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THE

PHILOSOPHY

OF

MATHEMATICS.


                          THE SCIENCE OF MATHEMATICS.
                                       |
                +----------------------+-------------------+
                |                                          |
                |                                          |
       ABSTRACT MATHEMATICS.                    CONCRETE MATHEMATICS.
                |                                          |
                |                                          |
                |                                  +-------+------+
                |                                  |              |
    ANALYSIS; _or_, _The Calculus_.             GEOMETRY.     MECHANICS.
                |                                  |
                |                                  |
        +-------+----------+               +-------+---------+
        |                  |               |                 |
    =Ordinary       =Transcendental   =Synthetic=        =Analytic=
    Analysis=;         Analysis=;     _or Special_       _or General_
  _or_, _Calculus   _or_, _Calculus    =Geometry=.       =Geometry=.
    of Direct         of Indirect          |                 |
   Functions_.        Functions_.          |                 |
        |                |                 |                 |
     +--+--+       +-----+--+         +----+----+        +---+---+
     |     |       |        |         |         |        |       |
     |     |       |        |         |         |        |       |
     |     |       |        |         |         |        |       |
     |     |       |        |         |         |        |       |
     | Algebra.    |    Calculus      |     Algebraic.   |   Of three
     |             |       of         |   Trigonometry.  |  dimensions.
     |             |   Variations.    |                  |
     |             |                  |                  |
  Arithmetic.   Differential       Graphical.          Of two
                and Integral      Descriptive        dimensions.
                 Calculus.         Geometry.




THE
PHILOSOPHY
OF
MATHEMATICS;

TRANSLATED FROM THE
COURS DE PHILOSOPHIE POSITIVE
OF
AUGUSTE COMTE,
BY
W. M. GILLESPIE,
PROFESSOR OF CIVIL ENGINEERING & ADJ. PROF. OF MATHEMATICS
IN UNION COLLEGE.

NEW YORK:
HARPER & BROTHERS, PUBLISHERS,
82 CLIFF STREET
1851.


Entered, according to Act of Congress, in the year one thousand
eight hundred and fifty-one, by

HARPER & BROTHERS.

in the Clerk's Office of the District Court of the Southern District
of New York.




PREFACE.


The pleasure and profit which the translator has received from the great
work here presented, have induced him to lay it before his
fellow-teachers and students of Mathematics in a more accessible form
than that in which it has hitherto appeared. The want of a comprehensive
map of the wide region of mathematical science--a bird's-eye view of its
leading features, and of the true bearings and relations of all its
parts--is felt by every thoughtful student. He is like the visitor to a
great city, who gets no just idea of its extent and situation till he
has seen it from some commanding eminence. To have a panoramic view of
the whole district--presenting at one glance all the parts in due
co-ordination, and the darkest nooks clearly shown--is invaluable to
either traveller or student. It is this which has been most perfectly
accomplished for mathematical science by the author whose work is here
presented.

Clearness and depth, comprehensiveness and precision, have never,
perhaps, been so remarkably united as in AUGUSTE COMTE. He views his
subject from an elevation which gives to each part of the complex whole
its true position and value, while his telescopic glance loses none of
the needful details, and not only itself pierces to the heart of the
matter, but converts its opaqueness into such transparent crystal, that
other eyes are enabled to see as deeply into it as his own.

Any mathematician who peruses this volume will need no other
justification of the high opinion here expressed; but others may
appreciate the following endorsements of well-known authorities. _Mill_,
in his "Logic," calls the work of M. Comte "by far the greatest yet
produced on the Philosophy of the sciences;" and adds, "of this
admirable work, one of the most admirable portions is that in which he
may truly be said to have created the Philosophy of the higher
Mathematics:" _Morell_, in his "Speculative Philosophy of Europe," says,
"The classification given of the sciences at large, and their regular
order of development, is unquestionably a master-piece of scientific
thinking, as simple as it is comprehensive;" and _Lewes_, in his
"Biographical History of Philosophy," names Comte "the Bacon of the
nineteenth century," and says, "I unhesitatingly record my conviction
that this is the greatest work of our age."

The complete work of M. Comte--his "_Cours de Philosophie
Positive_"--fills six large octavo volumes, of six or seven hundred
pages each, two thirds of the first volume comprising the purely
mathematical portion. The great bulk of the "Course" is the probable
cause of the fewness of those to whom even this section of it is known.
Its presentation in its present form is therefore felt by the translator
to be a most useful contribution to mathematical progress in this
country. The comprehensiveness of the style of the author--grasping all
possible forms of an idea in one Briarean sentence, armed at all points
against leaving any opening for mistake or forgetfulness--occasionally
verges upon cumbersomeness and formality. The translator has, therefore,
sometimes taken the liberty of breaking up or condensing a long
sentence, and omitting a few passages not absolutely necessary, or
referring to the peculiar "Positive philosophy" of the author; but he
has generally aimed at a conscientious fidelity to the original. It has
often been difficult to retain its fine shades and subtile distinctions
of meaning, and, at the same time, replace the peculiarly appropriate
French idioms by corresponding English ones. The attempt, however, has
always been made, though, when the best course has been at all doubtful,
the language of the original has been followed as closely as possible,
and, when necessary, smoothness and grace have been unhesitatingly
sacrificed to the higher attributes of clearness and precision.

Some forms of expression may strike the reader as unusual, but they have
been retained because they were characteristic, not of the mere language
of the original, but of its spirit. When a great thinker has clothed his
conceptions in phrases which are singular even in his own tongue, he who
professes to translate him is bound faithfully to preserve such forms of
speech, as far as is practicable; and this has been here done with
respect to such peculiarities of expression as belong to the author,
not as a foreigner, but as an individual--not because he writes in
French, but because he is Auguste Comte.

The young student of Mathematics should not attempt to read the whole of
this volume at once, but should peruse each portion of it in connexion
with the temporary subject of his special study: the first chapter of
the first book, for example, while he is studying Algebra; the first
chapter of the second book, when he has made some progress in Geometry;
and so with the rest. Passages which are obscure at the first reading
will brighten up at the second; and as his own studies cover a larger
portion of the field of Mathematics, he will see more and more clearly
their relations to one another, and to those which he is next to take
up. For this end he is urgently recommended to obtain a perfect
familiarity with the "Analytical Table of Contents," which maps out the
whole subject, the grand divisions of which are also indicated in the
Tabular View facing the title-page. Corresponding heads will be found in
the body of the work, the principal divisions being in SMALL CAPITALS,
and the subdivisions in _Italics_. For these details the translator
alone is responsible.




  ANALYTICAL TABLE OF CONTENTS.

  INTRODUCTION.

  Page

  GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE   17

  THE OBJECT OF MATHEMATICS                        18
    Measuring Magnitudes                           18
      Difficulties                                 19
      General Method                               20
      Illustrations                                21
        1. Falling Bodies                          21
        2. Inaccessible Distances                  23
        3. Astronomical Facts                      24

  TRUE DEFINITION OF MATHEMATICS                   25
    A Science, not an Art                          25

  ITS TWO FUNDAMENTAL DIVISIONS                    26
      Their different Objects                      27
      Their different Natures                      29
    _Concrete Mathematics_                         31
      Geometry and Mechanics                       32
    _Abstract Mathematics_                         33
      The Calculus, or Analysis                    33

  EXTENT OF ITS FIELD                              35
      Its Universality                             36
      Its Limitations                              37


  BOOK I.

  ANALYSIS.


  CHAPTER I.

  Page

  GENERAL VIEW OF MATHEMATICAL ANALYSIS            45

  THE TRUE IDEA OF AN EQUATION                     46
      Division of Functions into Abstract and
        Concrete                                   47
      Enumeration of Abstract Functions            50

  DIVISIONS OF THE CALCULUS                        53
    _The Calculus of Values, or Arithmetic_        57
      Its Extent                                   57
      Its true Nature                              59
    _The Calculus of Functions_                    61
      Two Modes of obtaining Equations             61
        1. By the Relations between the given
           Quantities                              61
        2. By the Relations between auxiliary
           Quantities                              64
    Corresponding Divisions of the Calculus of
      Functions                                    67


  CHAPTER II.

  ORDINARY ANALYSIS; OR, ALGEBRA.                  69

      Its Object                                   69
      Classification of Equations                  70

  ALGEBRAIC EQUATIONS                              71
      Their Classification                         71

  ALGEBRAIC RESOLUTION OF EQUATIONS                72
      Its Limits                                   72
      General Solution                             72
      What we know in Algebra                      74

  NUMERICAL RESOLUTION OF EQUATIONS                75
      Its limited Usefulness                       76
  Different Divisions of the two Systems           78

  THE THEORY OF EQUATIONS                          79

  THE METHOD OF INDETERMINATE COEFFICIENTS         80

  IMAGINARY QUANTITIES                             81

  NEGATIVE QUANTITIES                              81

  THE PRINCIPLE OF HOMOGENEITY                     84


  CHAPTER III.

  TRANSCENDENTAL ANALYSIS:

  Page

  ITS DIFFERENT CONCEPTIONS                        88

      Preliminary Remarks                          88
      Its early History                            89

  METHOD OF LEIBNITZ                               91
      Infinitely small Elements                    91
      _Examples_:
        1. Tangents                                93
        2. Rectification of an Arc                 94
        3. Quadrature of a Curve                   95
        4. Velocity in variable Motion             95
        5. Distribution of Heat                    96
      Generality of the Formulas                   97
      Demonstration of the Method                  98
        Illustration by Tangents                  102

  METHOD OF NEWTON                                103
      Method of Limits                            103
        _Examples_:
          1. Tangents                             104
          2. Rectifications                       105
      Fluxions and Fluents                        106

  METHOD OF LAGRANGE                              108
      Derived Functions                           108
      An extension of ordinary Analysis           108
      _Example_: Tangents                         109
    _Fundamental Identity of the three Methods_   110
    _Their comparative Value_                     113
      That of Leibnitz                            113
      That of Newton                              115
      That of Lagrange                            117


  CHAPTER IV.

  Page

  THE DIFFERENTIAL AND INTEGRAL CALCULUS          120

  ITS TWO FUNDAMENTAL DIVISIONS                   120

  THEIR RELATIONS TO EACH OTHER                   121
        1. Use of the Differential Calculus as
           preparatory to that of the Integral    123
        2. Employment of the Differential
           Calculus alone                         125
        3. Employment of the Integral Calculus
           alone                                  125
            Three Classes of Questions hence
              resulting                           126

  THE DIFFERENTIAL CALCULUS                       127
      Two Cases: Explicit and Implicit Functions  127
        Two sub-Cases: a single Variable or
          several                                 129
        Two other Cases: Functions separate or
          combined                                130
      Reduction of all to the Differentiation of
        the ten elementary Functions              131
      Transformation of derived Functions for
        new Variables                             132
      Different Orders of Differentiation         133
      Analytical Applications                     133

  THE INTEGRAL CALCULUS                           135
      Its fundamental Division: Explicit and
        Implicit Functions                        135
      Subdivisions: a single Variable or several  136
      Calculus of partial Differences             137
      Another Subdivision: different Orders of
        Differentiation                           138
      Another equivalent Distinction              140
      _Quadratures_                               142
        Integration of Transcendental Functions   143
          Integration by Parts                    143
        Integration of Algebraic Functions        143
      Singular Solutions                          144
      Definite Integrals                          146
      Prospects of the Integral Calculus          148


  CHAPTER V.

  Page

  THE CALCULUS OF VARIATIONS                      151

  PROBLEMS GIVING RISE TO IT                      151
      Ordinary Questions of Maxima and Minima     151
      A new Class of Questions                    152
        Solid of least Resistance;
          Brachystochrone; Isoperimeters          153

  ANALYTICAL NATURE OF THESE QUESTIONS            154

  METHODS OF THE OLDER GEOMETERS                  155

  METHOD OF LAGRANGE                              156
      Two Classes of Questions                    157
        1. Absolute Maxima and Minima             157
           Equations of Limits                    159
             A more general Consideration         159
        2. Relative Maxima and Minima             160
      Other Applications of the Method of
        Variations                                162

  ITS RELATIONS TO THE ORDINARY CALCULUS          163


  CHAPTER VI.

  THE CALCULUS OF FINITE DIFFERENCES              167

      Its general Character                       167
      Its true Nature                             168

  GENERAL THEORY OF SERIES                        170
      Its Identity with this Calculus             172

  PERIODIC OR DISCONTINUOUS FUNCTIONS             173

  APPLICATIONS OF THIS CALCULUS                   173
      Series                                      173
      Interpolation                               173
      Approximate Rectification, &c.              174


  BOOK II.

  GEOMETRY.

  CHAPTER I.

  Page

  A GENERAL VIEW OF GEOMETRY                      179

      The true Nature of Geometry                 179
      Two fundamental Ideas                       181
        1. The Idea of Space                      181
        2. Different kinds of Extension           182

  THE FINAL OBJECT OF GEOMETRY                    184
      Nature of Geometrical Measurement           185
        Of Surfaces and Volumes                   185
        Of curve Lines                            187
        Of right Lines                            189

  THE INFINITE EXTENT OF ITS FIELD                190
      Infinity of Lines                           190
      Infinity of Surfaces                        191
      Infinity of Volumes                         192
      Analytical Invention of Curves, &c.         193

  EXPANSION OF ORIGINAL DEFINITION                193
      Properties of Lines and Surfaces            195
      Necessity of their Study                    195
        1. To find the most suitable Property     195
        2. To pass from the Concrete to the
           Abstract                               197
      Illustrations:
        Orbits of the Planets                     198
        Figure of the Earth                       199

  THE TWO GENERAL METHODS OF GEOMETRY             202
      Their fundamental Difference                203
        1°. Different Questions with respect to
          the same Figure                         204
        2°. Similar Questions with respect to
          different Figures                       204
      Geometry of the Ancients                    204
      Geometry of the Moderns                     206
      Superiority of the Modern                   207
      The Ancient the base of the Modern          209


  CHAPTER II.

  ANCIENT OR SYNTHETIC GEOMETRY

  Page

  ITS PROPER EXTENT                               212
      Lines; Polygons; Polyhedrons                212
      Not to be farther restricted                213
      Improper Application of Analysis            214
      Attempted Demonstrations of Axioms          216

  GEOMETRY OF THE RIGHT LINE                      217

  GRAPHICAL SOLUTIONS                             218
      _Descriptive Geometry_                      220

  ALGEBRAICAL SOLUTIONS                           224
      _Trigonometry_                              225
        Two Methods of introducing Angles         226
          1. By Arcs                              226
          2. By trigonometrical Lines             226
        Advantages of the latter                  226
        Its Division of trigonometrical Questions 227
          1. Relations between Angles and
             trigonometrical Lines                228
          2. Relations between trigonometrical
             Lines and Sides                      228
        Increase of trigonometrical Lines         228
        Study of the Relations between them       230


  CHAPTER III.

  MODERN OR ANALYTICAL GEOMETRY

  Page

  THE ANALYTICAL REPRESENTATION OF FIGURES        232
      Reduction of Figure to Position             233
      Determination of the position of a Point    234

  PLANE CURVES                                    237
      Expression of Lines by Equations            237
      Expression of Equations by Lines            238
      Any change in the Line changes the Equation 240
      Every "Definition" of a Line is an Equation 241
      _Choice of Co-ordinates_                    245
        Two different points of View              245
          1. Representation of Lines by Equations 246
          2. Representation of Equations by Lines 246
        Superiority of the rectilinear System     248
          Advantages of perpendicular Axes        249

  SURFACES    251
      Determination of a Point in Space           251
      Expression of Surfaces by Equations         253
      Expression of Equations by Surfaces         253

  CURVES IN SPACE     255

  Imperfections of Analytical Geometry            258
      Relatively to Geometry                      258
      Relatively to Analysis                      258




THE

PHILOSOPHY OF MATHEMATICS.

INTRODUCTION.

GENERAL CONSIDERATIONS.


Although Mathematical Science is the most ancient and the most perfect
of all, yet the general idea which we ought to form of it has not yet
been clearly determined. Its definition and its principal divisions have
remained till now vague and uncertain. Indeed the plural name--"The
Mathematics"--by which we commonly designate it, would alone suffice to
indicate the want of unity in the common conception of it.

In truth, it was not till the commencement of the last century that the
different fundamental conceptions which constitute this great science
were each of them sufficiently developed to permit the true spirit of
the whole to manifest itself with clearness. Since that epoch the
attention of geometers has been too exclusively absorbed by the special
perfecting of the different branches, and by the application which they
have made of them to the most important laws of the universe, to allow
them to give due attention to the general system of the science.

But at the present time the progress of the special departments is no
longer so rapid as to forbid the contemplation of the whole. The science
of mathematics is now sufficiently developed, both in itself and as to
its most essential application, to have arrived at that state of
consistency in which we ought to strive to arrange its different parts
in a single system, in order to prepare for new advances. We may even
observe that the last important improvements of the science have
directly paved the way for this important philosophical operation, by
impressing on its principal parts a character of unity which did not
previously exist.

To form a just idea of the object of mathematical science, we may start
from the indefinite and meaningless definition of it usually given, in
calling it "_The science of magnitudes_," or, which is more definite,
"_The science which has for its object the measurement of magnitudes._"
Let us see how we can rise from this rough sketch (which is singularly
deficient in precision and depth, though, at bottom, just) to a
veritable definition, worthy of the importance, the extent, and the
difficulty of the science.


THE OBJECT OF MATHEMATICS.

_Measuring Magnitudes._ The question of _measuring_ a magnitude in
itself presents to the mind no other idea than that of the simple direct
comparison of this magnitude with another similar magnitude, supposed to
be known, which it takes for the _unit_ of comparison among all others
of the same kind. According to this definition, then, the science of
mathematics--vast and profound as it is with reason reputed to
be--instead of being an immense concatenation of prolonged mental
labours, which offer inexhaustible occupation to our intellectual
activity, would seem to consist of a simple series of mechanical
processes for obtaining directly the ratios of the quantities to be
measured to those by which we wish to measure them, by the aid of
operations of similar character to the superposition of lines, as
practiced by the carpenter with his rule.

The error of this definition consists in presenting as direct an object
which is almost always, on the contrary, very indirect. The _direct_
measurement of a magnitude, by superposition or any similar process, is
most frequently an operation quite impossible for us to perform; so that
if we had no other means for determining magnitudes than direct
comparisons, we should be obliged to renounce the knowledge of most of
those which interest us.

_Difficulties._ The force of this general observation will be understood
if we limit ourselves to consider specially the particular case which
evidently offers the most facility--that of the measurement of one
straight line by another. This comparison, which is certainly the most
simple which we can conceive, can nevertheless scarcely ever be effected
directly. In reflecting on the whole of the conditions necessary to
render a line susceptible of a direct measurement, we see that most
frequently they cannot be all fulfilled at the same time. The first and
the most palpable of these conditions--that of being able to pass over
the line from one end of it to the other, in order to apply the unit of
measurement to its whole length--evidently excludes at once by far the
greater part of the distances which interest us the most; in the first
place, all the distances between the celestial bodies, or from any one
of them to the earth; and then, too, even the greater number of
terrestrial distances, which are so frequently inaccessible. But even if
this first condition be found to be fulfilled, it is still farther
necessary that the length be neither too great nor too small, which
would render a direct measurement equally impossible. The line must also
be suitably situated; for let it be one which we could measure with the
greatest facility, if it were horizontal, but conceive it to be turned
up vertically, and it becomes impossible to measure it.

The difficulties which we have indicated in reference to measuring
lines, exist in a very much greater degree in the measurement of
surfaces, volumes, velocities, times, forces, &c. It is this fact which
makes necessary the formation of mathematical science, as we are going
to see; for the human mind has been compelled to renounce, in almost all
cases, the direct measurement of magnitudes, and to seek to determine
them _indirectly_, and it is thus that it has been led to the creation
of mathematics.

_General Method._ The general method which is constantly employed, and
evidently the only one conceivable, to ascertain magnitudes which do not
admit of a direct measurement, consists in connecting them with others
which are susceptible of being determined immediately, and by means of
which we succeed in discovering the first through the relations which
subsist between the two. Such is the precise object of mathematical
science viewed as a whole. In order to form a sufficiently extended idea
of it, we must consider that this indirect determination of magnitudes
may be indirect in very different degrees. In a great number of cases,
which are often the most important, the magnitudes, by means of which
the principal magnitudes sought are to be determined, cannot themselves
be measured directly, and must therefore, in their turn, become the
subject of a similar question, and so on; so that on many occasions the
human mind is obliged to establish a long series of intermediates
between the system of unknown magnitudes which are the final objects of
its researches, and the system of magnitudes susceptible of direct
measurement, by whose means we finally determine the first, with which
at first they appear to have no connexion.

_Illustrations._ Some examples will make clear any thing which may seem
too abstract in the preceding generalities.

1. _Falling Bodies._ Let us consider, in the first place, a natural
phenomenon, very simple, indeed, but which may nevertheless give rise to
a mathematical question, really existing, and susceptible of actual
applications--the phenomenon of the vertical fall of heavy bodies.

The mind the most unused to mathematical conceptions, in observing this
phenomenon, perceives at once that the two _quantities_ which it
presents--namely, the _height_ from which a body has fallen, and the
_time_ of its fall--are necessarily connected with each other, since
they vary together, and simultaneously remain fixed; or, in the language
of geometers, that they are "_functions_" of each other. The phenomenon,
considered under this point of view, gives rise then to a mathematical
question, which consists in substituting for the direct measurement of
one of these two magnitudes, when it is impossible, the measurement of
the other. It is thus, for example, that we may determine indirectly the
depth of a precipice, by merely measuring the time that a heavy body
would occupy in falling to its bottom, and by suitable procedures this
inaccessible depth will be known with as much precision as if it was a
horizontal line placed in the most favourable circumstances for easy and
exact measurement. On other occasions it is the height from which a body
has fallen which it will be easy to ascertain, while the time of the
fall could not be observed directly; then the same phenomenon would give
rise to the inverse question, namely, to determine the time from the
height; as, for example, if we wished to ascertain what would be the
duration of the vertical fall of a body falling from the moon to the
earth.

In this example the mathematical question is very simple, at least when
we do not pay attention to the variation in the intensity of gravity, or
the resistance of the fluid which the body passes through in its fall.
But, to extend the question, we have only to consider the same
phenomenon in its greatest generality, in supposing the fall oblique,
and in taking into the account all the principal circumstances. Then,
instead of offering simply two variable quantities connected with each
other by a relation easy to follow, the phenomenon will present a much
greater number; namely, the space traversed, whether in a vertical or
horizontal direction; the time employed in traversing it; the velocity
of the body at each point of its course; even the intensity and the
direction of its primitive impulse, which may also be viewed as
variables; and finally, in certain cases (to take every thing into the
account), the resistance of the medium and the intensity of gravity. All
these different quantities will be connected with one another, in such a
way that each in its turn may be indirectly determined by means of the
others; and this will present as many distinct mathematical questions as
there may be co-existing magnitudes in the phenomenon under
consideration. Such a very slight change in the physical conditions of a
problem may cause (as in the above example) a mathematical research, at
first very elementary, to be placed at once in the rank of the most
difficult questions, whose complete and rigorous solution surpasses as
yet the utmost power of the human intellect.

2. _Inaccessible Distances._ Let us take a second example from
geometrical phenomena. Let it be proposed to determine a distance which
is not susceptible of direct measurement; it will be generally conceived
as making part of a _figure_, or certain system of lines, chosen in such
a way that all its other parts may be observed directly; thus, in the
case which is most simple, and to which all the others may be finally
reduced, the proposed distance will be considered as belonging to a
triangle, in which we can determine directly either another side and two
angles, or two sides and one angle. Thence-forward, the knowledge of the
desired distance, instead of being obtained directly, will be the result
of a mathematical calculation, which will consist in deducing it from
the observed elements by means of the relation which connects it with
them. This calculation will become successively more and more
complicated, if the parts which we have supposed to be known cannot
themselves be determined (as is most frequently the case) except in an
indirect manner, by the aid of new auxiliary systems, the number of
which, in great operations of this kind, finally becomes very
considerable. The distance being once determined, the knowledge of it
will frequently be sufficient for obtaining new quantities, which will
become the subject of new mathematical questions. Thus, when we know at
what distance any object is situated, the simple observation of its
apparent diameter will evidently permit us to determine indirectly its
real dimensions, however inaccessible it may be, and, by a series of
analogous investigations, its surface, its volume, even its weight, and
a number of other properties, a knowledge of which seemed forbidden to
us.

3. _Astronomical Facts._ It is by such calculations that man has been
able to ascertain, not only the distances from the planets to the earth,
and, consequently, from each other, but their actual magnitude, their
true figure, even to the inequalities of their surface; and, what seemed
still more completely hidden from us, their respective masses, their
mean densities, the principal circumstances of the fall of heavy bodies
on the surface of each of them, &c.

By the power of mathematical theories, all these different results, and
many others relative to the different classes of mathematical phenomena,
have required no other direct measurements than those of a very small
number of straight lines, suitably chosen, and of a greater number of
angles. We may even say, with perfect truth, so as to indicate in a word
the general range of the science, that if we did not fear to multiply
calculations unnecessarily, and if we had not, in consequence, to
reserve them for the determination of the quantities which could not be
measured directly, the determination of all the magnitudes susceptible
of precise estimation, which the various orders of phenomena can offer
us, could be finally reduced to the direct measurement of a single
straight line and of a suitable number of angles.


TRUE DEFINITION OF MATHEMATICS.

We are now able to define mathematical science with precision, by
assigning to it as its object the _indirect_ measurement of magnitudes,
and by saying it constantly proposes _to determine certain magnitudes
from others by means of the precise relations existing between them_.

This enunciation, instead of giving the idea of only an _art_, as do all
the ordinary definitions, characterizes immediately a true _science_,
and shows it at once to be composed of an immense chain of intellectual
operations, which may evidently become very complicated, because of the
series of intermediate links which it will be necessary to establish
between the unknown quantities and those which admit of a direct
measurement; of the number of variables coexistent in the proposed
question; and of the nature of the relations between all these different
magnitudes furnished by the phenomena under consideration. According to
such a definition, the spirit of mathematics consists in always
regarding all the quantities which any phenomenon can present, as
connected and interwoven with one another, with the view of deducing
them from one another. Now there is evidently no phenomenon which cannot
give rise to considerations of this kind; whence results the naturally
indefinite extent and even the rigorous logical universality of
mathematical science. We shall seek farther on to circumscribe as
exactly as possible its real extension.

The preceding explanations establish clearly the propriety of the name
employed to designate the science which we are considering. This
denomination, which has taken to-day so definite a meaning by itself
signifies simply _science_ in general. Such a designation, rigorously
exact for the Greeks, who had no other real science, could be retained
by the moderns only to indicate the mathematics as _the_ science, beyond
all others--the science of sciences.

Indeed, every true science has for its object the determination of
certain phenomena by means of others, in accordance with the relations
which exist between them. Every _science_ consists in the co-ordination
of facts; if the different observations were entirely isolated, there
would be no science. We may even say, in general terms, that _science_
is essentially destined to dispense, so far as the different phenomena
permit it, with all direct observation, by enabling us to deduce from
the smallest possible number of immediate data the greatest possible
number of results. Is not this the real use, whether in speculation or
in action, of the _laws_ which we succeed in discovering among natural
phenomena? Mathematical science, in this point of view, merely pushes to
the highest possible degree the same kind of researches which are
pursued, in degrees more or less inferior, by every real science in its
respective sphere.


ITS TWO FUNDAMENTAL DIVISIONS.

We have thus far viewed mathematical science only as a whole, without
paying any regard to its divisions. We must now, in order to complete
this general view, and to form a just idea of the philosophical
character of the science, consider its fundamental division. The
secondary divisions will be examined in the following chapters.

This principal division, which we are about to investigate, can be
truly rational, and derived from the real nature of the subject, only so
far as it spontaneously presents itself to us, in making the exact
analysis of a complete mathematical question. We will, therefore, having
determined above what is the general object of mathematical labours, now
characterize with precision the principal different orders of inquiries,
of which they are constantly composed.

_Their different Objects._ The complete solution of every mathematical
question divides itself necessarily into two parts, of natures
essentially distinct, and with relations invariably determinate. We have
seen that every mathematical inquiry has for its object to determine
unknown magnitudes, according to the relations between them and known
magnitudes. Now for this object, it is evidently necessary, in the first
place, to ascertain with precision the relations which exist between the
quantities which we are considering. This first branch of inquiries
constitutes that which I call the _concrete_ part of the solution. When
it is finished, the question changes; it is now reduced to a pure
question of numbers, consisting simply in determining unknown numbers,
when we know what precise relations connect them with known numbers.
This second branch of inquiries is what I call the _abstract_ part of
the solution. Hence follows the fundamental division of general
mathematical science into _two_ great sciences--ABSTRACT MATHEMATICS,
and CONCRETE MATHEMATICS.

This analysis may be observed in every complete mathematical question,
however simple or complicated it may be. A single example will suffice
to make it intelligible.

Taking up again the phenomenon of the vertical fall of a heavy body, and
considering the simplest case, we see that in order to succeed in
determining, by means of one another, the height whence the body has
fallen, and the duration of its fall, we must commence by discovering
the exact relation of these two quantities, or, to use the language of
geometers, the _equation_ which exists between them. Before this first
research is completed, every attempt to determine numerically the value
of one of these two magnitudes from the other would evidently be
premature, for it would have no basis. It is not enough to know vaguely
that they depend on one another--which every one at once perceives--but
it is necessary to determine in what this dependence consists. This
inquiry may be very difficult, and in fact, in the present case,
constitutes incomparably the greater part of the problem. The true
scientific spirit is so modern, that no one, perhaps, before Galileo,
had ever remarked the increase of velocity which a body experiences in
its fall: a circumstance which excludes the hypothesis, towards which
our mind (always involuntarily inclined to suppose in every phenomenon
the most simple _functions_, without any other motive than its greater
facility in conceiving them) would be naturally led, that the height was
proportional to the time. In a word, this first inquiry terminated in
the discovery of the law of Galileo.

When this _concrete_ part is completed, the inquiry becomes one of quite
another nature. Knowing that the spaces passed through by the body in
each successive second of its fall increase as the series of odd
numbers, we have then a problem purely numerical and _abstract_; to
deduce the height from the time, or the time from the height; and this
consists in finding that the first of these two quantities, according to
the law which has been established, is a known multiple of the second
power of the other; from which, finally, we have to calculate the value
of the one when that of the other is given.

In this example the concrete question is more difficult than the
abstract one. The reverse would be the case if we considered the same
phenomenon in its greatest generality, as I have done above for another
object. According to the circumstances, sometimes the first, sometimes
the second, of these two parts will constitute the principal difficulty
of the whole question; for the mathematical law of the phenomenon may be
very simple, but very difficult to obtain, or it may be easy to
discover, but very complicated; so that the two great sections of
mathematical science, when we compare them as wholes, must be regarded
as exactly equivalent in extent and in difficulty, as well as in
importance, as we shall show farther on, in considering each of them
separately.

_Their different Natures._ These two parts, essentially distinct in
their _object_, as we have just seen, are no less so with regard to the
_nature_ of the inquiries of which they are composed.

The first should be called _concrete_, since it evidently depends on the
character of the phenomena considered, and must necessarily vary when we
examine new phenomena; while the second is completely independent of the
nature of the objects examined, and is concerned with only the
_numerical_ relations which they present, for which reason it should be
called _abstract_. The same relations may exist in a great number of
different phenomena, which, in spite of their extreme diversity, will
be viewed by the geometer as offering an analytical question
susceptible, when studied by itself, of being resolved once for all.
Thus, for instance, the same law which exists between the space and the
time of the vertical fall of a body in a vacuum, is found again in many
other phenomena which offer no analogy with the first nor with each
other; for it expresses the relation between the surface of a spherical
body and the length of its diameter; it determines, in like manner, the
decrease of the intensity of light or of heat in relation to the
distance of the objects lighted or heated, &c. The abstract part, common
to these different mathematical questions, having been treated in
reference to one of these, will thus have been treated for all; while
the concrete part will have necessarily to be again taken up for each
question separately, without the solution of any one of them being able
to give any direct aid, in that connexion, for the solution of the rest.

The abstract part of mathematics is, then, general in its nature; the
concrete part, special.

To present this comparison under a new point of view, we may say
concrete mathematics has a philosophical character, which is essentially
experimental, physical, phenomenal; while that of abstract mathematics
is purely logical, rational. The concrete part of every mathematical
question is necessarily founded on the consideration of the external
world, and could never be resolved by a simple series of intellectual
combinations. The abstract part, on the contrary, when it has been very
completely separated, can consist only of a series of logical
deductions, more or less prolonged; for if we have once found the
equations of a phenomenon, the determination of the quantities therein
considered, by means of one another, is a matter for reasoning only,
whatever the difficulties may be. It belongs to the understanding alone
to deduce from these equations results which are evidently contained in
them, although perhaps in a very involved manner, without there being
occasion to consult anew the external world; the consideration of which,
having become thenceforth foreign to the subject, ought even to be
carefully set aside in order to reduce the labour to its true peculiar
difficulty. The _abstract_ part of mathematics is then purely
instrumental, and is only an immense and admirable extension of natural
logic to a certain class of deductions. On the other hand, geometry and
mechanics, which, as we shall see presently, constitute the _concrete_
part, must be viewed as real natural sciences, founded on observation,
like all the rest, although the extreme simplicity of their phenomena
permits an infinitely greater degree of systematization, which has
sometimes caused a misconception of the experimental character of their
first principles.

We see, by this brief general comparison, how natural and profound is
our fundamental division of mathematical science.

We have now to circumscribe, as exactly as we can in this first sketch,
each of these two great sections.


CONCRETE MATHEMATICS.

_Concrete Mathematics_ having for its object the discovery of the
_equations_ of phenomena, it would seem at first that it must be
composed of as many distinct sciences as we find really distinct
categories among natural phenomena. But we are yet very far from having
discovered mathematical laws in all kinds of phenomena; we shall even
see, presently, that the greater part will very probably always hide
themselves from our investigations. In reality, in the present condition
of the human mind, there are directly but two great general classes of
phenomena, whose equations we constantly know; these are, firstly,
geometrical, and, secondly, mechanical phenomena. Thus, then, the
concrete part of mathematics is composed of GEOMETRY and RATIONAL
MECHANICS.

This is sufficient, it is true, to give to it a complete character of
logical universality, when we consider all phenomena from the most
elevated point of view of natural philosophy. In fact, if all the parts
of the universe were conceived as immovable, we should evidently have
only geometrical phenomena to observe, since all would be reduced to
relations of form, magnitude, and position; then, having regard to the
motions which take place in it, we would have also to consider
mechanical phenomena. Hence the universe, in the statical point of view,
presents only geometrical phenomena; and, considered dynamically, only
mechanical phenomena. Thus geometry and mechanics constitute the two
fundamental natural sciences, in this sense, that all natural effects
may be conceived as simple necessary results, either of the laws of
extension or of the laws of motion.

But although this conception is always logically possible, the
difficulty is to specialize it with the necessary precision, and to
follow it exactly in each of the general cases offered to us by the
study of nature; that is, to effectually reduce each principal question
of natural philosophy, for a certain determinate order of phenomena, to
the question of geometry or mechanics, to which we might rationally
suppose it should be brought. This transformation, which requires great
progress to have been previously made in the study of each class of
phenomena, has thus far been really executed only for those of
astronomy, and for a part of those considered by terrestrial physics,
properly so called. It is thus that astronomy, acoustics, optics, &c.,
have finally become applications of mathematical science to certain
orders of observations.[1] But these applications not being by their
nature rigorously circumscribed, to confound them with the science would
be to assign to it a vague and indefinite domain; and this is done in
the usual division, so faulty in so many other respects, of the
mathematics into "Pure" and "Applied."

  [Footnote 1: The investigation of the mathematical phenomena of the
  laws of heat by Baron Fourier has led to the establishment, in an
  entirely direct manner, of Thermological equations. This great
  discovery tends to elevate our philosophical hopes as to the future
  extensions of the legitimate applications of mathematical analysis,
  and renders it proper, in the opinion of author, to regard
  _Thermology_ as a third principal branch of concrete mathematics.]


ABSTRACT MATHEMATICS.

The nature of abstract mathematics (the general division of which will
be examined in the following chapter) is clearly and exactly determined.
It is composed of what is called the _Calculus_,[2] taking this word in
its greatest extent, which reaches from the most simple numerical
operations to the most sublime combinations of transcendental analysis.
The _Calculus_ has the solution of all questions relating to numbers
for its peculiar object. Its _starting point_ is, constantly and
necessarily, the knowledge of the precise relations, _i.e._, of the
_equations_, between the different magnitudes which are simultaneously
considered; that which is, on the contrary, the _stopping point_ of
concrete mathematics. However complicated, or however indirect these
relations may be, the final object of the calculus always is to obtain
from them the values of the unknown quantities by means of those which
are known. This _science_, although nearer perfection than any other, is
really little advanced as yet, so that this object is rarely attained in
a manner completely satisfactory.

  [Footnote 2: The translator has felt justified in employing this
  very convenient word (for which our language has no precise
  equivalent) as an English one, in its most extended sense, in spite
  of its being often popularly confounded with its Differential and
  Integral department.]

Mathematical analysis is, then, the true rational basis of the entire
system of our actual knowledge. It constitutes the first and the most
perfect of all the fundamental sciences. The ideas with which it
occupies itself are the most universal, the most abstract, and the most
simple which it is possible for us to conceive.

