you must measure in order to get its
length. So that if _B_ were a circular bob, and the rod of the pendulum
were very light, the distance _A B_ to the centre of the bob would be
the length of the pendulum. But if we were to fly to this conclusion,
we should be making the same error that Galileo made when he allowed a
ball to _roll_ down an inclined plane. He forgot that the motion was
not a simple one of a body down a plane, but was also a rolling motion.
The pendulum does not vibrate so as always to keep the bob immovable
with the top side _C_ always uppermost. On the contrary, at each beat
the bob rotates on its centre and makes, as it were, some swings of
its own. Therefore in the total motions of the pendulum this rotation
of the bob has to be taken into account. Of course, if the pendulum
were so arranged that the bob did not rotate, and the point _C_ were
always uppermost, as, for instance, if the pendulum consisted of two
parallel rods, _A B_ and _C D_, suspended from _A_ and _C_, then we
might consider the bob as that of a pendulum suspended from _E_, and
the pendulum would swing once in a second if _A B_ = _C D_ = _E F_
were equal to 39·14 inches, for by this arrangement there would be no
rotation of the bob. But as pendulums are generally made with the bob
rigidly fixed to the rod _E F_, the rotation must be taken into account.

[Illustration: FIG. 55.]

It wants some rather advanced mathematical knowledge to do this. But
in practice clockmakers take no account of it. The correction is not a
large one, so they make the rod as nearly true as they can, arrange a
screw on the bob to allow of adjustment, and then screw the bob up and
down until in practice the time of oscillation is found to be correct.

[Illustration: FIG. 56.]

The mode of suspension of a pendulum of the best class is that shown
in Fig. 56, which allows the pendulum to fall into its true position
without strain. _A_ is a tempered steel spring, which bends to and
fro at each oscillation. It is wonderful how long these springs can
be bent to and fro without breaking. Inasmuch as lengthening the
pendulum increases the time, so that the time of vibration _t_ varies
as the square of the length of the pendulum, a very small lengthening
of the pendulum causes a difference in the time. In practice, for
each thousandth of an inch that we lengthen the pendulum we make a
difference of about one second a day in the going of the clock. If we
cut a screw with eighteen threads to the inch on the bottom of the
pendulum rod, and put a circular nut on it, with the rim divided into
sixty parts, then each turn through one division will raise or lower
the bob by 1/1080th of an inch, and this first causes an alteration
of time of the clock by one second in the day. This is a convenient
arrangement in practice, for it affords an easy means of adjusting the
pendulum. We need only observe how many seconds the clock loses or
gains in the day, and then turn the nut through a corresponding number
of divisions in order to rectify the pendulum.

[Illustration: FIG. 57.]

Another needful correction of the pendulum is that due to changes in
temperature. If the rod of the pendulum be made of thoroughly dried
mahogany, soaked in a weak solution of shellac in spirits of wine,
and then dried, there will not be much variation either from heat or
moisture. But for clocks required to have great precision the pendulum
rod is usually made of metal. A rod of iron expands about 1/160000th
of its length for each degree Fahrenheit; and therefore for each degree
Fahrenheit a pendulum rod of 39·14 inches will expand about 1/4000
thousandths of an inch, and thus make a difference in the going of the
clock of about one-fourth of a second per day. The expansion will, of
course, make the clock go slower. It would be possible to correct this
expansion if some arrangement could be made, whenever it occurred, to
lift up the bob of the pendulum by an amount corresponding to it, as,
for instance, to make the bob of some material which expanded very much
more by heat than the material of which the pendulum rod was made.

[Illustration: FIG. 58.]

Thus if we hang on to the end of a pendulum of iron a bottle of iron
about seven inches long, and almost fill it with mercury, then, as
soon as the heat increases, the iron of the rod and of the bottle
expands, and the centre of oscillation of the pendulum is lowered.
But as the linear expansion of mercury contained in a bottle is about
five times that of iron, the mercury rises in the bottle, and thus the
expansion downwards of the pendulum rod is compensated by the expansion
upwards of the mercury in the bottle. The rod may be fastened to the
mouth of the bottle by a screw, so that as the bottle is turned round
it may be raised or lowered on the rod, and thus the length of the
pendulum may be adjusted. The bottle is made of steel tube, screwed
into a thin turned iron top and bottom. Of course no solder must be
used to unite the iron, for mercury dissolves solder. A little oil and
white-lead will make the screwed joints tight. This is an excellent
form of pendulum. Another plan is to use zinc as the metal which is to
counteract the expansion of the iron. The expansion of zinc is about
three times that of iron.

[Illustration: FIG. 59.]

Hence a zinc tube, about twenty inches long (shown shaded in Fig. 59),
is made to rest upon a disc fastened to the lower part of the iron
pendulum rod. On the top of the zinc rests a flat ring _A_, from which
is suspended an iron tube _A_, which carries the bob _B_. The expansion
of the zinc tube is large enough to compensate the expansion both of
the rod and the tube, and the bob consequently remains at the same
depth below the point of suspension, whatever be the temperature.

There is, however, a new method which is far superior to all these, and
this is due to the discovery by M. Guilliaume, of Paris, of a compound
of nickel and steel which expands so little that it can be compensated
by a bob of lead instead of by a bob of mercury. This material is sold
in England under the name of “invar.” An invar rod with a properly
proportioned lead bob makes an almost perfect pendulum, the expansion
of the invar and the lead going on together. The exact expansion of the
invar is given by the makers, who also supply information as to the
size and suspension of the bob proper to use with it.

It has been already shown that the uniformity of time of swing of a
pendulum is only true when the arc through which it swings is very
small. If the total swing from one side to another is not more than
about two inches very little difference in time-keeping is made by
putting a little more driving weight on the clock, and thus increasing
its arc of swing; but when the arc of swing becomes say three inches,
or one and a half inches on each side of the pendulum, then the time of
vibration is affected. At this distance each tenth of an inch increase
of swing makes the pendulum go slower by about a second a day.

The resistance of the air, of course, has a great influence on a
pendulum, and is one of the chief causes that bring it ultimately to
rest. Even the variations of pressure of the atmosphere which the
barometer shows as the weather varies have an effect on the going of
a clock. Attempts have been made by fixing barometers on to pendulums
with an ingenious system of counter balancing to counteract this, but
these refinements are not in common use, and are too complicated to be
susceptible of effective regulation.


APPENDIX TO CHAPTER IV.

It may be useful to give a simple form of proof of the law which
governs the time of oscillation of a pendulum whose length is given.

Unfortunately, it is impossible to give one so simple as to be
comprehended by those who know nothing whatever of mathematics. It
is, however, possible to give a proof that requires very little
mathematical knowledge.

We know that when a mass of matter is whirled round at the end of a
string it tends to fly outwards and puts a strain on the string. The
faster the speed at which the mass is whirled, the stronger will be the
strain on the string. Suppose that the length of the string equals R,
the velocity of the mass as it flies round equals V. Let _a_ be the
body whirled round by a string _o a_ from a centre at _O_. The body
always, of course, tends to fly on in a straight line from the point at
which it is at any instant. But that tendency is frustrated by the pull
of the string which constrains it to take a circular path. It is, of
course, all one whether the force that tends to pull the body inwards
towards _O_ is a string or an attractive force of any kind acting
through a distance without any string at all. Evidently if the body
keeps its place in the circle it must be because the centrifugal force
tending to whirl it out is equal to the centripetal or attractive force
tending to pull it in.

[Illustration: FIG. 60.]

The strain on the body, due to the force tending to pull it inwards, we
shall designate by F, meaning by F the number of feet of velocity that
would in one second be imparted to the body by the attractive force.

Suppose that at some given instant of time the body is at a point _a_.
At that instant its _direction_ will be along _a b_, tangential to the
circle at _a_, and that is the path it would take if the centripetal
or attractive force ceased to act just as the body got to _a_. In that
case the body would be whirled off like a stone from a sling along
the line _a b_, and would at the end of a given time, let us suppose a
second, arrive at _b_. But it is not so whirled off; it is attracted
towards _O_ and pulled inwards, and comes to _c_. Hence, then, the
attractive force acting during one second must have been sufficient to
pull the mass in from _b_ to _c_. But we know that if an accelerating
force (F) acts on a body for a second it produces a final velocity
equal to F at the end of the second, and an average velocity half F
during the second.

Hence, then, the space _b c_, by which the body has been pulled in,
is represented by half F, but _a b_, the space which the body would
have travelled forwards, will be represented by V, the velocity of the
body in a second; but if the motion be such that the distance _b c_
travelled in a second is very small, then the triangles _a b d_ and
_a b c_ are approximately similar, and the smaller _a b_ is the more
nearly similar they are. Whence then (a b)/(b c) = (a d)/(a b), that is
to say (a b)² = a d × b c.

But _a b_ represents the space which would have been traversed by the
body in one second at the rate it was going, and hence is equal to V;
_a d_ is the diameter of the circle, and hence equals 2 R; _b c_ is
the space through which the body has been drawn in the second by the
attractive force F, and therefore equals half F.

Whence then V² = 2 R × half F = R F.

We took a second as the limit of time during which the motion was to be
considered. Of course any other time could have been taken. Now what
is true of the motion of a body during a very short time is also true
of the body during the whole of its path, assuming that the path is a
circle, and that F remains constant, as it obviously will if the path
is a circle, and the velocity is uniform. Whence then we may generally
say that if a body is being whirled round at the end of a string the
strain F on the string is directly proportional to the square of the
velocity, and is inversely proportional to the length of the string.

The time of rotation, is of course = length of the path ÷ velocity

                  = (2πR)/V = (2πR)/√(R F) = 2π√(R/F).

Whence then we see that for motion in a circle of a mass under the
attraction of a centripetal force, or pull of a string, the time of
rotation will be uniform, provided that the centripetal force always
varies as the radius of the path. From this it is evident that a body
fixed on to an elastic thread where the pull varies as the extension
would make its rotations always in equal times. If your sling consists
of elastic, whirl as you will, you can only whirl the body round so
many times in a second, and no more. Any increase in your efforts only
makes the string stretch, and the circle get bigger. The velocity of
the body in its path of course increases, but the time it takes to go
once round is invariable.

It also follows that if a body hung by a string of length _l_, under
the action of gravity, be travelling in a circle round and round, then,
_if the circle is a small one compared with the length of the string_,
the inward acceleration _f_ towards the centre will be approximately
proportional to the radius _r_ of the circle, and the time of rotation
will be

                             t = 2π√(r/f).

But in this case _f_, the inward acceleration, is to _g_ the
acceleration downwards of gravity as A B:A P or

                 f/g = (A B)/(A P) = (A P)/(O P) = r/l.

[Illustration: FIG. 61.]

[Illustration: FIG. 62.]

Whence then the time of rotation of this body would be if the circle of
rotation was small

                              = 2π√(l/g).

And if you try you will find that this is so. For instance, take a
thread 39-1/7 inches long, that is 3·25 feet. Hang anything heavy from
one end of it, and cause it to swing round and round in a _small_
circle. Now _g_ the acceleration of gravity = 32·2 feet per second.
π the ratio of the circumference of a circle to its diameter = 3·14.
From which it follows that the time of rotation = 2 × 3·14√(3·25/32·2)
seconds = 2 seconds. But if we look at the rotating body sideways, it
appears to act as a pendulum; it matters nothing whether we swing it
round and round or to and fro. For in any case the accelerative force
tending to bring it back to a position of rest is always proportional
to the distance of displacement, and, therefore, its time of motion
must always be 2π√(l/g) and its motion harmonic.

The length of a seconds pendulum, that is a pendulum that makes its
double swing in two seconds, will therefore be

                        l = 4/((2π)²) × g feet

                          = (g × 12)/π² inches

                          = 39·14 inches.




CHAPTER V.


I have thus described the principal features of ordinary clocks. For
the details many treatises must be studied, and knowledge acquired
which is not in any books at all.

I now, however, pass to watches. It will be remembered that a verge
escapement consists of a crown wheel with teeth, engaging two pallets
fixed upon a verge, furnished with balls at its extremities.

As the crown wheel was urged forwards each pallet in succession was
pushed till it slipped over the tooth which was engaging it. Then
a tooth on the other side came into sharp collision with the other
pallet, and drove the verge the other way, and so on.

Now here we have a driving force, and a sort of pendulum. But how did
the verge act as a pendulum to measure time? It is not a body rocking
under the action of gravity, nor under the acceleration of a spring.
How then can it act as a regulator of time, and what is the period of
its swing?

The answer to this is, that it is under the acceleration of gravity,
but that gravity does not act freely on the bobs or weights, but only
through the driving weight and teeth. The impulse that drives the
verge is really also the accelerating force upon it, and the only
accelerating force upon it.

And the worst feature about the movement is, that as the teeth and
pallets move, the leverage of the teeth on the pallets alters, and
thus the bobs on the verge are under the influence not of a uniform or
duly regulated force, but of a constantly varying one, and one that
varies in a very complicated and erratic way. It would be hopeless to
expect much time-keeping from such a contrivance. The most that could
be expected would be by putting on a very big weight to reduce to
comparative insignificance the friction, and then hope that the swings
would be uniform, so that whatever went on in one swing would go on in
the next, and thus the time-keeping be regular.

But any course tending to diminish the driving force, such as the
thickening of the oil, would greatly affect the going. It was for this
reason that Huygens turned the verge into a pendulum by removing one of
the bobs, and letting gravity thus act on the other.