This peculiar nature of mathematical analysis enables us easily to
explain why, when it is properly employed, it is such a powerful
instrument, not only to give more precision to our real knowledge, which
is self-evident, but especially to establish an infinitely more perfect
co-ordination in the study of the phenomena which admit of that
application; for, our conceptions having been so generalized and
simplified that a single analytical question, abstractly resolved,
contains the _implicit_ solution of a great number of diverse physical
questions, the human mind must necessarily acquire by these means a
greater facility in perceiving relations between phenomena which at
first appeared entirely distinct from one another. We thus naturally see
arise, through the medium of analysis, the most frequent and the most
unexpected approximations between problems which at first offered no
apparent connection, and which we often end in viewing as identical.
Could we, for example, without the aid of analysis, perceive the least
resemblance between the determination of the direction of a curve at
each of its points and that of the velocity acquired by a body at every
instant of its variable motion? and yet these questions, however
different they may be, compose but one in the eyes of the geometer.

The high relative perfection of mathematical analysis is as easily
perceptible. This perfection is not due, as some have thought, to the
nature of the signs which are employed as instruments of reasoning,
eminently concise and general as they are. In reality, all great
analytical ideas have been formed without the algebraic signs having
been of any essential aid, except for working them out after the mind
had conceived them. The superior perfection of the science of the
calculus is due principally to the extreme simplicity of the ideas which
it considers, by whatever signs they may be expressed; so that there is
not the least hope, by any artifice of scientific language, of
perfecting to the same degree theories which refer to more complex
subjects, and which are necessarily condemned by their nature to a
greater or less logical inferiority.


THE EXTENT OF ITS FIELD.

Our examination of the philosophical character of mathematical science
would remain incomplete, if, after having viewed its object and
composition, we did not examine the real extent of its domain.

_Its Universality_. For this purpose it is indispensable to perceive,
first of all, that, in the purely logical point of view, this science is
by itself necessarily and rigorously universal; for there is no question
whatever which may not be finally conceived as consisting in determining
certain quantities from others by means of certain relations, and
consequently as admitting of reduction, in final analysis, to a simple
question of numbers. In all our researches, indeed, on whatever subject,
our object is to arrive at numbers, at quantities, though often in a
very imperfect manner and by very uncertain methods. Thus, taking an
example in the class of subjects the least accessible to mathematics,
the phenomena of living bodies, even when considered (to take the most
complicated case) in the state of disease, is it not manifest that all
the questions of therapeutics may be viewed as consisting in determining
the _quantities_ of the different agents which modify the organism, and
which must act upon it to bring it to its normal state, admitting, for
some of these quantities in certain cases, values which are equal to
zero, or negative, or even contradictory?

The fundamental idea of Descartes on the relation of the concrete to the
abstract in mathematics, has proven, in opposition to the superficial
distinction of metaphysics, that all ideas of quality may be reduced to
those of quantity. This conception, established at first by its immortal
author in relation to geometrical phenomena only, has since been
effectually extended to mechanical phenomena, and in our days to those
of heat. As a result of this gradual generalization, there are now no
geometers who do not consider it, in a purely theoretical sense, as
capable of being applied to all our real ideas of every sort, so that
every phenomenon is logically susceptible of being represented by an
_equation_; as much so, indeed, as is a curve or a motion, excepting the
difficulty of discovering it, and then of _resolving_ it, which may be,
and oftentimes are, superior to the greatest powers of the human mind.

_Its Limitations_. Important as it is to comprehend the rigorous
universality, in a logical point of view, of mathematical science, it is
no less indispensable to consider now the great real _limitations_,
which, through the feebleness of our intellect, narrow in a remarkable
degree its actual domain, in proportion as phenomena, in becoming
special, become complicated.

Every question may be conceived as capable of being reduced to a pure
question of numbers; but the difficulty of effecting such a
transformation increases so much with the complication of the phenomena
of natural philosophy, that it soon becomes insurmountable.

This will be easily seen, if we consider that to bring a question within
the field of mathematical analysis, we must first have discovered the
precise relations which exist between the quantities which are found in
the phenomenon under examination, the establishment of these equations
being the necessary starting point of all analytical labours. This must
evidently be so much the more difficult as we have to do with phenomena
which are more special, and therefore more complicated. We shall thus
find that it is only in _inorganic physics_, at the most, that we can
justly hope ever to obtain that high degree of scientific perfection.

The _first_ condition which is necessary in order that phenomena may
admit of mathematical laws, susceptible of being discovered, evidently
is, that their different quantities should admit of being expressed by
fixed numbers. We soon find that in this respect the whole of _organic
physics_, and probably also the most complicated parts of inorganic
physics, are necessarily inaccessible, by their nature, to our
mathematical analysis, by reason of the extreme numerical variability of
the corresponding phenomena. Every precise idea of fixed numbers is
truly out of place in the phenomena of living bodies, when we wish to
employ it otherwise than as a means of relieving the attention, and when
we attach any importance to the exact relations of the values assigned.

We ought not, however, on this account, to cease to conceive all
phenomena as being necessarily subject to mathematical laws, which we
are condemned to be ignorant of, only because of the too great
complication of the phenomena. The most complex phenomena of living
bodies are doubtless essentially of no other special nature than the
simplest phenomena of unorganized matter. If it were possible to isolate
rigorously each of the simple causes which concur in producing a single
physiological phenomenon, every thing leads us to believe that it would
show itself endowed, in determinate circumstances, with a kind of
influence and with a quantity of action as exactly fixed as we see it in
universal gravitation, a veritable type of the fundamental laws of
nature.

There is a _second_ reason why we cannot bring complicated phenomena
under the dominion of mathematical analysis. Even if we could ascertain
the mathematical law which governs each agent, taken by itself, the
combination of so great a number of conditions would render the
corresponding mathematical problem so far above our feeble means, that
the question would remain in most cases incapable of solution.

To appreciate this difficulty, let us consider how complicated
mathematical questions become, even those relating to the most simple
phenomena of unorganized bodies, when we desire to bring sufficiently
near together the abstract and the concrete state, having regard to all
the principal conditions which can exercise a real influence over the
effect produced. We know, for example, that the very simple phenomenon
of the flow of a fluid through a given orifice, by virtue of its gravity
alone, has not as yet any complete mathematical solution, when we take
into the account all the essential circumstances. It is the same even
with the still more simple motion of a solid projectile in a resisting
medium.

Why has mathematical analysis been able to adapt itself with such
admirable success to the most profound study of celestial phenomena?
Because they are, in spite of popular appearances, much more simple than
any others. The most complicated problem which they present, that of the
modification produced in the motions of two bodies tending towards each
other by virtue of their gravitation, by the influence of a third body
acting on both of them in the same manner, is much less complex than the
most simple terrestrial problem. And, nevertheless, even it presents
difficulties so great that we yet possess only approximate solutions of
it. It is even easy to see that the high perfection to which solar
astronomy has been able to elevate itself by the employment of
mathematical science is, besides, essentially due to our having
skilfully profited by all the particular, and, so to say, accidental
facilities presented by the peculiarly favourable constitution of our
planetary system. The planets which compose it are quite few in number,
and their masses are in general very unequal, and much less than that of
the sun; they are, besides, very distant from one another; they have
forms almost spherical; their orbits are nearly circular, and only
slightly inclined to each other, and so on. It results from all these
circumstances that the perturbations are generally inconsiderable, and
that to calculate them it is usually sufficient to take into the
account, in connexion with the action of the sun on each particular
planet, the influence of only one other planet, capable, by its size and
its proximity, of causing perceptible derangements.

If, however, instead of such a state of things, our solar system had
been composed of a greater number of planets concentrated into a less
space, and nearly equal in mass; if their orbits had presented very
different inclinations, and considerable eccentricities; if these bodies
had been of a more complicated form, such as very eccentric ellipsoids,
it is certain that, supposing the same law of gravitation to exist, we
should not yet have succeeded in subjecting the study of the celestial
phenomena to our mathematical analysis, and probably we should not even
have been able to disentangle the present principal law.

These hypothetical conditions would find themselves exactly realized in
the highest degree in _chemical_ phenomena, if we attempted to calculate
them by the theory of general gravitation.

On properly weighing the preceding considerations, the reader will be
convinced, I think, that in reducing the future extension of the great
applications of mathematical analysis, which are really possible, to
the field comprised in the different departments of inorganic physics, I
have rather exaggerated than contracted the extent of its actual domain.
Important as it was to render apparent the rigorous logical universality
of mathematical science, it was equally so to indicate the conditions
which limit for us its real extension, so as not to contribute to lead
the human mind astray from the true scientific direction in the study of
the most complicated phenomena, by the chimerical search after an
impossible perfection.

       *       *       *       *       *

Having thus exhibited the essential object and the principal composition
of mathematical science, as well as its general relations with the whole
body of natural philosophy, we have now to pass to the special
examination of the great sciences of which it is composed.

     _Note._--ANALYSIS and GEOMETRY are the two great heads under which
     the subject is about to be examined. To these _M. Comte_ adds
     Rational MECHANICS; but as it is not comprised in the usual idea of
     Mathematics, and as its discussion would be of but limited utility
     and interest, it is not included in the present translation.




BOOK I.

ANALYSIS.




BOOK I.

ANALYSIS.




CHAPTER I.

GENERAL VIEW OF MATHEMATICAL ANALYSIS.


In the historical development of mathematical science since the time of
Descartes, the advances of its abstract portion have always been
determined by those of its concrete portion; but it is none the less
necessary, in order to conceive the science in a manner truly logical,
to consider the Calculus in all its principal branches before proceeding
to the philosophical study of Geometry and Mechanics. Its analytical
theories, more simple and more general than those of concrete
mathematics, are in themselves essentially independent of the latter;
while these, on the contrary, have, by their nature, a continual need of
the former, without the aid of which they could make scarcely any
progress. Although the principal conceptions of analysis retain at
present some very perceptible traces of their geometrical or mechanical
origin, they are now, however, mainly freed from that primitive
character, which no longer manifests itself except in some secondary
points; so that it is possible (especially since the labours of
Lagrange) to present them in a dogmatic exposition, by a purely abstract
method, in a single and continuous system. It is this which will be
undertaken in the present and the five following chapters, limiting our
investigations to the most general considerations upon each principal
branch of the science of the calculus.

The definite object of our researches in concrete mathematics being the
discovery of the _equations_ which express the mathematical laws of the
phenomenon under consideration, and these equations constituting the
true starting point of the calculus, which has for its object to obtain
from them the determination of certain quantities by means of others, I
think it indispensable, before proceeding any farther, to go more deeply
than has been customary into that fundamental idea of _equation_, the
continual subject, either as end or as beginning, of all mathematical
labours. Besides the advantage of circumscribing more definitely the
true field of analysis, there will result from it the important
consequence of tracing in a more exact manner the real line of
demarcation between the concrete and the abstract part of mathematics,
which will complete the general exposition of the fundamental division
established in the introductory chapter.




THE TRUE IDEA OF AN EQUATION.


We usually form much too vague an idea of what an _equation_ is, when we
give that name to every kind of relation of equality between _any_ two
functions of the magnitudes which we are considering. For, though every
equation is evidently a relation of equality, it is far from being true
that, reciprocally, every relation of equality is a veritable
_equation_, of the kind of those to which, by their nature, the methods
of analysis are applicable.

This want of precision in the logical consideration of an idea which is
so fundamental in mathematics, brings with it the serious inconvenience
of rendering it almost impossible to explain, in general terms, the
great and fundamental difficulty which we find in establishing the
relation between the concrete and the abstract, and which stands out so
prominently in each great mathematical question taken by itself. If the
meaning of the word _equation_ was truly as extended as we habitually
suppose it to be in our definition of it, it is not apparent what great
difficulty there could really be, in general, in establishing the
equations of any problem whatsoever; for the whole would thus appear to
consist in a simple question of form, which ought never even to exact
any great intellectual efforts, seeing that we can hardly conceive of
any precise relation which is not immediately a certain relation of
equality, or which cannot be readily brought thereto by some very easy
transformations.

Thus, when we admit every species of _functions_ into the definition of
_equations_, we do not at all account for the extreme difficulty which
we almost always experience in putting a problem into an equation, and
which so often may be compared to the efforts required by the analytical
elaboration of the equation when once obtained. In a word, the ordinary
abstract and general idea of an _equation_ does not at all correspond to
the real meaning which geometers attach to that expression in the actual
development of the science. Here, then, is a logical fault, a defect of
correlation, which it is very important to rectify.


_Division of Functions into Abstract and Concrete._ To succeed in doing
so, I begin by distinguishing two sorts of _functions_, _abstract_ or
analytical functions, and _concrete_ functions. The first alone can
enter into veritable _equations_. We may, therefore, henceforth define
every _equation_, in an exact and sufficiently profound manner, as a
relation of equality between two _abstract_ functions of the magnitudes
under consideration. In order not to have to return again to this
fundamental definition, I must add here, as an indispensable complement,
without which the idea would not be sufficiently general, that these
abstract functions may refer not only to the magnitudes which the
problem presents of itself, but also to all the other auxiliary
magnitudes which are connected with it, and which we will often be able
to introduce, simply as a mathematical artifice, with the sole object of
facilitating the discovery of the equations of the phenomena. I here
anticipate summarily the result of a general discussion of the highest
importance, which will be found at the end of this chapter. We will now
return to the essential distinction of functions as abstract and
concrete.

This distinction may be established in two ways, essentially different,
but complementary of each other, _à priori_ and _à posteriori_; that is
to say, by characterizing in a general manner the peculiar nature of
each species of functions, and then by making the actual enumeration of
all the abstract functions at present known, at least so far as relates
to the elements of which they are composed.

_À priori_, the functions which I call _abstract_ are those which
express a manner of dependence between magnitudes, which can be
conceived between numbers alone, without there being need of indicating
any phenomenon whatever in which it is realized. I name, on the other
hand, _concrete_ functions, those for which the mode of dependence
expressed cannot be defined or conceived except by assigning a
determinate case of physics, geometry, mechanics, &c., in which it
actually exists.

Most functions in their origin, even those which are at present the most
purely _abstract_, have begun by being _concrete_; so that it is easy to
make the preceding distinction understood, by citing only the successive
different points of view under which, in proportion as the science has
become formed, geometers have considered the most simple analytical
functions. I will indicate powers, for example, which have in general
become abstract functions only since the labours of Vieta and Descartes.
The functions _x²_, _x³_, which in our present analysis are so well
conceived as simply _abstract_, were, for the geometers of antiquity,
perfectly _concrete_ functions, expressing the relation of the
superficies of a square, or the volume of a cube to the length of their
side. These had in their eyes such a character so exclusively, that it
was only by means of the geometrical definitions that they discovered
the elementary algebraic properties of these functions, relating to the
decomposition of the variable into two parts, properties which were at
that epoch only real theorems of geometry, to which a numerical meaning
was not attached until long afterward.

I shall have occasion to cite presently, for another reason, a new
example, very suitable to make apparent the fundamental distinction
which I have just exhibited; it is that of circular functions, both
direct and inverse, which at the present time are still sometimes
concrete, sometimes abstract, according to the point of view under
which they are regarded.

_À posteriori_, the general character which renders a function abstract
or concrete having been established, the question as to whether a
certain determinate function is veritably abstract, and therefore
susceptible of entering into true analytical equations, becomes a simple
question of fact, inasmuch as we are going to enumerate all the
functions of this species.


_Enumeration of Abstract Functions._ At first view this enumeration
seems impossible, the distinct analytical functions being infinite in
number. But when we divide them into _simple_ and _compound_, the
difficulty disappears; for, though the number of the different functions
considered in mathematical analysis is really infinite, they are, on the
contrary, even at the present day, composed of a very small number of
elementary functions, which can be easily assigned, and which are
evidently sufficient for deciding the abstract or concrete character of
any given function; which will be of the one or the other nature,
according as it shall be composed exclusively of these simple abstract
functions, or as it shall include others.

We evidently have to consider, for this purpose, only the functions of a
single variable, since those relative to several independent variables
are constantly, by their nature, more or less _compound_.

Let _x_ be the independent variable, _y_ the correlative variable which
depends upon it. The different simple modes of abstract dependence,
which we can now conceive between _y_ and _x_, are expressed by the ten
following elementary formulas, in which each function is coupled with
its _inverse_, that is, with that which would be obtained from the
direct function by referring _x_ to _y_, instead of referring _y_ to
_x_.

            FUNCTION.                   ITS NAME.

1st couple {1° _y_ = _a_ + _x_             _Sum._
           {2° _y_ = _a_ - _x_             _Difference._

2d couple  {1° _y_ = _ax_                  _Product._
           {2° _y_ = _a/x_                 _Quotient._

3d couple  {1° _y_ = _x^a_                 _Power._
           {2° _y_ = _[ath root]x_            _Root._

4th couple {1° _y_ = _a^x_                 _Exponential._
           {2° _y_ = _[log a]x_            _Logarithmic._

5th couple {1° _y_ = sin. _x_              _Direct Circular._
           {2° _y_ = arc(sin. = _x_).      _Inverse Circular._[3]

  [Footnote 3: With the view of increasing as much as possible the
  resources and the extent (now so insufficient) of mathematical
  analysis, geometers count this last couple of functions among the
  analytical elements. Although this inscription is strictly
  legitimate, it is important to remark that circular functions are
  not exactly in the same situation as the other abstract elementary
  functions. There is this very essential difference, that the
  functions of the four first couples are at the same time simple and
  abstract, while the circular functions, which may manifest each
  character in succession, according to the point of view under which
  they are considered and the manner in which they are employed,
  never present these two properties simultaneously.

  Some other concrete functions may be usefully introduced into the
  number of analytical elements, certain conditions being fulfilled.
  It is thus, for example, that the labours of M. Legendre and of M.
  Jacobi on _elliptical_ functions have truly enlarged the field of
  analysis; and the same is true of some definite integrals obtained
  by M. Fourier in the theory of heat.]

Such are the elements, very few in number, which directly compose all
the abstract functions known at the present day. Few as they are, they
are evidently sufficient to give rise to an infinite number of
analytical combinations.

No rational consideration rigorously circumscribes, _à priori_, the
preceding table, which is only the actual expression of the present
state of the science. Our analytical elements are at the present day
more numerous than they were for Descartes, and even for Newton and
Leibnitz: it is only a century since the last two couples have been
introduced into analysis by the labours of John Bernouilli and Euler.
Doubtless new ones will be hereafter admitted; but, as I shall show
towards the end of this chapter, we cannot hope that they will ever be
greatly multiplied, their real augmentation giving rise to very great
difficulties.

We can now form a definite, and, at the same time, sufficiently extended
idea of what geometers understand by a veritable _equation_. This
explanation is especially suited to make us understand how difficult it
must be really to establish the _equations_ of phenomena, since we have
effectually succeeded in so doing only when we have been able to
conceive the mathematical laws of these phenomena by the aid of
functions entirely composed of only the mathematical elements which I
have just enumerated. It is clear, in fact, that it is then only that
the problem becomes truly abstract, and is reduced to a pure question of
numbers, these functions being the only simple relations which we can
conceive between numbers, considered by themselves. Up to this period of
the solution, whatever the appearances may be, the question is still
essentially concrete, and does not come within the domain of the
_calculus_. Now the fundamental difficulty of this passage from the
_concrete_ to the _abstract_ in general consists especially in the
insufficiency of this very small number of analytical elements which we
possess, and by means of which, nevertheless, in spite of the little
real variety which they offer us, we must succeed in representing all
the precise relations which all the different natural phenomena can
manifest to us. Considering the infinite diversity which must
necessarily exist in this respect in the external world, we easily
understand how far below the true difficulty our conceptions must
frequently be found, especially if we add that as these elements of our
analysis have been in the first place furnished to us by the
mathematical consideration of the simplest phenomena, we have, _à
priori_, no rational guarantee of their necessary suitableness to
represent the mathematical law of every other class of phenomena. I will
explain presently the general artifice, so profoundly ingenious, by
which the human mind has succeeded in diminishing, in a remarkable
degree, this fundamental difficulty which is presented by the relation
of the concrete to the abstract in mathematics, without, however, its
having been necessary to multiply the number of these analytical
elements.




THE TWO PRINCIPAL DIVISIONS OF THE CALCULUS.


The preceding explanations determine with precision the true object and
the real field of abstract mathematics. I must now pass to the
examination of its principal divisions, for thus far we have considered
the calculus as a whole.

The first direct consideration to be presented on the composition of the
science of the _calculus_ consists in dividing it, in the first place,
into two principal branches, to which, for want of more suitable
denominations, I will give the names of _Algebraic calculus_, or
_Algebra_, and of _Arithmetical calculus_, or _Arithmetic_; but with
the caution to take these two expressions in their most extended logical
acceptation, in the place of the by far too restricted meaning which is
usually attached to them.

The complete solution of every question of the _calculus_, from the most
elementary up to the most transcendental, is necessarily composed of two
successive parts, whose nature is essentially distinct. In the first,
the object is to transform the proposed equations, so as to make
apparent the manner in which the unknown quantities are formed by the
known ones: it is this which constitutes the _algebraic_ question. In
the second, our object is to _find the values_ of the formulas thus
obtained; that is, to determine directly the values of the numbers
sought, which are already represented by certain explicit functions of
given numbers: this is the _arithmetical_ question.[4] It is apparent
that, in every solution which is truly rational, it necessarily follows
the algebraical question, of which it forms the indispensable
complement, since it is evidently necessary to know the mode of
generation of the numbers sought for before determining their actual
values for each particular case. Thus the stopping-place of the
algebraic part of the solution becomes the starting point of the
arithmetical part.

  [Footnote 4: Suppose, for example, that a question gives the
  following equation between an unknown magnitude x, and two known
  magnitudes, _a_ and _b_,

  _x³_ + 3_ax_ = 2_b_,

  as is the case in the problem of the trisection of an angle. We see
  at once that the dependence between _x_ on the one side, and _ab_ on
  the other, is completely determined; but, so long as the equation
  preserves its primitive form, we do not at all perceive in what
  manner the unknown quantity is derived from the data. This must be
  discovered, however, before we can think of determining its value.
  Such is the object of the algebraic part of the solution. When, by a
  series of transformations which have successively rendered that
  derivation more and more apparent, we have arrived at presenting the
  proposed equation under the form

  _x_ = ∛(_b_ + √(_b²_ + _a³_)) + ∛(_b_ - √(_b²_ + _a³_)),

  the work of _algebra_ is finished; and even if we could not perform
  the arithmetical operations indicated by that formula, we would
  nevertheless have obtained a knowledge very real, and often very
  important. The work of _arithmetic_ will now consist in taking that
  formula for its starting point, and finding the number _x_ when the
  values of the numbers _a_ and _b_ are given.]

We thus see that the _algebraic_ calculus and the _arithmetical_
calculus differ essentially in their object. They differ no less in the
point of view under which they regard quantities; which are considered
in the first as to their _relations_, and in the second as to their
_values_. The true spirit of the calculus, in general, requires this
distinction to be maintained with the most severe exactitude, and the
line of demarcation between the two periods of the solution to be
rendered as clear and distinct as the proposed question permits. The
attentive observation of this precept, which is too much neglected, may
be of much assistance, in each particular question, in directing the
efforts of our mind, at any moment of the solution, towards the real
corresponding difficulty. In truth, the imperfection of the science of
the calculus obliges us very often (as will be explained in the next
chapter) to intermingle algebraic and arithmetical considerations in the
solution of the same question. But, however impossible it may be to
separate clearly the two parts of the labour, yet the preceding
indications will always enable us to avoid confounding them.

In endeavouring to sum up as succinctly as possible the distinction just
established, we see that ALGEBRA may be defined, in general, as having
for its object the _resolution of equations_; taking this expression in
its full logical meaning, which signifies the transformation of
_implicit_ functions into equivalent _explicit_ ones. In the same way,
ARITHMETIC may be defined as destined to _the determination of the
values of functions_. Henceforth, therefore, we will briefly say that
ALGEBRA is the _Calculus of Functions_, and ARITHMETIC the _Calculus of
Values_.

We can now perceive how insufficient and even erroneous are the ordinary
definitions. Most generally, the exaggerated importance attributed to
Signs has led to the distinguishing the two fundamental branches of the
science of the Calculus by the manner of designating in each the
subjects of discussion, an idea which is evidently absurd in principle
and false in fact. Even the celebrated definition given by Newton,
characterizing _Algebra_ as _Universal Arithmetic_, gives certainly a
very false idea of the nature of algebra and of that of arithmetic.[5]

  [Footnote 5: I have thought that I ought to specially notice this
  definition, because it serves as the basis of the opinion which
  many intelligent persons, unacquainted with mathematical science,
  form of its abstract part, without considering that at the time of
  this definition mathematical analysis was not sufficiently
  developed to enable the general character of each of its principal
  parts to be properly apprehended, which explains why Newton could
  at that time propose a definition which at the present day he would
  certainly reject.]

Having thus established the fundamental division of the calculus into
two principal branches, I have now to compare in general terms the
extent, the importance, and the difficulty of these two sorts of
calculus, so as to have hereafter to consider only the _Calculus of
Functions_, which is to be the principal subject of our study.




THE CALCULUS OF VALUES, OR ARITHMETIC.


_Its Extent._ The _Calculus of Values, or Arithmetic_, would appear, at
first view, to present a field as vast as that of _algebra_, since it
would seem to admit as many distinct questions as we can conceive
different algebraic formulas whose values are to be determined. But a
very simple reflection will show the difference. Dividing functions into
_simple_ and _compound_, it is evident that when we know how to
determine the _value_ of simple functions, the consideration of compound
functions will no longer present any difficulty. In the algebraic point
of view, a compound function plays a very different part from that of
the elementary functions of which it consists, and from this, indeed,
proceed all the principal difficulties of analysis. But it is very
different with the Arithmetical Calculus. Thus the number of truly
distinct arithmetical operations is only that determined by the number
of the elementary abstract functions, the very limited list of which has
been given above. The determination of the values of these ten functions
necessarily gives that of all the functions, infinite in number, which
are considered in the whole of mathematical analysis, such at least as
it exists at present. There can be no new arithmetical operations
without the creation of really new analytical elements, the number of
which must always be extremely small. The field of _arithmetic_ is,
then, by its nature, exceedingly restricted, while that of algebra is
rigorously indefinite.

It is, however, important to remark, that the domain of the _calculus of
values_ is, in reality, much more extensive than it is commonly
represented; for several questions truly _arithmetical_, since they
consist of determinations of values, are not ordinarily classed as such,
because we are accustomed to treat them only as incidental in the midst
of a body of analytical researches more or less elevated, the too high
opinion commonly formed of the influence of signs being again the
principal cause of this confusion of ideas. Thus not only the
construction of a table of logarithms, but also the calculation of
trigonometrical tables, are true arithmetical operations of a higher
kind. We may also cite as being in the same class, although in a very
distinct and more elevated order, all the methods by which we determine
directly the value of any function for each particular system of values
attributed to the quantities on which it depends, when we cannot express
in general terms the explicit form of that function. In this point of
view the _numerical_ solution of questions which we cannot resolve
algebraically, and even the calculation of "Definite Integrals," whose
general integrals we do not know, really make a part, in spite of all
appearances, of the domain of _arithmetic_, in which we must necessarily
comprise all that which has for its object the _determination of the
values of functions_. The considerations relative to this object are, in
fact, constantly homogeneous, whatever the _determinations_ in question,
and are always very distinct from truly _algebraic_ considerations.

To complete a just idea of the real extent of the calculus of values, we
must include in it likewise that part of the general science of the
calculus which now bears the name of the _Theory of Numbers_, and which
is yet so little advanced. This branch, very extensive by its nature,
but whose importance in the general system of science is not very
great, has for its object the discovery of the properties inherent in
different numbers by virtue of their values, and independent of any
particular system of numeration. It forms, then, a sort of
_transcendental arithmetic_; and to it would really apply the definition
proposed by Newton for algebra.

The entire domain of arithmetic is, then, much more extended than is
commonly supposed; but this _calculus of values_ will still never be
more than a point, so to speak, in comparison with the _calculus of
functions_, of which mathematical science essentially consists. This
comparative estimate will be still more apparent from some
considerations which I have now to indicate respecting the true nature
of arithmetical questions in general, when they are more profoundly
examined.


_Its true Nature._ In seeking to determine with precision in what
_determinations of values_ properly consist, we easily recognize that
they are nothing else but veritable _transformations_ of the functions
to be valued; transformations which, in spite of their special end, are
none the less essentially of the same nature as all those taught by
analysis. In this point of view, the _calculus of values_ might be
simply conceived as an appendix, and a particular application of the
_calculus of functions_, so that _arithmetic_ would disappear, so to
say, as a distinct section in the whole body of abstract mathematics.

In order thoroughly to comprehend this consideration, we must observe
that, when we propose to determine the _value_ of an unknown number
whose mode of formation is given, it is, by the mere enunciation of the
arithmetical question, already defined and expressed under a certain
form; and that in _determining its value_ we only put its expression
under another determinate form, to which we are accustomed to refer the
exact notion of each particular number by making it re-enter into the
regular system of _numeration_. The determination of values consists so
completely of a simple _transformation_, that when the primitive
expression of the number is found to be already conformed to the regular
system of numeration, there is no longer any determination of value,
properly speaking, or, rather, the question is answered by the question
itself. Let the question be to add the two numbers _one_ and _twenty_,
we answer it by merely repeating the enunciation of the question,[6] and
nevertheless we think that we have _determined the value_ of the sum.
This signifies that in this case the first expression of the function
had no need of being transformed, while it would not be thus in adding
twenty-three and fourteen, for then the sum would not be immediately
expressed in a manner conformed to the rank which it occupies in the
fixed and general scale of numeration.


  [Footnote 6: This is less strictly true in the English system of
  numeration than in the French, since "twenty-one" is our more usual
  mode of expressing this number.]

To sum up as comprehensively as possible the preceding views, we may
say, that to determine the _value_ of a number is nothing else than
putting its primitive expression under the form

  _a_ + _bz_ + _cz²_ + _dz³_ + _ez⁴_ . . . . . + _pz^m_,

_z_ being generally equal to 10, and the coefficients _a_, _b_, _c_,
_d_, &c., being subjected to the conditions of being whole numbers less
than _z_; capable of becoming equal to zero; but never negative. Every
arithmetical question may thus be stated as consisting in putting under
such a form any abstract function whatever of different quantities,
which are supposed to have themselves a similar form already. We might
then see in the different operations of arithmetic only simple
particular cases of certain algebraic transformations, excepting the
special difficulties belonging to conditions relating to the nature of
the coefficients.

It clearly follows that abstract mathematics is essentially composed of
the _Calculus of Functions_, which had been already seen to be its most
important, most extended, and most difficult part. It will henceforth be
the exclusive subject of our analytical investigations. I will therefore
no longer delay on the _Calculus of Values_, but pass immediately to the
examination of the fundamental division of the _Calculus of Functions_.




THE CALCULUS OF FUNCTIONS, OR ALGEBRA.


_Principle of its Fundamental Division._ We have determined, at the
beginning of this chapter, wherein properly consists the difficulty
which we experience in putting mathematical questions into _equations_.
It is essentially because of the insufficiency of the very small number
of analytical elements which we possess, that the relation of the
concrete to the abstract is usually so difficult to establish. Let us
endeavour now to appreciate in a philosophical manner the general
process by which the human mind has succeeded, in so great a number of
important cases, in overcoming this fundamental obstacle to _The
establishment of Equations_.


1. _By the Creation of new Functions._ In looking at this important
question from the most general point of view, we are led at once to the
conception of one means of facilitating the establishment of the
equations of phenomena. Since the principal obstacle in this matter
comes from the too small number of our analytical elements, the whole
question would seem to be reduced to creating new ones. But this means,
though natural, is really illusory; and though it might be useful, it is
certainly insufficient.

In fact, the creation of an elementary abstract function, which shall be
veritably new, presents in itself the greatest difficulties. There is
even something contradictory in such an idea; for a new analytical
element would evidently not fulfil its essential and appropriate
conditions, if we could not immediately _determine its value_. Now, on
the other hand, how are we to _determine the value_ of a new function
which is truly _simple_, that is, which is not formed by a combination
of those already known? That appears almost impossible. The introduction
into analysis of another elementary abstract function, or rather of
another couple of functions (for each would be always accompanied by its
_inverse_), supposes then, of necessity, the simultaneous creation of a
new arithmetical operation, which is certainly very difficult.

If we endeavour to obtain an idea of the means which the human mind
employs for inventing new analytical elements, by the examination of the
procedures by the aid of which it has actually conceived those which we
already possess, our observations leave us in that respect in an entire
uncertainty, for the artifices which it has already made use of for that
purpose are evidently exhausted. To convince ourselves of it, let us
consider the last couple of simple functions which has been introduced
into analysis, and at the formation of which we have been present, so
to speak, namely, the fourth couple; for, as I have explained, the fifth
couple does not strictly give veritable new analytical elements. The
function _a^x_, and, consequently, its inverse, have been formed by
conceiving, under a new point of view, a function which had been a long
time known, namely, powers--when the idea of them had become
sufficiently generalized. The consideration of a power relatively to the
variation of its exponent, instead of to the variation of its base, was
sufficient to give rise to a truly novel simple function, the variation
following then an entirely different route. But this artifice, as simple
as ingenious, can furnish nothing more; for, in turning over in the same
manner all our present analytical elements, we end in only making them
return into one another.

We have, then, no idea as to how we could proceed to the creation of new
elementary abstract functions which would properly satisfy all the
necessary conditions. This is not to say, however, that we have at
present attained the effectual limit established in that respect by the
bounds of our intelligence. It is even certain that the last special
improvements in mathematical analysis have contributed to extend our
resources in that respect, by introducing within the domain of the
calculus certain definite integrals, which in some respects supply the
place of new simple functions, although they are far from fulfilling all
the necessary conditions, which has prevented me from inserting them in
the table of true analytical elements. But, on the whole, I think it
unquestionable that the number of these elements cannot increase except
with extreme slowness. It is therefore not from these sources that the
human mind has drawn its most powerful means of facilitating, as much
as is possible, the establishment of equations.


2. _By the Conception of Equations between certain auxiliary
Quantities._ This first method being set aside, there remains evidently
but one other: it is, seeing the impossibility of finding directly the
equations between the quantities under consideration, to seek for
corresponding ones between other auxiliary quantities, connected with
the first according to a certain determinate law, and from the relation
between which we may return to that between the primitive magnitudes.
Such is, in substance, the eminently fruitful conception, which the
human mind has succeeded in establishing, and which constitutes its most
admirable instrument for the mathematical explanation of natural
phenomena; the _analysis_, called _transcendental_.

As a general philosophical principle, the auxiliary quantities, which
are introduced in the place of the primitive magnitudes, or concurrently
with them, in order to facilitate the establishment of equations, might
be derived according to any law whatever from the immediate elements of
the question. This conception has thus a much more extensive reach than
has been commonly attributed to it by even the most profound geometers.
It is extremely important for us to view it in its whole logical extent,
for it will perhaps be by establishing a general mode of _derivation_
different from that to which we have thus far confined ourselves
(although it is evidently very far from being the only possible one)
that we shall one day succeed in essentially perfecting mathematical
analysis as a whole, and consequently in establishing more powerful
means of investigating the laws of nature than our present processes,
which are unquestionably susceptible of becoming exhausted.

But, regarding merely the present constitution of the science, the only
auxiliary quantities habitually introduced in the place of the primitive
quantities in the _Transcendental Analysis_ are what are called, 1⁰,
_infinitely small_ elements, the _differentials_ (of different orders)
of those quantities, if we regard this analysis in the manner of
LEIBNITZ; or, 2⁰, the _fluxions_, the limits of the ratios of the
simultaneous increments of the primitive quantities compared with one
another, or, more briefly, the _prime and ultimate ratios_ of these
increments, if we adopt the conception of NEWTON; or, 3⁰, the
_derivatives_, properly so called, of those quantities, that is, the
coefficients of the different terms of their respective increments,
according to the conception of LAGRANGE.

These three principal methods of viewing our present transcendental
analysis, and all the other less distinctly characterized ones which
have been successively proposed, are, by their nature, necessarily
identical, whether in the calculation or in the application, as will be
explained in a general manner in the third chapter. As to their relative
value, we shall there see that the conception of Leibnitz has thus far,
in practice, an incontestable superiority, but that its logical
character is exceedingly vicious; while that the conception of Lagrange,
admirable by its simplicity, by its logical perfection, by the
philosophical unity which it has established in mathematical analysis
(till then separated into two almost entirely independent worlds),
presents, as yet, serious inconveniences in the applications, by
retarding the progress of the mind. The conception of Newton occupies
nearly middle ground in these various relations, being less rapid, but
more rational than that of Leibnitz; less philosophical, but more
applicable than that of Lagrange.

This is not the place to explain the advantages of the introduction of
this kind of auxiliary quantities in the place of the primitive
magnitudes. The third chapter is devoted to this subject. At present I
limit myself to consider this conception in the most general manner, in
order to deduce therefrom the fundamental division of the _calculus of
functions_ into two systems essentially distinct, whose dependence, for
the complete solution of any one mathematical question, is invariably
determinate.

In this connexion, and in the logical order of ideas, the transcendental
analysis presents itself as being necessarily the first, since its
general object is to facilitate the establishment of equations, an
operation which must evidently precede the _resolution_ of those
equations, which is the object of the ordinary analysis. But though it
is exceedingly important to conceive in this way the true relations of
these two systems of analysis, it is none the less proper, in conformity
with the regular usage, to study the transcendental analysis after
ordinary analysis; for though the former is, at bottom, by itself
logically independent of the latter, or, at least, may be essentially
disengaged from it, yet it is clear that, since its employment in the
solution of questions has always more or less need of being completed by
the use of the ordinary analysis, we would be constrained to leave the
questions in suspense if this latter had not been previously studied.