For watches, however, a different plan was contrived. One end of a
slender spiral spring was affixed to the verge. The other end of the
spring was made fast to the clock frame. The verge was now, therefore,
chiefly under the action of the acceleration of the spring. To make the
acceleration of the teeth of the ’scape wheel less embarrassing, the
teeth were so shaped as only to give a short push at stated intervals,
and not interfere with the free swing of the verge under the alternate
to-and-fro accelerations and retardations of the spring. By this means
the verge became in every way an excellent pendulum, not dependent on
gravity, and permitting the watch to be held in any position.

The verge thus fitted was turned into a wheel, and became a “balance
wheel.” It was compensated for heat expansion by a cunning use of the
unequal expansion of brass and steel, in a manner analogous to the way
this unequal expansion of metals had been employed to compensate the
pendulum, and became the beautiful and accurate time-measurer that we
see to-day, with its pivots mounted in jewels to diminish friction, and
with screws round the rims of the balance wheel to enable the centre of
gravity to be exactly adjusted to its centre of rotation, and with a
delicate hair-spring of tempered steel that is a marvel of microscopic
work.

But the escapement of the early watches left much to be desired. In
order to make it clear how imperfect that early escapement was, we
have to turn back and remember what has been said about the dead beat
escapement.

It will then be remembered that it was shown that for small arcs the
pendulum would keep good time provided you let it have as much swing as
it wanted to use up the force which the escapement had applied to it,
_but not otherwise_, so the pendulums only acted really well when the
impulse was given about the middle of the swing, and they were free to
go on and stop when they pleased, and turn back at the end of it.

This essential condition was fairly approximated to in the dead beat
escapement of clocks which left them at the end of their swing with
only a very slight friction to impede their free motion.

But when you come to deal with a watch the case is quite different.
Here the escapement is of a great size compared with the balance wheel,
and the friction even of the most dead beat watch escapement that could
be contrived was so big compared with the forces acting on the balance
wheel as seriously to derange its motion, and render it far from a
perfect time-keeper.

Now about this time—I am speaking of the early part of the eighteenth
century—a demand of a very exceptional character arose for a really
perfect watch. The demand did not arise from gentlemen who wanted
to keep appointments to play at ombre at their clubs, or even
from merchants to time their counting house hours. For these the
old-fashioned watch did very well. The demand came from mariners.
But the seamen did not want to know the time merely to arrange the
hours for meals on the ship or to determine when the watch was to be
relieved, but for a far more important purpose, namely, to find out by
observation of the heavens their place upon the ocean when far out of
sight of the land. It will be very interesting to see how this problem
arose, and how the patient industry and ingenuity of man has solved it.

The ancient navigators never went very far from the shore, for, once
out of sight of land, a ship was out of all means of knowing where she
was. On clear days and nights the compass, and the sun and stars would
tell the mariner the _direction_ he was sailing in, but it was quite a
problem to determine where he was on the surface of the earth.

[Illustration: FIG. 63.]

Let us consider the problem. Suppose for convenience that the earth is
divided up into “squares,” as nearly, at least, as you can consider a
globe to be so marked out. Let us suppose that it has been agreed to
draw on it from pole to pole 360 lines of longitude, commencing with
one through say Greenwich Observatory as a starting-point, and going
right round the earth till you come back to Greenwich again. Also
suppose that there have been drawn a series of circles parallel to the
equator, but going up at equal distances apart towards the poles. Let
us have 179 of these circles, so as to leave 180 spaces, _a_ to _b_,
_b_ to _c_, etc., from pole to pole. This will divide the earth up like
a bird-cage into squares, as if we had robed it in a well-fitting
Scotch plaid. The length measured along the equator of the side _p q_
of each square at the equator is taken as exactly sixty nautical miles
(apart from a small error of measurement, which makes it in actual
practice 59·96). This is equal to sixty-nine and a quarter English
statute miles. The side of the square leading towards the poles _q s_
would also be sixty nautical miles were it not that the earth is not
truly spherical, which introduces a slight error. We may, however,
roughly say that at the equator each square measures sixty nautical
miles each way.

[Illustration: FIG. 64.]

As we get towards the poles the squares become rectangular figures,
with the heights of latitude still sixty nautical miles, but the widths
becoming smaller. Thus in England our squares measure _p q_ = 37
nautical miles and _q s_ = 60 nautical miles.

Now of course we can see at once that it is easy at any place on the
earth’s surface to find your _latitude_ by a simple observation of the
sun at noon, if you know the day of the year, and have got a nautical
almanac. For by an instrument called a sextant you can measure the
angle he appears to be above the horizon, and then, as you know from
a nautical almanac the angle he is above the equator, you can soon
determine your place _A_ on the globe. Or at night, if you measure the
angular distance that the polar star _P_ is from the zenith, or point
exactly over your head—that is, the angle _P O Z_—you can subtract it
from a right angle and get your latitude, _A O E_, at once.

[Illustration: FIG. 65.]

But how are you to determine your longitude? The pole-star, or sun,
or any other star won’t help you, for as the earth is moving they
keep shifting, and at one time or another appear exactly in the same
position to everyone on the same parallel of latitude, as it is easy to
see. The fact is that you are on a ball turning round. You know easily
what latitude you are on, but you cannot tell your longitude unless
you can tell how many hours and minutes you get to a position before
Greenwich gets to the same position. If when a particular star got to
Greenwich a gong were sounded which could be heard all over the earth,
then of course, by seeing what stars were overhead, everyone would
know their longitude at once. Perhaps by means of the new electric
waves this will before long be done, and the Greenwich hours will be
sounded all over the world for the use of mariners. But till this is
accomplished all that can be done is to keep an accurate clock on
board, so as always to give you Greenwich time.

Early attempts were made to take a pendulum clock to sea, suspending it
so as to avoid disturbance to its motion by the rocking of the ship.
These proved vain.

It therefore became desirable that a watch with a balance wheel
should be contrived to go with a degree of accuracy in some respects
comparable with the accuracy of a pendulum clock. To encourage
inventors an Act of Parliament was passed in the thirteenth year of
Queen Anne’s reign (chapter xv.) (1713) promising a reward of £20,000
to anyone who would invent a method of finding the longitude at sea
true to half a degree—that is, true to thirty geographical miles.

If the finding of the longitude were to be accomplished by the
invention of an accurate watch, then this involved the use of a watch
that should not, in several months’ going, have an error of more than
two minutes, which is the time which the earth takes to turn through
half a degree of longitude.

This was the problem which John Harrison, a carpenter, of Yorkshire,
made it his life business to solve. His efforts lasted over forty
years, but at the end he succeeded in winning the prize.

These instruments have been much improved by subsequent inventors, and
have resulted in the construction of the modern ship’s chronometer,
a large watch about six inches in diameter, mounted on axles, in a
mahogany box. Several of these are taken to sea by every ship.

The peculiarity of the chronometer is its escapement.

Let _A B_ be the scape wheel, and _C D_ a small lever attached to _C_,
the pivot on which the balance wheel and spring is fastened. Let _E G_
be a lever, with a tooth _F_ which engages the teeth of the scape wheel
and prevents it moving round. Let _H_ be a spring holding the lever _E
G_ up to its work.

[Illustration: FIG. 66.]

The lever has a spring _K E_ fastened to it at the point _K_. This
spring is very delicate. If the lever _C D_ is turned so that the
little projection _M_ on it strikes the spring _E_ from left to right,
then, as the spring rests on the lever, the whole lever is pushed over,
and the teeth of the scape wheel set free. At that instant, however,
the escapement is so arranged that the arm _C D_ is just opposite
the tooth _D_ of the scape wheel, so that the scape wheel, instead
of running away, leaps with its tooth _D_ on to the lever _C D_ and
swings the balance wheel round. The balance wheel is free to twist as
much as it pleases, but the moment it has twisted so much that the
projection _M_ passes the spring _E_, then the lever _G E_ flies back
to its place, and the scape wheel is again checked. Meanwhile the
balance wheel flies round till at last it is brought to rest by the
balance spring. It then recoils and sets out on its return path. This
time, however, the projection _M_ merely flips aside the spring _E_
and the balance wheel goes back, till again it is brought to rest and
returns. As soon as the lever comes opposite _D_ the projection _M_
then again hits the spring _E_, and releases the catch at _F_, and
another tooth of the scape wheel goes by.

There then you have a completely free escapement, and consequently an
accurate one. Many watches are made with these escapements, but they
are more expensive than those in common use.

There is but little remaining in a watch that is not in a clock, for
the wheel-trains and general arrangements are very similar.

It is possible to apply the chronometer’s detached escapement to a
clock. This was done by several clock-makers in the eighteenth and
early part of the nineteenth century. One method of doing it is as
follows:

_A_ is a block of metal fitted to the bottom of the pendulum, _B_ a
light lever pivoted on it. _C_ is the scape wheel, with four teeth;
_D_ a tooth of the scape wheel, which hops on to the projection of the
pendulum the moment that the impact of the point _E_ of the lever _B E_
has pushed aside the lever _G F_, and thus released the scape wheel.
The advantage is that it is a very easy escapement to make. But it is
in reality a detached (that is to say, a completely free) chronometer
escapement, as can easily be seen.

[Illustration: FIG. 67.]

Turret-clocks are open to considerable disadvantages, for the wind
blowing on the hands gives rise to considerable pressure, so that the
clocks are sometimes urging the hands against the wind, sometimes are
being helped by the wind. And this inequality of driving force makes
the pendulum at some times make a bigger arc of swing than at others.

But we saw above that though difference of arc of swing ought to make
no difference in the time of swing of the pendulum, yet this was only
strictly true if the arc of swing were a cycloid.

But as for practical convenience we are obliged to make it a circle,
it follows, as we saw, that for every tenth of an inch of increase of
swing of an ordinary seconds pendulum about a second a day of error is
introduced. To remove this difficulty a gravity escapement was invented
by Mudge in the eighteenth century, improved by Bloxam, a barrister,
and perfected by the late Lord Grimthorpe. The idea was to make the
scape wheel, instead of directly driving the pendulum, lift a weight,
which, being subsequently released, drove the pendulum. The consequence
was, that inequalities in wind pressure, which affected the driving
force of the scape wheel, would not act on the pendulum, which would
be always driven by the uniform fall of a fixed and definite weight. A
movement of this kind has been fixed in the great clock at Westminster,
and has gone admirably. A description of its details will be found in
the _Encyclopædia Britannica_, written by Lord Grimthorpe himself.

All sorts of eccentric clocks and watches have been proposed. For
instance, it seems wonderful to see a pair of hands fitted to the
centre of a transparent sheet of glass go round and keep time with
apparently nothing to drive them.

But the mystery is simple. The seeming sheet of glass is not one sheet,
but four. The two centre sheets move round invisibly, carrying the hour
hand and minute hand with them, being urged by little rollers below
on which they rest. When you touch the glass the outside sheets appear
at rest, and you do not suspect that it is other than a single sheet.
But beware of dust, for if dust gets on the inner plate you detect the
trick. In this way a mechanical hand was made that wrote down answers
to questions. This plan can be applied to all sorts of tricks.

Sir William Congreve, an ingenious inventor, proposed to make a clock
that measured time by letting a ball roll down an incline. When it got
to the bottom it hit a lever, which released a spring and tipped the
plane up again, so that the ball now ran down the other way. It is a
poor time-keeper, and the idea was not original, for a ball had been
previously designed for the same purpose.

Sometimes clocks are constructed by attaching pendulums to bronze
figures, which have so small a movement that the eye is unable to
detect it. The figure appears to be at rest, but is in reality slowly
rocking to and fro. It is necessary to make the movement as small as
about one four hundredth of an inch in half a second, if the movement
is to escape human observation. For a movement of one two hundredth
of an inch per second is about the largest that will certainly remain
unperceived.

In mediæval times clocks were constructed with all sorts of queer
devices. The people of the upper town at Basle having quarrelled with
those of the lower town, fought and beat them. To commemorate this
victory they put on the old bridge at the upper town a clock provided
with an iron head, that slowly put out and drew in a long tongue of
derision. This clock may still be seen in the museum. It is as though
the council of the city of London put a clock of derision at Temple Bar
to put out its tongue at the County Council.

I do not propose here to describe the striking mechanism of clocks.
There are several different ways of arranging it. They are rather
complicated to follow out, but they all resolve themselves into a
few simple principles. As the hour hand revolves it carries a cam so
arranged as to be deeper cut away for the twelfth hour, less for the
eleventh, and so on. When the minute hand comes to the hour it releases
the striking mechanism, which, urged by a weight, begins to revolve,
and, driving an arm carrying a pin, raises a hammer, which goes on
striking away as the arm revolves. This would continue for ever if
it were not that at the same moment an arm is liberated which falls
against the cam. At each stroke the arm is (by the striking apparatus)
raised a bit back into position. When it comes back into position it
stops the striking. It thus acts as a counter, or reckoner of the blows
given, stopping the movement when the clock has struck sufficiently.
If the counting mechanism fails to act, we have the phenomenon which
occasionally occurs of a “Grandfather” clock striking the whole of the
hours for the week without stopping.

A chiming clock is simpler still. For here we have a barrel covered
with pins, like the barrel in a musical box. As the pins go round they
raise hammers which fall against bells. The barrel is wound up and
driven by a spring or weight. When the clock comes to the hour, the
barrel is released, and rotating, plays the tune.