_Corresponding Divisions of the Calculus of Functions._ It follows from
the preceding considerations that the _Calculus of Functions_, or
_Algebra_ (taking this word in its most extended meaning), is composed
of two distinct fundamental branches, one of which has for its immediate
object the _resolution_ of equations, when they are directly established
between the magnitudes themselves which are under consideration; and the
other, starting from equations (generally much easier to form) between
quantities indirectly connected with those of the problem, has for its
peculiar and constant destination the deduction, by invariable
analytical methods, of the corresponding equations between the direct
magnitudes which we are considering; which brings the question within
the domain of the preceding calculus.

The former calculus bears most frequently the name of _Ordinary
Analysis_, or of _Algebra_, properly so called. The second constitutes
what is called the _Transcendental Analysis_, which has been designated
by the different denominations of _Infinitesimal Calculus_, _Calculus of
Fluxions and of Fluents_, _Calculus of Vanishing Quantities_, the
_Differential and Integral Calculus_, &c., according to the point of
view in which it has been conceived.

In order to remove every foreign consideration, I will propose to name
it CALCULUS OF INDIRECT FUNCTIONS, giving to ordinary analysis the title
of CALCULUS OF DIRECT FUNCTIONS. These expressions, which I form
essentially by generalizing and epitomizing the ideas of Lagrange, are
simply intended to indicate with precision the true general character
belonging to each of these two forms of analysis.

Having now established the fundamental division of mathematical
analysis, I have next to consider separately each of its two parts,
commencing with the _Calculus of Direct Functions_, and reserving more
extended developments for the different branches of the _Calculus of
Indirect Functions_.




CHAPTER II.

ORDINARY ANALYSIS, OR ALGEBRA.


The _Calculus of direct Functions_, or _Algebra_, is (as was shown at
the end of the preceding chapter) entirely sufficient for the solution
of mathematical questions, when they are so simple that we can form
directly the equations between the magnitudes themselves which we are
considering, without its being necessary to introduce in their place, or
conjointly with them, any system of auxiliary quantities _derived_ from
the first. It is true that in the greatest number of important cases its
use requires to be preceded and prepared by that of the _Calculus of
indirect Functions_, which is intended to facilitate the establishment
of equations. But, although algebra has then only a secondary office to
perform, it has none the less a necessary part in the complete solution
of the question, so that the _Calculus of direct Functions_ must
continue to be, by its nature, the fundamental base of all mathematical
analysis. We must therefore, before going any further, consider in a
general manner the logical composition of this calculus, and the degree
of development to which it has at the present day arrived.


_Its Object._ The final object of this calculus being the _resolution_
(properly so called) of _equations_, that is, the discovery of the
manner in which the unknown quantities are formed from the known
quantities, in accordance with the _equations_ which exist between them,
it naturally presents as many different departments as we can conceive
truly distinct classes of equations. Its appropriate extent is
consequently rigorously indefinite, the number of analytical functions
susceptible of entering into equations being in itself quite unlimited,
although they are composed of only a very small number of primitive
elements.


_Classification of Equations._ The rational classification of equations
must evidently be determined by the nature of the analytical elements of
which their numbers are composed; every other classification would be
essentially arbitrary. Accordingly, analysts begin by dividing equations
with one or more variables into two principal classes, according as they
contain functions of only the first three couples (see the table in
chapter i., page 51), or as they include also exponential or circular
functions. The names of _Algebraic_ functions and _Transcendental_
functions, commonly given to these two principal groups of analytical
elements, are undoubtedly very inappropriate. But the universally
established division between the corresponding equations is none the
less very real in this sense, that the resolution of equations
containing the functions called _transcendental_ necessarily presents
more difficulties than those of the equations called _algebraic_. Hence
the study of the former is as yet exceedingly imperfect, so that
frequently the resolution of the most simple of them is still unknown to
us,[7] and our analytical methods have almost exclusive reference to the
elaboration of the latter.

  [Footnote 7: Simple as may seem, for example, the equation

  _a^x_ + _b^x_ = _c^x_,

  we do not yet know how to resolve it, which may give some idea of
  the extreme imperfection of this part of algebra.]




ALGEBRAIC EQUATIONS.


Considering now only these _Algebraic_ equations, we must observe, in
the first place, that although they may often contain _irrational_
functions of the unknown quantities as well as _rational_ functions, we
can always, by more or less easy transformations, make the first case
come under the second, so that it is with this last that analysts have
had to occupy themselves exclusively in order to resolve all sorts of
_algebraic_ equations.


_Their Classification._ In the infancy of algebra, these equations were
classed according to the number of their terms. But this classification
was evidently faulty, since it separated cases which were really
similar, and brought together others which had nothing in common besides
this unimportant characteristic.[8] It has been retained only for
equations with two terms, which are, in fact, capable of being resolved
in a manner peculiar to themselves.

  [Footnote 8: The same error was afterward committed, in the infancy
  of the infinitesimal calculus, in relation to the integration of
  differential equations.]

The classification of equations by what is called their _degrees_, is,
on the other hand, eminently natural, for this distinction rigorously
determines the greater or less difficulty of their _resolution_. This
gradation is apparent in the cases of all the equations which can be
resolved; but it may be indicated in a general manner independently of
the fact of the resolution. We need only consider that the most general
equation of each degree necessarily comprehends all those of the
different inferior degrees, as must also the formula which determines
the unknown quantity. Consequently, however slight we may suppose the
difficulty peculiar to the _degree_ which we are considering, since it
is inevitably complicated in the execution with those presented by all
the preceding degrees, the resolution really offers more and more
obstacles, in proportion as the degree of the equation is elevated.




ALGEBRAIC RESOLUTION OF EQUATIONS.


_Its Limits._ The resolution of algebraic equations is as yet known to
us only in the four first degrees, such is the increase of difficulty
noticed above. In this respect, algebra has made no considerable
progress since the labours of Descartes and the Italian analysts of the
sixteenth century, although in the last two centuries there has been
perhaps scarcely a single geometer who has not busied himself in trying
to advance the resolution of equations. The general equation of the
fifth degree itself has thus far resisted all attacks.

The constantly increasing complication which the formulas for resolving
equations must necessarily present, in proportion as the degree
increases (the difficulty of using the formula of the fourth degree
rendering it almost inapplicable), has determined analysts to renounce,
by a tacit agreement, the pursuit of such researches, although they are
far from regarding it as impossible to obtain the resolution of
equations of the fifth degree, and of several other higher ones.


_General Solution._ The only question of this kind which would be really
of great importance, at least in its logical relations, would be the
general resolution of algebraic equations of any degree whatsoever. Now,
the more we meditate on this subject, the more we are led to think, with
Lagrange, that it really surpasses the scope of our intelligence. We
must besides observe that the formula which would express the _root_ of
an equation of the _m^{th}_ degree would necessarily include radicals of
the _m^{th}_ order (or functions of an equivalent multiplicity), because
of the _m_ determinations which it must admit. Since we have seen,
besides, that this formula must also embrace, as a particular case, that
formula which corresponds to every lower degree, it follows that it
would inevitably also contain radicals of the next lower degree, the
next lower to that, &c., so that, even if it were possible to discover
it, it would almost always present too great a complication to be
capable of being usefully employed, unless we could succeed in
simplifying it, at the same time retaining all its generality, by the
introduction of a new class of analytical elements of which we yet have
no idea. We have, then, reason to believe that, without having already
here arrived at the limits imposed by the feeble extent of our
intelligence, we should not be long in reaching them if we actively and
earnestly prolonged this series of investigations.

It is, besides, important to observe that, even supposing we had
obtained the resolution of _algebraic_ equations of any degree whatever,
we would still have treated only a very small part of _algebra_,
properly so called, that is, of the calculus of direct functions,
including the resolution of all the equations which can be formed by the
known analytical functions.

Finally, we must remember that, by an undeniable law of human nature,
our means for conceiving new questions being much more powerful than our
resources for resolving them, or, in other words, the human mind being
much more ready to inquire than to reason, we shall necessarily always
remain _below_ the difficulty, no matter to what degree of development
our intellectual labour may arrive. Thus, even though we should some day
discover the complete resolution of all the analytical equations at
present known, chimerical as the supposition is, there can be no doubt
that, before attaining this end, and probably even as a subsidiary
means, we would have already overcome the difficulty (a much smaller
one, though still very great) of conceiving new analytical elements, the
introduction of which would give rise to classes of equations of which,
at present, we are completely ignorant; so that a similar imperfection
in algebraic science would be continually reproduced, in spite of the
real and very important increase of the absolute mass of our knowledge.


_What we know in Algebra._ In the present condition of algebra, the
complete resolution of the equations of the first four degrees, of any
binomial equations, of certain particular equations of the higher
degrees, and of a very small number of exponential, logarithmic, or
circular equations, constitute the fundamental methods which are
presented by the calculus of direct functions for the solution of
mathematical problems. But, limited as these elements are, geometers
have nevertheless succeeded in treating, in a truly admirable manner, a
very great number of important questions, as we shall find in the course
of the volume. The general improvements introduced within a century into
the total system of mathematical analysis, have had for their principal
object to make immeasurably useful this little knowledge which we have,
instead of tending to increase it. This result has been so fully
obtained, that most frequently this calculus has no real share in the
complete solution of the question, except by its most simple parts;
those which have reference to equations of the two first degrees, with
one or more variables.




NUMERICAL RESOLUTION OF EQUATIONS.


The extreme imperfection of algebra, with respect to the resolution of
equations, has led analysts to occupy themselves with a new class of
questions, whose true character should be here noted. They have busied
themselves in filling up the immense gap in the resolution of algebraic
equations of the higher degrees, by what they have named the _numerical
resolution_ of equations. Not being able to obtain, in general, the
_formula_ which expresses what explicit function of the given quantities
the unknown one is, they have sought (in the absence of this kind of
resolution, the only one really _algebraic_) to determine, independently
of that formula, at least the _value_ of each unknown quantity, for
various designated systems of particular values attributed to the given
quantities. By the successive labours of analysts, this incomplete and
illegitimate operation, which presents an intimate mixture of truly
algebraic questions with others which are purely arithmetical, has been
rendered possible in all cases for equations of any degree and even of
any form. The methods for this which we now possess are sufficiently
general, although the calculations to which they lead are often so
complicated as to render it almost impossible to execute them. We have
nothing else to do, then, in this part of algebra, but to simplify the
methods sufficiently to render them regularly applicable, which we may
hope hereafter to effect. In this condition of the calculus of direct
functions, we endeavour, in its application, so to dispose the proposed
questions as finally to require only this numerical resolution of the
equations.


_Its limited Usefulness._ Valuable as is such a resource in the absence
of the veritable solution, it is essential not to misconceive the true
character of these methods, which analysts rightly regard as a very
imperfect algebra. In fact, we are far from being always able to reduce
our mathematical questions to depend finally upon only the _numerical_
resolution of equations; that can be done only for questions quite
isolated or truly final, that is, for the smallest number. Most
questions, in fact, are only preparatory, and intended to serve as an
indispensable preparation for the solution of other questions. Now, for
such an object, it is evident that it is not the actual _value_ of the
unknown quantity which it is important to discover, but the _formula_,
which shows how it is derived from the other quantities under
consideration. It is this which happens, for example, in a very
extensive class of cases, whenever a certain question includes at the
same time several unknown quantities. We have then, first of all, to
separate them. By suitably employing the simple and general method so
happily invented by analysts, and which consists in referring all the
other unknown quantities to one of them, the difficulty would always
disappear if we knew how to obtain the algebraic resolution of the
equations under consideration, while the _numerical_ solution would then
be perfectly useless. It is only for want of knowing the _algebraic_
resolution of equations with a single unknown quantity, that we are
obliged to treat _Elimination_ as a distinct question, which forms one
of the greatest special difficulties of common algebra. Laborious as are
the methods by the aid of which we overcome this difficulty, they are
not even applicable, in an entirely general manner, to the elimination
of one unknown quantity between two equations of any form whatever.

In the most simple questions, and when we have really to resolve only a
single equation with a single unknown quantity, this _numerical_
resolution is none the less a very imperfect method, even when it is
strictly sufficient. It presents, in fact, this serious inconvenience of
obliging us to repeat the whole series of operations for the slightest
change which may take place in a single one of the quantities
considered, although their relations to one another remain unchanged;
the calculations made for one case not enabling us to dispense with any
of those which relate to a case very slightly different. This happens
because of our inability to abstract and treat separately that purely
algebraic part of the question which is common to all the cases which
result from the mere variation of the given numbers.

According to the preceding considerations, the calculus of direct
functions, viewed in its present state, divides into two very distinct
branches, according as its subject is the _algebraic_ resolution of
equations or their _numerical_ resolution. The first department, the
only one truly satisfactory, is unhappily very limited, and will
probably always remain so; the second, too often insufficient, has, at
least, the advantage of a much greater generality. The necessity of
clearly distinguishing these two parts is evident, because of the
essentially different object proposed in each, and consequently the
peculiar point of view under which quantities are therein considered.


_Different Divisions of the two Methods of Resolution._ If, moreover, we
consider these parts with reference to the different methods of which
each is composed, we find in their logical distribution an entirely
different arrangement. In fact, the first part must be divided according
to the nature of the equations which we are able to resolve, and
independently of every consideration relative to the _values_ of the
unknown quantities. In the second part, on the contrary, it is not
according to the _degrees_ of the equations that the methods are
naturally distinguished, since they are applicable to equations of any
degree whatever; it is according to the numerical character of the
_values_ of the unknown quantities; for, in calculating these numbers
directly, without deducing them from general formulas, different means
would evidently be employed when the numbers are not susceptible of
having their values determined otherwise than by a series of
approximations, always incomplete, or when they can be obtained with
entire exactness. This distinction of _incommensurable_ and of
_commensurable_ roots, which require quite different principles for
their determination, important as it is in the numerical resolution of
equations, is entirely insignificant in the algebraic resolution, in
which the _rational_ or _irrational_ nature of the numbers which are
obtained is a mere accident of the calculation, which cannot exercise
any influence over the methods employed; it is, in a word, a simple
arithmetical consideration. We may say as much, though in a less degree,
of the division of the commensurable roots themselves into _entire_ and
_fractional_. In fine, the case is the same, in a still greater degree,
with the most general classification of roots, as _real_ and
_imaginary_. All these different considerations, which are preponderant
as to the numerical resolution of equations, and which are of no
importance in their algebraic resolution, render more and more sensible
the essentially distinct nature of these two principal parts of algebra.




THE THEORY OF EQUATIONS.


These two departments, which constitute the immediate object of the
calculus of direct functions, are subordinate to a third one, purely
speculative, from which both of them borrow their most powerful
resources, and which has been very exactly designated by the general
name of _Theory of Equations_, although it as yet relates only to
_Algebraic_ equations. The numerical resolution of equations, because of
its generality, has special need of this rational foundation.

This last and important branch of algebra is naturally divided into two
orders of questions, viz., those which refer to the _composition_ of
equations, and those which concern their _transformation_; these latter
having for their object to modify the roots of an equation without
knowing them, in accordance with any given law, providing that this law
is uniform in relation to all the parts.[9]

  [Footnote 9: The fundamental principle on which reposes the theory
  of equations, and which is so frequently applied in all
  mathematical analysis--the decomposition of algebraic, rational,
  and entire functions, of any degree whatever, into factors of the
  first degree--is never employed except for functions of a single
  variable, without any one having examined if it ought to be
  extended to functions of several variables. The general
  impossibility of such a decomposition is demonstrated by the author
  in detail, but more properly belongs to a special treatise.]




THE METHOD OF INDETERMINATE COEFFICIENTS.


To complete this rapid general enumeration of the different essential
parts of the calculus of direct functions, I must, lastly, mention
expressly one of the most fruitful and important theories of algebra
proper, that relating to the transformation of functions into series by
the aid of what is called the _Method of indeterminate Coefficients_.
This method, so eminently analytical, and which must be regarded as one
of the most remarkable discoveries of Descartes, has undoubtedly lost
some of its importance since the invention and the development of the
infinitesimal calculus, the place of which it might so happily take in
some particular respects. But the increasing extension of the
transcendental analysis, although it has rendered this method much less
necessary, has, on the other hand, multiplied its applications and
enlarged its resources; so that by the useful combination between the
two theories, which has finally been effected, the use of the method of
indeterminate coefficients has become at present much more extensive
than it was even before the formation of the calculus of indirect
functions.

       *       *       *       *       *

Having thus sketched the general outlines of algebra proper, I have now
to offer some considerations on several leading points in the calculus
of direct functions, our ideas of which may be advantageously made more
clear by a philosophical examination.




IMAGINARY QUANTITIES.


The difficulties connected with several peculiar symbols to which
algebraic calculations sometimes lead, and especially to the expressions
called _imaginary_, have been, I think, much exaggerated through purely
metaphysical considerations, which have been forced upon them, in the
place of regarding these abnormal results in their true point of view as
simple analytical facts. Viewing them thus, we readily see that, since
the spirit of mathematical analysis consists in considering magnitudes
in reference to their relations only, and without any regard to their
determinate value, analysts are obliged to admit indifferently every
kind of expression which can be engendered by algebraic combinations.
The interdiction of even one expression because of its apparent
singularity would destroy the generality of their conceptions. The
common embarrassment on this subject seems to me to proceed essentially
from an unconscious confusion between the idea of _function_ and the
idea of _value_, or, what comes to the same thing, between the
_algebraic_ and the _arithmetical_ point of view. A thorough examination
would show mathematical analysis to be much more clear in its nature
than even mathematicians commonly suppose.




NEGATIVE QUANTITIES.


As to negative quantities, which have given rise to so many misplaced
discussions, as irrational as useless, we must distinguish between their
_abstract_ signification and their _concrete_ interpretation, which have
been almost always confounded up to the present day. Under the first
point of view, the theory of negative quantities can be established in a
complete manner by a single algebraical consideration. The necessity of
admitting such expressions is the same as for imaginary quantities, as
above indicated; and their employment as an analytical artifice, to
render the formulas more comprehensive, is a mechanism of calculation
which cannot really give rise to any serious difficulty. We may
therefore regard the abstract theory of negative quantities as leaving
nothing essential to desire; it presents no obstacles but those
inappropriately introduced by sophistical considerations.

It is far from being so, however, with their concrete theory. This
consists essentially in that admirable property of the signs + and-, of
representing analytically the oppositions of directions of which certain
magnitudes are susceptible. This _general theorem_ on the relation of
the concrete to the abstract in mathematics is one of the most beautiful
discoveries which we owe to the genius of Descartes, who obtained it as
a simple result of properly directed philosophical observation. A great
number of geometers have since striven to establish directly its general
demonstration, but thus far their efforts have been illusory. Their vain
metaphysical considerations and heterogeneous minglings of the abstract
and the concrete have so confused the subject, that it becomes necessary
to here distinctly enunciate the general fact. It consists in this: if,
in any equation whatever, expressing the relation of certain quantities
which are susceptible of opposition of directions, one or more of those
quantities come to be reckoned in a direction contrary to that which
belonged to them when the equation was first established, it will not be
necessary to form directly a new equation for this second state of the
phenomena; it will suffice to change, in the first equation, the sign of
each of the quantities which shall have changed its direction; and the
equation, thus modified, will always rigorously coincide with that which
we would have arrived at in recommencing to investigate, for this new
case, the analytical law of the phenomenon. The general theorem consists
in this constant and necessary coincidence. Now, as yet, no one has
succeeded in directly proving this; we have assured ourselves of it only
by a great number of geometrical and mechanical verifications, which
are, it is true, sufficiently multiplied, and especially sufficiently
varied, to prevent any clear mind from having the least doubt of the
exactitude and the generality of this essential property, but which, in
a philosophical point of view, do not at all dispense with the research
for so important an explanation. The extreme extent of the theorem must
make us comprehend both the fundamental difficulties of this research
and the high utility for the perfecting of mathematical science which
would belong to the general conception of this great truth. This
imperfection of theory, however, has not prevented geometers from making
the most extensive and the most important use of this property in all
parts of concrete mathematics.

It follows from the above general enunciation of the fact, independently
of any demonstration, that the property of which we speak must never be
applied to magnitudes whose directions are continually varying, without
giving rise to a simple opposition of direction; in that case, the sign
with which every result of calculation is necessarily affected is not
susceptible of any concrete interpretation, and the attempts sometimes
made to establish one are erroneous. This circumstance occurs, among
other occasions, in the case of a radius vector in geometry, and
diverging forces in mechanics.




PRINCIPLE OF HOMOGENEITY.


A second general theorem on the relation of the concrete to the abstract
is that which is ordinarily designated under the name of _Principle of
Homogeneity_. It is undoubtedly much less important in its applications
than the preceding, but it particularly merits our attention as having,
by its nature, a still greater extent, since it is applicable to all
phenomena without distinction, and because of the real utility which it
often possesses for the verification of their analytical laws. I can,
moreover, exhibit a direct and general demonstration of it which seems
to me very simple. It is founded on this single observation, which is
self-evident, that the exactitude of every relation between any concrete
magnitudes whatsoever is independent of the value of the _units_ to
which they are referred for the purpose of expressing them in numbers.
For example, the relation which exists between the three sides of a
right-angled triangle is the same, whether they are measured by yards,
or by miles, or by inches.

It follows from this general consideration, that every equation which
expresses the analytical law of any phenomenon must possess this
property of being in no way altered, when all the quantities which are
found in it are made to undergo simultaneously the change corresponding
to that which their respective units would experience. Now this change
evidently consists in all the quantities of each sort becoming at once
_m_ times smaller, if the unit which corresponds to them becomes _m_
times greater, or reciprocally. Thus every equation which represents any
concrete relation whatever must possess this characteristic of remaining
the same, when we make _m_ times greater all the quantities which it
contains, and which express the magnitudes between which the relation
exists; excepting always the numbers which designate simply the mutual
_ratios_ of these different magnitudes, and which therefore remain
invariable during the change of the units. It is this property which
constitutes the law of Homogeneity in its most extended signification,
that is, of whatever analytical functions the equations may be composed.

But most frequently we consider only the cases in which the functions
are such as are called _algebraic_, and to which the idea of _degree_ is
applicable. In this case we can give more precision to the general
proposition by determining the analytical character which must be
necessarily presented by the equation, in order that this property may
be verified. It is easy to see, then, that, by the modification just
explained, all the _terms_ of the first degree, whatever may be their
form, rational or irrational, entire or fractional, will become _m_
times greater; all those of the second degree, _m²_ times; those of the
third, _m³_ times, &c. Thus the terms of the same degree, however
different may be their composition, varying in the same manner, and the
terms of different degrees varying in an unequal proportion, whatever
similarity there may be in their composition, it will be necessary, to
prevent the equation from being disturbed, that all the terms which it
contains should be of the same degree. It is in this that properly
consists the ordinary theorem of _Homogeneity_, and it is from this
circumstance that the general law has derived its name, which, however,
ceases to be exactly proper for all other functions.

In order to treat this subject in its whole extent, it is important to
observe an essential condition, to which attention must be paid in
applying this property when the phenomenon expressed by the equation
presents magnitudes of different natures. Thus it may happen that the
respective units are completely independent of each other, and then the
theorem of Homogeneity will hold good, either with reference to all the
corresponding classes of quantities, or with regard to only a single one
or more of them. But it will happen on other occasions that the
different units will have fixed relations to one another, determined by
the nature of the question; then it will be necessary to pay attention
to this subordination of the units in verifying the homogeneity, which
will not exist any longer in a purely algebraic sense, and the precise
form of which will vary according to the nature of the phenomena. Thus,
for example, to fix our ideas, when, in the analytical expression of
geometrical phenomena, we are considering at once lines, areas, and
volumes, it will be necessary to observe that the three corresponding
units are necessarily so connected with each other that, according to
the subordination generally established in that respect, when the first
becomes _m_ times greater, the second becomes _m²_ times, and the third
_m³_ times. It is with such a modification that homogeneity will exist
in the equations, in which, if they are _algebraic_, we will have to
estimate the degree of each term by doubling the exponents of the
factors which correspond to areas, and tripling those of the factors
relating to volumes.

       *       *       *       *       *

Such are the principal general considerations relating to the _Calculus
of Direct Functions_. We have now to pass to the philosophical
examination of the _Calculus of Indirect Functions_, the much superior
importance and extent of which claim a fuller development.




CHAPTER III.

TRANSCENDENTAL ANALYSIS:

DIFFERENT MODES OF VIEWING IT.


We determined, in the second chapter, the philosophical character of the
transcendental analysis, in whatever manner it may be conceived,
considering only the general nature of its actual destination as a part
of mathematical science. This analysis has been presented by geometers
under several points of view, really distinct, although necessarily
equivalent, and leading always to identical results. They may be reduced
to three principal ones; those of LEIBNITZ, of NEWTON, and of LAGRANGE,
of which all the others are only secondary modifications. In the present
state of science, each of these three general conceptions offers
essential advantages which pertain to it exclusively, without our having
yet succeeded in constructing a single method uniting all these
different characteristic qualities. This combination will probably be
hereafter effected by some method founded upon the conception of
Lagrange when that important philosophical labour shall have been
accomplished, the study of the other conceptions will have only a
historic interest; but, until then, the science must be considered as in
only a provisional state, which requires the simultaneous consideration
of all the various modes of viewing this calculus. Illogical as may
appear this multiplicity of conceptions of one identical subject, still,
without them all, we could form but a very insufficient idea of this
analysis, whether in itself, or more especially in relation to its
applications. This want of system in the most important part of
mathematical analysis will not appear strange if we consider, on the one
hand, its great extent and its superior difficulty, and, on the other,
its recent formation.




ITS EARLY HISTORY.


If we had to trace here the systematic history of the successive
formation of the transcendental analysis, it would be necessary
previously to distinguish carefully from the _calculus of indirect
functions_, properly so called, the original idea of the _infinitesimal
method_, which can be conceived by itself, independently of any
_calculus_. We should see that the first germ of this idea is found in
the procedure constantly employed by the Greek geometers, under the name
of the _Method of Exhaustions_, as a means of passing from the
properties of straight lines to those of curves, and consisting
essentially in substituting for the curve the auxiliary consideration of
an inscribed or circumscribed polygon, by means of which they rose to
the curve itself, taking in a suitable manner the limits of the
primitive ratios. Incontestable as is this filiation of ideas, it would
be giving it a greatly exaggerated importance to see in this method of
exhaustions the real equivalent of our modern methods, as some geometers
have done; for the ancients had no logical and general means for the
determination of these limits, and this was commonly the greatest
difficulty of the question; so that their solutions were not subjected
to abstract and invariable rules, the uniform application of which would
lead with certainty to the knowledge sought; which is, on the contrary,
the principal characteristic of our transcendental analysis. In a word,
there still remained the task of generalizing the conceptions used by
the ancients, and, more especially, by considering it in a manner purely
abstract, of reducing it to a complete system of calculation, which to
them was impossible.

The first idea which was produced in this new direction goes back to the
great geometer Fermat, whom Lagrange has justly presented as having
blocked out the direct formation of the transcendental analysis by his
method for the determination of _maxima_ and _minima_, and for the
finding of _tangents_, which consisted essentially in introducing the
auxiliary consideration of the correlative increments of the proposed
variables, increments afterward suppressed as equal to zero when the
equations had undergone certain suitable transformations. But, although
Fermat was the first to conceive this analysis in a truly abstract
manner, it was yet far from being regularly formed into a general and
distinct calculus having its own notation, and especially freed from the
superfluous consideration of terms which, in the analysis of Fermat,
were finally not taken into the account, after having nevertheless
greatly complicated all the operations by their presence. This is what
Leibnitz so happily executed, half a century later, after some
intermediate modifications of the ideas of Fermat introduced by Wallis,
and still more by Barrow; and he has thus been the true creator of the
transcendental analysis, such as we now employ it. This admirable
discovery was so ripe (like all the great conceptions of the human
intellect at the moment of their manifestation), that Newton, on his
side, had arrived, at the same time, or a little earlier, at a method
exactly equivalent, by considering this analysis under a very different
point of view, which, although more logical in itself, is really less
adapted to give to the common fundamental method all the extent and the
facility which have been imparted to it by the ideas of Leibnitz.
Finally, Lagrange, putting aside the heterogeneous considerations which
had guided Leibnitz and Newton, has succeeded in reducing the
transcendental analysis, in its greatest perfection, to a purely
algebraic system, which only wants more aptitude for its practical
applications.

After this summary glance at the general history of the transcendental
analysis, we will proceed to the dogmatic exposition of the three
principal conceptions, in order to appreciate exactly their
characteristic properties, and to show the necessary identity of the
methods which are thence derived. Let us begin with that of Leibnitz.




METHOD OF LEIBNITZ.


_Infinitely small Elements._ This consists in introducing into the
calculus, in order to facilitate the establishment of equations, the
infinitely small elements of which all the quantities, the relations
between which are sought, are considered to be composed. These elements
or _differentials_ will have certain relations to one another, which are
constantly and necessarily more simple and easy to discover than those
of the primitive quantities, and by means of which we will be enabled
(by a special calculus having for its peculiar object the elimination of
these auxiliary infinitesimals) to go back to the desired equations,
which it would have been most frequently impossible to obtain directly.
This indirect analysis may have different degrees of indirectness; for,
when there is too much difficulty in forming immediately the equation
between the differentials of the magnitudes under consideration, a
second application of the same general artifice will have to be made,
and these differentials be treated, in their turn, as new primitive
quantities, and a relation be sought between their infinitely small
elements (which, with reference to the final objects of the question,
will be _second differentials_), and so on; the same transformation
admitting of being repeated any number of times, on the condition of
finally eliminating the constantly increasing number of infinitesimal
quantities introduced as auxiliaries.

A person not yet familiar with these considerations does not perceive at
once how the employment of these auxiliary quantities can facilitate the
discovery of the analytical laws of phenomena; for the infinitely small
increments of the proposed magnitudes being of the same species with
them, it would seem that their relations should not be obtained with
more ease, inasmuch as the greater or less value of a quantity cannot,
in fact, exercise any influence on an inquiry which is necessarily
independent, by its nature, of every idea of value. But it is easy,
nevertheless, to explain very clearly, and in a quite general manner,
how far the question must be simplified by such an artifice. For this
purpose, it is necessary to begin by distinguishing _different orders_
of infinitely small quantities, a very precise idea of which may be
obtained by considering them as being either the successive powers of
the same primitive infinitely small quantity, or as being quantities
which may be regarded as having finite ratios with these powers; so
that, to take an example, the second, third, &c., differentials of any
one variable are classed as infinitely small quantities of the second
order, the third, &c., because it is easy to discover in them finite
multiples of the second, third, &c., powers of a certain first
differential. These preliminary ideas being established, the spirit of
the infinitesimal analysis consists in constantly neglecting the
infinitely small quantities in comparison with finite quantities, and
generally the infinitely small quantities of any order whatever in
comparison with all those of an inferior order. It is at once apparent
how much such a liberty must facilitate the formation of equations
between the differentials of quantities, since, in the place of these
differentials, we can substitute such other elements as we may choose,
and as will be more simple to consider, only taking care to conform to
this single condition, that the new elements differ from the preceding
ones only by quantities infinitely small in comparison with them. It is
thus that it will be possible, in geometry, to treat curved lines as
composed of an infinity of rectilinear elements, curved surfaces as
formed of plane elements, and, in mechanics, variable motions as an
infinite series of uniform motions, succeeding one another at infinitely
small intervals of time.


EXAMPLES. Considering the importance of this admirable conception, I
think that I ought here to complete the illustration of its fundamental
character by the summary indication of some leading examples.


1. _Tangents._ Let it be required to determine, for each point of a
plane curve, the equation of which is given, the direction of its
tangent; a question whose general solution was the primitive object of
the inventors of the transcendental analysis. We will consider the
tangent as a secant joining two points infinitely near to each other;
and then, designating by _dy_ and _dx_ the infinitely small differences
of the co-ordinates of those two points, the elementary principles of
geometry will immediately give the equation _t_ = _dy_/_dx_ for the
trigonometrical tangent of the angle which is made with the axis of the
abscissas by the desired tangent, this being the most simple way of
fixing its position in a system of rectilinear co-ordinates. This
equation, common to all curves, being established, the question is
reduced to a simple analytical problem, which will consist in
eliminating the infinitesimals _dx_ and _dy_, which were introduced as
auxiliaries, by determining in each particular case, by means of the
equation of the proposed curve, the ratio of _dy_ to _dx_, which will be
constantly done by uniform and very simple methods.


2. _Rectification of an Arc._ In the second place, suppose that we wish
to know the length of the arc of any curve, considered as a function of
the co-ordinates of its extremities. It would be impossible to establish
directly the equation between this arc s and these co-ordinates, while
it is easy to find the corresponding relation between the differentials
of these different magnitudes. The most simple theorems of elementary
geometry will in fact give at once, considering the infinitely small arc
_ds_ as a right line, the equations

  _ds²_ = _dy²_ + _dx²_, or _ds²_ = _dx²_ + _dy²_ + _dz²_,

according as the curve is of single or double curvature. In either case,
the question is now entirely within the domain of analysis, which, by
the elimination of the differentials (which is the peculiar object of
the calculus of indirect functions), will carry us back from this
relation to that which exists between the finite quantities themselves
under examination.


3. _Quadrature of a Curve._ It would be the same with the quadrature of
curvilinear areas. If the curve is a plane one, and referred to
rectilinear co-ordinates, we will conceive the area A comprised between
this curve, the axis of the abscissas, and two extreme co-ordinates, to
increase by an infinitely small quantity _d_A, as the result of a
corresponding increment of the abscissa. The relation between these two
differentials can be immediately obtained with the greatest facility by
substituting for the curvilinear element of the proposed area the
rectangle formed by the extreme ordinate and the element of the
abscissa, from which it evidently differs only by an infinitely small
quantity of the second order. This will at once give, whatever may be
the curve, the very simple differential equation

  _d_A = _ydx_,

from which, when the curve is defined, the calculus of indirect
functions will show how to deduce the finite equation, which is the
immediate object of the problem.


4. _Velocity in Variable Motion._ In like manner, in Dynamics, when we
desire to know the expression for the velocity acquired at each instant
by a body impressed with a motion varying according to any law, we will
consider the motion as being uniform during an infinitely small element
of the time _t_, and we will thus immediately form the differential
equation _de_ = _vdt_, in which _v_ designates the velocity acquired
when the body has passed over the space _e_; and thence it will be easy
to deduce, by simple and invariable analytical procedures, the formula
which would give the velocity in each particular motion, in accordance
with the corresponding relation between the time and the space; or,
reciprocally, what this relation would be if the mode of variation of
the velocity was supposed to be known, whether with respect to the space
or to the time.


5. _Distribution of Heat._ Lastly, to indicate another kind of
questions, it is by similar steps that we are able, in the study of
thermological phenomena, according to the happy conception of M.
Fourier, to form in a very simple manner the general differential
equation which expresses the variable distribution of heat in any body
whatever, subjected to any influences, by means of the single and
easily-obtained relation, which represents the uniform distribution of
heat in a right-angled parallelopipedon, considering (geometrically)
every other body as decomposed into infinitely small elements of a
similar form, and (thermologically) the flow of heat as constant during
an infinitely small element of time. Henceforth, all the questions which
can be presented by abstract thermology will be reduced, as in geometry
and mechanics, to mere difficulties of analysis, which will always
consist in the elimination of the differentials introduced as
auxiliaries to facilitate the establishment of the equations.

Examples of such different natures are more than sufficient to give a
clear general idea of the immense scope of the fundamental conception of
the transcendental analysis as formed by Leibnitz, constituting, as it
undoubtedly does, the most lofty thought to which the human mind has as
yet attained.

It is evident that this conception was indispensable to complete the
foundation of mathematical science, by enabling us to establish, in a
broad and fruitful manner, the relation of the concrete to the abstract.
In this respect it must be regarded as the necessary complement of the
great fundamental idea of Descartes on the general analytical
representation of natural phenomena: an idea which did not begin to be
worthily appreciated and suitably employed till after the formation of
the infinitesimal analysis, without which it could not produce, even in
geometry, very important results.


_Generality of the Formulas._ Besides the admirable facility which is
given by the transcendental analysis for the investigation of the
mathematical laws of all phenomena, a second fundamental and inherent
property, perhaps as important as the first, is the extreme generality
of the differential formulas, which express in a single equation each
determinate phenomenon, however varied the subjects in relation to which
it is considered. Thus we see, in the preceding examples, that a single
differential equation gives the tangents of all curves, another their
rectifications, a third their quadratures; and in the same way, one
invariable formula expresses the mathematical law of every variable
motion; and, finally, a single equation constantly represents the
distribution of heat in any body and for any case. This generality,
which is so exceedingly remarkable, and which is for geometers the basis
of the most elevated considerations, is a fortunate and necessary
consequence of the very spirit of the transcendental analysis,
especially in the conception of Leibnitz. Thus the infinitesimal
analysis has not only furnished a general method for indirectly forming
equations which it would have been impossible to discover in a direct
manner, but it has also permitted us to consider, for the mathematical
study of natural phenomena, a new order of more general laws, which
nevertheless present a clear and precise signification to every mind
habituated to their interpretation. By virtue of this second
characteristic property, the entire system of an immense science, such
as geometry or mechanics, has been condensed into a small number of
analytical formulas, from which the human mind can deduce, by certain
and invariable rules, the solution of all particular problems.