If you want to make a clock wake you up in the morning it can be
done by making the striking arrangement hammer away with no counting
mechanism to stop it until the weight has run down. If, not content
with that, you want the sheets pulled off the bed or the bed tilted
up, or a can of water emptied over the person who will not rise, a
mechanical device known as a relay must be used. It is very simple.
What is wanted is that, after the lapse of a time which a clock
must measure, a considerable force must be exerted to pull off the
bedclothes. It would be absurd to make the clock exercise this pull.
It is obviously better to attach the clothes by a hook to a rope which
passes over a pulley, and from which hangs a weight. A pin secures the
weight from falling, the pin being withdrawn by the clock. The work is
thus done by the weight when released by the clock.

In like manner, if you have a telegraph designed to print messages at
a distance, you do not send along the wires the whole force necessary
for doing the printing. You only send impulses, which, like triggers,
release the forces by which the letters are to be stamped.

Electric clocks of many kinds have been invented. The principle of an
electric escapement is similar to that of an ordinary escapement.

[Illustration: FIG. 68.]

The reader no doubt knows that, when a circuit of wire is joined or
completed leading to a source of electricity, electricity flows through
the wire.

If the wire is wound round a piece of iron, then, whenever the circuit
is joined, a current is set in motion, and the iron becomes an
electro-magnet. When the circuit is severed the iron ceases to be a
magnet.

If put at a proper position it would at each time an iron pendulum
approached give it a small impulse provided that at that instant
the current is turned on. This can easily be made to be done by the
pendulum itself. For just as the pendulum is coming back to the
central position a flipper _P_ attached to the rod can be caused
to make contact with a piece of metal fixed on its path. Then the
electro-magnet, becoming magnetised, exerts a pull on the iron
pendulum. On the return beat of the pendulum the other side of the
flipper _R_ strikes the obstruction. But if that side _R_ is covered
with ebonite or some non-conducting material no current will be set
in motion, and the electro-magnet will not (as it would otherwise do)
retard the pendulum. Such a pendulum has therefore an impulse given to
it every second beat.

Such pendulums do not act very well, because it is difficult to keep
metallic surfaces like _Q_ clean, and therefore misses often occur.
Besides, the strength of the current varies with the goodness of the
contact and with other things.

What is now preferred is to make an arrangement by which an electric
current winds the clock up every minute or so. By this means the
impulse which drives the clock is not a varying electric one, but is
a steady weight. The most successful clocks have been made on these
principles.

The advantage of electricity is, that by means of the current that
actuates the clock, or winds it up, you can at regular intervals set
the hands in motion of a great number of clocks.

So that only one going clock with a pendulum is needed. The other
clocks distributed over the building have only faces and hands, and
a very few simple wheels, to which a slight push is given by an
electro-magnet, say, every minute or so. The system is therefore well
adapted for offices and hotels.

In America, by means of electric contacts, clocks have been arranged to
put gramophones into action. You will remember that it was pointed out
that if a wire were dragged over a file a sound would be produced due
to the little taps made as the wire clicked against the rough cuts on
the file, and that the tone of the note depended on the fineness of the
cuts, and hence the rapidity of the little taps. You can imagine that,
if the roughnesses were properly arranged, we might get the tones to
vary, and thus imitate speech. This is the principle of the gramophone.
The roughnesses are produced by a tool, which, vibrating under the
influence of human speech, makes small cuts in a soft material. This
is hardened, and then, when another wire is dragged over the cuts, the
voice is reproduced.

In this way clocks are made to speak and tell the children when dinner
is ready and when to go to bed. On this simple plan, too, dolls can be
made to speak.

The modern methods of clock and watch-making are very different from
those in use in olden days. In former times the pivots were turned up
by hand on small lathes, and even the teeth of the wheels were filed
out. Each hole in the clock or watch frame was drilled out separately,
and each wheel separately fitted in, so that the watch was gradually
built up as one would build a house. Each wheel, of course, only fitted
its own watch, and the parts of watches were not interchangeable.

This has now all been altered. By means of elaborate machinery the
whole of the work of cutting out every wheel and the making of every
single part is done by tools moved independently of the will of the
workman, whose only duty is to sit still and see the things made. He
is, as it were, the slave of the machine, watching it and answering to
its calls. Or shall we rather say that he is the machine’s employer and
master? He has here a servant who never tires nor ever disobeys him.
All the machine requires is that its cutting edges should be exactly
true and sharp and microscopically perfect; then it will cut away and
make wheel after wheel. It oils itself. It only wants the man to act as
superintendent, and stop it if any cutting edge gets unduly worn. For
this purpose he measures the work it is doing from time to time with a
microscope to see that it is good and true and exact.

When all the parts have thus been made you have perhaps a hundred
boxes, each with a thousand watch parts in it, each part exactly like
its fellows. You take one wheel or bit from each box indiscriminately,
and you then have the materials for a watch, screws, fittings, pins,
and all. All you have now got to do is simply to screw them all
together, like putting together a puzzle. Everything fits; there is no
snipping or filing.

In such a watch if a bit gets broken you simply send for another bit of
the same kind and fit it into its place.

Motor cars, bicycles, and many other machines are, or ought to be, made
in this manner, so that if a driver at York breaks a part of the car he
simply sends to London for another. It comes and fits into its place at
once. But for this sort of plan you must do work true to much less than
a thousandth of an inch, and, of course, no one must want to indulge
his individual fancy as to the shape or appearance of the watch. The
whole advantage consists in dead uniformity. But the cheapness is
surprising. You can have a better watch now for 30_s._ than could have
been got for £30 twenty years ago.

Artistic people are in the habit of condemning this uniformity as
though it were inartistic and degrading. In truth, it is not degrading
to get a machine to do what you want at the expense of as little labour
as possible. You pay 30_s._ for the watch, but you have £28 10_s._
left to spend on pictures.

Only one ought not to confuse industry with art. Watches made in this
way have no pretence to be artistic products. They are simply useful.
To rule them all over with machine lines or to put hideous machine
ornament on them is purely and simply base and degrading. Let your
_ornament_ be hand work, your utility machine work.

Thus then I have endeavoured to give a very brief sketch of the modes
of measuring time, and incidentally to introduce my readers to those
laws of motion which are the foundation of so large a part of modern
science.

It only remains that I should shortly describe modern apparatus by
means of which it is possible to measure with accuracy periods of time
so short as to appear impossible. But when you see how it is done the
method seems easy enough. It is still by means of a pendulum, only a
pendulum beating time not once, but hundreds and even thousands of
times in a second.

And such pendulums, instead of being difficult to make, are remarkably
simple, and present no difficulty whatever. For we have only to use the
tuning fork which has been previously described.

The tuning fork consists of a piece of steel bent into a U shape. The
arms are set vibrating so as alternately to approach and recede from
one another.

The reason why there are two arms is that, if they come together and
recede, they balance, and hence the instrument as a whole does not
shake on its base. This balance of moving parts of a rapidly moving
machine is very important. Some motor cars are arranged so that
the engines are “balanced,” and the moving parts come in and out
simultaneously, leaving the centre of gravity unchanged whatever be the
position of the motion. This makes the vibration of the car very small.

The tuning fork is therefore balanced. Being elastic, it obeys Hook’s
law, “As the force, so the deflection.” And therefore, as we have seen,
the vibrations of the fork are isochronous.

A fork with arms about six or seven inches long will make about fifty
or sixty vibrations in a second. How are we to record those vibrations,
and how keep the tuning fork vibrating?

[Illustration: FIG. 69.]

A train of wheels is almost an impossibility, not perhaps so impossible
as might be supposed, but still very difficult. So a different method
is adopted. A little wire projects from one tuning fork arm. A piece of
glazed paper is gently smoked by means of a wax taper, and is stretched
round a well-made brass drum. The tuning fork is then put so that the
little wire just touches the paper. The tuning fork is then made to
vibrate by a blow, and while it is vibrating the drum is revolved.
Thus a wavy line is formed on the drum by the wire on the tuning fork.
If the tuning fork made fifty complete vibrations to and fro in a
second there would be one hundred such indentations, fifty to the right
and fifty to the left, and by these the time can be measured as you
would measure a length upon a rule.

[Illustration: FIG. 70.]

If an arm _a b_ be fitted to move sideways when a little string _c
d_ is pulled, and be also provided with a small wire, so as to touch
the drum, then it also will trace a straight line on the drum as the
wire lightly scratches away the thin coating of smoke. Now, if it is
suddenly jerked and flips back, then a little indentation will be
made in the line, and if when we are to measure a rapid lapse of time
a jerk is given at the beginning, and another jerk at the end of it,
we should get a diagram like that in the adjoining figure, where _a_
is the trace of the tuning fork, _b_ that of the indicating arm. The
time which has elapsed between the jerk which produced the indentation
_c_ and that which produced the indentation _d_ will be about three
and three-quarter double indentations of the tuning fork line, thus
indicating three and three-quarter fiftieths of a second. It is easy
to see how delicate this means of measurement can be made. With small
tuning forks we can easily measure times to a thousandth part of a
second, and much less if desired.

The jerk may be given by electricity if it is wished. When the current
is joined a little electro-magnet pulls a bit of iron and gives a pull
to the string. So extremely rapid is the flight of electricity that no
appreciable time is lost in its transit through the wires, so that the
impulse may be given from a distance. Thus we may arrange that when a
cannon ball leaves a gun an electric impulse shall be given. When it
reaches and hits a target another electric impulse is given. These make
nicks in the tracing line on the drum from which we can easily compute
the time that has elapsed between the leaving of the mouth of the gun
and the arrival of the shot at its destination.

[Illustration: FIG. 71.]

Such an apparatus is used in modern gunnery experiments. It is an
elaborate one, but is based on the principle above described.

Drums are sometimes driven by clockwork, and tuning forks are also
often kept vibrating by electricity, thus constituting very rapidly
moving electric clocks. The arrangement is simple. An electro-magnet
_E_ is put in the vicinity of the arm of the tuning fork. A small piece
of wire from the arm is in contact with a piece of metal _Q_, from
which a wire runs to the electro-magnet, thence to a battery, and from
the battery to the tuning fork, through which the current runs to the
wire _R_. When the fork vibrates the arm, being bent outwards, makes
the wire _R_ touch _Q_. This at once causes the electro-magnet to give
a small pull to the steel arm of the tuning fork, and thus assists the
swing of the arm. The whole arrangement is exactly analogous to an
electric clock, as may be seen by comparing Fig. 71 with Fig. 68.

There is another method of measuring rapid intervals of time which also
merits attention. It is to let a body drop at the commencement of the
period of time to be measured, and mark how far it falls in the time,
and then find the time from the equation given previously,

                             S = 1/2 g t².

It is practically done by letting a piece of smoked glass fall and
making a small pointer make two dots upon it, one at the beginning,
another at the end, of the time to be measured.

An interesting adaptation of this method can serve as a basis of a
curious toy.

Take a crossbow, with a bolt with a spike on it; fix it firmly in a
vice so that the barrel points at a spot _a_ on a wooden wall. On the
spot _a_ hang a cardboard figure of a cat on to a nail so contrived
that when an electro-magnet acts the nail is pulled aside, and the
cat drops. Thus let _a_ be the cat, _b_ the loop by which it is hung
over the nail _c_, that is fixed to another piece of iron furnished
with a hinge at _c_, so that when the electric current is turned on
the nail _c_ is withdrawn and the cat drops. Carry the wires from the
electro-magnet and battery to the crossbow, and so arrange them that
when the bolt leaves the muzzle one is pressed against the other, and
contact made.

Now here you have an apparatus such that exactly as the bolt leaves the
crossbow, the cat drops. Now what will happen?

[Illustration: FIG. 72.]

When the bolt leaves the bow it is subject to two motions, one a motion
of projection at a uniform pace in the direction of _b a_ from the bow
to the target.

But it is also subject to another force, namely that of gravity, which
acts on it vertically, and deflects it _in a vertical direction_
exactly as much and as fast as a body would do if dropped from rest at
the same instant as the bolt leaves the bow. But the cat is such a
body. Hence, then, since by the electric arrangement they are both let
go together, they will both drop simultaneously, and thus will always
be on the same level, and when the bolt reaches the wooden wall and
has fallen vertically from _a_ to _c_, the cat will also have fallen
vertically from _a_ to _c_, and the bolt will pin him to the wall. It
does not matter how far you take the bow from the wall, nor how strong
the bow is, nor how heavy the bolt is, nor how heavy the cat is, nor
whether _a b_ is horizontal or pointing upwards or downwards.

[Illustration: FIG. 73.]

In every case, if only the barrel is pointed directly at the cat, then
the bolt and cat fall simultaneously and at the same rate, and the bolt
will pin the cat to the wall.

In trying the experiment the bolt should be pretty heavy, say half a
pound, and have a good spike; but if carefully done the experiment will
succeed every time. It enables you also to measure the speed of flight
of the bolt. For if the distance of the bow from the wall be thirty
feet, and the cat have fallen three feet when it is struck, then the
time of fall is T² = √((2S)/g) = √(6/g) = ·43 seconds. But the bolt
in this time went thirty feet; hence its velocity was thirty feet in
·43 seconds, or seventy feet per second.

Of course if you make the bolt heavier the velocity of projection will
become slower, the time longer, and hence the cat will fall further
before it is transfixed by the bolt.

My task is now at a close. I have endeavoured not merely to give a
description of clocks and various apparatus for measuring time, but to
explain the fundamental principles of mechanics which lie at the root
of the subject.

May I end with a word of advice to parents?

There is a certain number of boys, but only a certain number, who have
a real love for mechanical science. Such boys should be encouraged
in every way by the possession of tools and apparatus, but in the
selection of this apparatus the following principles should be borne in
mind:—

_First_, that almost everything a boy wants can be made with wood, and
metal, and wire, and string, _if_ he has someone to give him a little
instruction how to do it. A bent bit of steel jammed in a vice makes an
excellent tuning fork.