_Demonstration of the Method._ To complete the general exposition of the
conception of Leibnitz, there remains to be considered the demonstration
of the logical procedure to which it leads, and this, unfortunately, is
the most imperfect part of this beautiful method.

In the beginning of the infinitesimal analysis, the most celebrated
geometers rightly attached more importance to extending the immortal
discovery of Leibnitz and multiplying its applications than to
rigorously establishing the logical bases of its operations. They
contented themselves for a long time by answering the objections of
second-rate geometers by the unhoped-for solution of the most difficult
problems; doubtless persuaded that in mathematical science, much more
than in any other, we may boldly welcome new methods, even when their
rational explanation is imperfect, provided they are fruitful in
results, inasmuch as its much easier and more numerous verifications
would not permit any error to remain long undiscovered. But this state
of things could not long exist, and it was necessary to go back to the
very foundations of the analysis of Leibnitz in order to prove, in a
perfectly general manner, the rigorous exactitude of the procedures
employed in this method, in spite of the apparent infractions of the
ordinary rules of reasoning which it permitted.

Leibnitz, urged to answer, had presented an explanation entirely
erroneous, saying that he treated infinitely small quantities as
_incomparables_, and that he neglected them in comparison with finite
quantities, "like grains of sand in comparison with the sea:" a view
which would have completely changed the nature of his analysis, by
reducing it to a mere approximative calculus, which, under this point of
view, would be radically vicious, since it would be impossible to
foresee, in general, to what degree the successive operations might
increase these first errors, which could thus evidently attain any
amount. Leibnitz, then, did not see, except in a very confused manner,
the true logical foundations of the analysis which he had created. His
earliest successors limited themselves, at first, to verifying its
exactitude by showing the conformity of its results, in particular
applications, to those obtained by ordinary algebra or the geometry of
the ancients; reproducing, according to the ancient methods, so far as
they were able, the solutions of some problems after they had been once
obtained by the new method, which alone was capable of discovering them
in the first place.

When this great question was considered in a more general manner,
geometers, instead of directly attacking the difficulty, preferred to
elude it in some way, as Euler and D'Alembert, for example, have done,
by demonstrating the necessary and constant conformity of the conception
of Leibnitz, viewed in all its applications, with other fundamental
conceptions of the transcendental analysis, that of Newton especially,
the exactitude of which was free from any objection. Such a general
verification is undoubtedly strictly sufficient to dissipate any
uncertainty as to the legitimate employment of the analysis of Leibnitz.
But the infinitesimal method is so important--it offers still, in almost
all its applications, such a practical superiority over the other
general conceptions which have been successively proposed--that there
would be a real imperfection in the philosophical character of the
science if it could not justify itself, and needed to be logically
founded on considerations of another order, which would then cease to be
employed.

It was, then, of real importance to establish directly and in a general
manner the necessary rationality of the infinitesimal method. After
various attempts more or less imperfect, a distinguished geometer,
Carnot, presented at last the true direct logical explanation of the
method of Leibnitz, by showing it to be founded on the principle of the
necessary compensation of errors, this being, in fact, the precise and
luminous manifestation of what Leibnitz had vaguely and confusedly
perceived. Carnot has thus rendered the science an essential service,
although, as we shall see towards the end of this chapter, all this
logical scaffolding of the infinitesimal method, properly so called, is
very probably susceptible of only a provisional existence, inasmuch as
it is radically vicious in its nature. Still, we should not fail to
notice the general system of reasoning proposed by Carnot, in order to
directly legitimate the analysis of Leibnitz. Here is the substance of
it:

In establishing the differential equation of a phenomenon, we
substitute, for the immediate elements of the different quantities
considered, other simpler infinitesimals, which differ from them
infinitely little in comparison with them; and this substitution
constitutes the principal artifice of the method of Leibnitz, which
without it would possess no real facility for the formation of
equations. Carnot regards such an hypothesis as really producing an
error in the equation thus obtained, and which for this reason he calls
_imperfect_; only, it is clear that this error must be infinitely small.
Now, on the other hand, all the analytical operations, whether of
differentiation or of integration, which are performed upon these
differential equations, in order to raise them to finite equations by
eliminating all the infinitesimals which have been introduced as
auxiliaries, produce as constantly, by their nature, as is easily seen,
other analogous errors, so that an exact compensation takes place, and
the final equations, in the words of Carnot, become _perfect_. Carnot
views, as a certain and invariable indication of the actual
establishment of this necessary compensation, the complete elimination
of the various infinitely small quantities, which is always, in fact,
the final object of all the operations of the transcendental analysis;
for if we have committed no other infractions of the general rules of
reasoning than those thus exacted by the very nature of the
infinitesimal method, the infinitely small errors thus produced cannot
have engendered other than infinitely small errors in all the equations,
and the relations are necessarily of a rigorous exactitude as soon as
they exist between finite quantities alone, since the only errors then
possible must be finite ones, while none such can have entered. All this
general reasoning is founded on the conception of infinitesimal
quantities, regarded as indefinitely decreasing, while those from which
they are derived are regarded as fixed.


_Illustration by Tangents._ Thus, to illustrate this abstract exposition
by a single example, let us take up again the question of _tangents_,
which is the most easy to analyze completely. We will regard the
equation _t_ = _dy/dx_, obtained above, as being affected with an
infinitely small error, since it would be perfectly rigorous only for
the secant. Now let us complete the solution by seeking, according to
the equation of each curve, the ratio between the differentials of the
co-ordinates. If we suppose this equation to be _y_ = _ax²_, we shall
evidently have

  _dy_ = 2_axdx_ + _adx²_.

In this formula we shall have to neglect the term _dx²_ as an infinitely
small quantity of the second order. Then the combination of the two
_imperfect_ equations.

  _t_ = _dy/dx_, _dy_ = 2_ax(dx)_,

being sufficient to eliminate entirely the infinitesimals, the finite
result, _t_ = 2_ax_, will necessarily be rigorously correct, from the
effect of the exact compensation of the two errors committed; since, by
its finite nature, it cannot be affected by an infinitely small error,
and this is, nevertheless, the only one which it could have, according
to the spirit of the operations which have been executed.

It would be easy to reproduce in a uniform manner the same reasoning
with reference to all the other general applications of the analysis of
Leibnitz.

This ingenious theory is undoubtedly more subtile than solid, when we
examine it more profoundly; but it has really no other radical logical
fault than that of the infinitesimal method itself, of which it is, it
seems to me, the natural development and the general explanation, so
that it must be adopted for as long a time as it shall be thought proper
to employ this method directly.

       *       *       *       *       *

I pass now to the general exposition of the two other fundamental
conceptions of the transcendental analysis, limiting myself in each to
its principal idea, the philosophical character of the analysis having
been sufficiently determined above in the examination of the conception
of Leibnitz, which I have specially dwelt upon because it admits of
being most easily grasped as a whole, and most rapidly described.




METHOD OF NEWTON.


Newton has successively presented his own method of conceiving the
transcendental analysis under several different forms. That which is at
present the most commonly adopted was designated by Newton, sometimes
under the name of the _Method of prime and ultimate Ratios_, sometimes
under that of the _Method of Limits_.


_Method of Limits._ The general spirit of the transcendental analysis,
from this point of view, consists in introducing as auxiliaries, in the
place of the primitive quantities, or concurrently with them, in order
to facilitate the establishment of equations, the _limits of the ratios_
of the simultaneous increments of these quantities; or, in other words,
the _final ratios_ of these increments; limits or final ratios which can
be easily shown to have a determinate and finite value. A special
calculus, which is the equivalent of the infinitesimal calculus, is then
employed to pass from the equations between these limits to the
corresponding equations between the primitive quantities themselves.

The power which is given by such an analysis, of expressing with more
ease the mathematical laws of phenomena, depends in general on this,
that since the calculus applies, not to the increments themselves of the
proposed quantities, but to the limits of the ratios of those
increments, we can always substitute for each increment any other
magnitude more easy to consider, provided that their final ratio is the
ratio of equality, or, in other words, that the limit of their ratio is
unity. It is clear, indeed, that the calculus of limits would be in no
way affected by this substitution. Starting from this principle, we find
nearly the equivalent of the facilities offered by the analysis of
Leibnitz, which are then merely conceived under another point of view.
Thus curves will be regarded as the _limits_ of a series of rectilinear
polygons, variable motions as the _limits_ of a collection of uniform
motions of constantly diminishing durations, and so on.


EXAMPLES. 1. _Tangents._ Suppose, for example, that we wish to determine
the direction of the tangent to a curve; we will regard it as the limit
towards which would tend a secant, which should turn about the given
point so that its second point of intersection should indefinitely
approach the first. Representing the differences of the co-ordinates of
the two points by Δ_y_ and Δ_x_, we would have at each instant, for the
trigonometrical tangent of the angle which the secant makes with the
axis of abscissas,

  _t_ = Δ_y_/Δ_x_;

from which, taking the limits, we will obtain, relatively to the tangent
itself, this general formula of transcendental analysis,

  _t_ = _L_(Δ_y_/Δ_x_),

the characteristic _L_ being employed to designate the limit. The
calculus of indirect functions will show how to deduce from this formula
in each particular case, when the equation of the curve is given, the
relation between _t_ and _x_, by eliminating the auxiliary quantities
which have been introduced. If we suppose, in order to complete the
solution, that the equation of the proposed curve is _y_ = _ax²_, we
shall evidently have

  Δ_y_ = 2_ax_Δ_x_ + _a_(Δ_x_)²,

from which we shall obtain

  Δ_y_/Δ_x_ = 2_ax_ + _a_Δ_x_.

Now it is clear that the _limit_ towards which the second number tends,
in proportion as Δ_x_ diminishes, is 2_ax_. We shall therefore find, by
this method, _t_ = 2_ax_, as we obtained it for the same case by the
method of Leibnitz.

2. _Rectifications._ In like manner, when the rectification of a curve
is desired, we must substitute for the increment of the arc s the chord
of this increment, which evidently has such a connexion with it that the
limit of their ratio is unity; and then we find (pursuing in other
respects the same plan as with the method of Leibnitz) this general
equation of rectifications:

  (_LΔs_/Δ_x_)² = 1 + (_LΔy_/Δ_x_)²,
  or (_LΔs_/Δ_x_)² = 1 + (_LΔy_/Δ_x_)² + (_LΔz_/Δ_x_)²,

according as the curve is plane or of double curvature. It will now be
necessary, for each particular curve, to pass from this equation to that
between the arc and the abscissa, which depends on the transcendental
calculus properly so called.

We could take up, with the same facility, by the method of limits, all
the other general questions, the solution of which has been already
indicated according to the infinitesimal method.

Such is, in substance, the conception which Newton formed for the
transcendental analysis, or, more precisely, that which Maclaurin and
D'Alembert have presented as the most rational basis of that analysis,
in seeking to fix and to arrange the ideas of Newton upon that subject.


_Fluxions and Fluents._ Another distinct form under which Newton has
presented this same method should be here noticed, and deserves
particularly to fix our attention, as much by its ingenious clearness in
some cases as by its having furnished the notation best suited to this
manner of viewing the transcendental analysis, and, moreover, as having
been till lately the special form of the calculus of indirect functions
commonly adopted by the English geometers. I refer to the calculus of
_fluxions_ and of _fluents_, founded on the general idea of
_velocities_.

To facilitate the conception of the fundamental idea, let us consider
every curve as generated by a point impressed with a motion varying
according to any law whatever. The different quantities which the curve
can present, the abscissa, the ordinate, the arc, the area, &c., will be
regarded as simultaneously produced by successive degrees during this
motion. The _velocity_ with which each shall have been described will be
called the _fluxion_ of that quantity, which will be inversely named its
_fluent_. Henceforth the transcendental analysis will consist, according
to this conception, in forming directly the equations between the
fluxions of the proposed quantities, in order to deduce therefrom, by a
special calculus, the equations between the fluents themselves. What
has been stated respecting curves may, moreover, evidently be applied to
any magnitudes whatever, regarded, by the aid of suitable images, as
produced by motion.

It is easy to understand the general and necessary identity of this
method with that of limits complicated with the foreign idea of motion.
In fact, resuming the case of the curve, if we suppose, as we evidently
always may, that the motion of the describing point is uniform in a
certain direction, that of the abscissa, for example, then the fluxion
of the abscissa will be constant, like the element of the time; for all
the other quantities generated, the motion cannot be conceived to be
uniform, except for an infinitely small time. Now the velocity being in
general according to its mechanical conception, the ratio of each space
to the time employed in traversing it, and this time being here
proportional to the increment of the abscissa, it follows that the
fluxions of the ordinate, of the arc, of the area, &c., are really
nothing else (rejecting the intermediate consideration of time) than the
final ratios of the increments of these different quantities to the
increment of the abscissa. This method of fluxions and fluents is, then,
in reality, only a manner of representing, by a comparison borrowed from
mechanics, the method of prime and ultimate ratios, which alone can be
reduced to a calculus. It evidently, then, offers the same general
advantages in the various principal applications of the transcendental
analysis, without its being necessary to present special proofs of
this.




METHOD OF LAGRANGE.


_Derived Functions._ The conception of Lagrange, in its admirable
simplicity, consists in representing the transcendental analysis as a
great algebraic artifice, by which, in order to facilitate the
establishment of equations, we introduce, in the place of the primitive
functions, or concurrently with them, their _derived_ functions; that
is, according to the definition of Lagrange, the coefficient of the
first term of the increment of each function, arranged according to the
ascending powers of the increment of its variable. The special calculus
of indirect functions has for its constant object, here as well as in
the conceptions of Leibnitz and of Newton, to eliminate these
_derivatives_ which have been thus employed as auxiliaries, in order to
deduce from their relations the corresponding equations between the
primitive magnitudes.


_An Extension of ordinary Analysis._ The transcendental analysis is,
then, nothing but a simple though very considerable extension of
ordinary analysis. Geometers have long been accustomed to introduce in
analytical investigations, in the place of the magnitudes themselves
which they wished to study, their different powers, or their logarithms,
or their sines, &c., in order to simplify the equations, and even to
obtain them more easily. This successive _derivation_ is an artifice of
the same nature, only of greater extent, and procuring, in consequence,
much more important resources for this common object.

But, although we can readily conceive, _à priori_, that the auxiliary
consideration of these derivatives _may_ facilitate the establishment
of equations, it is not easy to explain why this _must_ necessarily
follow from this mode of derivation rather than from any other
transformation. Such is the weak point of the great idea of Lagrange.
The precise advantages of this analysis cannot as yet be grasped in an
abstract manner, but only shown by considering separately each principal
question, so that the verification is often exceedingly laborious.


EXAMPLE. _Tangents._ This manner of conceiving the transcendental
analysis may be best illustrated by its application to the most simple
of the problems above examined--that of tangents.

Instead of conceiving the tangent as the prolongation of the infinitely
small element of the curve, according to the notion of Leibnitz--or as
the limit of the secants, according to the ideas of Newton--Lagrange
considers it, according to its simple geometrical character, analogous
to the definitions of the ancients, to be a right line such that no
other right line can pass through the point of contact between it and
the curve. Then, to determine its direction, we must seek the general
expression of its distance from the curve, measured in any direction
whatever--in that of the ordinate, for example--and dispose of the
arbitrary constant relating to the inclination of the right line, which
will necessarily enter into that expression, in such a way as to
diminish that separation as much as possible. Now this distance, being
evidently equal to the difference of the two ordinates of the curve and
of the right line, which correspond to the same new abscissa _x_ + _h_,
will be represented by the formula

  (_f'_(_x_) - _t_)_h_ + _qh²_ + _rh³_ + etc.,

in which _t_ designates, as above, the unknown trigonometrical tangent
of the angle which the required line makes with the axis of abscissas,
and _f'_(_x_) the derived function of the ordinate _f_(_x_). This being
understood, it is easy to see that, by disposing of _t_ so as to make
the first term of the preceding formula equal to zero, we will render
the interval between the two lines the least possible, so that any other
line for which _t_ did not have the value thus determined would
necessarily depart farther from the proposed curve. We have, then, for
the direction of the tangent sought, the general expression _t_ =
_f'_(_x_), a result exactly equivalent to those furnished by the
Infinitesimal Method and the Method of Limits. We have yet to find
_f'_(_x_) in each particular curve, which is a mere question of
analysis, quite identical with those which are presented, at this stage
of the operations, by the other methods.

After these considerations upon the principal general conceptions, we
need not stop to examine some other theories proposed, such as Euler's
_Calculus of Vanishing Quantities_, which are really modifications--more
or less important, and, moreover, no longer used--of the preceding
methods.

I have now to establish the comparison and the appreciation of these
three fundamental methods. Their _perfect and necessary conformity_ is
first to be proven in a general manner.




FUNDAMENTAL IDENTITY OF THE THREE METHODS.


It is, in the first place, evident from what precedes, considering these
three methods as to their actual destination, independently of their
preliminary ideas, that they all consist in the same general logical
artifice, which has been characterized in the first chapter; to wit,
the introduction of a certain system of auxiliary magnitudes, having
uniform relations to those which are the special objects of the inquiry,
and substituted for them expressly to facilitate the analytical
expression of the mathematical laws of the phenomena, although they have
finally to be eliminated by the aid of a special calculus. It is this
which has determined me to regularly define the transcendental analysis
as _the calculus of indirect functions_, in order to mark its true
philosophical character, at the same time avoiding any discussion upon
the best manner of conceiving and applying it. The general effect of
this analysis, whatever the method employed, is, then, to bring every
mathematical question much more promptly within the power of the
_calculus_, and thus to diminish considerably the serious difficulty
which is usually presented by the passage from the concrete to the
abstract. Whatever progress we may make, we can never hope that the
calculus will ever be able to grasp every question of natural
philosophy, geometrical, or mechanical, or thermological, &c.,
immediately upon its birth, which would evidently involve a
contradiction. Every problem will constantly require a certain
preliminary labour to be performed, in which the calculus can be of no
assistance, and which, by its nature, cannot be subjected to abstract
and invariable rules; it is that which has for its special object the
establishment of equations, which form the indispensable starting point
of all analytical researches. But this preliminary labour has been
remarkably simplified by the creation of the transcendental analysis,
which has thus hastened the moment at which the solution admits of the
uniform and precise application of general and abstract methods; by
reducing, in each case, this special labour to the investigation of
equations between the auxiliary magnitudes; from which the calculus then
leads to equations directly referring to the proposed magnitudes, which,
before this admirable conception, it had been necessary to establish
directly and separately. Whether these indirect equations are
_differential_ equations, according to the idea of Leibnitz, or
equations of _limits_, conformably to the conception of Newton, or,
lastly, _derived_ equations, according to the theory of Lagrange, the
general procedure is evidently always the same.

But the coincidence of these three principal methods is not limited to
the common effect which they produce; it exists, besides, in the very
manner of obtaining it. In fact, not only do all three consider, in the
place of the primitive magnitudes, certain auxiliary ones, but, still
farther, the quantities thus introduced as subsidiary are exactly
identical in the three methods, which consequently differ only in the
manner of viewing them. This can be easily shown by taking for the
general term of comparison any one of the three conceptions, especially
that of Lagrange, which is the most suitable to serve as a type, as
being the freest from foreign considerations. Is it not evident, by the
very definition of _derived functions_, that they are nothing else than
what Leibnitz calls _differential coefficients_, or the ratios of the
differential of each function to that of the corresponding variable,
since, in determining the first differential, we will be obliged, by the
very nature of the infinitesimal method, to limit ourselves to taking
the only term of the increment of the function which contains the first
power of the infinitely small increment of the variable? In the same
way, is not the derived function, by its nature, likewise the necessary
_limit_ towards which tends the ratio between the increment of the
primitive function and that of its variable, in proportion as this last
indefinitely diminishes, since it evidently expresses what that ratio
becomes when we suppose the increment of the variable to equal zero?
That which is designated by _dx_/_dy_ in the method of Leibnitz; that
which ought to be noted as _L_(Δ_y_/Δ_x_) in that of Newton; and that
which Lagrange has indicated by _f'_(_x_), is constantly one same
function, seen from three different points of view, the considerations
of Leibnitz and Newton properly consisting in making known two general
necessary properties of the derived function. The transcendental
analysis, examined abstractedly and in its principle, is then always the
same, whatever may be the conception which is adopted, and the
procedures of the calculus of indirect functions are necessarily
identical in these different methods, which in like manner must, for any
application whatever, lead constantly to rigorously uniform results.




COMPARATIVE VALUE OF THE THREE METHODS.


If now we endeavour to estimate the comparative value of these three
equivalent conceptions, we shall find in each advantages and
inconveniences which are peculiar to it, and which still prevent
geometers from confining themselves to any one of them, considered as
final.


_That of Leibnitz._ The conception of Leibnitz presents incontestably,
in all its applications, a very marked superiority, by leading in a much
more rapid manner, and with much less mental effort, to the formation
of equations between the auxiliary magnitudes. It is to its use that we
owe the high perfection which has been acquired by all the general
theories of geometry and mechanics. Whatever may be the different
speculative opinions of geometers with respect to the infinitesimal
method, in an abstract point of view, all tacitly agree in employing it
by preference, as soon as they have to treat a new question, in order
not to complicate the necessary difficulty by this purely artificial
obstacle proceeding from a misplaced obstinacy in adopting a less
expeditious course. Lagrange himself, after having reconstructed the
transcendental analysis on new foundations, has (with that noble
frankness which so well suited his genius) rendered a striking and
decisive homage to the characteristic properties of the conception of
Leibnitz, by following it exclusively in the entire system of his
_Méchanique Analytique_. Such a fact renders any comments unnecessary.

But when we consider the conception of Leibnitz in itself and in its
logical relations, we cannot escape admitting, with Lagrange, that it is
radically vicious in this, that, adopting its own expressions, the
notion of infinitely small quantities is a _false idea_, of which it is
in fact impossible to obtain a clear conception, however we may deceive
ourselves in that matter. Even if we adopt the ingenious idea of the
compensation of errors, as above explained, this involves the radical
inconvenience of being obliged to distinguish in mathematics two classes
of reasonings, those which are perfectly rigorous, and those in which we
designedly commit errors which subsequently have to be compensated. A
conception which leads to such strange consequences is undoubtedly very
unsatisfactory in a logical point of view.

To say, as do some geometers, that it is possible in every case to
reduce the infinitesimal method to that of limits, the logical character
of which is irreproachable, would evidently be to elude the difficulty
rather than to remove it; besides, such a transformation almost entirely
strips the conception of Leibnitz of its essential advantages of
facility and rapidity.

Finally, even disregarding the preceding important considerations, the
infinitesimal method would no less evidently present by its nature the
very serious defect of breaking the unity of abstract mathematics, by
creating a transcendental analysis founded on principles so different
from those which form the basis of the ordinary analysis. This division
of analysis into two worlds almost entirely independent of each other,
tends to hinder the formation of truly general analytical conceptions.
To fully appreciate the consequences of this, we should have to go back
to the state of the science before Lagrange had established a general
and complete harmony between these two great sections.


_That of Newton._ Passing now to the conception of Newton, it is evident
that by its nature it is not exposed to the fundamental logical
objections which are called forth by the method of Leibnitz. The notion
of _limits_ is, in fact, remarkable for its simplicity and its
precision. In the transcendental analysis presented in this manner, the
equations are regarded as exact from their very origin, and the general
rules of reasoning are as constantly observed as in ordinary analysis.
But, on the other hand, it is very far from offering such powerful
resources for the solution of problems as the infinitesimal method. The
obligation which it imposes, of never considering the increments of
magnitudes separately and by themselves, nor even in their ratios, but
only in the limits of those ratios, retards considerably the operations
of the mind in the formation of auxiliary equations. We may even say
that it greatly embarrasses the purely analytical transformations. Thus
the transcendental analysis, considered separately from its
applications, is far from presenting in this method the extent and the
generality which have been imprinted upon it by the conception of
Leibnitz. It is very difficult, for example, to extend the theory of
Newton to functions of several independent variables. But it is
especially with reference to its applications that the relative
inferiority of this theory is most strongly marked.

Several Continental geometers, in adopting the method of Newton as the
more logical basis of the transcendental analysis, have partially
disguised this inferiority by a serious inconsistency, which consists in
applying to this method the notation invented by Leibnitz for the
infinitesimal method, and which is really appropriate to it alone. In
designating by _dy_/_dx_ that which logically ought, in the theory of
limits, to be denoted by _L_(Δ_y_/Δ_x_), and in extending to all the
other analytical conceptions this displacement of signs, they intended,
undoubtedly, to combine the special advantages of the two methods; but,
in reality, they have only succeeded in causing a vicious confusion
between them, a familiarity with which hinders the formation of clear
and exact ideas of either. It would certainly be singular, considering
this usage in itself, that, by the mere means of signs, it could be
possible to effect a veritable combination between two theories so
distinct as those under consideration.

Finally, the method of limits presents also, though in a less degree,
the greater inconvenience, which I have above noted in reference to the
infinitesimal method, of establishing a total separation between the
ordinary and the transcendental analysis; for the idea of _limits_,
though clear and rigorous, is none the less in itself, as Lagrange has
remarked, a foreign idea, upon which analytical theories ought not to be
dependent.


_That of Lagrange._ This perfect unity of analysis, and this purely
abstract character of its fundamental notions, are found in the highest
degree in the conception of Lagrange, and are found there alone; it is,
for this reason, the most rational and the most philosophical of all.
Carefully removing every heterogeneous consideration, Lagrange has
reduced the transcendental analysis to its true peculiar character, that
of presenting a very extensive class of analytical transformations,
which facilitate in a remarkable degree the expression of the conditions
of various problems. At the same time, this analysis is thus necessarily
presented as a simple extension of ordinary analysis; it is only a
higher algebra. All the different parts of abstract mathematics,
previously so incoherent, have from that moment admitted of being
conceived as forming a single system.

Unhappily, this conception, which possesses such fundamental properties,
independently of its so simple and so lucid notation, and which is
undoubtedly destined to become the final theory of transcendental
analysis, because of its high philosophical superiority over all the
other methods proposed, presents in its present state too many
difficulties in its applications, as compared with the conception of
Newton, and still more with that of Leibnitz, to be as yet exclusively
adopted. Lagrange himself has succeeded only with great difficulty in
rediscovering, by his method, the principal results already obtained by
the infinitesimal method for the solution of the general questions of
geometry and mechanics; we may judge from that what obstacles would be
found in treating in the same manner questions which were truly new and
important. It is true that Lagrange, on several occasions, has shown
that difficulties call forth, from men of genius, superior efforts,
capable of leading to the greatest results. It was thus that, in trying
to adapt his method to the examination of the curvature of lines, which
seemed so far from admitting its application, he arrived at that
beautiful theory of contacts which has so greatly perfected that
important part of geometry. But, in spite of such happy exceptions, the
conception of Lagrange has nevertheless remained, as a whole,
essentially unsuited to applications.

The final result of the general comparison which I have too briefly
sketched, is, then, as already suggested, that, in order to really
understand the transcendental analysis, we should not only consider it
in its principles according to the three fundamental conceptions of
Leibnitz, of Newton, and of Lagrange, but should besides accustom
ourselves to carry out almost indifferently, according to these three
principal methods, and especially according to the first and the last,
the solution of all important questions, whether of the pure calculus of
indirect functions or of its applications. This is a course which I
could not too strongly recommend to all those who desire to judge
philosophically of this admirable creation of the human mind, as well as
to those who wish to learn to make use of this powerful instrument with
success and with facility. In all the other parts of mathematical
science, the consideration of different methods for a single class of
questions may be useful, even independently of its historical interest,
but it is not indispensable; here, on the contrary, it is strictly
necessary.

Having determined with precision, in this chapter, the philosophical
character of the calculus of indirect functions, according to the
principal fundamental conceptions of which it admits, we have next to
consider, in the following chapter, the logical division and the general
composition of this calculus.




CHAPTER IV.


THE DIFFERENTIAL AND INTEGRAL CALCULUS.


ITS TWO FUNDAMENTAL DIVISIONS.


The _calculus of indirect functions_, in accordance with the
considerations explained in the preceding chapter, is necessarily
divided into two parts (or, more properly, is decomposed into two
different _calculi_ entirely distinct, although intimately connected by
their nature), according as it is proposed to find the relations between
the auxiliary magnitudes (the introduction of which constitutes the
general spirit of this calculus) by means of the relations between the
corresponding primitive magnitudes; or, conversely, to try to discover
these direct equations by means of the indirect equations originally
established. Such is, in fact, constantly the double object of the
transcendental analysis.

These two systems have received different names, according to the point
of view under which this analysis has been regarded. The infinitesimal
method, properly so called, having been the most generally employed for
the reasons which have been given, almost all geometers employ
habitually the denominations of _Differential Calculus_ and of _Integral
Calculus_, established by Leibnitz, and which are, in fact, very
rational consequences of his conception. Newton, in accordance with his
method, named the first the _Calculus of Fluxions_, and the second the
_Calculus of Fluents_, expressions which were commonly employed in
England. Finally, following the eminently philosophical theory founded
by Lagrange, one would be called the _Calculus of Derived Functions_,
and the other the _Calculus of Primitive Functions_. I will continue to
make use of the terms of Leibnitz, as being more convenient for the
formation of secondary expressions, although I ought, in accordance with
the suggestions made in the preceding chapter, to employ concurrently
all the different conceptions, approaching as nearly as possible to that
of Lagrange.




THEIR RELATIONS TO EACH OTHER.


The differential calculus is evidently the logical basis of the integral
calculus; for we do not and cannot know how to integrate directly any
other differential expressions than those produced by the
differentiation of the ten simple functions which constitute the general
elements of our analysis. The art of integration consists, then,
essentially in bringing all the other cases, as far as is possible, to
finally depend on only this small number of fundamental integrations.

In considering the whole body of the transcendental analysis, as I have
characterized it in the preceding chapter, it is not at first apparent
what can be the peculiar utility of the differential calculus,
independently of this necessary relation with the integral calculus,
which seems as if it must be, by itself, the only one directly
indispensable. In fact, the elimination of the _infinitesimals_ or of
the _derivatives_, introduced as auxiliaries to facilitate the
establishment of equations, constituting, as we have seen, the final and
invariable object of the calculus of indirect functions, it is natural
to think that the calculus which teaches how to deduce from the
equations between these auxiliary magnitudes, those which exist between
the primitive magnitudes themselves, ought strictly to suffice for the
general wants of the transcendental analysis without our perceiving, at
the first glance, what special and constant part the solution of the
inverse question can have in such an analysis. It would be a real error,
though a common one, to assign to the differential calculus, in order to
explain its peculiar, direct, and necessary influence, the destination
of forming the differential equations, from which the integral calculus
then enables us to arrive at the finite equations; for the primitive
formation of differential equations is not and cannot be, properly
speaking, the object of any calculus, since, on the contrary, it forms
by its nature the indispensable starting point of any calculus whatever.
How, in particular, could the differential calculus, which in itself is
reduced to teaching the means of _differentiating_ the different
equations, be a general procedure for establishing them? That which in
every application of the transcendental analysis really facilitates the
formation of equations, is the infinitesimal _method_, and not the
infinitesimal _calculus_, which is perfectly distinct from it, although
it is its indispensable complement. Such a consideration would, then,
give a false idea of the special destination which characterizes the
differential calculus in the general system of the transcendental
analysis.

But we should nevertheless very imperfectly conceive the real peculiar
importance of this first branch of the calculus of indirect functions,
if we saw in it only a simple preliminary labour, having no other
general and essential object than to prepare indispensable foundations
for the integral calculus. As the ideas on this matter are generally
confused, I think that I ought here to explain in a summary manner this
important relation as I view it, and to show that in every application
of the transcendental analysis a primary, direct, and necessary part is
constantly assigned to the differential calculus.


1. _Use of the Differential Calculus as preparatory to that of the
Integral._ In forming the differential equations of any phenomenon
whatever, it is very seldom that we limit ourselves to introduce
differentially only those magnitudes whose relations are sought. To
impose that condition would be to uselessly diminish the resources
presented by the transcendental analysis for the expression of the
mathematical laws of phenomena. Most frequently we introduce into the
primitive equations, through their differentials, other magnitudes whose
relations are already known or supposed to be so, and without the
consideration of which it would be frequently impossible to establish
equations. Thus, for example, in the general problem of the
rectification of curves, the differential equation,

  _ds_² = _dy_² + _dx_², or _ds_² = _dx_² + _dy_² + _dz_²,

is not only established between the desired function s and the
independent variable _x_, to which it is referred, but, at the same
time, there have been introduced, as indispensable intermediaries, the
differentials of one or two other functions, _y_ and _z_, which are
among the data of the problem; it would not have been possible to form
directly the equation between _ds_ and _dx_, which would, besides, be
peculiar to each curve considered. It is the same for most questions.
Now in these cases it is evident that the differential equation is not
immediately suitable for integration. It is previously necessary that
the differentials of the functions supposed to be known, which have
been employed as intermediaries, should be entirely eliminated, in order
that equations may be obtained between the differentials of the
functions which alone are sought and those of the really independent
variables, after which the question depends on only the integral
calculus. Now this preparatory elimination of certain differentials, in
order to reduce the infinitesimals to the smallest number possible,
belongs simply to the differential calculus; for it must evidently be
done by determining, by means of the equations between the functions
supposed to be known, taken as intermediaries, the relations of their
differentials, which is merely a question of differentiation. Thus, for
example, in the case of rectifications, it will be first necessary to
calculate _dy_, or _dy_ and _dz_, by differentiating the equation or the
equations of each curve proposed; after eliminating these expressions,
the general differential formula above enunciated will then contain only
_ds_ and _dx_; having arrived at this point, the elimination of the
infinitesimals can be completed only by the integral calculus.

Such is, then, the general office necessarily belonging to the
differential calculus in the complete solution of the questions which
exact the employment of the transcendental analysis; to produce, as far
as is possible, the elimination of the infinitesimals, that is, to
reduce in each case the primitive differential equations so that they
shall contain only the differentials of the really independent
variables, and those of the functions sought, by causing to disappear,
by elimination, the differentials of all the other known functions which
may have been taken as intermediaries at the time of the formation of
the differential equations of the problem which is under consideration.


2. _Employment of the Differential Calculus alone._ For certain
questions, which, although few in number, have none the less, as we
shall see hereafter, a very great importance, the magnitudes which are
sought enter directly, and not by their differentials, into the
primitive differential equations, which then contain differentially only
the different known functions employed as intermediaries, in accordance
with the preceding explanation. These cases are the most favourable of
all; for it is evident that the differential calculus is then entirely
sufficient for the complete elimination of the infinitesimals, without
the question giving rise to any integration. This is what occurs, for
example, in the problem of _tangents_ in geometry; in that of
_velocities_ in mechanics, &c.


3. _Employment of the Integral Calculus alone._ Finally, some other
questions, the number of which is also very small, but the importance of
which is no less great, present a second exceptional case, which is in
its nature exactly the converse of the preceding. They are those in
which the differential equations are found to be immediately ready for
integration, because they contain, at their first formation, only the
infinitesimals which relate to the functions sought, or to the really
independent variables, without its being necessary to introduce,
differentially, other functions as intermediaries. If in these new cases
we introduce these last functions, since, by hypothesis, they will enter
directly and not by their differentials, ordinary algebra will suffice
to eliminate them, and to bring the question to depend on only the
integral calculus. The differential calculus will then have no special
part in the complete solution of the problem, which will depend entirely
upon the integral calculus. The general question of _quadratures_ offers
an important example of this, for the differential equation being then
_dA = ydx_, will become immediately fit for integration as soon as we
shall have eliminated, by means of the equation of the proposed curve,
the intermediary function _y_, which does not enter into it
differentially. The same circumstances exist in the problem of
_cubatures_, and in some others equally important.


_Three classes of Questions hence resulting._ As a general result of the
previous considerations, it is then necessary to divide into three
classes the mathematical questions which require the use of the
transcendental analysis; the _first_ class comprises the problems
susceptible of being entirely resolved by means of the differential
calculus alone, without any need of the integral calculus; the _second_,
those which are, on the contrary, entirely dependent upon the integral
calculus, without the differential calculus having any part in their
solution; lastly, in the _third_ and the most extensive, which
constitutes the normal case, the two others being only exceptional, the
differential and the integral calculus have each in their turn a
distinct and necessary part in the complete solution of the problem, the
former making the primitive differential equations undergo a preparation
which is indispensable for the application of the latter. Such are
exactly their general relations, of which too indefinite and inexact
ideas are generally formed.

       *       *       *       *       *

Let us now take a general survey of the logical composition of each
calculus, beginning with the differential.




THE DIFFERENTIAL CALCULUS.


In the exposition of the transcendental analysis, it is customary to
intermingle with the purely analytical part (which reduces itself to the
treatment of the abstract principles of differentiation and integration)
the study of its different principal applications, especially those
which concern geometry. This confusion of ideas, which is a consequence
of the actual manner in which the science has been developed, presents,
in the dogmatic point of view, serious inconveniences in this respect,
that it makes it difficult properly to conceive either analysis or
geometry. Having to consider here the most rational co-ordination which
is possible, I shall include, in the following sketch, only the calculus
of indirect functions properly so called, reserving for the portion of
this volume which relates to the philosophical study of _concrete_
mathematics the general examination of its great geometrical and
mechanical applications.


_Two Cases: explicit and implicit Functions._ The fundamental division
of the differential calculus, or of the general subject of
differentiation, consists in distinguishing two cases, according as the
analytical functions which are to be differentiated are _explicit_ or
_implicit_; from which flow two parts ordinarily designated by the names
of differentiation _of formulas_ and differentiation _of equations_. It
is easy to understand, _à priori_, the importance of this
classification. In fact, such a distinction would be illusory if the
ordinary analysis was perfect; that is, if we knew how to resolve all
equations algebraically, for then it would be possible to render every
_implicit_ function _explicit_; and, by differentiating it in that
state alone, the second part of the differential calculus would be
immediately comprised in the first, without giving rise to any new
difficulty. But the algebraical resolution of equations being, as we
have seen, still almost in its infancy, and as yet impossible for most
cases, it is plain that the case is very different, since we have,
properly speaking, to differentiate a function without knowing it,
although it is determinate. The differentiation of implicit functions
constitutes then, by its nature, a question truly distinct from that
presented by explicit functions, and necessarily more complicated. It is
thus evident that we must commence with the differentiation of formulas,
and reduce the differentiation of equations to this primary case by
certain invariable analytical considerations, which need not be here
mentioned.

These two general cases of differentiation are also distinct in another
point of view equally necessary, and too important to be left unnoticed.
The relation which is obtained between the differentials is constantly
more indirect, in comparison with that of the finite quantities, in the
differentiation of implicit functions than in that of explicit
functions. We know, in fact, from the considerations presented by
Lagrange on the general formation of differential equations, that, on
the one hand, the same primitive equation may give rise to a greater or
less number of derived equations of very different forms, although at
bottom equivalent, depending upon which of the arbitrary constants is
eliminated, which is not the case in the differentiation of explicit
formulas; and that, on the other hand, the unlimited system of the
different primitive equations, which correspond to the same derived
equation, presents a much more profound analytical variety than that of
the different functions, which admit of one same explicit differential,
and which are distinguished from each other only by a constant term.
Implicit functions must therefore be regarded as being in reality still
more modified by differentiation than explicit functions. We shall again
meet with this consideration relatively to the integral calculus, where
it acquires a preponderant importance.


_Two Sub-cases: A single Variable or several Variables._ Each of the two
fundamental parts of the Differential Calculus is subdivided into two
very distinct theories, according as we are required to differentiate
functions of a single variable or functions of several independent
variables. This second case is, by its nature, quite distinct from the
first, and evidently presents more complication, even in considering
only explicit functions, and still more those which are implicit. As to
the rest, one of these cases is deduced from the other in a general
manner, by the aid of an invariable and very simple principle, which
consists in regarding the total differential of a function which is
produced by the simultaneous increments of the different independent
variables which it contains, as the sum of the partial differentials
which would be produced by the separate increment of each variable in
turn, if all the others were constant. It is necessary, besides,
carefully to remark, in connection with this subject, a new idea which
is introduced by the distinction of functions into those of one variable
and of several; it is the consideration of these different special
derived functions, relating to each variable separately, and the number
of which increases more and more in proportion as the order of the
derivation becomes higher, and also when the variables become more
numerous. It results from this that the differential relations belonging
to functions of several variables are, by their nature, both much more
indirect, and especially much more indeterminate, than those relating to
functions of a single variable. This is most apparent in the case of
implicit functions, in which, in the place of the simple arbitrary
constants which elimination causes to disappear when we form the proper
differential equations for functions of a single variable, it is the
arbitrary functions of the proposed variables which are then eliminated;
whence must result special difficulties when these equations come to be
integrated.

Finally, to complete this summary sketch of the different essential
parts of the differential calculus proper, I should add, that in the
differentiation of implicit functions, whether of a single variable or
of several, it is necessary to make another distinction; that of the
case in which it is required to differentiate at once different
functions of this kind, _combined_ in certain primitive equations, from
that in which all these functions are _separate_.

The functions are evidently, in fact, still more implicit in the first
case than in the second, if we consider that the same imperfection of
ordinary analysis, which forbids our converting every implicit function
into an equivalent explicit function, in like manner renders us unable
to separate the functions which enter simultaneously into any system of
equations. It is then necessary to differentiate, not only without
knowing how to resolve the primitive equations, but even without being
able to effect the proper eliminations among them, thus producing a new
difficulty.


_Reduction of the whole to the Differentiation of the ten elementary
Functions._ Such, then, are the natural connection and the logical
distribution of the different principal theories which compose the
general system of differentiation. Since the differentiation of implicit
functions is deduced from that of explicit functions by a single
constant principle, and the differentiation of functions of several
variables is reduced by another fixed principle to that of functions of
a single variable, the whole of the differential calculus is finally
found to rest upon the differentiation of explicit functions with a
single variable, the only one which is ever executed directly. Now it is
easy to understand that this first theory, the necessary basis of the
entire system, consists simply in the differentiation of the ten simple
functions, which are the uniform elements of all our analytical
combinations, and the list of which has been given in the first chapter,
on page 51; for the differentiation of compound functions is evidently
deduced, in an immediate and necessary manner, from that of the simple
functions which compose them. It is, then, to the knowledge of these ten
fundamental differentials, and to that of the two general principles
just mentioned, which bring under it all the other possible cases, that
the whole system of differentiation is properly reduced. We see, by the
combination of these different considerations, how simple and how
perfect is the entire system of the differential calculus. It certainly
constitutes, in its logical relations, the most interesting spectacle
which mathematical analysis can present to our understanding.


_Transformation of derived Functions for new Variables._ The general
sketch which I have just summarily drawn would nevertheless present an
important deficiency, if I did not here distinctly indicate a final
theory, which forms, by its nature, the indispensable complement of the
system of differentiation. It is that which has for its object the
constant transformation of derived functions, as a result of determinate
changes in the independent variables, whence results the possibility of
referring to new variables all the general differential formulas
primitively established for others. This question is now resolved in the
most complete and the most simple manner, as are all those of which the
differential calculus is composed. It is easy to conceive the general
importance which it must have in any of the applications of the
transcendental analysis, the fundamental resources of which it may be
considered as augmenting, by permitting us to choose (in order to form
the differential equations, in the first place, with more ease) that
system of independent variables which may appear to be the most
advantageous, although it is not to be finally retained. It is thus, for
example, that most of the principal questions of geometry are resolved
much more easily by referring the lines and surfaces to _rectilinear_
co-ordinates, and that we may, nevertheless, have occasion to express
these lines, etc., analytically by the aid of _polar_ co-ordinates, or
in any other manner. We will then be able to commence the differential
solution of the problem by employing the rectilinear system, but only as
an intermediate step, from which, by the general theory here referred
to, we can pass to the final system, which sometimes could not have been
considered directly.


_Different Orders of Differentiation._ In the logical classification of
the differential calculus which has just been given, some may be
inclined to suggest a serious omission, since I have not subdivided each
of its four essential parts according to another general consideration,
which seems at first view very important; namely, that of the higher or
lower order of differentiation. But it is easy to understand that this
distinction has no real influence in the differential calculus, inasmuch
as it does not give rise to any new difficulty. If, indeed, the
differential calculus was not rigorously complete, that is, if we did
not know how to differentiate at will any function whatever, the
differentiation to the second or higher order of each determinate
function might engender special difficulties. But the perfect
universality of the differential calculus plainly gives us the assurance
of being able to differentiate, to any order whatever, all known
functions whatever, the question reducing itself to a constantly
repeated differentiation of the first order. This distinction,
unimportant as it is for the differential calculus, acquires, however, a
very great importance in the integral calculus, on account of the
extreme imperfection of the latter.


_Analytical Applications._ Finally, though this is not the place to
consider the various applications of the differential calculus, yet an
exception may be made for those which consist in the solution of
questions which are purely analytical, which ought, indeed, to be
logically treated in continuation of a system of differentiation,
because of the evident homogeneity of the considerations involved. These
questions may be reduced to three essential ones.

Firstly, the _development into series_ of functions of one or more
variables, or, more generally, the transformation of functions, which
constitutes the most beautiful and the most important application of the
differential calculus to general analysis, and which comprises, besides
the fundamental series discovered by Taylor, the remarkable series
discovered by Maclaurin, John Bernouilli, Lagrange, &c.:

Secondly, the general _theory of maxima and minima_ values for any
functions whatever, of one or more variables; one of the most
interesting problems which analysis can present, however elementary it
may now have become, and to the complete solution of which the
differential calculus naturally applies:

Thirdly, the general determination of the true value of functions which
present themselves under an _indeterminate_ appearance for certain
hypotheses made on the values of the corresponding variables; which is
the least extensive and the least important of the three.

The first question is certainly the principal one in all points of view;
it is also the most susceptible of receiving a new extension hereafter,
especially by conceiving, in a broader manner than has yet been done,
the employment of the differential calculus in the transformation of
functions, on which subject Lagrange has left some valuable hints.

       *       *       *       *       *

Having thus summarily, though perhaps too briefly, considered the chief
points in the differential calculus, I now proceed to an equally rapid
exposition of a systematic outline of the Integral Calculus, properly so
called, that is, the abstract subject of integration.




THE INTEGRAL CALCULUS.


_Its Fundamental Division._ The fundamental division of the Integral
Calculus is founded on the same principle as that of the Differential
Calculus, in distinguishing the integration of _explicit_ differential
formulas, and the integration of _implicit_ differentials or of
differential equations. The separation of these two cases is even much
more profound in relation to integration than to differentiation. In the
differential calculus, in fact, this distinction rests, as we have seen,
only on the extreme imperfection of ordinary analysis. But, on the other
hand, it is easy to see that, even though all equations could be
algebraically resolved, differential equations would none the less
constitute a case of integration quite distinct from that presented by
the explicit differential formulas; for, limiting ourselves, for the
sake of simplicity, to the first order, and to a single function _y_ of
a single variable _x_, if we suppose any differential equation between
_x_, _y_, and _dy/dx_, to be resolved with reference to _dy/dx_, the
expression of the derived function being then generally found to contain
the primitive function itself, which is the object of the inquiry, the
question of integration will not have at all changed its nature, and the
solution will not really have made any other progress than that of
having brought the proposed differential equation to be of only the
first degree relatively to the derived function, which is in itself of
little importance. The differential would not then be determined in a
manner much less _implicit_ than before, as regards the integration,
which would continue to present essentially the same characteristic
difficulty. The algebraic resolution of equations could not make the
case which we are considering come within the simple integration of
explicit differentials, except in the special cases in which the
proposed differential equation did not contain the primitive function
itself, which would consequently permit us, by resolving it, to find
_dy/dx_ in terms of _x_ only, and thus to reduce the question to the
class of quadratures. Still greater difficulties would evidently be
found in differential equations of higher orders, or containing
simultaneously different functions of several independent variables.

The integration of differential equations is then necessarily more
complicated than that of explicit differentials, by the elaboration of
which last the integral calculus has been created, and upon which the
others have been made to depend as far as it has been possible. All the
various analytical methods which have been proposed for integrating
differential equations, whether it be the separation of the variables,
the method of multipliers, &c., have in fact for their object to reduce
these integrations to those of differential formulas, the only one
which, by its nature, can be undertaken directly. Unfortunately,
imperfect as is still this necessary base of the whole integral
calculus, the art of reducing to it the integration of differential
equations is still less advanced.


_Subdivisions: one variable or several._ Each of these two fundamental
branches of the integral calculus is next subdivided into two others (as
in the differential calculus, and for precisely analogous reasons),
according as we consider functions with a _single variable_, or
functions with _several independent variables_.

This distinction is, like the preceding one, still more important for
integration than for differentiation. This is especially remarkable in
reference to differential equations. Indeed, those which depend on
several independent variables may evidently present this characteristic
and much more serious difficulty, that the desired function may be
differentially defined by a simple relation between its different
special derivatives relative to the different variables taken
separately. Hence results the most difficult and also the most extensive
branch of the integral calculus, which is commonly named the _Integral
Calculus of partial differences_, created by D'Alembert, and in which,
according to the just appreciation of Lagrange, geometers ought to have
seen a really new calculus, the philosophical character of which has not
yet been determined with sufficient exactness. A very striking
difference between this case and that of equations with a single
independent variable consists, as has been already observed, in the
arbitrary functions which take the place of the simple arbitrary
constants, in order to give to the corresponding integrals all the
proper generality.

It is scarcely necessary to say that this higher branch of
transcendental analysis is still entirely in its infancy, since, even in
the most simple case, that of an equation of the first order between the
partial derivatives of a single function with two independent variables,
we are not yet completely able to reduce the integration to that of the
ordinary differential equations. The integration of functions of several
variables is much farther advanced in the case (infinitely more simple
indeed) in which it has to do with only explicit differential formulas.
We can then, in fact, when these formulas fulfil the necessary
conditions of integrability, always reduce their integration to
quadratures.


_Other Subdivisions: different Orders of Differentiation._ A new general
distinction, applicable as a subdivision to the integration of explicit
or implicit differentials, with one variable or several, is drawn from
the _higher or lower order of the differentials_: a distinction which,
as we have above remarked, does not give rise to any special question in
the differential calculus.

Relatively to _explicit differentials_, whether of one variable or of
several, the necessity of distinguishing their different orders belongs
only to the extreme imperfection of the integral calculus. In fact, if
we could always integrate every differential formula of the first order,
the integration of a formula of the second order, or of any other, would
evidently not form a new question, since, by integrating it at first in
the first degree, we would arrive at the differential expression of the
immediately preceding order, from which, by a suitable series of
analogous integrations, we would be certain of finally arriving at the
primitive function, the final object of these operations. But the little
knowledge which we possess on integration of even the first order causes
quite another state of affairs, so that a higher order of differentials
produces new difficulties; for, having differential formulas of any
order above the first, it may happen that we may be able to integrate
them, either once, or several times in succession, and that we may still
be unable to go back to the primitive functions, if these preliminary
labours have produced, for the differentials of a lower order,
expressions whose integrals are not known. This circumstance must occur
so much the oftener (the number of known integrals being still very
small), seeing that these successive integrals are generally very
different functions from the derivatives which have produced them.

With reference to _implicit differentials_, the distinction of orders is
still more important; for, besides the preceding reason, the influence
of which is evidently analogous in this case, and is even greater, it is
easy to perceive that the higher order of the differential equations
necessarily gives rise to questions of a new nature. In fact, even if we
could integrate every equation of the first order relating to a single
function, that would not be sufficient for obtaining the final integral
of an equation of any order whatever, inasmuch as every differential
equation is not reducible to that of an immediately inferior order.
Thus, for example, if we have given any relation between _x_, _y_,
_dx/dy_, and _d_²_y_/_dx_², to determine a function _y_ of a variable
_x_, we shall not be able to deduce from it at once, after effecting a
first integration, the corresponding differential relation between _x_,
_y_, and _dy/dx_, from which, by a second integration, we could ascend
to the primitive equations. This would not necessarily take place, at
least without introducing new auxiliary functions, unless the proposed
equation of the second order did not contain the required function _y_,
together with its derivatives. As a general principle, differential
equations will have to be regarded as presenting cases which are more
and more _implicit_, as they are of a higher order, and which cannot be
made to depend on one another except by special methods, the
investigation of which consequently forms a new class of questions, with
respect to which we as yet know scarcely any thing, even for functions
of a single variable.[10]

  [Footnote 10: The only important case of this class which has thus
  far been completely treated is the general integration of _linear_
  equations of any order whatever, with constant coefficients. Even
  this case finally depends on the algebraic resolution of equations
  of a degree equal to the order of differentiation.]

_Another equivalent distinction._ Still farther, when we examine more
profoundly this distinction of different orders of differential
equations, we find that it can be always made to come under a final
general distinction, relative to differential equations, which remains
to be noticed. Differential equations with one or more independent
variables may contain simply a single function, or (in a case evidently
more complicated and more implicit, which corresponds to the
differentiation of simultaneous implicit functions) we may have to
determine at the same time several functions from the differential
equations in which they are found united, together with their different
derivatives. It is clear that such a state of the question necessarily
presents a new special difficulty, that of separating the different
functions desired, by forming for each, from the proposed differential
equations, an isolated differential equation which does not contain the
other functions or their derivatives. This preliminary labour, which is
analogous to the elimination of algebra, is evidently indispensable
before attempting any direct integration, since we cannot undertake
generally (except by special artifices which are very rarely applicable)
to determine directly several distinct functions at once.

Now it is easy to establish the exact and necessary coincidence of this
new distinction with the preceding one respecting the order of
differential equations. We know, in fact, that the general method for
isolating functions in simultaneous differential equations consists
essentially in forming differential equations, separately in relation to
each function, and of an order equal to the sum of all those of the
different proposed equations. This transformation can always be
effected. On the other hand, every differential equation of any order in
relation to a single function might evidently always be reduced to the
first order, by introducing a suitable number of auxiliary differential
equations, containing at the same time the different anterior
derivatives regarded as new functions to be determined. This method has,
indeed, sometimes been actually employed with success, though it is not
the natural one.

Here, then, are two necessarily equivalent orders of conditions in the
general theory of differential equations; the simultaneousness of a
greater or smaller number of functions, and the higher or lower order of
differentiation of a single function. By augmenting the order of the
differential equations, we can isolate all the functions; and, by
artificially multiplying the number of the functions, we can reduce all
the equations to the first order. There is, consequently, in both cases,
only one and the same difficulty from two different points of sight.
But, however we may conceive it, this new difficulty is none the less
real, and constitutes none the less, by its nature, a marked separation
between the integration of equations of the first order and that of
equations of a higher order. I prefer to indicate the distinction under
this last form as being more simple, more general, and more logical.


_Quadratures._ From the different considerations which have been
indicated respecting the logical dependence of the various principal
parts of the integral calculus, we see that the integration of explicit
differential formulas of the first order and of a single variable is the
necessary basis of all other integrations, which we never succeed in
effecting but so far as we reduce them to this elementary case,
evidently the only one which, by its nature, is capable of being treated
directly. This simple fundamental integration is often designated by the
convenient expression of _quadratures_, seeing that every integral of
this kind, S_f_(_x_)_dx_, may, in fact, be regarded as representing the
area of a curve, the equation of which in rectilinear co-ordinates would
be _y_ = _f_(_x_). Such a class of questions corresponds, in the
differential calculus, to the elementary case of the differentiation of
explicit functions of a single variable. But the integral question is,
by its nature, very differently complicated, and especially much more
extensive than the differential question. This latter is, in fact,
necessarily reduced, as we have seen, to the differentiation of the ten
simple functions, the elements of all which are considered in analysis.
On the other hand, the integration of compound functions does not
necessarily follow from that of the simple functions, each combination
of which may present special difficulties with respect to the integral
calculus. Hence results the naturally indefinite extent, and the so
varied complication of the question of _quadratures_, upon which, in
spite of all the efforts of analysts, we still possess so little
complete knowledge.

In decomposing this question, as is natural, according to the different
forms which may be assumed by the derivative function, we distinguish
the case of _algebraic_ functions and that of _transcendental_
functions.

_Integration of Transcendental Functions._ The truly analytical
integration of transcendental functions is as yet very little advanced,
whether for _exponential_, or for _logarithmic_, or for _circular_
functions. But a very small number of cases of these three different
kinds have as yet been treated, and those chosen from among the
simplest; and still the necessary calculations are in most cases
extremely laborious. A circumstance which we ought particularly to
remark in its philosophical connection is, that the different procedures
of quadrature have no relation to any general view of integration, and
consist of simple artifices very incoherent with each other, and very
numerous, because of the very limited extent of each.

One of these artifices should, however, here be noticed, which, without
being really a method of integration, is nevertheless remarkable for its
generality; it is the procedure invented by John Bernouilli, and known
under the name of _integration by parts_, by means of which every
integral may be reduced to another which is sometimes found to be more
easy to be obtained. This ingenious relation deserves to be noticed for
another reason, as having suggested the first idea of that
transformation of integrals yet unknown, which has lately received a
greater extension, and of which M. Fourier especially has made so new
and important a use in the analytical questions produced by the theory
of heat.

_Integration of Algebraic Functions._ As to the integration of algebraic
functions, it is farther advanced. However, we know scarcely any thing
in relation to irrational functions, the integrals of which have been
obtained only in extremely limited cases, and particularly by rendering
them rational. The integration of rational functions is thus far the
only theory of the integral calculus which has admitted of being treated
in a truly complete manner; in a logical point of view, it forms, then,
its most satisfactory part, but perhaps also the least important. It is
even essential to remark, in order to have a just idea of the extreme
imperfection of the integral calculus, that this case, limited as it is,
is not entirely resolved except for what properly concerns integration
viewed in an abstract manner; for, in the execution, the theory finds
its progress most frequently quite stopped, independently of the
complication of the calculations, by the imperfection of ordinary
analysis, seeing that it makes the integration finally depend upon the
algebraic resolution of equations, which greatly limits its use.

To grasp in a general manner the spirit of the different procedures
which are employed in quadratures, we must observe that, by their
nature, they can be primitively founded only on the differentiation of
the ten simple functions. The results of this, conversely considered,
establish as many direct theorems of the integral calculus, the only
ones which can be directly known. All the art of integration afterwards
consists, as has been said in the beginning of this chapter, in reducing
all the other quadratures, so far as is possible, to this small number
of elementary ones, which unhappily we are in most cases unable to
effect.

_Singular Solutions._ In this systematic enumeration of the various
essential parts of the integral calculus, considered in their logical
relations, I have designedly neglected (in order not to break the chain
of sequence) to consider a very important theory, which forms implicitly
a portion of the general theory of the integration of differential
equations, but which I ought here to notice separately, as being, so to
speak, outside of the integral calculus, and being nevertheless of the
greatest interest, both by its logical perfection and by the extent of
its applications. I refer to what are called _Singular Solutions_ of
differential equations, called sometimes, but improperly, _particular_
solutions, which have been the subject of very remarkable investigations
by Euler and Laplace, and of which Lagrange especially has presented
such a beautiful and simple general theory. Clairaut, who first had
occasion to remark their existence, saw in them a paradox of the
integral calculus, since these solutions have the peculiarity of
satisfying the differential equations without being comprised in the
corresponding general integrals. Lagrange has since explained this
paradox in the most ingenious and most satisfactory manner, by showing
how such solutions are always derived from the general integral by the
variation of the arbitrary constants. He was also the first to suitably
appreciate the importance of this theory, and it is with good reason
that he devoted to it so full a development in his "Calculus of
Functions." In a logical point of view, this theory deserves all our
attention by the character of perfect generality which it admits of,
since Lagrange has given invariable and very simple procedures for
finding the _singular_ solution of any differential equation which is
susceptible of it; and, what is no less remarkable, these procedures
require no integration, consisting only of differentiations, and are
therefore always applicable. Differentiation has thus become, by a
happy artifice, a means of compensating, in certain circumstances, for
the imperfection of the integral calculus. Indeed, certain problems
especially require, by their nature, the knowledge of these _singular_
solutions; such, for example, in geometry, are all the questions in
which a curve is to be determined from any property of its tangent or
its osculating circle. In all cases of this kind, after having expressed
this property by a differential equation, it will be, in its analytical
relations, the _singular_ equation which will form the most important
object of the inquiry, since it alone will represent the required curve;
the general integral, which thenceforth it becomes unnecessary to know,
designating only the system of the tangents, or of the osculating
circles of this curve. We may hence easily understand all the importance
of this theory, which seems to me to be not as yet sufficiently
appreciated by most geometers.

_Definite Integrals._ Finally, to complete our review of the vast
collection of analytical researches of which is composed the integral
calculus, properly so called, there remains to be mentioned one theory,
very important in all the applications of the transcendental analysis,
which I have had to leave outside of the system, as not being really
destined for veritable integration, and proposing, on the contrary, to
supply the place of the knowledge of truly analytical integrals, which
are most generally unknown. I refer to the determination of _definite
integrals_.

The expression, always possible, of integrals in infinite series, may at
first be viewed as a happy general means of compensating for the extreme
imperfection of the integral calculus. But the employment of such
series, because of their complication, and of the difficulty of
discovering the law of their terms, is commonly of only moderate utility
in the algebraic point of view, although sometimes very essential
relations have been thence deduced. It is particularly in the
arithmetical point of view that this procedure acquires a great
importance, as a means of calculating what are called _definite
integrals_, that is, the values of the required functions for certain
determinate values of the corresponding variables.

An inquiry of this nature exactly corresponds, in transcendental
analysis, to the numerical resolution of equations in ordinary analysis.
Being generally unable to obtain the veritable integral--named by
opposition the _general_ or _indefinite_ integral; that is, the function
which, differentiated, has produced the proposed differential
formula--analysts have been obliged to employ themselves in determining
at least, without knowing this function, the particular numerical values
which it would take on assigning certain designated values to the
variables. This is evidently resolving the arithmetical question without
having previously resolved the corresponding algebraic one, which most
generally is the most important one. Such an analysis is, then, by its
nature, as imperfect as we have seen the numerical resolution of
equations to be. It presents, like this last, a vicious confusion of
arithmetical and algebraic considerations, whence result analogous
inconveniences both in the purely logical point of view and in the
applications. We need not here repeat the considerations suggested in
our third chapter. But it will be understood that, unable as we almost
always are to obtain the true integrals, it is of the highest importance
to have been able to obtain this solution, incomplete and necessarily
insufficient as it is. Now this has been fortunately attained at the
present day for all cases, the determination of the value of definite
integrals having been reduced to entirely general methods, which leave
nothing to desire, in a great number of cases, but less complication in
the calculations, an object towards which are at present directed all
the special transformations of analysts. Regarding now this sort of
_transcendental arithmetic_ as perfect, the difficulty in the
applications is essentially reduced to making the proposed research
depend, finally, on a simple determination of definite integrals, which
evidently cannot always be possible, whatever analytical skill may be
employed in effecting such a transformation.


_Prospects of the Integral Calculus._ From the considerations indicated
in this chapter, we see that, while the differential calculus
constitutes by its nature a limited and perfect system, to which nothing
essential remains to be added, the integral calculus, or the simple
system of integration, presents necessarily an inexhaustible field for
the activity of the human mind, independently of the indefinite
applications of which the transcendental analysis is evidently
susceptible. The general argument by which I have endeavoured, in the
second chapter, to make apparent the impossibility of ever discovering
the algebraic solution of equations of any degree and form whatsoever,
has undoubtedly infinitely more force with regard to the search for a
single method of integration, invariably applicable to all cases. "It
is," says Lagrange, "one of those problems whose general solution we
cannot hope for." The more we meditate on this subject, the more we
will be convinced that such a research is utterly chimerical, as being
far above the feeble reach of our intelligence; although the labours of
geometers must certainly augment hereafter the amount of our knowledge
respecting integration, and thus create methods of greater generality.
The transcendental analysis is still too near its origin--there is
especially too little time since it has been conceived in a truly
rational manner--for us now to be able to have a correct idea of what it
will hereafter become. But, whatever should be our legitimate hopes, let
us not forget to consider, before all, the limits which are imposed by
our intellectual constitution, and which, though not susceptible of a
precise determination, have none the less an incontestable reality.

I am induced to think that, when geometers shall have exhausted the most
important applications of our present transcendental analysis, instead
of striving to impress upon it, as now conceived, a chimerical
perfection, they will rather create new resources by changing the mode
of derivation of the auxiliary quantities introduced in order to
facilitate the establishment of equations, and the formation of which
might follow an infinity of other laws besides the very simple relation
which has been chosen, according to the conception suggested in the
first chapter. The resources of this nature appear to me susceptible of
a much greater fecundity than those which would consist of merely
pushing farther our present calculus of indirect functions. It is a
suggestion which I submit to the geometers who have turned their
thoughts towards the general philosophy of analysis.

Finally, although, in the summary exposition which was the object of
this chapter, I have had to exhibit the condition of extreme
imperfection which still belongs to the integral calculus, the student
would have a false idea of the general resources of the transcendental
analysis if he gave that consideration too great an importance. It is
with it, indeed, as with ordinary analysis, in which a very small amount
of fundamental knowledge respecting the resolution of equations has been
employed with an immense degree of utility. Little advanced as geometers
really are as yet in the science of integrations, they have nevertheless
obtained, from their scanty abstract conceptions, the solution of a
multitude of questions of the first importance in geometry, in
mechanics, in thermology, &c. The philosophical explanation of this
double general fact results from the necessarily preponderating
importance and grasp of _abstract_ branches of knowledge, the least of
which is naturally found to correspond to a crowd of _concrete_
researches, man having no other resource for the successive extension of
his intellectual means than in the consideration of ideas more and more
abstract, and still positive.

       *       *       *       *       *

In order to finish the complete exposition of the philosophical
character of the transcendental analysis, there remains to be considered
a final conception, by which the immortal Lagrange has rendered this
analysis still better adapted to facilitate the establishment of
equations in the most difficult problems, by considering a class of
equations still more _indirect_ than the ordinary differential
equations. It is the _Calculus_, or, rather, the _Method of Variations_;
the general appreciation of which will be our next subject.




CHAPTER V.

THE CALCULUS OF VARIATIONS.


In order to grasp with more ease the philosophical character of the
_Method of Variations_, it will be well to begin by considering in a
summary manner the special nature of the problems, the general
resolution of which has rendered necessary the formation of this
hyper-transcendental analysis. It is still too near its origin, and its
applications have been too few, to allow us to obtain a sufficiently
clear general idea of it from a purely abstract exposition of its
fundamental theory.




PROBLEMS GIVING RISE TO IT.


The mathematical questions which have given birth to the _Calculus of
Variations_ consist generally in the investigation of the _maxima_ and
_minima_ of certain indeterminate integral formulas, which express the
analytical law of such or such a phenomenon of geometry or mechanics,
considered independently of any particular subject. Geometers for a long
time designated all the questions of this character by the common name
of _Isoperimetrical Problems_, which, however, is really suitable to
only the smallest number of them.


_Ordinary Questions of Maxima and Minima._ In the common theory of
_maxima_ and _minima_, it is proposed to discover, with reference to a
given function of one or more variables, what particular values must be
assigned to these variables, in order that the corresponding value of
the proposed function may be a _maximum_ or a _minimum_ with respect to
those values which immediately precede and follow it; that is, properly
speaking, we seek to know at what instant the function ceases to
increase and commences to decrease, or reciprocally. The differential
calculus is perfectly sufficient, as we know, for the general resolution
of this class of questions, by showing that the values of the different
variables, which suit either the maximum or minimum, must always reduce
to zero the different first derivatives of the given function, taken
separately with reference to each independent variable, and by
indicating, moreover, a suitable characteristic for distinguishing the
maximum from the minimum; consisting, in the case of a function of a
single variable, for example, in the derived function of the second
order taking a negative value for the maximum, and a positive value for
the minimum. Such are the well-known fundamental conditions belonging to
the greatest number of cases.


_A new Class of Questions._ The construction of this general theory
having necessarily destroyed the chief interest which questions of this
kind had for geometers, they almost immediately rose to the
consideration of a new order of problems, at once much more important
and of much greater difficulty--those of _isoperimeters_. It is, then,
no longer _the values of the variables_ belonging to the maximum or the
minimum of a given function that it is required to determine. It is _the
form of the function itself_ which is required to be discovered, from
the condition of the maximum or of the minimum of a certain definite
integral, merely indicated, which depends upon that function.


_Solid of least Resistance._ The oldest question of this nature is that
of _the solid of least resistance_, treated by Newton in the second book
of the Principia, in which he determines what ought to be the meridian
curve of a solid of revolution, in order that the resistance experienced
by that body in the direction of its axis may be the least possible. But
the course pursued by Newton, from the nature of his special method of
transcendental analysis, had not a character sufficiently simple,
sufficiently general, and especially sufficiently analytical, to attract
geometers to this new order of problems. To effect this, the application
of the infinitesimal method was needed; and this was done, in 1695, by
John Bernouilli, in proposing the celebrated problem of the
_Brachystochrone_.

This problem, which afterwards suggested such a long series of analogous
questions, consists in determining the curve which a heavy body must
follow in order to descend from one point to another in the shortest
possible time. Limiting the conditions to the simple fall in a vacuum,
the only case which was at first considered, it is easily found that the
required curve must be a reversed cycloid with a horizontal base, and
with its origin at the highest point. But the question may become
singularly complicated, either by taking into account the resistance of
the medium, or the change in the intensity of gravity.


_Isoperimeters._ Although this new class of problems was in the first
place furnished by mechanics, it is in geometry that the principal
investigations of this character were subsequently made. Thus it was
proposed to discover which, among all the curves of the same contour
traced between two given points, is that whose area is a maximum or
minimum, whence has come the name of _Problem of Isoperimeters_; or it
was required that the maximum or minimum should belong to the surface
produced by the revolution of the required curve about an axis, or to
the corresponding volume; in other cases, it was the vertical height of
the center of gravity of the unknown curve, or of the surface and of the
volume which it might generate, which was to become a maximum or
minimum, &c. Finally, these problems were varied and complicated almost
to infinity by the Bernouillis, by Taylor, and especially by Euler,
before Lagrange reduced their solution to an abstract and entirely
general method, the discovery of which has put a stop to the enthusiasm
of geometers for such an order of inquiries. This is not the place for
tracing the history of this subject. I have only enumerated some of the
simplest principal questions, in order to render apparent the original
general object of the method of variations.


_Analytical Nature of these Problems._ We see that all these problems,
considered in an analytical point of view, consist, by their nature, in
determining what form a certain unknown function of one or more
variables ought to have, in order that such or such an integral,
dependent upon that function, shall have, within assigned limits, a
value which is a maximum or a minimum with respect to all those which it
would take if the required function had any other form whatever.

Thus, for example, in the problem of the _brachystochrone_, it is well
known that if _y_ = _f(z)_, _x_ = π(_z_), are the rectilinear equations
of the required curve, supposing the axes of _x_ and of _y_ to be
horizontal, and the axis of _z_ to be vertical, the time of the fall of
a heavy body in that curve from the point whose ordinate is _z₁_, to
that whose ordinate is _z₂_, is expressed in general terms by the
definite integral

  ∫_{_z₂_}^{_z₁_}√(1 + (_f'(z))²_ + (π'(_z_))²/(2_gz_))_dz._

It is, then, necessary to find what the two unknown functions _f_ and π
must be, in order that this integral may be a minimum.

In the same way, to demand what is the curve among all plane
isoperimetrical curves, which includes the greatest area, is the same
thing as to propose to find, among all the functions _f(x)_ which can
give a certain constant value to the integral

  ∫_dx_√(1 + (_f'(x)_ )²),

that one which renders the integral ∫_f(x)dx_, taken between the same
limits, a maximum. It is evidently always so in other questions of this
class.


_Methods of the older Geometers._ In the solutions which geometers
before Lagrange gave of these problems, they proposed, in substance, to
reduce them to the ordinary theory of maxima and minima. But the means
employed to effect this transformation consisted in special simple
artifices peculiar to each case, and the discovery of which did not
admit of invariable and certain rules, so that every really new question
constantly reproduced analogous difficulties, without the solutions
previously obtained being really of any essential aid, otherwise than by
their discipline and training of the mind. In a word, this branch of
mathematics presented, then, the necessary imperfection which always
exists when the part common to all questions of the same class has not
yet been distinctly grasped in order to be treated in an abstract and
thenceforth general manner.




METHOD OF LAGRANGE.


Lagrange, in endeavouring to bring all the different problems of
isoperimeters to depend upon a common analysis, organized into a
distinct calculus, was led to conceive a new kind of differentiation, to
which he has applied the characteristic δ, reserving the characteristic
_d_ for the common differentials. These differentials of a new species,
which he has designated under the name of _Variations_, consist of the
infinitely small increments which the integrals receive, not by virtue
of analogous increments on the part of the corresponding variables, as
in the ordinary transcendental analysis, but by supposing that the
_form_ of the function placed under the sign of integration undergoes an
infinitely small change. This distinction is easily conceived with
reference to curves, in which we see the ordinate, or any other variable
of the curve, admit of two sorts of differentials, evidently very
different, according as we pass from one point to another infinitely
near it on the same curve, or to the corresponding point of the
infinitely near curve produced by a certain determinate modification of
the first curve.[11] It is moreover clear, that the relative
_variations_ of different magnitudes connected with each other by any
laws whatever are calculated, all but the characteristic, almost exactly
in the same manner as the differentials. Finally, from the general
notion of _variations_ are in like manner deduced the fundamental
principles of the algorithm proper to this method, consisting simply in
the evidently permissible liberty of transposing at will the
characteristics specially appropriated to variations, before or after
those which correspond to the ordinary differentials.

  [Footnote 11: Leibnitz had already considered the comparison of one
  curve with an other infinitely near to it, calling it
  "_Differentiatio de curva in curvam_." But this comparison had no
  analogy with the conception of Lagrange, the curves of Leibnitz
  being embraced in the same general equation, from which they were
  deduced by the simple change of an arbitrary constant.]

This abstract conception having been once formed, Lagrange was able to
reduce with ease, and in the most general manner, all the problems of
_Isoperimeters_ to the simple ordinary theory of _maxima_ and _minima_.
To obtain a clear idea of this great and happy transformation, we must
previously consider an essential distinction which arises in the
different questions of isoperimeters.


_Two Classes of Questions._ These investigations must, in fact, be
divided into two general classes, according as the maxima and minima
demanded are _absolute_ or _relative_, to employ the abridged
expressions of geometers.


_Questions of the first Class._ The _first case_ is that in which the
indeterminate definite integrals, the maximum or minimum of which is
sought, are not subjected, by the nature of the problem, to any
condition; as happens, for example, in the problem of the
_brachystochrone_, in which the choice is to be made between all
imaginable curves. The _second_ case takes place when, on the contrary,
the variable integrals can vary only according to certain conditions,
which usually consist in other definite integrals (which depend, in like
manner, upon the required functions) always retaining the same given
value; as, for example, in all the geometrical questions relating to
real _isoperimetrical_ figures, and in which, by the nature of the
problem, the integral relating to the length of the curve, or to the
area of the surface, must remain constant during the variation of that
integral which is the object of the proposed investigation.