_Second_, that he wants not toy tools, but good tools. If an expert
wants a good tool, how much more a beginner.

_Third_, that he ought to have a reasonably dry and comfortable place
to work in, and the help and advice of the village carpenter or
blacksmith.

_Fourth_, that he ought not to be allowed to potter with his tools, but
to make something really sensible and useful, and not begin a dozen
things and finish none.

_Fifth_, that the making of apparatus to show scientific facts is more
useful than making bootjacks for his father or workboxes for his mother.

And, _lastly_, that a little money spent in this way will keep many a
young rascal from worrying his sisters and stoning the cat; and when
the inevitable time comes at which he must face the young man’s first
trial, THE EXAMINER, he will often thank his stars that he learned in
play the fundamental formula S = 1/2 g t², and that he knows the
nature of “harmonic motion,” the two most important principles in the
measurement of time.


THE END.




APPENDIX ON THE SHAPE OF THE TEETH OF WHEELS.


[Illustration: FIG. 74.]

The teeth of wheels for watches and clocks need particular care in
shaping, and it may be of interest if I describe briefly the principles
upon which these wheels are made. What is required is that the motion
shall not be communicated by jerks as the teeth successively engage one
another, but that the motion shall be perfectly smooth. The problem
therefore becomes this: How are we to arrange the teeth of the wheels
so that as one of them turns and drives the other round the leverage
or turning power exercised by the driving wheel on the driven wheel
shall always be uniform? Now if the teeth were simple spikes one can
easily see that this would not be the case. For instance, as the arm _a
c_ turned round, driving before it the arm _b d_, the point _c_ would
scrape along, and the leverage between the two teeth would constantly
alter. Evidently some other construction must be adopted. Before we can
determine what it is to be, we must inquire what the leverage would
be between two rods, _a c_ and _d b_, mounted on pivots at _a_ and
_d_. The answer to this question is, that when a lever such as _a c_
presses with its end against another, _d b_, the power is exercised in
a direction _c e_ at right angles to _d b_. Hence the leverage between
the two arms is in the ratio of _a e_ to _d c_. The system is just as
if we had a lever _a e_ united to a lever _d c_ by a rigid rod _e c_ at
right angles to both of them.

[Illustration: FIG. 75.]

Whence then the ratio of the power is as _a e_ is to _d c_.

[Illustration: FIG. 76.]

But since the triangles _a e f_, _d c f_, are similar, _a e_ is to _d
c_ as _a f_ to _f d_. Whence then we get this general proposition: If
one body mounted on an axis is pressing upon another body mounted
on an axis, the pressure exerted between them is always exercised in
a direction, shown by the dotted line, at right angles to the two
surfaces in contact; and the ratio of the leverage is found by drawing
a line from one axis to the other, so as to cut the line of direction
of pressure in _f_. The leverage of one on the other is then as _a f_
to _f d_. Our problem has now become the following: Given a rod _b d_,
suppose that it is pressed upon by a curved surface mounted on an axis
at _a_. Then the direction of the pressure that the curved surface
(called in engineering language a cam) will exercise on the rod _b d_
is shown by the dotted line; and the ratio of the driving power to the
driven power is as _d f_ to _a f_. Now how can we shape the cam so that
as it moves round, and different parts of its surface come successively
into contact with _b c_, the ratio of the leverage is always the
same; that is to say, the ratio of _a f_ to _f d_ shall always be
constant; that is to say, the line drawn through the point of contact
perpendicular to the curve at that point, shall always pass through the
point _f_?

[Illustration: FIG. 77.]

[Illustration: FIG. 78.]

Evidently, if this is to be so, the point _d_ must be on a semicircle,
whose diameter is _f b_, for in that case the angle _f d b_ will always
be a right angle.

[Illustration: FIG. 79.]

The surface must then be so arranged that, whatever be the position of
the cam and of the rod _b d_, the point of contact between them must
always be on the semicircle _f c d_; that is to say, as the cam moves
round the axis _a_ its shape must be such that a line drawn from _f_ to
the point where it cuts the circle _f d b_ is always perpendicular to
the curve.

Now suppose that we move a circle whose centre is at _a_, and radius _a
f_, so as to roll the circle _f d b_ by simple surface friction round
its centre _o_, then any point _d_ on it would mark out a curve on a
piece of paper attached to the moving circle whose centre is at _a_,
and the direction of motion of the curve would always be such that the
point _d_ on it would at any instant be describing a circle round _f_,
and the direction of the curve would thus at any point always be at
right angles to the line _d f_ for the time being.

[Illustration: FIG. 80.]

This curve, caused by the rolling of one circle on another, is called
an epicycloid. Hence, then, for a clock, if we make the pinion wheel
with straight spokes and the driving wheel with its teeth cut in the
form of epicycloids, caused by rolling a circle with a diameter equal
to the radius of the pinion upon the driving wheel, we shall get a
uniform ratio of leverage one upon the other.

The circles with radii _a f_, _b f_, are called the “pitch circles,”
and these radii are in the ratio of the movement that is required for
the wheels, usually six to one or eight to one, as the case may be. The
sides of the teeth of the pinion wheels are straight lines radiating
from the centre, and rounded off at the ends so as to avoid accidental
jambing. The teeth of the cogwheel have epicycloidal sides. The tips
are cut off so as to be out of the way, and spaces are left between
them for the width of the leaves of the pinion wheel.

[Illustration: FIG. 81.]

Both pinion wheels and cogwheels are cut by cutters rotating at a
high speed, about 3,500 times in a minute, the cutters being carefully
shaped for the pinion wheels with straight edges, for the cogwheels
in epicycloids. It is a pretty thing to see a wheel-cutting engine at
work, the cutter flying round with a hum, cutting the rim of a brass
wheel into teeth, the brass coming off in flakes thinner than fine
hairs and falling in fine dust. When a tooth is cut, the wheel is moved
round one division of an apparatus called a “dividing plate,” so as to
present a new part of the wheel to the cutter. Of course, the cutter
and wheels must all be properly proportioned. Cutters are sold in sets
duly shaped for the work they have to do. Wheel-cutting is a special
branch of the clockmaking industry. The reason the speed of cutting is
so high is because it is desired to take off small portions of metal
at a time, and thus not strain the wheel and the cutting machinery. If
bigger cuts were made, then the machine would have to go slower, for
it is a principle in the construction of cutting machinery that the
speed of the cut must always be proportioned to the depth of it. If
you want to take deep cuts you must move the cutting edge slowly, and
_vice versâ_. The most modern method of making cogwheels of brass, and
the best, is to stamp them out of solid sheet metal at a single punch
of a punching machine, and cheap watches are always made in this way.
In fact, the whole method of watch and clock-making is undergoing a
transformation.