The _Calculus of Variations_ gives immediately the general solution of
questions of the former class; for it evidently follows, from the
ordinary theory of maxima and minima, that the required relation must
reduce to zero the _variation_ of the proposed integral with reference
to each independent variable; which gives the condition common to both
the maximum and the minimum: and, as a characteristic for distinguishing
the one from the other, that the variation of the second order of the
same integral must be negative for the maximum and positive for the
minimum. Thus, for example, in the problem of the brachystochrone, we
will have, in order to determine the nature of the curve sought, the
equation of condition

  δ∫_{_z₂_}^{_z₁_}√([1 + (_f'(z)_)² + (π'(_z_))²]/(2_gz_))_dz_ = 0,

which, being decomposed into two, with respect to the two unknown
functions _f_ and π, which are independent of each other, will
completely express the analytical definition of the required curve. The
only difficulty peculiar to this new analysis consists in the
elimination of the characteristic δ, for which the calculus of
variations furnishes invariable and complete rules, founded, in general,
on the method of "integration by parts," from which Lagrange has thus
derived immense advantage. The constant object of this first analytical
elaboration (which this is not the place for treating in detail) is to
arrive at real differential equations, which can always be done; and
thereby the question comes under the ordinary transcendental analysis,
which furnishes the solution, at least so far as to reduce it to pure
algebra if the integration can be effected. The general object of the
method of variations is to effect this transformation, for which
Lagrange has established rules, which are simple, invariable, and
certain of success.


_Equations of Limits._ Among the greatest special advantages of the
method of variations, compared with the previous isolated solutions of
isoperimetrical problems, is the important consideration of what
Lagrange calls _Equations of Limits_, which were entirely neglected
before him, though without them the greater part of the particular
solutions remained necessarily incomplete. When the limits of the
proposed integrals are to be fixed, their variations being zero, there
is no occasion for noticing them. But it is no longer so when these
limits, instead of being rigorously invariable, are only subjected to
certain conditions; as, for example, if the two points between which the
required curve is to be traced are not fixed, and have only to remain
upon given lines or surfaces. Then it is necessary to pay attention to
the variation of their co-ordinates, and to establish between them the
relations which correspond to the equations of these lines or of these
surfaces.


_A more general consideration._ This essential consideration is only the
final complement of a more general and more important consideration
relative to the variations of different independent variables. If these
variables are really independent of one another, as when we compare
together all the imaginable curves susceptible of being traced between
two points, it will be the same with their variations, and,
consequently, the terms relating to each of these variations will have
to be separately equal to zero in the general equation which expresses
the maximum or the minimum. But if, on the contrary, we suppose the
variables to be subjected to any fixed conditions, it will be necessary
to take notice of the resulting relation between their variations, so
that the number of the equations into which this general equation is
then decomposed is always equal to only the number of the variables
which remain truly independent. It is thus, for example, that instead of
seeking for the shortest path between any two points, in choosing it
from among all possible ones, it may be proposed to find only what is
the shortest among all those which may be taken on any given surface; a
question the general solution of which forms certainly one of the most
beautiful applications of the method of variations.

_Questions of the second Class._ Problems in which such modifying
conditions are considered approach very nearly, in their nature, to the
second general class of applications of the method of variations,
characterized above as consisting in the investigation of _relative_
maxima and minima. There is, however, this essential difference between
the two cases, that in this last the modification is expressed by an
integral which depends upon the function sought, while in the other it
is designated by a finite equation which is immediately given. It is
hence apparent that the investigation of _relative_ maxima and minima is
constantly and necessarily more complicated than that of _absolute_
maxima and minima. Luckily, a very important general theory, discovered
by the genius of the great Euler before the invention of the Calculus of
Variations, gives a uniform and very simple means of making one of
these two classes of questions dependent on the other. It consists in
this, that if we add to the integral which is to be a maximum or a
minimum, a constant and indeterminate multiple of that one which, by the
nature of the problem, is to remain constant, it will be sufficient to
seek, by the general method of Lagrange above indicated, the _absolute_
maximum or minimum of this whole expression. It can be easily conceived,
indeed, that the part of the complete variation which would proceed from
the last integral must be equal to zero (because of the constant
character of this last) as well as the portion due to the first
integral, which disappears by virtue of the maximum or minimum state.
These two conditions evidently unite to produce, in that respect,
effects exactly alike.

Such is a sketch of the general manner in which the method of variation
is applied to all the different questions which compose what is called
the _Theory of Isoperimeters_. It will undoubtedly have been remarked in
this summary exposition how much use has been made in this new analysis
of the second fundamental property of the transcendental analysis
noticed in the third chapter, namely, the generality of the
infinitesimal expressions for the representation of the same geometrical
or mechanical phenomenon, in whatever body it may be considered. Upon
this generality, indeed, are founded, by their nature, all the solutions
due to the method of variations. If a single formula could not express
the length or the area of any curve whatever; if another fixed formula
could not designate the time of the fall of a heavy body, according to
whatever line it may descend, &c., how would it have been possible to
resolve questions which unavoidably require, by their nature, the
simultaneous consideration of all the cases which can be determined in
each phenomenon by the different subjects which exhibit it.


_Other Applications of this Method._ Notwithstanding the extreme
importance of the theory of isoperimeters, and though the method of
variations had at first no other object than the logical and general
solution of this order of problems, we should still have but an
incomplete idea of this beautiful analysis if we limited its destination
to this. In fact, the abstract conception of two distinct natures of
differentiation is evidently applicable not only to the cases for which
it was created, but also to all those which present, for any reason
whatever, two different manners of making the same magnitudes vary. It
is in this way that Lagrange himself has made, in his "_Méchanique
Analytique_," an extensive and important application of his calculus of
variations, by employing it to distinguish the two sorts of changes
which are naturally presented by the questions of rational mechanics for
the different points which are considered, according as we compare the
successive positions which are occupied, in virtue of its motion, by the
same point of each body in two consecutive instants, or as we pass from
one point of the body to another in the same instant. One of these
comparisons produces ordinary differentials; the other gives rise to
_variations_, which, there as every where, are only differentials taken
under a new point of view. Such is the general acceptation in which we
should conceive the Calculus of Variations, in order suitably to
appreciate the importance of this admirable logical instrument, the
most powerful that the human mind has as yet constructed.

The method of variations being only an immense extension of the general
transcendental analysis, I have no need of proving specially that it is
susceptible of being considered under the different fundamental points
of view which the calculus of indirect functions, considered as a whole,
admits of. Lagrange invented the Calculus of Variations in accordance
with the infinitesimal conception, and, indeed, long before he undertook
the general reconstruction of the transcendental analysis. When he had
executed this important reformation, he easily showed how it could also
be applied to the Calculus of Variations, which he expounded with all
the proper development, according to his theory of derivative functions.
But the more that the use of the method of variations is difficult of
comprehension, because of the higher degree of abstraction of the ideas
considered, the more necessary is it, in its application, to economize
the exertions of the mind, by adopting the most direct and rapid
analytical conception, namely, that of Leibnitz. Accordingly, Lagrange
himself has constantly preferred it in the important use which he has
made of the Calculus of Variations in his "Analytical Mechanics." In
fact, there does not exist the least hesitation in this respect among
geometers.




ITS RELATIONS TO THE ORDINARY CALCULUS.


In order to make as clear as possible the philosophical character of the
Calculus of Variations, I think that I should, in conclusion, briefly
indicate a consideration which seems to me important, and by which I can
approach it to the ordinary transcendental analysis in a higher degree
than Lagrange seems to me to have done.[12]

  [Footnote 12: I propose hereafter to develop this new
  consideration, in a special work upon the _Calculus of Variations_,
  intended to present this hyper-transcendental analysis in a new
  point of view, which I think adapted to extend its general range.]

We noticed in the preceding chapter the formation of the _calculus of
partial differences_, created by D'Alembert, as having introduced into
the transcendental analysis a new elementary idea; the notion of two
kinds of increments, distinct and independent of one another, which a
function of two variables may receive by virtue of the change of each
variable separately. It is thus that the vertical ordinate of a surface,
or any other magnitude which is referred to it, varies in two manners
which are quite distinct, and which may follow the most different laws,
according as we increase either the one or the other of the two
horizontal co-ordinates. Now such a consideration seems to me very
nearly allied, by its nature, to that which serves as the general basis
of the method of variations. This last, indeed, has in reality done
nothing but transfer to the independent variables themselves the
peculiar conception which had been already adopted for the functions of
these variables; a modification which has remarkably enlarged its use. I
think, therefore, that so far as regards merely the fundamental
conceptions, we may consider the calculus created by D'Alembert as
having established a natural and necessary transition between the
ordinary infinitesimal calculus and the calculus of variations; such a
derivation of which seems to be adapted to make the general notion more
clear and simple.

According to the different considerations indicated in this chapter, the
method of variations presents itself as the highest degree of perfection
which the analysis of indirect functions has yet attained. In its
primitive state, this last analysis presented itself as a powerful
general means of facilitating the mathematical study of natural
phenomena, by introducing, for the expression of their laws, the
consideration of auxiliary magnitudes, chosen in such a manner that
their relations are necessarily more simple and more easy to obtain than
those of the direct magnitudes. But the formation of these differential
equations was not supposed to admit of any general and abstract rules.
Now the Analysis of Variations, considered in the most philosophical
point of view, may be regarded as essentially destined, by its nature,
to bring within the reach of the calculus the actual establishment of
the differential equations; for, in a great number of important and
difficult questions, such is the general effect of the _varied_
equations, which, still more _indirect_ than the simple differential
equations with respect to the special objects of the investigation, are
also much more easy to form, and from which we may then, by invariable
and complete analytical methods, the object of which is to eliminate the
new order of auxiliary infinitesimals which have been introduced, deduce
those ordinary differential equations which it would often have been
impossible to establish directly. The method of variations forms, then,
the most sublime part of that vast system of mathematical analysis,
which, setting out from the most simple elements of algebra, organizes,
by an uninterrupted succession of ideas, general methods more and more
powerful, for the study of natural philosophy, and which, in its whole,
presents the most incomparably imposing and unequivocal monument of the
power of the human intellect.

We must, however, also admit that the conceptions which are habitually
considered in the method of variations being, by their nature, more
indirect, more general, and especially more abstract than all others,
the employment of such a method exacts necessarily and continuously the
highest known degree of intellectual exertion, in order never to lose
sight of the precise object of the investigation, in following
reasonings which offer to the mind such uncertain resting-places, and in
which signs are of scarcely any assistance. We must undoubtedly
attribute in a great degree to this difficulty the little real use which
geometers, with the exception of Lagrange, have as yet made of such an
admirable conception.




CHAPTER VI.

THE CALCULUS OF FINITE DIFFERENCES.


The different fundamental considerations indicated in the five preceding
chapters constitute, in reality, all the essential bases of a complete
exposition of mathematical analysis, regarded in the philosophical point
of view. Nevertheless, in order not to neglect any truly important
general conception relating to this analysis, I think that I should here
very summarily explain the veritable character of a kind of calculus
which is very extended, and which, though at bottom it really belongs to
ordinary analysis, is still regarded as being of an essentially distinct
nature. I refer to the _Calculus of Finite Differences_, which will be
the special subject of this chapter.


_Its general Character._ This calculus, created by Taylor, in his
celebrated work entitled _Methodus Incrementorum_, consists essentially
in the consideration of the finite increments which functions receive as
a consequence of analogous increments on the part of the corresponding
variables. These increments or _differences_, which take the
characteristic Δ, to distinguish them from _differentials_, or
infinitely small increments, may be in their turn regarded as new
functions, and become the subject of a second similar consideration, and
so on; from which results the notion of differences of various
successive orders, analogous, at least in appearance, to the consecutive
orders of differentials. Such a calculus evidently presents, like the
calculus of indirect functions, two general classes of questions:

1°. To determine the successive differences of all the various
analytical functions of one or more variables, as the result of a
definite manner of increase of the independent variables, which are
generally supposed to augment in arithmetical progression.

2°. Reciprocally, to start from these differences, or, more generally,
from any equations established between them, and go back to the
primitive functions themselves, or to their corresponding relations.

Hence follows the decomposition of this calculus into two distinct ones,
to which are usually given the names of the _Direct_, and the _Inverse
Calculus of Finite Differences_, the latter being also sometimes called
the _Integral Calculus of Finite Differences_. Each of these would,
also, evidently admit of a logical distribution similar to that given in
the fourth chapter for the differential and the integral calculus.


_Its true Nature._ There is no doubt that Taylor thought that by such a
conception he had founded a calculus of an entirely new nature,
absolutely distinct from ordinary analysis, and more general than the
calculus of Leibnitz, although resting on an analogous consideration. It
is in this way, also, that almost all geometers have viewed the analysis
of Taylor; but Lagrange, with his usual profundity, clearly perceived
that these properties belonged much more to the forms and to the
notations employed by Taylor than to the substance of his theory. In
fact, that which constitutes the peculiar character of the analysis of
Leibnitz, and makes of it a truly distinct and superior calculus, is the
circumstance that the derived functions are in general of an entirely
different nature from the primitive functions, so that they may give
rise to more simple and more easily formed relations: whence result the
admirable fundamental properties of the transcendental analysis, which
have been already explained. But it is not so with the _differences_
considered by Taylor; for these differences are, by their nature,
functions essentially similar to those which have produced them, a
circumstance which renders them unsuitable to facilitate the
establishment of equations, and prevents their leading to more general
relations. Every equation of finite differences is truly, at bottom, an
equation directly relating to the very magnitudes whose successive
states are compared. The scaffolding of new signs, which produce an
illusion respecting the true character of these equations, disguises it,
however, in a very imperfect manner, since it could always be easily
made apparent by replacing the _differences_ by the equivalent
combinations of the primitive magnitudes, of which they are really only
the abridged designations. Thus the calculus of Taylor never has
offered, and never can offer, in any question of geometry or of
mechanics, that powerful general aid which we have seen to result
necessarily from the analysis of Leibnitz. Lagrange has, moreover, very
clearly proven that the pretended analogy observed between the calculus
of differences and the infinitesimal calculus was radically vicious, in
this way, that the formulas belonging to the former calculus can never
furnish, as particular cases, those which belong to the latter, the
nature of which is essentially distinct.

From these considerations I am led to think that the calculus of finite
differences is, in general, improperly classed with the transcendental
analysis proper, that is, with the calculus of indirect functions. I
consider it, on the contrary, in accordance with the views of Lagrange,
to be only a very extensive and very important branch of ordinary
analysis, that is to say, of that which I have named the calculus of
direct functions, the equations which it considers being always, in
spite of the notation, simple _direct_ equations.




GENERAL THEORY OF SERIES.


To sum up as briefly as possible the preceding explanation, the calculus
of Taylor ought to be regarded as having constantly for its true object
the general theory of _Series_, the most simple cases of which had alone
been considered before that illustrious geometer. I ought, properly, to
have mentioned this important theory in treating, in the second chapter,
of Algebra proper, of which it is such an extensive branch. But, in
order to avoid a double reference to it, I have preferred to notice it
only in the consideration of the calculus of finite differences, which,
reduced to its most simple general expression, is nothing but a complete
logical study of questions relating to _series_.

Every _Series_, or succession of numbers deduced from one another
according to any constant law, necessarily gives rise to these two
fundamental questions:

1°. The law of the series being supposed known, to find the expression
for its general term, so as to be able to calculate immediately any term
whatever without being obliged to form successively all the preceding
terms.

2°. In the same circumstances, to determine the _sum_ of any number of
terms of the series by means of their places, so that it can be known
without the necessity of continually adding these terms together.

These two fundamental questions being considered to be resolved, it may
be proposed, reciprocally, to find the law of a series from the form of
its general term, or the expression of the sum. Each of these different
problems has so much the more extent and difficulty, as there can be
conceived a greater number of different _laws_ for the series, according
to the number of preceding terms on which each term directly depends,
and according to the function which expresses that dependence. We may
even consider series with several variable indices, as Laplace has done
in his "Analytical Theory of Probabilities," by the analysis to which he
has given the name of _Theory of Generating Functions_, although it is
really only a new and higher branch of the calculus of finite
differences or of the general theory of series.

These general views which I have indicated give only an imperfect idea
of the truly infinite extent and variety of the questions to which
geometers have risen by means of this single consideration of series, so
simple in appearance and so limited in its origin. It necessarily
presents as many different cases as the algebraic resolution of
equations, considered in its whole extent; and it is, by its nature,
much more complicated, so much, indeed, that it always needs this last
to conduct it to a complete solution. We may, therefore, anticipate what
must still be its extreme imperfection, in spite of the successive
labours of several geometers of the first order. We do not, indeed,
possess as yet the complete and logical solution of any but the most
simple questions of this nature.


_Its identity with this Calculus._ It is now easy to conceive the
necessary and perfect identity, which has been already announced,
between the calculus of finite differences and the theory of series
considered in all its bearings. In fact, every differentiation after the
manner of Taylor evidently amounts to finding the _law_ of formation of
a series with one or with several variable indices, from the expression
of its general term; in the same way, every analogous integration may be
regarded as having for its object the summation of a series, the general
term of which would be expressed by the proposed difference. In this
point of view, the various problems of the calculus of differences,
direct or inverse, resolved by Taylor and his successors, have really a
very great value, as treating of important questions relating to series.
But it is very doubtful if the form and the notation introduced by
Taylor really give any essential facility in the solution of questions
of this kind. It would be, perhaps, more advantageous for most cases,
and certainly more logical, to replace the _differences_ by the terms
themselves, certain combinations of which they represent. As the
calculus of Taylor does not rest on a truly distinct fundamental idea,
and has nothing peculiar to it but its system of signs, there could
never really be any important advantage in considering it as detached
from ordinary analysis, of which it is, in reality, only an immense
branch. This consideration of _differences_, most generally useless,
even if it does not cause complication, seems to me to retain the
character of an epoch in which, analytical ideas not being sufficiently
familiar to geometers, they were naturally led to prefer the special
forms suitable for simple numerical comparisons.




PERIODIC OR DISCONTINUOUS FUNCTIONS.


However that may be, I must not finish this general appreciation of the
calculus of finite differences without noticing a new conception to
which it has given birth, and which has since acquired a great
importance. It is the consideration of those periodic or discontinuous
functions which preserve the same value for an infinite series of values
of the corresponding variables, subjected to a certain law, and which
must be necessarily added to the integrals of the equations of finite
differences in order to render them sufficiently general, as simple
arbitrary constants are added to all quadratures in order to complete
their generality. This idea, primitively introduced by Euler, has since
been the subject of extended investigation by M. Fourier, who has made
new and important applications of it in his mathematical theory of heat.




APPLICATIONS OF THIS CALCULUS.


_Series._ Among the principal general applications which have been made
of the calculus of finite differences, it would be proper to place in
the first rank, as the most extended and the most important, the
solution of questions relating to series; if, as has been shown, the
general theory of series ought not to be considered as constituting, by
its nature, the actual foundation of the calculus of Taylor.


_Interpolations._ This great class of problems being then set aside, the
most essential of the veritable applications of the analysis of Taylor
is, undoubtedly, thus far, the general method of _interpolations_, so
frequently and so usefully employed in the investigation of the
empirical laws of natural phenomena. The question consists, as is well
known, in intercalating between certain given numbers other intermediate
numbers, subjected to the same law which we suppose to exist between the
first. We can abundantly verify, in this principal application of the
calculus of Taylor, how truly foreign and often inconvenient is the
consideration of _differences_ with respect to the questions which
depend on that analysis. Indeed, Lagrange has replaced the formulas of
interpolation, deduced from the ordinary algorithm of the calculus of
finite differences, by much simpler general formulas, which are now
almost always preferred, and which have been found directly, without
making any use of the notion of _differences_, which only complicates
the question.


_Approximate Rectification, &c._ A last important class of applications
of the calculus of finite differences, which deserves to be
distinguished from the preceding, consists in the eminently useful
employment made of it in geometry for determining by approximation the
length and the area of any curve, and in the same way the cubature of a
body of any form whatever. This procedure (which may besides be
conceived abstractly as depending on the same analytical investigation
as the question of interpolation) frequently offers a valuable
supplement to the entirely logical geometrical methods which often lead
to integrations, which we do not yet know how to effect, or to
calculations of very complicated execution.

       *       *       *       *       *

Such are the various principal considerations to be noticed with respect
to the calculus of finite differences. This examination completes the
proposed philosophical outline of ABSTRACT MATHEMATICS.


CONCRETE MATHEMATICS will now be the subject of a similar labour. In it
we shall particularly devote ourselves to examining how it has been
possible (supposing the general science of the calculus to be perfect),
by invariable procedures, to reduce to pure questions of analysis all
the problems which can be presented by _Geometry_ and _Mechanics_, and
thus to impress on these two fundamental bases of natural philosophy a
degree of precision and especially of unity; in a word, a character of
high perfection, which could be communicated to them by such a course
alone.




BOOK II.

GEOMETRY.




BOOK II.

GEOMETRY.




CHAPTER I.

GENERAL VIEW OF GEOMETRY.


_Its true Nature._ After the general exposition of the philosophical
character of concrete mathematics, compared with that of abstract
mathematics, given in the introductory chapter, it need not here be
shown in a special manner that geometry must be considered as a true
natural science, only much more simple, and therefore much more perfect,
than any other. This necessary perfection of geometry, obtained
essentially by the application of mathematical analysis, which it so
eminently admits, is apt to produce erroneous views of the real nature
of this fundamental science, which most minds at present conceive to be
a purely logical science quite independent of observation. It is
nevertheless evident, to any one who examines with attention the
character of geometrical reasonings, even in the present state of
abstract geometry, that, although the facts which are considered in it
are much more closely united than those relating to any other science,
still there always exists, with respect to every body studied by
geometers, a certain number of primitive phenomena, which, since they
are not established by any reasoning, must be founded on observation
alone, and which form the necessary basis of all the deductions.

The scientific superiority of geometry arises from the phenomena which
it considers being necessarily the most universal and the most simple of
all. Not only may all the bodies of nature give rise to geometrical
inquiries, as well as mechanical ones, but still farther, geometrical
phenomena would still exist, even though all the parts of the universe
should be considered as immovable. Geometry is then, by its nature, more
general than mechanics. At the same time, its phenomena are more simple,
for they are evidently independent of mechanical phenomena, while these
latter are always complicated with the former. The same relations hold
good in comparing geometry with abstract thermology.

For these reasons, in our classification we have made geometry the first
part of concrete mathematics; that part the study of which, in addition
to its own importance, serves as the indispensable basis of all the
rest.

Before considering directly the philosophical study of the different
orders of inquiries which constitute our present geometry, we should
obtain a clear and exact idea of the general destination of that
science, viewed in all its bearings. Such is the object of this chapter.


_Definition._ Geometry is commonly defined in a very vague and entirely
improper manner, as being _the science of extension_. An improvement on
this would be to say that geometry has for its object the _measurement_
of extension; but such an explanation would be very insufficient,
although at bottom correct, and would be far from giving any idea of the
true general character of geometrical science.

To do this, I think that I should first explain _two fundamental ideas_,
which, very simple in themselves, have been singularly obscured by the
employment of metaphysical considerations.


_The Idea of Space._ The first is that of _Space_. This conception
properly consists simply in this, that, instead of considering extension
in the bodies themselves, we view it in an indefinite medium, which we
regard as containing all the bodies of the universe. This notion is
naturally suggested to us by observation, when we think of the
_impression_ which a body would leave in a fluid in which it had been
placed. It is clear, in fact, that, as regards its geometrical
relations, such an _impression_ may be substituted for the body itself,
without altering the reasonings respecting it. As to the physical nature
of this indefinite _space_, we are spontaneously led to represent it to
ourselves, as being entirely analogous to the actual medium in which we
live; so that if this medium was liquid instead of gaseous, our
geometrical _space_ would undoubtedly be conceived as liquid also. This
circumstance is, moreover, only very secondary, the essential object of
such a conception being only to make us view extension separately from
the bodies which manifest it to us. We can easily understand in advance
the importance of this fundamental image, since it permits us to study
geometrical phenomena in themselves, abstraction being made of all the
other phenomena which constantly accompany them in real bodies, without,
however, exerting any influence over them. The regular establishment of
this general abstraction must be regarded as the first step which has
been made in the rational study of geometry, which would have been
impossible if it had been necessary to consider, together with the form
and the magnitude of bodies, all their other physical properties. The
use of such an hypothesis, which is perhaps the most ancient
philosophical conception created by the human mind, has now become so
familiar to us, that we have difficulty in exactly estimating its
importance, by trying to appreciate the consequences which would result
from its suppression.


_Different Kinds of Extension._ The second preliminary geometrical
conception which we have to examine is that of the different kinds of
extension, designated by the words _volume_, _surface_, _line_, and even
_point_, and of which the ordinary explanation is so unsatisfactory.[13]

  [Footnote 13: Lacroix has justly criticised the expression of
  _solid_, commonly used by geometers to designate a _volume_. It is
  certain, in fact, that when we wish to consider separately a
  certain portion of indefinite space, conceived as gaseous, we
  mentally solidify its exterior envelope, so that a _line_ and a
  _surface_ are habitually, to our minds, just as _solid_ as a
  _volume_. It may also be remarked that most generally, in order
  that bodies may penetrate one another with more facility, we are
  obliged to imagine the interior of the _volumes_ to be hollow,
  which renders still more sensible the impropriety of the word
  _solid_.]

Although it is evidently impossible to conceive any extension absolutely
deprived of any one of the three fundamental dimensions, it is no less
incontestable that, in a great number of occasions, even of immediate
utility, geometrical questions depend on only two dimensions, considered
separately from the third, or on a single dimension, considered
separately from the two others. Again, independently of this direct
motive, the study of extension with a single dimension, and afterwards
with two, clearly presents itself as an indispensable preliminary for
facilitating the study of complete bodies of three dimensions, the
immediate theory of which would be too complicated. Such are the two
general motives which oblige geometers to consider separately extension
with regard to one or to two dimensions, as well as relatively to all
three together.

The general notions of _surface_ and of _line_ have been formed by the
human mind, in order that it may be able to think, in a permanent
manner, of extension in two directions, or in one only. The hyperbolical
expressions habitually employed by geometers to define these notions
tend to convey false ideas of them; but, examined in themselves, they
have no other object than to permit us to reason with facility
respecting these two kinds of extension, making complete abstraction of
that which ought not to be taken into consideration. Now for this it is
sufficient to conceive the dimension which we wish to eliminate as
becoming gradually smaller and smaller, the two others remaining the
same, until it arrives at such a degree of tenuity that it can no longer
fix the attention. It is thus that we naturally acquire the real idea of
a _surface_, and, by a second analogous operation, the idea of a _line_,
by repeating for breadth what we had at first done for thickness.
Finally, if we again repeat the same operation, we arrive at the idea of
a _point_, or of an extension considered only with reference to its
place, abstraction being made of all magnitude, and designed
consequently to determine positions.

_Surfaces_ evidently have, moreover, the general property of exactly
circumscribing volumes; and in the same way, _lines_, in their turn,
circumscribe _surfaces_ and are limited by _points_. But this
consideration, to which too much importance is often given, is only a
secondary one.

Surfaces and lines are, then, in reality, always conceived with three
dimensions; it would be, in fact, impossible to represent to one's self
a surface otherwise than as an extremely thin plate, and a line
otherwise than as an infinitely fine thread. It is even plain that the
degree of tenuity attributed by each individual to the dimensions of
which he wishes to make abstraction is not constantly identical, for it
must depend on the degree of subtilty of his habitual geometrical
observations. This want of uniformity has, besides, no real
inconvenience, since it is sufficient, in order that the ideas of
surface and of line should satisfy the essential condition of their
destination, for each one to represent to himself the dimensions which
are to be neglected as being smaller than all those whose magnitude his
daily experience gives him occasion to appreciate.

We hence see how devoid of all meaning are the fantastic discussions of
metaphysicians upon the foundations of geometry. It should also be
remarked that these primordial ideas are habitually presented by
geometers in an unphilosophical manner, since, for example, they explain
the notions of the different sorts of extent in an order absolutely the
inverse of their natural dependence, which often produces the most
serious inconveniences in elementary instruction.




THE FINAL OBJECT OF GEOMETRY.


These preliminaries being established, we can proceed directly to the
general definition of geometry, continuing to conceive this science as
having for its final object the _measurement_ of extension.

It is necessary in this matter to go into a thorough explanation,
founded on the distinction of the three kinds of extension, since the
notion of _measurement_ is not exactly the same with reference to
surfaces and volumes as to lines.


_Nature of Geometrical Measurement._ If we take the word _measurement_
in its direct and general mathematical acceptation, which signifies
simply the determination of the value of the _ratios_ between any
homogeneous magnitudes, we must consider, in geometry, that the
_measurement_ of surfaces and of volumes, unlike that of lines, is never
conceived, even in the most simple and the most favourable cases, as
being effected directly. The comparison of two lines is regarded as
direct; that of two surfaces or of two volumes is, on the contrary,
always indirect. Thus we conceive that two lines may be superposed; but
the superposition of two surfaces, or, still more so, of two volumes, is
evidently impossible in most cases; and, even when it becomes rigorously
practicable, such a comparison is never either convenient or exact. It
is, then, very necessary to explain wherein properly consists the truly
geometrical measurement of a surface or of a volume.


_Measurement of Surfaces and of Volumes._ For this we must consider
that, whatever may be the form of a body, there always exists a certain
number of lines, more or less easy to be assigned, the length of which
is sufficient to define exactly the magnitude of its surface or of its
volume. Geometry, regarding these lines as alone susceptible of being
directly measured, proposes to deduce, from the simple determination of
them, the ratio of the surface or of the volume sought, to the unity of
surface, or to the unity of volume. Thus the general object of
geometry, with respect to surfaces and to volumes, is properly to reduce
all comparisons of surfaces or of volumes to simple comparisons of
lines.

Besides the very great facility which such a transformation evidently
offers for the measurement of volumes and of surfaces, there results
from it, in considering it in a more extended and more scientific
manner, the general possibility of reducing to questions of lines all
questions relating to volumes and to surfaces, considered with reference
to their magnitude. Such is often the most important use of the
geometrical expressions which determine surfaces and volumes in
functions of the corresponding lines.

It is true that direct comparisons between surfaces or between volumes
are sometimes employed; but such measurements are not regarded as
geometrical, but only as a supplement sometimes necessary, although too
rarely applicable, to the insufficiency or to the difficulty of truly
rational methods. It is thus that we often determine the volume of a
body, and in certain cases its surface, by means of its weight. In the
same way, on other occasions, when we can substitute for the proposed
volume an equivalent liquid volume, we establish directly the comparison
of the two volumes, by profiting by the property possessed by liquid
masses, of assuming any desired form. But all means of this nature are
purely mechanical, and rational geometry necessarily rejects them.

To render more sensible the difference between these modes of
determination and true geometrical measurements, I will cite a single
very remarkable example; the manner in which Galileo determined the
ratio of the ordinary cycloid to that of the generating circle. The
geometry of his time was as yet insufficient for the rational solution
of such a problem. Galileo conceived the idea of discovering that ratio
by a direct experiment. Having weighed as exactly as possible two plates
of the same material and of equal thickness, one of them having the form
of a circle and the other that of the generated cycloid, he found the
weight of the latter always triple that of the former; whence he
inferred that the area of the cycloid is triple that of the generating
circle, a result agreeing with the veritable solution subsequently
obtained by Pascal and Wallis. Such a success evidently depends on the
extreme simplicity of the ratio sought; and we can understand the
necessary insufficiency of such expedients, even when they are actually
practicable.

We see clearly, from what precedes, the nature of that part of geometry
relating to _volumes_ and that relating to _surfaces_. But the character
of the geometry of _lines_ is not so apparent, since, in order to
simplify the exposition, we have considered the measurement of lines as
being made directly. There is, therefore, needed a complementary
explanation with respect to them.


_Measurement of curved Lines._ For this purpose, it is sufficient to
distinguish between the right line and curved lines, the measurement of
the first being alone regarded as direct, and that of the other as
always indirect. Although superposition is sometimes strictly
practicable for curved lines, it is nevertheless evident that truly
rational geometry must necessarily reject it, as not admitting of any
precision, even when it is possible. The geometry of lines has, then,
for its general object, to reduce in every case the measurement of
curved lines to that of right lines; and consequently, in the most
extended point of view, to reduce to simple questions of right lines all
questions relating to the magnitude of any curves whatever. To
understand the possibility of such a transformation, we must remark,
that in every curve there always exist certain right lines, the length
of which must be sufficient to determine that of the curve. Thus, in a
circle, it is evident that from the length of the radius we must be able
to deduce that of the circumference; in the same way, the length of an
ellipse depends on that of its two axes; the length of a cycloid upon
the diameter of the generating circle, &c.; and if, instead of
considering the whole of each curve, we demand, more generally, the
length of any arc, it will be sufficient to add to the different
rectilinear parameters, which determine the whole curve, the chord of
the proposed arc, or the co-ordinates of its extremities. To discover
the relation which exists between the length of a curved line and that
of similar right lines, is the general problem of the part of geometry
which relates to the study of lines.

Combining this consideration with those previously suggested with
respect to volumes and to surfaces, we may form a very clear idea of the
science of geometry, conceived in all its parts, by assigning to it, for
its general object, the final reduction of the comparisons of all kinds
of extent, volumes, surfaces, or lines, to simple comparisons of right
lines, the only comparisons regarded as capable of being made directly,
and which indeed could not be reduced to any others more easy to effect.
Such a conception, at the same time, indicates clearly the veritable
character of geometry, and seems suited to show at a single glance its
utility and its perfection.


_Measurement of right Lines._ In order to complete this fundamental
explanation, I have yet to show how there can be, in geometry, a special
section relating to the right line, which seems at first incompatible
with the principle that the measurement of this class of lines must
always be regarded as direct.

It is so, in fact, as compared with that of curved lines, and of all the
other objects which geometry considers. But it is evident that the
estimation of a right line cannot be viewed as direct except so far as
the linear unit can be applied to it. Now this often presents
insurmountable difficulties, as I had occasion to show, for another
reason, in the introductory chapter. We must, then, make the measurement
of the proposed right line depend on other analogous measurements
capable of being effected directly. There is, then, necessarily a
primary distinct branch of geometry, exclusively devoted to the right
line; its object is to determine certain right lines from others by
means of the relations belonging to the figures resulting from their
assemblage. This preliminary part of geometry, which is almost
imperceptible in viewing the whole of the science, is nevertheless
susceptible of a great development. It is evidently of especial
importance, since all other geometrical measurements are referred to
those of right lines, and if they could not be determined, the solution
of every question would remain unfinished.

Such, then, are the various fundamental parts of rational geometry,
arranged according to their natural dependence; the geometry of _lines_
being first considered, beginning with the right line; then the geometry
of _surfaces_, and, finally, that of _solids_.




INFINITE EXTENT OF ITS FIELD.


Having determined with precision the general and final object of
geometrical inquiries, the science must now be considered with respect
to the field embraced by each of its three fundamental sections.

Thus considered, geometry is evidently susceptible, by its nature, of an
extension which is rigorously infinite; for the measurement of lines, of
surfaces, or of volumes presents necessarily as many distinct questions
as we can conceive different figures subjected to exact definitions; and
their number is evidently infinite.

Geometers limited themselves at first to consider the most simple
figures which were directly furnished them by nature, or which were
deduced from these primitive elements by the least complicated
combinations. But they have perceived, since Descartes, that, in order
to constitute the science in the most philosophical manner, it was
necessary to make it apply to all imaginable figures. This abstract
geometry will then inevitably comprehend as particular cases all the
different real figures which the exterior world could present. It is
then a fundamental principle in truly rational geometry to consider, as
far as possible, all figures which can be rigorously conceived.

The most superficial examination is enough to convince us that these
figures present a variety which is quite infinite.


_Infinity of Lines._ With respect to curved _lines_, regarding them as
generated by the motion of a point governed by a certain law, it is
plain that we shall have, in general, as many different curves as we
conceive different laws for this motion, which may evidently be
determined by an infinity of distinct conditions; although it may
sometimes accidentally happen that new generations produce curves which
have been already obtained. Thus, among plane curves, if a point moves
so as to remain constantly at the same distance from a fixed point, it
will generate a _circle_; if it is the sum or the difference of its
distances from two fixed points which remains constant, the curve
described will be an _ellipse_ or an _hyperbola_; if it is their
product, we shall have an entirely different curve; if the point departs
equally from a fixed point and from a fixed line, it will describe a
_parabola_; if it revolves on a circle at the same time that this circle
rolls along a straight line, we shall have a _cycloid_; if it advances
along a straight line, while this line, fixed at one of its extremities,
turns in any manner whatever, there will result what in general terms
are called _spirals_, which of themselves evidently present as many
perfectly distinct curves as we can suppose different relations between
these two motions of translation and of rotation, &c. Each of these
different curves may then furnish new ones, by the different general
constructions which geometers have imagined, and which give rise to
evolutes, to epicycloids, to caustics, &c. Finally, there exists a still
greater variety among curves of double curvature.


_Infinity of Surfaces._ As to _surfaces_, the figures are necessarily
more different still, considering them as generated by the motion of
lines. Indeed, the figure may then vary, not only in considering, as in
curves, the different infinitely numerous laws to which the motion of
the generating line may be subjected, but also in supposing that this
line itself may change its nature; a circumstance which has nothing
analogous in curves, since the points which describe them cannot have
any distinct figure. Two classes of very different conditions may then
cause the figures of surfaces to vary, while there exists only one for
lines. It is useless to cite examples of this doubly infinite
multiplicity of surfaces. It would be sufficient to consider the extreme
variety of the single group of surfaces which may be generated by a
right line, and which comprehends the whole family of cylindrical
surfaces, that of conical surfaces, the most general class of
developable surfaces, &c.


_Infinity of Volumes._ With respect to _volumes_, there is no occasion
for any special consideration, since they are distinguished from each
other only by the surfaces which bound them.

In order to complete this sketch, it should be added that surfaces
themselves furnish a new general means of conceiving new curves, since
every curve may be regarded as produced by the intersection of two
surfaces. It is in this way, indeed, that the first lines which we may
regard as having been truly invented by geometers were obtained, since
nature gave directly the straight line and the circle. We know that the
ellipse, the parabola, and the hyperbola, the only curves completely
studied by the ancients, were in their origin conceived only as
resulting from the intersection of a cone with circular base by a plane
in different positions. It is evident that, by the combined employment
of these different general means for the formation of lines and of
surfaces, we could produce a rigorously infinitely series of distinct
forms in starting from only a very small number of figures directly
furnished by observation.


_Analytical invention of Curves, &c._ Finally, all the various direct
means for the invention of figures have scarcely any farther importance,
since rational geometry has assumed its final character in the hands of
Descartes. Indeed, as we shall see more fully in chapter iii., the
invention of figures is now reduced to the invention of equations, so
that nothing is more easy than to conceive new lines and new surfaces,
by changing at will the functions introduced into the equations. This
simple abstract procedure is, in this respect, infinitely more fruitful
than all the direct resources of geometry, developed by the most
powerful imagination, which should devote itself exclusively to that
order of conceptions. It also explains, in the most general and the most
striking manner, the necessarily infinite variety of geometrical forms,
which thus corresponds to the diversity of analytical functions. Lastly,
it shows no less clearly that the different forms of surfaces must be
still more numerous than those of lines, since lines are represented
analytically by equations with two variables, while surfaces give rise
to equations with three variables, which necessarily present a greater
diversity.

The preceding considerations are sufficient to show clearly the
rigorously infinite extent of each of the three general sections of
geometry.




EXPANSION OF ORIGINAL DEFINITION.


To complete the formation of an exact and sufficiently extended idea of
the nature of geometrical inquiries, it is now indispensable to return
to the general definition above given, in order to present it under a
new point of view, without which the complete science would be only very
imperfectly conceived.

When we assign as the object of geometry the _measurement_ of all sorts
of lines, surfaces, and volumes, that is, as has been explained, the
reduction of all geometrical comparisons to simple comparisons of right
lines, we have evidently the advantage of indicating a general
destination very precise and very easy to comprehend. But if we set
aside every definition, and examine the actual composition of the
science of geometry, we will at first be induced to regard the preceding
definition as much too narrow; for it is certain that the greater part
of the investigations which constitute our present geometry do not at
all appear to have for their object the _measurement_ of extension. In
spite of this fundamental objection, I will persist in retaining this
definition; for, in fact, if, instead of confining ourselves to
considering the different questions of geometry isolatedly, we endeavour
to grasp the leading questions, in comparison with which all others,
however important they may be, must be regarded as only secondary, we
will finally recognize that the measurement of lines, of surfaces, and
of volumes, is the invariable object, sometimes _direct_, though most
often _indirect_, of all geometrical labours.

This general proposition being fundamental, since it can alone give our
definition all its value, it is indispensable to enter into some
developments upon this subject.




PROPERTIES OF LINES AND SURFACES.


When we examine with attention the geometrical investigations which do
not seem to relate to the _measurement_ of extent, we find that they
consist essentially in the study of the different _properties_ of each
line or of each surface; that is, in the knowledge of the different
modes of generation, or at least of definition, peculiar to each figure
considered. Now we can easily establish in the most general manner the
necessary relation of such a study to the question of _measurement_, for
which the most complete knowledge of the properties of each form is an
indispensable preliminary. This is concurrently proven by two
considerations, equally fundamental, although quite distinct in their
nature.


NECESSITY OF THEIR STUDY: 1. _To find the most suitable Property._ The
_first_, purely scientific, consists in remarking that, if we did not
know any other characteristic property of each line or surface than that
one according to which geometers had first conceived it, in most cases
it would be impossible to succeed in the solution of questions relating
to its _measurement_. In fact, it is easy to understand that the
different definitions which each figure admits of are not all equally
suitable for such an object, and that they even present the most
complete oppositions in that respect. Besides, since the primitive
definition of each figure was evidently not chosen with this condition
in view, it is clear that we must not expect, in general, to find it the
most suitable; whence results the necessity of discovering others, that
is, of studying as far as is possible the _properties_ of the proposed
figure. Let us suppose, for example, that the circle is defined to be
"the curve which, with the same contour, contains the greatest area."
This is certainly a very characteristic property, but we would evidently
find insurmountable difficulties in trying to deduce from such a
starting point the solution of the fundamental questions relating to the
rectification or to the quadrature of this curve. It is clear, in
advance, that the property of having all its points equally distant from
a fixed point must evidently be much better adapted to inquiries of this
nature, even though it be not precisely the most suitable. In like
manner, would Archimedes ever have been able to discover the quadrature
of the parabola if he had known no other property of that curve than
that it was the section of a cone with a circular base, by a plane
parallel to its generatrix? The purely speculative labours of preceding
geometers, in transforming this first definition, were evidently
indispensable preliminaries to the direct solution of such a question.
The same is true, in a still greater degree, with respect to surfaces.
To form a just idea of this, we need only compare, as to the question of
cubature or quadrature, the common definition of the sphere with that
one, no less characteristic certainly, which would consist in regarding
a spherical body, as that one which, with the same area, contains the
greatest volume.

No more examples are needed to show the necessity of knowing, so far as
is possible, all the properties of each line or of each surface, in
order to facilitate the investigation of rectifications, of quadratures,
and of cubatures, which constitutes the final object of geometry. We may
even say that the principal difficulty of questions of this kind
consists in employing in each case the property which is best adapted
to the nature of the proposed problem. Thus, while we continue to
indicate, for more precision, the measurement of extension as the
general destination of geometry, this first consideration, which goes to
the very bottom of the subject, shows clearly the necessity of including
in it the study, as thorough as possible, of the different generations
or definitions belonging to the same form.


2. _To pass from the Concrete to the Abstract._ A second consideration,
of at least equal importance, consists in such a study being
indispensable for organizing in a rational manner the relation of the
abstract to the concrete in geometry.

The science of geometry having to consider all imaginable figures which
admit of an exact definition, it necessarily results from this, as we
have remarked, that questions relating to any figures presented by
nature are always implicitly comprised in this abstract geometry,
supposed to have attained its perfection. But when it is necessary to
actually pass to concrete geometry, we constantly meet with a
fundamental difficulty, that of knowing to which of the different
abstract types we are to refer, with sufficient approximation, the real
lines or surfaces which we have to study. Now it is for the purpose of
establishing such a relation that it is particularly indispensable to
know the greatest possible number of properties of each figure
considered in geometry.

In fact, if we always confined ourselves to the single primitive
definition of a line or of a surface, supposing even that we could then
_measure_ it (which, according to the first order of considerations,
would generally be impossible), this knowledge would remain almost
necessarily barren in the application, since we should not ordinarily
know how to recognize that figure in nature when it presented itself
there; to ensure that, it would be necessary that the single
characteristic, according to which geometers had conceived it, should be
precisely that one whose verification external circumstances would
admit: a coincidence which would be purely fortuitous, and on which we
could not count, although it might sometimes take place. It is, then,
only by multiplying as much as possible the characteristic properties of
each abstract figure, that we can be assured, in advance, of recognizing
it in the concrete state, and of thus turning to account all our
rational labours, by verifying in each case the definition which is
susceptible of being directly proven. This definition is almost always
the only one in given circumstances, and varies, on the other hand, for
the same figure, with different circumstances; a double reason for its
previous determination.


_Illustration: Orbits of the Planets._ The geometry of the heavens
furnishes us with a very memorable example in this matter, well suited
to show the general necessity of such a study. We know that the ellipse
was discovered by Kepler to be the curve which the planets describe
about the sun, and the satellites about their planets. Now would this
fundamental discovery, which re-created astronomy, ever have been
possible, if geometers had been always confined to conceiving the
ellipse only as the oblique section of a circular cone by a plane? No
such definition, it is evident, would admit of such a verification. The
most general property of the ellipse, that the sum of the distances from
any of its points to two fixed points is a constant quantity, is
undoubtedly much more susceptible, by its nature, of causing the curve
to be recognized in this case, but still is not directly suitable. The
only characteristic which can here be immediately verified is that which
is derived from the relation which exists in the ellipse between the
length of the focal distances and their direction; the only relation
which admits of an astronomical interpretation, as expressing the law
which connects the distance from the planet to the sun, with the time
elapsed since the beginning of its revolution. It was, then, necessary
that the purely speculative labours of the Greek geometers on the
properties of the conic sections should have previously presented their
generation under a multitude of different points of view, before Kepler
could thus pass from the abstract to the concrete, in choosing from
among all these different characteristics that one which could be most
easily proven for the planetary orbits.


_Illustration: Figure of the Earth._ Another example of the same order,
but relating to surfaces, occurs in considering the important question
of the figure of the earth. If we had never known any other property of
the sphere than its primitive character of having all its points equally
distant from an interior point, how would we ever have been able to
discover that the surface of the earth was spherical? For this, it was
necessary previously to deduce from this definition of the sphere some
properties capable of being verified by observations made upon the
surface alone, such as the constant ratio which exists between the
length of the path traversed in the direction of any meridian of a
sphere going towards a pole, and the angular height of this pole above
the horizon at each point. Another example, but involving a much longer
series of preliminary speculations, is the subsequent proof that the
earth is not rigorously spherical, but that its form is that of an
ellipsoid of revolution.

After such examples, it would be needless to give any others, which any
one besides may easily multiply. All of them prove that, without a very
extended knowledge of the different properties of each figure, the
relation of the abstract to the concrete, in geometry, would be purely
accidental, and that the science would consequently want one of its most
essential foundations.

Such, then, are two general considerations which fully demonstrate the
necessity of introducing into geometry a great number of investigations
which have not the _measurement_ of extension for their direct object;
while we continue, however, to conceive such a measurement as being the
final destination of all geometrical science. In this way we can retain
the philosophical advantages of the clearness and precision of this
definition, and still include in it, in a very logical though indirect
manner, all known geometrical researches, in considering those which do
not seem to relate to the measurement of extension, as intended either
to prepare for the solution of the final questions, or to render
possible the application of the solutions obtained.

Having thus recognized, as a general principle, the close and necessary
connexion of the study of the properties of lines and surfaces with
those researches which constitute the final object of geometry, it is
evident that geometers, in the progress of their labours, must by no
means constrain themselves to keep such a connexion always in view.
Knowing, once for all, how important it is to vary as much as possible
the manner of conceiving each figure, they should pursue that study,
without considering of what immediate use such or such a special
property may be for rectifications, quadratures, and cubatures. They
would uselessly fetter their inquiries by attaching a puerile importance
to the continued establishment of that co-ordination.

This general exposition of the general object of geometry is so much the
more indispensable, since, by the very nature of the subject, this study
of the different properties of each line and of each surface necessarily
composes by far the greater part of the whole body of geometrical
researches. Indeed, the questions directly relating to rectifications,
to quadratures, and to cubatures, are evidently, by themselves, very few
in number for each figure considered. On the other hand, the study of
the properties of the same figure presents an unlimited field to the
activity of the human mind, in which it may always hope to make new
discoveries. Thus, although geometers have occupied themselves for
twenty centuries, with more or less activity undoubtedly, but without
any real interruption, in the study of the conic sections, they are far
from regarding that so simple subject as being exhausted; and it is
certain, indeed, that in continuing to devote themselves to it, they
would not fail to find still unknown properties of those different
curves. If labours of this kind have slackened considerably for a
century past, it is not because they are completed, but only, as will be
presently explained, because the philosophical revolution in geometry,
brought about by Descartes, has singularly diminished the importance of
such researches.

It results from the preceding considerations that not only is the field
of geometry necessarily infinite because of the variety of figures to
be considered, but also in virtue of the diversity of the points of view
under the same figure may be regarded. This last conception is, indeed,
that which gives the broadest and most complete idea of the whole body
of geometrical researches. We see that studies of this kind consist
essentially, for each line or for each surface, in connecting all the
geometrical phenomena which it can present, with a single fundamental
phenomenon, regarded as the primitive definition.




THE TWO GENERAL METHODS OF GEOMETRY.


Having now explained in a general and yet precise manner the final
object of geometry, and shown how the science, thus defined, comprehends
a very extensive class of researches which did not at first appear
necessarily to belong to it, there remains to be considered the _method_
to be followed for the formation of this science. This discussion is
indispensable to complete this first sketch of the philosophical
character of geometry. I shall here confine myself to indicating the
most general consideration in this matter, developing and summing up
this important fundamental idea in the following chapters.

Geometrical questions may be treated according to _two methods_ so
different, that there result from them two sorts of geometry, so to say,
the philosophical character of which does not seem to me to have yet
been properly apprehended. The expressions of _Synthetical Geometry_ and
_Analytical Geometry_, habitually employed to designate them, give a
very false idea of them. I would much prefer the purely historical
denominations of _Geometry of the Ancients_ and _Geometry of the
Moderns_, which have at least the advantage of not causing their true
character to be misunderstood. But I propose to employ henceforth the
regular expressions of _Special Geometry_ and _General Geometry_, which
seem to me suited to characterize with precision the veritable nature of
the two methods.


_Their fundamental Difference._ The fundamental difference between the
manner in which we conceive Geometry since Descartes, and the manner in
which the geometers of antiquity treated geometrical questions, is not
the use of the Calculus (or Algebra), as is commonly thought to be the
case. On the one hand, it is certain that the use of the calculus was
not entirely unknown to the ancient geometers, since they used to make
continual and very extensive applications of the theory of proportions,
which was for them, as a means of deduction, a sort of real, though very
imperfect and especially extremely limited equivalent for our present
algebra. The calculus may even be employed in a much more complete
manner than they have used it, in order to obtain certain geometrical
solutions, which will still retain all the essential character of the
ancient geometry; this occurs very frequently with respect to those
problems of geometry of two or of three dimensions, which are commonly
designated under the name of _determinate_. On the other hand, important
as is the influence of the calculus in our modern geometry, various
solutions obtained without algebra may sometimes manifest the peculiar
character which distinguishes it from the ancient geometry, although
analysis is generally indispensable. I will cite, as an example, the
method of Roberval for tangents, the nature of which is essentially
modern, and which, however, leads in certain cases to complete
solutions, without any aid from the calculus. It is not, then, the
instrument of deduction employed which is the principal distinction
between the two courses which the human mind can take in geometry.

The real fundamental difference, as yet imperfectly apprehended, seems
to me to consist in the very nature of the questions considered. In
truth, geometry, viewed as a whole, and supposed to have attained entire
perfection, must, as we have seen on the one hand, embrace all
imaginable figures, and, on the other, discover all the properties of
each figure. It admits, from this double consideration, of being treated
according to two essentially distinct plans; either, 1°, by grouping
together all the questions, however different they may be, which relate
to the same figure, and isolating those relating to different bodies,
whatever analogy there may exist between them; or, 2°, on the contrary,
by uniting under one point of view all similar inquiries, to whatever
different figures they may relate, and separating the questions relating
to the really different properties of the same body. In a word, the
whole body of geometry may be essentially arranged either with reference
to the _bodies_ studied or to the _phenomena_ to be considered. The
first plan, which is the most natural, was that of the ancients; the
second, infinitely more rational, is that of the moderns since
Descartes.


_Geometry of the Ancients._ Indeed, the principal characteristics of the
ancient geometry is that they studied, one by one, the different lines
and the different surfaces, not passing to the examination of a new
figure till they thought they had exhausted all that there was
interesting in the figures already known. In this way of proceeding,
when they undertook the study of a new curve, the whole of the labour
bestowed on the preceding ones could not offer directly any essential
assistance, otherwise than by the geometrical practice to which it had
trained the mind. Whatever might be the real similarity of the questions
proposed as to two different figures, the complete knowledge acquired
for the one could not at all dispense with taking up again the whole of
the investigation for the other. Thus the progress of the mind was never
assured; so that they could not be certain, in advance, of obtaining any
solution whatever, however analogous the proposed problem might be to
questions which had been already resolved. Thus, for example, the
determination of the tangents to the three conic sections did not
furnish any rational assistance for drawing the tangent to any other new
curve, such as the conchoid, the cissoid, &c. In a word, the geometry of
the ancients was, according to the expression proposed above,
essentially special.


_Geometry of the Moderns._ In the system of the moderns, geometry is, on
the contrary, eminently _general_, that is to say, relating to any
figures whatever. It is easy to understand, in the first place, that all
geometrical expressions of any interest may be proposed with reference
to all imaginable figures. This is seen directly in the fundamental
problems--of rectifications, quadratures, and cubatures--which
constitute, as has been shown, the final object of geometry. But this
remark is no less incontestable, even for investigations which relate to
the different _properties_ of lines and of surfaces, and of which the
most essential, such as the question of tangents or of tangent planes,
the theory of curvatures, &c., are evidently common to all figures
whatever. The very few investigations which are truly peculiar to
particular figures have only an extremely secondary importance. This
being understood, modern geometry consists essentially in abstracting,
in order to treat it by itself, in an entirely general manner, every
question relating to the same geometrical phenomenon, in whatever bodies
it may be considered. The application of the universal theories thus
constructed to the special determination of the phenomenon which is
treated of in each particular body, is now regarded as only a subaltern
labour, to be executed according to invariable rules, and the success of
which is certain in advance. This labour is, in a word, of the same
character as the numerical calculation of an analytical formula. There
can be no other merit in it than that of presenting in each case the
solution which is necessarily furnished by the general method, with all
the simplicity and elegance which the line or the surface considered can
admit of. But no real importance is attached to any thing but the
conception and the complete solution of a new question belonging to any
figure whatever. Labours of this kind are alone regarded as producing
any real advance in science. The attention of geometers, thus relieved
from the examination of the peculiarities of different figures, and
wholly directed towards general questions, has been thereby able to
elevate itself to the consideration of new geometrical conceptions,
which, applied to the curves studied by the ancients, have led to the
discovery of important properties which they had not before even
suspected. Such is geometry, since the radical revolution produced by
Descartes in the general system of the science.


_The Superiority of the modern Geometry._ The mere indication of the
fundamental character of each of the two geometries is undoubtedly
sufficient to make apparent the immense necessary superiority of modern
geometry. We may even say that, before the great conception of
Descartes, rational geometry was not truly constituted upon definitive
bases, whether in its abstract or concrete relations. In fact, as
regards science, considered speculatively, it is clear that, in
continuing indefinitely to follow the course of the ancients, as did the
moderns before Descartes, and even for a little while afterwards, by
adding some new curves to the small number of those which they had
studied, the progress thus made, however rapid it might have been, would
still be found, after a long series of ages, to be very inconsiderable
in comparison with the general system of geometry, seeing the infinite
variety of the forms which would still have remained to be studied. On
the contrary, at each question resolved according to the method of the
moderns, the number of geometrical problems to be resolved is then, once
for all, diminished by so much with respect to all possible bodies.
Another consideration is, that it resulted, from their complete want of
general methods, that the ancient geometers, in all their
investigations, were entirely abandoned to their own strength, without
ever having the certainty of obtaining, sooner or later, any solution
whatever. Though this imperfection of the science was eminently suited
to call forth all their admirable sagacity, it necessarily rendered
their progress extremely slow; we can form some idea of this by the
considerable time which they employed in the study of the conic
sections. Modern geometry, making the progress of our mind certain,
permits us, on the contrary, to make the greatest possible use of the
forces of our intelligence, which the ancients were often obliged to
waste on very unimportant questions.

A no less important difference between the two systems appears when we
come to consider geometry in the concrete point of view. Indeed, we have
already remarked that the relation of the abstract to the concrete in
geometry can be founded upon rational bases only so far as the
investigations are made to bear directly upon all imaginable figures. In
studying lines, only one by one, whatever may be the number, always
necessarily very small, of those which we shall have considered, the
application of such theories to figures really existing in nature will
never have any other than an essentially accidental character, since
there is nothing to assure us that these figures can really be brought
under the abstract types considered by geometers.

Thus, for example, there is certainly something fortuitous in the happy
relation established between the speculations of the Greek geometers
upon the conic sections and the determination of the true planetary
orbits. In continuing geometrical researches upon the same plan, there
was no good reason for hoping for similar coincidences; and it would
have been possible, in these special studies, that the researches of
geometers should have been directed to abstract figures entirely
incapable of any application, while they neglected others, susceptible
perhaps of an important and immediate application. It is clear, at
least, that nothing positively guaranteed the necessary applicability of
geometrical speculations. It is quite another thing in the modern
geometry. From the single circumstance that in it we proceed by general
questions relating to any figures whatever, we have in advance the
evident certainty that the figures really existing in the external world
could in no case escape the appropriate theory if the geometrical
phenomenon which it considers presents itself in them.

From these different considerations, we see that the ancient system of
geometry wears essentially the character of the infancy of the science,
which did not begin to become completely rational till after the
philosophical resolution produced by Descartes. But it is evident, on
the other hand, that geometry could not be at first conceived except in
this _special_ manner. _General_ geometry would not have been possible,
and its necessity could not even have been felt, if a long series of
special labours on the most simple figures had not previously furnished
bases for the conception of Descartes, and rendered apparent the
impossibility of persisting indefinitely in the primitive geometrical
philosophy.


_The Ancient the Base of the Modern._ From this last consideration we
must infer that, although the geometry which I have called _general_
must be now regarded as the only true dogmatical geometry, and that to
which we shall chiefly confine ourselves, the other having no longer
much more than an historical interest, nevertheless it is not possible
to entirely dispense with _special_ geometry in a rational exposition of
the science. We undoubtedly need not borrow directly from ancient
geometry all the results which it has furnished; but, from the very
nature of the subject, it is necessarily impossible entirely to dispense
with the ancient method, which will always serve as the preliminary
basis of the science, dogmatically as well as historically. The reason
of this is easy to understand. In fact, _general_ geometry being
essentially founded, as we shall soon establish, upon the employment of
the calculus in the transformation of geometrical into analytical
considerations, such a manner of proceeding could not take possession of
the subject immediately at its origin. We know that the application of
mathematical analysis, from its nature, can never commence any science
whatever, since evidently it cannot be employed until the science has
already been sufficiently cultivated to establish, with respect to the
phenomena considered, some _equations_ which can serve as starting
points for the analytical operations. These fundamental equations being
once discovered, analysis will enable us to deduce from them a multitude
of consequences which it would have been previously impossible even to
suspect; it will perfect the science to an immense degree, both with
respect to the generality of its conceptions and to the complete
co-ordination established between them. But mere mathematical analysis
could never be sufficient to form the bases of any natural science, not
even to demonstrate them anew when they have once been established.
Nothing can dispense with the direct study of the subject, pursued up to
the point of the discovery of precise relations.

We thus see that the geometry of the ancients will always have, by its
nature, a primary part, absolutely necessary and more or less extensive,
in the complete system of geometrical knowledge. It forms a rigorously
indispensable introduction to _general_ geometry. But it is to this that
it must be limited in a completely dogmatic exposition. I will consider,
then, directly, in the following chapter, this _special_ or
_preliminary_ geometry restricted to exactly its necessary limits, in
order to occupy myself thenceforth only with the philosophical
examination of _general_ or _definitive_ geometry, the only one which is
truly rational, and which at present essentially composes the science.




CHAPTER II.

ANCIENT OR SYNTHETIC GEOMETRY.


The geometrical method of the ancients necessarily constituting a
preliminary department in the dogmatical system of geometry, designed to
furnish _general_ geometry with indispensable foundations, it is now
proper to begin with determining wherein strictly consists this
preliminary function of _special_ geometry, thus reduced to the
narrowest possible limits.




ITS PROPER EXTENT.


_Lines; Polygons; Polyhedrons._ In considering it under this point of
view, it is easy to recognize that we might restrict it to the study of
the right line alone for what concerns the geometry of _lines_; to the
_quadrature_ of rectilinear plane areas; and, lastly, to the _cubature_
of bodies terminated by plane faces. The elementary propositions
relating to these three fundamental questions form, in fact, the
necessary starting point of all geometrical inquiries; they alone cannot
be obtained except by a direct study of the subject; while, on the
contrary, the complete theory of all other figures, even that of the
circle, and of the surfaces and volumes which are connected with it, may
at the present day be completely comprehended in the domain of _general_
or _analytical_ geometry; these primitive elements at once furnishing
_equations_ which are sufficient to allow of the application of the
calculus to geometrical questions, which would not have been possible
without this previous condition.

It results from this consideration that, in common practice, we give to
_elementary_ geometry more extent than would be rigorously necessary to
it; since, besides the right line, polygons, and polyhedrons, we also
include in it the circle and the "round" bodies; the study of which
might, however, be as purely analytical as that, for example, of the
conic sections. An unreflecting veneration for antiquity contributes to
maintain this defect in method; but the best reason which can be given
for it is the serious inconvenience for ordinary instruction which there
would be in postponing, to so distant an epoch of mathematical
education, the solution of several essential questions, which are
susceptible of a direct and continual application to a great number of
important uses. In fact, to proceed in the most rational manner, we
should employ the integral calculus in obtaining the interesting results
relating to the length or the area of the circle, or to the quadrature
of the sphere, &c., which have been determined by the ancients from
extremely simple considerations. This inconvenience would be of little
importance with regard to the persons destined to study the whole of
mathematical science, and the advantage of proceeding in a perfectly
logical order would have a much greater comparative value. But the
contrary case being the more frequent, theories so essential have
necessarily been retained in elementary geometry. Perhaps the conic
sections, the cycloid, &c., might be advantageously added in such cases.


_Not to be farther restricted._ While this preliminary portion of
geometry, which cannot be founded on the application of the calculus,
is reduced by its nature to a very limited series of fundamental
researches, relating to the right line, polygonal areas, and
polyhedrons, it is certain, on the other hand, that we cannot restrict
it any more; although, by a veritable abuse of the spirit of analysis,
it has been recently attempted to present the establishment of the
principal theorems of elementary geometry under an algebraical point of
view. Thus some have pretended to demonstrate, by simple abstract
considerations of mathematical analysis, the constant relation which
exists between the three angles of a rectilinear triangle, the
fundamental proposition of the theory of similar triangles, that of
parallelopipedons, &c.; in a word, precisely the only geometrical
propositions which cannot be obtained except by a direct study of the
subject, without the calculus being susceptible of having any part in
it. Such aberrations are the unreflecting exaggerations of that natural
and philosophical tendency which leads us to extend farther and farther
the influence of analysis in mathematical studies. In mechanics, the
pretended analytical demonstrations of the parallelogram of forces are
of similar character.

The viciousness of such a manner of proceeding follows from the
principles previously presented. We have already, in fact, recognized
that, since the calculus is not, and cannot be, any thing but a means of
deduction, it would indicate a radically false idea of it to wish to
employ it in establishing the elementary foundations of any science
whatever; for on what would the analytical reasonings in such an
operation repose? A labour of this nature, very far from really
perfecting the philosophical character of a science, would constitute a
return towards the metaphysical age, in presenting real facts as mere
logical abstractions.

When we examine in themselves these pretended analytical demonstrations
of the fundamental propositions of elementary geometry, we easily verify
their necessary want of meaning. They are all founded on a vicious
manner of conceiving the principle of _homogeneity_, the true general
idea of which was explained in the second chapter of the preceding book.
These demonstrations suppose that this principle does not allow us to
admit the coexistence in the same equation of numbers obtained by
different concrete comparisons, which is evidently false, and contrary
to the constant practice of geometers. Thus it is easy to recognize
that, by employing the law of homogeneity in this arbitrary and
illegitimate acceptation, we could succeed in "demonstrating," with
quite as much apparent rigour, propositions whose absurdity is manifest
at the first glance. In examining attentively, for example, the
procedure by the aid of which it has been attempted to prove
analytically that the sum of the three angles of any rectilinear
triangle is constantly equal to two right angles, we see that it is
founded on this preliminary principle that, if two triangles have two of
their angles respectively equal, the third angle of the one will
necessarily be equal to the third angle of the other. This first point
being granted, the proposed relation is immediately deduced from it in a
very exact and simple manner. Now the analytical consideration by which
this previous proposition has been attempted to be established, is of
such a nature that, if it could be correct, we could rigorously deduce
from it, in reproducing it conversely, this palpable absurdity, that two
sides of a triangle are sufficient, without any angle, for the entire
determination of the third side. We may make analogous remarks on all
the demonstrations of this sort, the sophisms of which will be thus
verified in a perfectly apparent manner.

The more reason that we have here to consider geometry as being at the
present day essentially analytical, the more necessary was it to guard
against this abusive exaggeration of mathematical analysis, according to
which all geometrical observation would be dispensed with, in
establishing upon pure algebraical abstractions the very foundations of
this natural science.


_Attempted Demonstrations of Axioms, &c._ Another indication that
geometers have too much overlooked the character of a natural science
which is necessarily inherent in geometry, appears from their vain
attempts, so long made, to _demonstrate_ rigorously, not by the aid of
the calculus, but by means of certain constructions, several fundamental
propositions of elementary geometry. Whatever may be effected, it will
evidently be impossible to avoid sometimes recurring to simple and
direct observation in geometry as a means of establishing various
results. While, in this science, the phenomena which are considered are,
by virtue of their extreme simplicity, much more closely connected with
one another than those relating to any other physical science, some must
still be found which cannot be deduced, and which, on the contrary,
serve as starting points. It may be admitted that the greatest logical
perfection of the science is to reduce these to the smallest number
possible, but it would be absurd to pretend to make them completely
disappear. I avow, moreover, that I find fewer real inconveniences in
extending, a little beyond what would be strictly necessary, the number
of these geometrical notions thus established by direct observation,
provided they are sufficiently simple, than in making them the subjects
of complicated and indirect demonstrations, even when these
demonstrations may be logically irreproachable.

The true dogmatic destination of the geometry of the ancients, reduced
to its least possible indispensable developments, having thus been
characterized as exactly as possible, it is proper to consider summarily
each of the principal parts of which it must be composed. I think that I
may here limit myself to considering the first and the most extensive of
these parts, that which has for its object the study of _the right
line_; the two other sections, namely, the _quadrature of polygons_ and
the _cubature of polyhedrons_, from their limited extent, not being
capable of giving rise to any philosophical consideration of any
importance, distinct from those indicated in the preceding chapter with
respect to the measure of areas and of volumes in general.




GEOMETRY OF THE RIGHT LINE.


The final question which we always have in view in the study of the
right line, properly consists in determining, by means of one another,
the different elements of any right-lined figure whatever; which enables
us always to know indirectly the length and position of a right line, in
whatever circumstances it may be placed. This fundamental problem is
susceptible of two general solutions, the nature of which is quite
distinct, the one _graphical_, the other _algebraic_. The first, though
very imperfect, is that which must be first considered, because it is
spontaneously derived from the direct study of the subject; the second,
much more perfect in the most important respects, cannot be studied till
afterwards, because it is founded upon the previous knowledge of the
other.




GRAPHICAL SOLUTIONS.


The graphical solution consists in constructing at will the proposed
figure, either with the same dimensions, or, more usually, with
dimensions changed in any ratio whatever. The first mode need merely be
mentioned as being the most simple and the one which would first occur
to the mind, for it is evidently, by its nature, almost entirely
incapable of application. The second is, on the contrary, susceptible of
being most extensively and most usefully applied. We still make an
important and continual use of it at the present day, not only to
represent with exactness the forms of bodies and their relative
positions, but even for the actual determination of geometrical
magnitudes, when we do not need great precision. The ancients, in
consequence of the imperfection of their geometrical knowledge, employed
this procedure in a much more extensive manner, since it was for a long
time the only one which they could apply, even in the most important
precise determinations. It was thus, for example, that Aristarchus of
Samos estimated the relative distance from the sun and from the moon to
the earth, by making measurements on a triangle constructed as exactly
as possible, so as to be similar to the right-angled triangle formed by
the three bodies at the instant when the moon is in quadrature, and when
an observation of the angle at the earth would consequently be
sufficient to define the triangle. Archimedes himself, although he was
the first to introduce calculated determinations into geometry, several
times employed similar means. The formation of trigonometry did not
cause this method to be entirely abandoned, although it greatly
diminished its use; the Greeks and the Arabians continued to employ it
for a great number of researches, in which we now regard the use of the
calculus as indispensable.

This exact reproduction of any figure whatever on a different scale
cannot present any great theoretical difficulty when all the parts of
the proposed figure lie in the same plane. But if we suppose, as most
frequently happens, that they are situated in different planes, we see,
then, a new order of geometrical considerations arise. The artificial
figure, which is constantly plane, not being capable, in that case, of
being a perfectly faithful image of the real figure, it is necessary
previously to fix with precision the mode of representation, which gives
rise to different systems of _Projection_.

It then remains to be determined according to what laws the geometrical
phenomena correspond in the two figures. This consideration generates a
new series of geometrical investigations, the final object of which is
properly to discover how we can replace constructions in relief by plane
constructions. The ancients had to resolve several elementary questions
of this kind for various cases in which we now employ spherical
trigonometry, principally for different problems relating to the
celestial sphere. Such was the object of their _analemmas_, and of the
other plane figures which for a long time supplied the place of the
calculus. We see by this that the ancients really knew the elements of
what we now name _Descriptive Geometry_, although they did not conceive
it in a distinct and general manner.

I think it proper briefly to indicate in this place the true
philosophical character of this "Descriptive Geometry;" although, being
essentially a science of application, it ought not to be included within
the proper domain of this work.




DESCRIPTIVE GEOMETRY.


All questions of geometry of three dimensions necessarily give rise,
when we consider their graphical solution, to a common difficulty which
is peculiar to them; that of substituting for the different
constructions in relief, which are necessary to resolve them directly,
and which it is almost always impossible to execute, simple equivalent
plane constructions, by means of which we finally obtain the same
results. Without this indispensable transformation, every solution of
this kind would be evidently incomplete and really inapplicable in
practice, although theoretically the constructions in space are usually
preferable as being more direct. It was in order to furnish general
means for always effecting such a transformation that _Descriptive
Geometry_ was created, and formed into a distinct and homogeneous
system, by the illustrious MONGE. He invented, in the first place, a
uniform method of representing bodies by figures traced on a single
plane, by the aid of _projections_ on two different planes, usually
perpendicular to each other, and one of which is supposed to turn about
their common intersection so as to coincide with the other produced; in
this system, or in any other equivalent to it, it is sufficient to
regard points and lines as being determined by their projections, and
surfaces by the projections of their generating lines. This being
established, Monge--analyzing with profound sagacity the various partial
labours of this kind which had before been executed by a number of
incongruous procedures, and considering also, in a general and direct
manner, in what any questions of that nature must consist--found that
they could always be reduced to a very small number of invariable
abstract problems, capable of being resolved separately, once for all,
by uniform operations, relating essentially some to the contacts and
others to the intersections of surfaces. Simple and entirely general
methods for the graphical solution of these two orders of problems
having been formed, all the geometrical questions which may arise in any
of the various arts of construction--stone-cutting, carpentry,
perspective, dialling, fortification, &c.--can henceforth be treated as
simple particular cases of a single theory, the invariable application
of which will always necessarily lead to an exact solution, which may be
facilitated in practice by profiting by the peculiar circumstances of
each case.

       *       *       *       *       *

This important creation deserves in a remarkable degree to fix the
attention of those philosophers who consider all that the human species
has yet effected as a first step, and thus far the only really complete
one, towards that general renovation of human labours, which must
imprint upon all our arts a character of precision and of rationality,
so necessary to their future progress. Such a revolution must, in fact,
inevitably commence with that class of industrial labours, which is
essentially connected with that science which is the most simple, the
most perfect, and the most ancient. It cannot fail to extend hereafter,
though with less facility, to all other practical operations. Indeed
Monge himself, who conceived the true philosophy of the arts better than
any one else, endeavoured to sketch out a corresponding system for the
mechanical arts.

Essential as the conception of descriptive geometry really is, it is
very important not to deceive ourselves with respect to its true
destination, as did those who, in the excitement of its first discovery,
saw in it a means of enlarging the general and abstract domain of
rational geometry. The result has in no way answered to these mistaken
hopes. And, indeed, is it not evident that descriptive geometry has no
special value except as a science of application, and as forming the
true special theory of the geometrical arts? Considered in its abstract
relations, it could not introduce any truly distinct order of
geometrical speculations. We must not forget that, in order that a
geometrical question should fall within the peculiar domain of
descriptive geometry, it must necessarily have been previously resolved
by speculative geometry, the solutions of which then, as we have seen,
always need to be prepared for practice in such a way as to supply the
place of constructions in relief by plane constructions; a substitution
which really constitutes the only characteristic function of descriptive
geometry.

It is proper, however, to remark here, that, with regard to intellectual
education, the study of descriptive geometry possesses an important
philosophical peculiarity, quite independent of its high industrial
utility. This is the advantage which it so pre-eminently offers--in
habituating the mind to consider very complicated geometrical
combinations in space, and to follow with precision their continual
correspondence with the figures which are actually traced--of thus
exercising to the utmost, in the most certain and precise manner, that
important faculty of the human mind which is properly called
"imagination," and which consists, in its elementary and positive
acceptation, in representing to ourselves, clearly and easily, a vast
and variable collection of ideal objects, as if they were really before
us.