Before the time of the great engineering development which took place
towards the end of the eighteenth century, the making of machines was
a sort of fine art. Cogwheels were cut by hand. The circumference
was marked out by means of compasses. Then holes were drilled round
the rim, and teeth cut out leading into them, and shaped by means of
special files constructed for the purpose (Fig. 82). Big machinery was
all shaped out at the forge and by the file. The engineers complained
that you could not get big work made true even to the eighth of an
inch. But watches and clocks were beautifully made, though only at the
cost of hours of patient measuring and filing. The taste for ornament
still existed. The wheels and backs of watches were chased over with
the most beautiful patterns; the frames of the clocks were wrought
into beautiful figures and forms. Even astronomical instruments were
embellished.

[Illustration: FIG. 82.]

Then came the era of severe accuracy. Men of science took the
government of machine-making whose feelings were repugnant to art in
any form. They hated to see any effort expended in ornament. With
severely utilitarian aims, they banished all appearance of beauty from
instruments and tools of all sorts, so that our modern machines, from a
steam engine down to a watch, are now models of precise but perfectly
unornamented workmanship. They are the fitting implements of a nation
that wears trousers and tall hats. One has only to compare an old
vessel of war, with its sculptured prow and streamers, with a modern
ironclad to take note of the difference. The art of ornamentation
is now little more than a spasmodic imitation of the past, with a
historical interest only. As a living entity it has almost ceased to
exist.

But in precision of manufacture the present age is without a rival in
the history of the world. People believe no longer in the old methods
of scraping and filing, and hand-work directly exercised on metal is
rapidly falling into desuetude. It is possible, of course, with a file
and scraper and days of labour to get two flat surfaces of metal so
perfect that when put together one will lift the other like a sucker
on a stone, but it is waste labour. A small machine will do it as well
in a few minutes. No longer is a watch built up as one would build a
house, fitting part to part. By expensive machines thousands of watch
parts are stamped and cut out to pattern, and then a watch is made by
taking one of each indiscriminately and just putting them into their
places. Comparatively unskilled workmen can do this. Where the skill
is wanted is to design and make the machinery and watch the cutters,
measuring them with microscopic gauges from time to time, and at once
remedying them if an edge is found to be a ten-thousandth part of an
inch out of place. So that the labour of man is becoming more and more
a labour of design and of supervision. Machines are the slaves that do
the work, for in a good machine we have an eye and an arm that never
tires, and only needs to be kept in working order. But machines are not
artistic, and thus art is lost while precision is gained. At present
a desperate attempt is being made to supply by means of machinery
the craving of the human mind for art. But it is destined to failure.
Art of this kind is generally produced by the same brain that designs
machines, and therefore presents an appearance of rigid accuracy and
uniformity, which, while essential to an engine, is out of place in an
artistic product.

The great manufacturers of our Midlands do not seem to understand that
there is no object in making a towel-horse as geometrically accurate as
a turning lathe. It will apparently be years before they learn to put
art and precision each in the place where it is wanted—precision in the
works of the watch, art in the face and the case of it; machine work in
the inside of a watch, hand work on the outside. When the public taste
is educated so as to see the odious character of the jumble of Gothic,
Egyptian, and meaningless ornament on such an article as the case of
an American organ, one step will have been made towards the revival of
artistic taste.

But to propose as a means of reviving art that we should discontinue
the use of machinery or abandon our modern cutters of precision to go
back to a hack-saw and file is ridiculous, and could only be suggested
by men quite destitute of scientific ideas. Art and precision each has
its place: there is room for both; let neither intrude on the domain of
the other.




INDEX.


  Acceleration, 73, 77

  Almagest, 53

  Anchor escapement, 120

  Ancient science, 50

  Aristotle’s ideas, 23, 52

  Attwood’s machine, 83


  Babylon, temple of, 24

  Balance wheel, 159


  Candles to measure time, 46

  Chaldean day, 15

  Chaucer, 56

  Chronographs, 179

  Chronometer, 165

  Chronometer escapement, 166

  Clock movement, 123

  Copernicus, 56

  Crossbow experiment, 183

  Crown wheel, 115

  Cycloid, the, 109


  Dante’s Inferno, 54

  Day, length of, 29

  Dead beat escapement, 135

  Density, 12

  Driving weight, 127, 141


  Earth, a sphere, 21

  Earth’s motion, 57, 69

  Earth not at rest, 67

  Egg-boiler, 43

  Electric clocks, 179

  Epicycloidal wheels, 191

  Escapements, anchor, 120
    crown, 115
    chronometer, 166
    dead beat, 135
    gravity, 169


  Falling bodies, laws of, 62

  Force, 76

  Forces, revolution of, 89

  Fusee, the, 117


  Galileo’s “Dialogues,” 58
    clock, 111

  Grandfather’s clock, 119

  Gravity, action of, 13, 65

  Gravity escapement, 169

  Greek day, 16


  Harmonic motion, 97

  Hooke’s law, 71


  Isochronism of springs, 93


  Lamps to measure time, 46

  Latitude and longitude, 161
    finding, 163


  Mass, nature of, 10

  Mercury clock, 45

  Modern methods, 177, 197

  Moments, 101

  Moon’s appearance, 17

  Motion, reliability of, 57

  Musical notes, 95


  North pole, days at, 33


  Oscillations, law of, 151


  Parabola, the, 87

  Pendulum, the, 103, 145
    suspension, 145
    mercury, 147
    gridiron, 149
    theory of, 155
    free, 133

  Pisa, leaning tower of, 61

  Planets, names of, 11

  Pulse measurer, 99


  Ratchet wheels, 129

  Roman clocks, 40


  Sand-glasses, 41

  Space, nature of, 8

  Speed of falling bodies, 79

  Spring balance, 107

  Stevinus’ theory, 81

  Style of sun-dials, 35

  Sun-dials, 27
    to make, 48

  Synchronous clocks, 175


  Time, 13

  Toothed wheels, 125, 137

  Tuning fork, 94, 181


  Velocities, composition of, 85


  Water pressure, 37

  Water clocks, 39

  Watches, 156

  Week days, names of, 24

  Wheels, shape of teeth, 190

  Wheel-cutting machines, 193

  Winding drum, 131

  Winter sun, 31


  Zodiac, 18


BRADBURY, AGNEW, & CO. LD., PRINTERS, LONDON AND TONBRIDGE.




      *      *      *      *      *      *




Transcriber’s note:

Punctuation, hyphenation, and spelling were made consistent when a
predominant preference was found in this book; otherwise they were not
changed.

Simple typographical errors were corrected; occasional unpaired
quotation marks were retained.

Ambiguous hyphens at the ends of lines were retained.

Square roots are represented as √(values).

Index not checked for proper alphabetization or correct page references.

Page 16: “six o’clock” was printed as “six clock”; changed here.