Finally, to complete the indication of the general nature of descriptive
geometry by determining its logical character, we have to observe that,
while it belongs to the geometry of the ancients by the character of its
solutions, on the other hand it approaches the geometry of the moderns
by the nature of the questions which compose it. These questions are in
fact eminently remarkable for that generality which, as we saw in the
preceding chapter, constitutes the true fundamental character of modern
geometry; for the methods used are always conceived as applicable to any
figures whatever, the peculiarity of each having only a purely secondary
influence. The solutions of descriptive geometry are then graphical,
like most of those of the ancients, and at the same time general, like
those of the moderns.

       *       *       *       *       *

After this important digression, we will pursue the philosophical
examination of _special_ geometry, always considered as reduced to its
least possible development, as an indispensable introduction to
_general_ geometry. We have now sufficiently considered the _graphical_
solution of the fundamental problem relating to the right line--that
is, the determination of the different elements of any right-lined
figure by means of one another--and have now to examine in a special
manner the _algebraic_ solution.




ALGEBRAIC SOLUTIONS.


This kind of solution, the evident superiority of which need not here be
dwelt upon, belongs necessarily, by the very nature of the question, to
the system of the ancient geometry, although the logical method which is
employed causes it to be generally, but very improperly, separated from
it. We have thus an opportunity of verifying, in a very important
respect, what was established generally in the preceding chapter, that
it is not by the employment of the calculus that the modern geometry is
essentially to be distinguished from the ancient. The ancients are in
fact the true inventors of the present trigonometry, spherical as well
as rectilinear; it being only much less perfect in their hands, on
account of the extreme inferiority of their algebraical knowledge. It
is, then, really in this chapter, and not, as it might at first be
thought, in those which we shall afterwards devote to the philosophical
examination of _general_ geometry, that it is proper to consider the
character of this important preliminary theory, which is usually, though
improperly, included in what is called _analytical geometry_, but which
is really only a complement of _elementary geometry_ properly so called.

Since all right-lined figures can be decomposed into triangles, it is
evidently sufficient to know how to determine the different elements of
a triangle by means of one another, which reduces _polygonometry_ to
simple _trigonometry_.




TRIGONOMETRY.


The difficulty in resolving algebraically such a question as the above,
consists essentially in forming, between the angles and the sides of a
triangle, three distinct equations; which, when once obtained, will
evidently reduce all trigonometrical problems to mere questions of
analysis.


_How to introduce Angles._ In considering the establishment of these
equations in the most general manner, we immediately meet with a
fundamental distinction with respect to the manner of introducing the
angles into the calculation, according as they are made to enter
_directly_, by themselves or by the circular arcs which are proportional
to them; or _indirectly_, by the chords of these arcs, which are hence
called their _trigonometrical lines_. Of these two systems of
trigonometry the second was of necessity the only one originally
adopted, as being the only practicable one, since the condition of
geometry made it easy enough to find exact relations between the sides
of the triangles and the trigonometrical lines which represent the
angles, while it would have been absolutely impossible at that epoch to
establish equations between the sides and the angles themselves.


_Advantages of introducing Trigonometrical Lines._ At the present day,
since the solution can be obtained by either system indifferently, that
motive for preference no longer exists; but geometers have none the less
persisted in following from choice the system primitively admitted from
necessity; for, the same reason which enabled these trigonometrical
equations to be obtained with much more facility, must, in like manner,
as it is still more easy to conceive _à priori_, render these equations
much more simple, since they then exist only between right lines,
instead of being established between right lines and arcs of circles.
Such a consideration has so much the more importance, as the question
relates to formulas which are eminently elementary, and destined to be
continually employed in all parts of mathematical science, as well as in
all its various applications.

It may be objected, however, that when an angle is given, it is, in
reality, always given by itself, and not by its trigonometrical lines;
and that when it is unknown, it is its angular value which is properly
to be determined, and not that of any of its trigonometrical lines. It
seems, according to this, that such lines are only useless
intermediaries between the sides and the angles, which have to be
finally eliminated, and the introduction of which does not appear
capable of simplifying the proposed research. It is indeed important to
explain, with more generality and precision than is customary, the great
real utility of this manner of proceeding.


_Division of Trigonometry into two Parts._ It consists in the fact that
the introduction of these auxiliary magnitudes divides the entire
question of trigonometry into two others essentially distinct, one of
which has for its object to pass from the angles to their
trigonometrical lines, or the converse, and the other of which proposes
to determine the sides of the triangles by the trigonometrical lines of
their angles, or the converse. Now the first of these two fundamental
questions is evidently susceptible, by its nature, of being entirely
treated and reduced to numerical tables once for all, in considering all
possible angles, since it depends only upon those angles, and not at all
upon the particular triangles in which they may enter in each case;
while the solution of the second question must necessarily be renewed,
at least in its arithmetical relations, for each new triangle which it
is necessary to resolve. This is the reason why the first portion of the
complete work, which would be precisely the most laborious, is no longer
taken into the account, being always done in advance; while, if such a
decomposition had not been performed, we would evidently have found
ourselves under the obligation of recommencing the entire calculation in
each particular case. Such is the essential property of the present
trigonometrical system, which in fact would really present no actual
advantage, if it was necessary to calculate continually the
trigonometrical line of each angle to be considered, or the converse;
the intermediate agency introduced would then be more troublesome than
convenient.

In order to clearly comprehend the true nature of this conception, it
will be useful to compare it with a still more important one, designed
to produce an analogous effect either in its algebraic, or, still more,
in its arithmetical relations--the admirable theory of _logarithms_. In
examining in a philosophical manner the influence of this theory, we see
in fact that its general result is to decompose all imaginable
arithmetical operations into two distinct parts. The first and most
complicated of these is capable of being executed in advance once for
all (since it depends only upon the numbers to be considered, and not at
all upon the infinitely different combinations into which they can
enter), and consists in considering all numbers as assignable powers of
a constant number. The second part of the calculation, which must of
necessity be recommenced for each new formula which is to have its
value determined, is thenceforth reduced to executing upon these
exponents correlative operations which are infinitely more simple. I
confine myself here to merely indicating this resemblance, which any one
can carry out for himself.

We must besides observe, as a property (secondary at the present day,
but all-important at its origin) of the trigonometrical system adopted,
the very remarkable circumstance that the determination of angles by
their trigonometrical lines, or the converse, admits of an arithmetical
solution (the only one which is directly indispensable for the special
destination of trigonometry) without the previous resolution of the
corresponding algebraic question. It is doubtless to such a peculiarity
that the ancients owed the possibility of knowing trigonometry. The
investigation conceived in this way was so much the more easy, inasmuch
as tables of chords (which the ancients naturally took as the
trigonometrical lines) had been previously constructed for quite a
different object, in the course of the labours of Archimedes on the
rectification of the circle, from which resulted the actual
determination of a certain series of chords; so that when Hipparchus
subsequently invented trigonometry, he could confine himself to
completing that operation by suitable intercalations; which shows
clearly the connexion of ideas in that matter.


_The Increase of such Trigonometrical Lines._ To complete this
philosophical sketch of trigonometry, it is proper now to observe that
the extension of the same considerations which lead us to replace angles
or arcs of circles by straight lines, with the view of simplifying our
equations, must also lead us to employ concurrently several
trigonometrical lines, instead of confining ourselves to one only (as
did the ancients), so as to perfect this system by choosing that one
which will be algebraically the most convenient on each occasion. In
this point of view, it is clear that the number of these lines is in
itself no ways limited; provided that they are determined by the arc,
and that they determine it, whatever may be the law according to which
they are derived from it, they are suitable to be substituted for it in
the equations. The Arabians, and subsequently the moderns, in confining
themselves to the most simple constructions, have carried to four or
five the number of _direct_ trigonometrical lines, which might be
extended much farther.

But instead of recurring to geometrical formations, which would finally
become very complicated, we conceive with the utmost facility as many
new trigonometrical lines as the analytical transformations may require,
by means of a remarkable artifice, which is not usually apprehended in a
sufficiently general manner. It consists in not directly multiplying the
trigonometrical lines appropriate to each arc considered, but in
introducing new ones, by considering this arc as indirectly determined
by all lines relating to an arc which is a very simple function of the
first. It is thus, for example, that, in order to calculate an angle
with more facility, we will determine, instead of its sine, the sine of
its half, or of its double, &c. Such a creation of _indirect_
trigonometrical lines is evidently much more fruitful than all the
direct geometrical methods for obtaining new ones. We may accordingly
say that the number of trigonometrical lines actually employed at the
present day by geometers is in reality unlimited, since at every
instant, so to say, the transformations of analysis may lead us to
augment it by the method which I have just indicated. Special names,
however, have been given to those only of these _indirect_ lines which
refer to the complement of the primitive arc, the others not occurring
sufficiently often to render such denominations necessary; a
circumstance which has caused a common misconception of the true extent
of the system of trigonometry.


_Study of their Mutual Relations._ This multiplicity of trigonometrical
lines evidently gives rise to a third fundamental question in
trigonometry, the study of the relations which exist between these
different lines; since, without such a knowledge, we could not make use,
for our analytical necessities, of this variety of auxiliary magnitudes,
which, however, have no other destination. It is clear, besides, from
the consideration just indicated, that this essential part of
trigonometry, although simply preparatory, is, by its nature,
susceptible of an indefinite extension when we view it in its entire
generality, while the two others are circumscribed within rigorously
defined limits.

It is needless to add that these three principal parts of trigonometry
have to be studied in precisely the inverse order from that in which we
have seen them necessarily derived from the general nature of the
subject; for the third is evidently independent of the two others, and
the second, of that which was first presented--the resolution of
triangles, properly so called--which must for that reason be treated in
the last place; which rendered so much the more important the
consideration of their natural succession and logical relations to one
another.

It is useless to consider here separately _spherical trigonometry_,
which cannot give rise to any special philosophical consideration;
since, essential as it is by the importance and the multiplicity of its
uses, it can be treated at the present day only as a simple application
of rectilinear trigonometry, which furnishes directly its fundamental
equations, by substituting for the spherical triangle the corresponding
trihedral angle.

This summary exposition of the philosophy of trigonometry has been here
given in order to render apparent, by an important example, that
rigorous dependence and those successive ramifications which are
presented by what are apparently the most simple questions of elementary
geometry.

       *       *       *       *       *

Having thus examined the peculiar character of _special_ geometry
reduced to its only dogmatic destination, that of furnishing to general
geometry an indispensable preliminary basis, we have now to give all our
attention to the true science of geometry, considered as a whole, in the
most rational manner. For that purpose, it is necessary to carefully
examine the great original idea of Descartes, upon which it is entirely
founded. This will be the object of the following chapter.




CHAPTER III.

MODERN OR ANALYTICAL GEOMETRY.


_General_ (or _Analytical_) geometry being entirely founded upon the
transformation of geometrical considerations into equivalent analytical
considerations, we must begin with examining directly and in a thorough
manner the beautiful conception by which Descartes has established in a
uniform manner the constant possibility of such a co-relation. Besides
its own extreme importance as a means of highly perfecting geometrical
science, or, rather, of establishing the whole of it on rational bases,
the philosophical study of this admirable conception must have so much
the greater interest in our eyes from its characterizing with perfect
clearness the general method to be employed in organizing the relations
of the abstract to the concrete in mathematics, by the analytical
representation of natural phenomena. There is no conception, in the
whole philosophy of mathematics which better deserves to fix all our
attention.




ANALYTICAL REPRESENTATION OF FIGURES.


In order to succeed in expressing all imaginable geometrical phenomena
by simple analytical relations, we must evidently, in the first place,
establish a general method for representing analytically the subjects
themselves in which these phenomena are found, that is, the lines or the
surfaces to be considered. The _subject_ being thus habitually
considered in a purely analytical point of view, we see how it is
thenceforth possible to conceive in the same manner the various
_accidents_ of which it is susceptible.

In order to organize the representation of geometrical figures by
analytical equations, we must previously surmount a fundamental
difficulty; that of reducing the general elements of the various
conceptions of geometry to simply numerical ideas; in a word, that of
substituting in geometry pure considerations of _quantity_ for all
considerations of _quality_.


_Reduction of Figure to Position._ For this purpose let us observe, in
the first place, that all geometrical ideas relate necessarily to these
three universal categories: the _magnitude_, the _figure_, and the
_position_ of the extensions to be considered. As to the first, there is
evidently no difficulty; it enters at once into the ideas of numbers.
With relation to the second, it must be remarked that it will always
admit of being reduced to the third. For the figure of a body evidently
results from the mutual position of the different points of which it is
composed, so that the idea of position necessarily comprehends that of
figure, and every circumstance of figure can be translated by a
circumstance of position. It is in this way, in fact, that the human
mind has proceeded in order to arrive at the analytical representation
of geometrical figures, their conception relating directly only to
positions. All the elementary difficulty is then properly reduced to
that of referring ideas of situation to ideas of magnitude. Such is the
direct destination of the preliminary conception upon which Descartes
has established the general system of analytical geometry.

His philosophical labour, in this relation, has consisted simply in the
entire generalization of an elementary operation, which we may regard as
natural to the human mind, since it is performed spontaneously, so to
say, in all minds, even the most uncultivated. Thus, when we have to
indicate the situation of an object without directly pointing it out,
the method which we always adopt, and evidently the only one which can
be employed, consists in referring that object to others which are
known, by assigning the magnitude of the various geometrical elements,
by which we conceive it connected with the known objects. These elements
constitute what Descartes, and after him all geometers, have called the
_co-ordinates_ of each point considered. They are necessarily two in
number, if it is known in advance in what plane the point is situated;
and three, if it may be found indifferently in any region of space. As
many different constructions as can be imagined for determining the
position of a point, whether on a plane or in space, so many distinct
systems of co-ordinates may be conceived; they are consequently
susceptible of being multiplied to infinity. But, whatever may be the
system adopted, we shall always have reduced the ideas of situation to
simple ideas of magnitude, so that we will consider the change in the
position of a point as produced by mere numerical variations in the
values of its co-ordinates.


_Determination of the Position of a Point._ Considering at first only
the least complicated case, that of _plane geometry_, it is in this way
that we usually determine the position of a point on a plane, by its
distances from two fixed right lines considered as known, which are
called _axes_, and which are commonly supposed to be perpendicular to
each other. This system is that most frequently adopted, because of its
simplicity; but geometers employ occasionally an infinity of others.
Thus the position of a point on a plane may be determined, 1°, by its
distances from two fixed points; or, 2°, by its distance from a single
fixed point, and the direction of that distance, estimated by the
greater or less angle which it makes with a fixed right line, which
constitutes the system of what are called _polar_ co-ordinates, the most
frequently used after the system first mentioned; or, 3°, by the angles
which the right lines drawn from the variable point to two fixed points
make with the right line which joins these last; or, 4°, by the
distances from that point to a fixed right line and a fixed point, &c.
In a word, there is no geometrical figure whatever from which it is not
possible to deduce a certain system of co-ordinates more or less
susceptible of being employed.

A general observation, which it is important to make in this connexion,
is, that every system of co-ordinates is equivalent to determining a
point, in plane geometry, by the intersection of two lines, each of
which is subjected to certain fixed conditions of determination; a
single one of these conditions remaining variable, sometimes the one,
sometimes the other, according to the system considered. We could not,
indeed, conceive any other means of constructing a point than to mark it
by the meeting of two lines. Thus, in the most common system, that of
_rectilinear co-ordinates_, properly so called, the point is determined
by the intersection of two right lines, each of which remains constantly
parallel to a fixed axis, at a greater or less distance from it; in the
_polar_ system, the position of the point is marked by the meeting of a
circle, of variable radius and fixed centre, with a movable right line
compelled to turn about this centre: in other systems, the required
point might be designated by the intersection of two circles, or of any
other two lines, &c. In a word, to assign the value of one of the
co-ordinates of a point in any system whatever, is always necessarily
equivalent to determining a certain line on which that point must be
situated. The geometers of antiquity had already made this essential
remark, which served as the base of their method of geometrical _loci_,
of which they made so happy a use to direct their researches in the
resolution of _determinate_ problems, in considering separately the
influence of each of the two conditions by which was defined each point
constituting the object, direct or indirect, of the proposed question.
It was the general systematization of this method which was the
immediate motive of the labours of Descartes, which led him to create
analytical geometry.

After having clearly established this preliminary conception--by means
of which ideas of position, and thence, implicitly, all elementary
geometrical conceptions are capable of being reduced to simple numerical
considerations--it is easy to form a direct conception, in its entire
generality, of the great original idea of Descartes, relative to the
analytical representation of geometrical figures: it is this which forms
the special object of this chapter. I will continue to consider at
first, for more facility, only geometry of two dimensions, which alone
was treated by Descartes; and will afterwards examine separately, under
the same point of view, the theory of surfaces and curves of double
curvature.




PLANE CURVES.


_Expression of Lines by Equations._ In accordance with the manner of
expressing analytically the position of a point on a plane, it can be
easily established that, by whatever property any line may be defined,
that definition always admits of being replaced by a corresponding
equation between the two variable co-ordinates of the point which
describes this line; an equation which will be thenceforth the
analytical representation of the proposed line, every phenomenon of
which will be translated by a certain algebraic modification of its
equation. Thus, if we suppose that a point moves on a plane without its
course being in any manner determined, we shall evidently have to regard
its co-ordinates, to whatever system they may belong, as two variables
entirely independent of one another. But if, on the contrary, this point
is compelled to describe a certain line, we shall necessarily be
compelled to conceive that its co-ordinates, in all the positions which
it can take, retain a certain permanent and precise relation to each
other, which is consequently susceptible of being expressed by a
suitable equation; which will become the very clear and very rigorous
analytical definition of the line under consideration, since it will
express an algebraical property belonging exclusively to the
co-ordinates of all the points of this line. It is clear, indeed, that
when a point is not subjected to any condition, its situation is not
determined except in giving at once its two co-ordinates, independently
of each other; while, when the point must continue upon a defined line,
a single co-ordinate is sufficient for completely fixing its position.
The second co-ordinate is then a determinate _function_ of the first;
or, in other words, there must exist between them a certain _equation_,
of a nature corresponding to that of the line on which the point is
compelled to remain. In a word, each of the co-ordinates of a point
requiring it to be situated on a certain line, we conceive reciprocally
that the condition, on the part of a point, of having to belong to a
line defined in any manner whatever, is equivalent to assigning the
value of one of the two co-ordinates; which is found in that case to be
entirely dependent on the other. The analytical relation which expresses
this dependence may be more or less difficult to discover, but it must
evidently be always conceived to exist, even in the cases in which our
present means may be insufficient to make it known. It is by this simple
consideration that we may demonstrate, in an entirely general
manner--independently of the particular verifications on which this
fundamental conception is ordinarily established for each special
definition of a line--the necessity of the analytical representation of
lines by equations.


_Expression of Equations by Lines._ Taking up again the same reflections
in the inverse direction, we could show as easily the geometrical
necessity of the representation of every equation of two variables, in a
determinate system of co-ordinates, by a certain line; of which such a
relation would be, in the absence of any other known property, a very
characteristic definition, the scientific destination of which will be
to fix the attention directly upon the general course of the solutions
of the equation, which will thus be noted in the most striking and the
most simple manner. This picturing of equations is one of the most
important fundamental advantages of analytical geometry, which has
thereby reacted in the highest degree upon the general perfecting of
analysis itself; not only by assigning to purely abstract researches a
clearly determined object and an inexhaustible career, but, in a still
more direct relation, by furnishing a new philosophical medium for
analytical meditation which could not be replaced by any other. In fact,
the purely algebraic discussion of an equation undoubtedly makes known
its solutions in the most precise manner, but in considering them only
one by one, so that in this way no general view of them could be
obtained, except as the final result of a long and laborious series of
numerical comparisons. On the other hand, the geometrical _locus_ of the
equation, being only designed to represent distinctly and with perfect
clearness the summing up of all these comparisons, permits it to be
directly considered, without paying any attention to the details which
have furnished it. It can thereby suggest to our mind general analytical
views, which we should have arrived at with much difficulty in any other
manner, for want of a means of clearly characterizing their object. It
is evident, for example, that the simple inspection of the logarithmic
curve, or of the curve _y_ = sin. _x_, makes us perceive much more
distinctly the general manner of the variations of logarithms with
respect to their numbers, or of sines with respect to their arcs, than
could the most attentive study of a table of logarithms or of natural
sines. It is well known that this method has become entirely elementary
at the present day, and that it is employed whenever it is desired to
get a clear idea of the general character of the law which reigns in a
series of precise observations of any kind whatever.


_Any Change in the Line causes a Change in the Equation._ Returning to
the representation of lines by equations, which is our principal object,
we see that this representation is, by its nature, so faithful, that the
line could not experience any modification, however slight it might be,
without causing a corresponding change in the equation. This perfect
exactitude even gives rise oftentimes to special difficulties; for
since, in our system of analytical geometry, the mere displacements of
lines affect the equations, as well as their real variations in
magnitude or form, we should be liable to confound them with one another
in our analytical expressions, if geometers had not discovered an
ingenious method designed expressly to always distinguish them. This
method is founded on this principle, that although it is impossible to
change analytically at will the position of a line with respect to the
axes of the co-ordinates, we can change in any manner whatever the
situation of the axes themselves, which evidently amounts to the same;
then, by the aid of the very simple general formula by which this
transformation of the axes is produced, it becomes easy to discover
whether two different equations are the analytical expressions of only
the same line differently situated, or refer to truly distinct
geometrical loci; since, in the former case, one of them will pass into
the other by suitably changing the axes or the other constants of the
system of co-ordinates employed. It must, moreover, be remarked on this
subject, that general inconveniences of this nature seem to be
absolutely inevitable in analytical geometry; for, since the ideas of
position are, as we have seen, the only geometrical ideas immediately
reducible to numerical considerations, and the conceptions of figure
cannot be thus reduced, except by seeing in them relations of situation,
it is impossible for analysis to escape confounding, at first, the
phenomena of figure with simple phenomena of position, which alone are
directly expressed by the equations.


_Every Definition of a Line is an Equation._ In order to complete the
philosophical explanation of the fundamental conception which serves as
the base of analytical geometry, I think that I should here indicate a
new general consideration, which seems to me particularly well adapted
for putting in the clearest point of view this necessary representation
of lines by equations with two variables. It consists in this, that not
only, as we have shown, must every defined line necessarily give rise to
a certain equation between the two co-ordinates of any one of its
points, but, still farther, every definition of a line may be regarded
as being already of itself an equation of that line in a suitable system
of co-ordinates.

It is easy to establish this principle, first making a preliminary
logical distinction with respect to different kinds of definitions. The
rigorously indispensable condition of every definition is that of
distinguishing the object defined from all others, by assigning to it a
property which belongs to it exclusively. But this end may be generally
attained in two very different ways; either by a definition which is
simply _characteristic_, that is, indicative of a property which,
although truly exclusive, does not make known the mode of generation of
the object; or by a definition which is really _explanatory_, that is,
which characterizes the object by a property which expresses one of its
modes of generation. For example, in considering the circle as the line,
which, under the same contour, contains the greatest area, we have
evidently a definition of the first kind; while in choosing the property
of its having all its points equally distant from a fixed point, we have
a definition of the second kind. It is, besides, evident, as a general
principle, that even when any object whatever is known at first only by
a _characteristic_ definition, we ought, nevertheless, to regard it as
susceptible of _explanatory_ definitions, which the farther study of the
object would necessarily lead us to discover.

This being premised, it is clear that the general observation above
made, which represents every definition of a line as being necessarily
an equation of that line in a certain system of co-ordinates, cannot
apply to definitions which are simply _characteristic_; it is to be
understood only of definitions which are truly _explanatory_. But, in
considering only this class, the principle is easy to prove. In fact, it
is evidently impossible to define the generation of a line without
specifying a certain relation between the two simple motions of
translation or of rotation, into which the motion of the point which
describes it will be decomposed at each instant. Now if we form the most
general conception of what constitutes _a system of co-ordinates_, and
admit all possible systems, it is clear that such a relation will be
nothing else but the _equation_ of the proposed line, in a system of
co-ordinates of a nature corresponding to that of the mode of generation
considered. Thus, for example, the common definition of the _circle_ may
evidently be regarded as being immediately the _polar equation_ of this
curve, taking the centre of the circle for the pole. In the same way,
the elementary definition of the _ellipse_ or of the _hyperbola_--as
being the curve generated by a point which moves in such a manner that
the sum or the difference of its distances from two fixed points remains
constant--gives at once, for either the one or the other curve, the
equation _y_ + _x_ = _c_, taking for the system of co-ordinates that in
which the position of a point would be determined by its distances from
two fixed points, and choosing for these poles the two given foci. In
like manner, the common definition of any _cycloid_ would furnish
directly, for that curve, the equation _y_ = _mx_; adopting as the
co-ordinates of each point the arc which it marks upon a circle of
invariable radius, measuring from the point of contact of that circle
with a fixed line, and the rectilinear distance from that point of
contact to a certain origin taken on that right line. We can make
analogous and equally easy verifications with respect to the customary
definitions of spirals, of epicycloids, &c. We shall constantly find
that there exists a certain system of co-ordinates, in which we
immediately obtain a very simple equation of the proposed line, by
merely writing algebraically the condition imposed by the mode of
generation considered.

Besides its direct importance as a means of rendering perfectly apparent
the necessary representation of every line by an equation, the preceding
consideration seems to me to possess a true scientific utility, in
characterizing with precision the principal general difficulty which
occurs in the actual establishment of these equations, and in
consequently furnishing an interesting indication with respect to the
course to be pursued in inquiries of this kind, which, by their nature,
could not admit of complete and invariable rules. In fact, since any
definition whatever of a line, at least among those which indicate a
mode of generation, furnishes directly the equation of that line in a
certain system of co-ordinates, or, rather, of itself constitutes that
equation, it follows that the difficulty which we often experience in
discovering the equation of a curve, by means of certain of its
characteristic properties, a difficulty which is sometimes very great,
must proceed essentially only from the commonly imposed condition of
expressing this curve analytically by the aid of a designated system of
co-ordinates, instead of admitting indifferently all possible systems.
These different systems cannot be regarded in analytical geometry as
being all equally suitable; for various reasons, the most important of
which will be hereafter discussed, geometers think that curves should
almost always be referred, as far as is possible, to _rectilinear
co-ordinates_, properly so called. Now we see, from what precedes, that
in many cases these particular co-ordinates will not be those with
reference to which the equation of the curve will be found to be
directly established by the proposed definition. The principal
difficulty presented by the formation of the equation of a line really
consists, then, in general, in a certain transformation of co-ordinates.
It is undoubtedly true that this consideration does not subject the
establishment of these equations to a truly complete general method, the
success of which is always certain; which, from the very nature of the
subject, is evidently chimerical: but such a view may throw much useful
light upon the course which it is proper to adopt, in order to arrive at
the end proposed. Thus, after having in the first place formed the
preparatory equation, which is spontaneously derived from the definition
which we are considering, it will be necessary, in order to obtain the
equation belonging to the system of co-ordinates which must be finally
admitted, to endeavour to express in a function of these last
co-ordinates those which naturally correspond to the given mode of
generation. It is upon this last labour that it is evidently impossible
to give invariable and precise precepts. We can only say that we shall
have so many more resources in this matter as we shall know more of true
analytical geometry, that is, as we shall know the algebraical
expression of a greater number of different algebraical phenomena.




CHOICE OF CO-ORDINATES.


In order to complete the philosophical exposition of the conception
which serves as the base of analytical geometry, I have yet to notice
the considerations relating to the choice of the system of co-ordinates
which is in general the most suitable. They will give the rational
explanation of the preference unanimously accorded to the ordinary
rectilinear system; a preference which has hitherto been rather the
effect of an empirical sentiment of the superiority of this system, than
the exact result of a direct and thorough analysis.


_Two different Points of View._ In order to decide clearly between all
the different systems of co-ordinates, it is indispensable to
distinguish with care the two general points of view, the converse of
one another, which belong to analytical geometry; namely, the relation
of algebra to geometry, founded upon the representation of lines by
equations; and, reciprocally, the relation of geometry to algebra,
founded on the representation of equations by lines.

It is evident that in every investigation of general geometry these two
fundamental points of view are of necessity always found combined,
since we have always to pass alternately, and at insensible intervals,
so to say, from geometrical to analytical considerations, and from
analytical to geometrical considerations. But the necessity of here
temporarily separating them is none the less real; for the answer to the
question of method which we are examining is, in fact, as we shall see
presently, very far from being the same in both these relations, so that
without this distinction we could not form any clear idea of it.


1. _Representation of Lines by Equations._ Under _the first point of
view_--the representation of lines by equations--the only reason which
could lead us to prefer one system of co-ordinates to another would be
the greater simplicity of the equation of each line, and greater
facility in arriving at it. Now it is easy to see that there does not
exist, and could not be expected to exist, any system of co-ordinates
deserving in that respect a constant preference over all others. In
fact, we have above remarked that for each geometrical definition
proposed we can conceive a system of co-ordinates in which the equation
of the line is obtained at once, and is necessarily found to be also
very simple; and this system, moreover, inevitably varies with the
nature of the characteristic property under consideration. The
rectilinear system could not, therefore, be constantly the most
advantageous for this object, although it may often be very favourable;
there is probably no system which, in certain particular cases, should
not be preferred to it, as well as to every other.


2. _Representation of Equations by Lines._ It is by no means so,
however, under the _second point of view_. We can, indeed, easily
establish, as a general principle, that the ordinary rectilinear system
must necessarily be better adapted than any other to the representation
of equations by the corresponding geometrical _loci_; that is to say,
that this representation is constantly more simple and more faithful in
it than in any other.

Let us consider, for this object, that, since every system of
co-ordinates consists in determining a point by the intersection of two
lines, the system adapted to furnish the most suitable geometrical
_loci_ must be that in which these two lines are the simplest possible;
a consideration which confines our choice to the _rectilinear_ system.
In truth, there is evidently an infinite number of systems which deserve
that name, that is to say, which employ only right lines to determine
points, besides the ordinary system which assigns the distances from two
fixed lines as co-ordinates; such, for example, would be that in which
the co-ordinates of each point should be the two angles which the right
lines, which go from that point to two fixed points, make with the right
line, which joins these last points: so that this first consideration is
not rigorously sufficient to explain the preference unanimously given to
the common system. But in examining in a more thorough manner the nature
of every system of co-ordinates, we also perceive that each of the two
lines, whose meeting determines the point considered, must necessarily
offer at every instant, among its different conditions of determination,
a single variable condition, which gives rise to the corresponding
co-ordinate, all the rest being fixed, and constituting the _axes_ of
the system, taking this term in its most extended mathematical
acceptation. The variation is indispensable, in order that we may be
able to consider all possible positions; and the fixity is no less so,
in order that there may exist means of comparison. Thus, in all
_rectilinear_ systems, each of the two right lines will be subjected to
a fixed condition, and the ordinate will result from the variable
condition.


_Superiority of rectilinear Co-ordinates._ From these considerations it
is evident, as a general principle, that the most favourable system for
the construction of geometrical _loci_ will necessarily be that in which
the variable condition of each right line shall be the simplest
possible; the fixed condition being left free to be made complex, if
necessary to attain that object. Now, of all possible manners of
determining two movable right lines, the easiest to follow geometrically
is certainly that in which, the direction of each right line remaining
invariable, it only approaches or recedes, more or less, to or from a
constant axis. It would be, for example, evidently more difficult to
figure to one's self clearly the changes of place of a point which is
determined by the intersection of two right lines, which each turn
around a fixed point, making a greater or smaller angle with a certain
axis, as in the system of co-ordinates previously noticed. Such is the
true general explanation of the fundamental property possessed by the
common rectilinear system, of being better adapted than any other to the
geometrical representation of equations, inasmuch as it is that one in
which it is the easiest to conceive the change of place of a point
resulting from the change in the value of its co-ordinates. In order to
feel clearly all the force of this consideration, it would be sufficient
to carefully compare this system with the polar system, in which this
geometrical image, so simple and so easy to follow, of two right lines
moving parallel, each one of them, to its corresponding axis, is
replaced by the complicated picture of an infinite series of concentric
circles, cut by a right line compelled to turn about a fixed point. It
is, moreover, easy to conceive in advance what must be the extreme
importance to analytical geometry of a property so profoundly
elementary, which, for that reason, must be recurring at every instant,
and take a progressively increasing value in all labours of this kind.


_Perpendicularity of the Axes._ In pursuing farther the consideration
which demonstrates the superiority of the ordinary system of
co-ordinates over any other as to the representation of equations, we
may also take notice of the utility for this object of the common usage
of taking the two axes perpendicular to each other, whenever possible,
rather than with any other inclination. As regards the representation of
lines by equations, this secondary circumstance is no more universally
proper than we have seen the general nature of the system to be; since,
according to the particular occasion, any other inclination of the axes
may deserve our preference in that respect. But, in the inverse point of
view, it is easy to see that rectangular axes constantly permit us to
represent equations in a more simple and even more faithful manner; for,
with oblique axes, space being divided by them into regions which no
longer have a perfect identity, it follows that, if the geometrical
_locus_ of the equation extends into all these regions at once, there
will be presented, by reason merely of this inequality of the angles,
differences of figure which do not correspond to any analytical
diversity, and will necessarily alter the rigorous exactness of the
representation, by being confounded with the proper results of the
algebraic comparisons. For example, an equation like: _x^m_ + _y^m_ =
_c_, which, by its perfect symmetry, should evidently give a curve
composed of four identical quarters, will be represented, on the
contrary, if we take axes not rectangular, by a geometric _locus_, the
four parts of which will be unequal. It is plain that the only means of
avoiding all inconveniences of this kind is to suppose the angle of the
two axes to be a right angle.

The preceding discussion clearly shows that, although the ordinary
system of rectilinear co-ordinates has no constant superiority over all
others in one of the two fundamental points of view which are
continually combined in analytical geometry, yet as, on the other hand,
it is not constantly inferior, its necessary and absolute greater
aptitude for the representation of equations must cause it to generally
receive the preference; although it may evidently happen, in some
particular cases, that the necessity of simplifying equations and of
obtaining them more easily may determine geometers to adopt a less
perfect system. The rectilinear system is, therefore, the one by means
of which are ordinarily constructed the most essential theories of
general geometry, intended to express analytically the most important
geometrical phenomena. When it is thought necessary to choose some
other, the polar system is almost always the one which is fixed upon,
this system being of a nature sufficiently opposite to that of the
rectilinear system to cause the equations, which are too complicated
with respect to the latter, to become, in general, sufficiently simple
with respect to the other. Polar co-ordinates, moreover, have often the
advantage of admitting of a more direct and natural concrete
signification; as is the case in mechanics, for the geometrical
questions to which the theory of circular movement gives rise, and in
almost all the cases of celestial geometry.

       *       *       *       *       *

In order to simplify the exposition, we have thus far considered the
fundamental conception of analytical geometry only with respect to
_plane curves_, the general study of which was the only object of the
great philosophical renovation produced by Descartes. To complete this
important explanation, we have now to show summarily how this elementary
idea was extended by Clairaut, about a century afterwards, to the
general study of _surfaces_ and _curves of double curvature_. The
considerations which have been already given will permit me to limit
myself on this subject to the rapid examination of what is strictly
peculiar to this new case.




SURFACES.


_Determination of a Point in Space._ The complete analytical
determination of a point in space evidently requires the values of three
co-ordinates to be assigned; as, for example, in the system which is
generally adopted, and which corresponds to the _rectilinear_ system of
plane geometry, distances from the point to three fixed planes, usually
perpendicular to one another; which presents the point as the
intersection of three planes whose direction is invariable. We might
also employ the distances from the movable point to three fixed points,
which would determine it by the intersection of three spheres with a
common centre. In like manner, the position of a point would be defined
by giving its distance from a fixed point, and the direction of that
distance, by means of the two angles which this right line makes with
two invariable axes; this is the _polar_ system of geometry of three
dimensions; the point is then constructed by the intersection of a
sphere having a fixed centre, with two right cones with circular bases,
whose axes and common summit do not change. In a word, there is
evidently, in this case at least, the same infinite variety among the
various possible systems of co-ordinates which we have already observed
in geometry of two dimensions. In general, we have to conceive a point
as being always determined by the intersection of any three surfaces
whatever, as it was in the former case by that of two lines: each of
these three surfaces has, in like manner, all its conditions of
determination constant, excepting one, which gives rise to the
corresponding co-ordinates, whose peculiar geometrical influence is thus
to constrain the point to be situated upon that surface.

This being premised, it is clear that if the three co-ordinates of a
point are entirely independent of one another, that point can take
successively all possible positions in space. But if the point is
compelled to remain upon a certain surface defined in any manner
whatever, then two co-ordinates are evidently sufficient for determining
its situation at each instant, since the proposed surface will take the
place of the condition imposed by the third co-ordinate. We must then,
in this case, under the analytical point of view, necessarily conceive
this last co-ordinate as a determinate function of the two others, these
latter remaining perfectly independent of each other. Thus there will be
a certain equation between the three variable co-ordinates, which will
be permanent, and which will be the only one, in order to correspond to
the precise degree of indetermination in the position of the point.


_Expression of Surfaces by Equations._ This equation, more or less easy
to be discovered, but always possible, will be the analytical definition
of the proposed surface, since it must be verified for all the points of
that surface, and for them alone. If the surface undergoes any change
whatever, even a simple change of place, the equation must undergo a
more or less serious corresponding modification. In a word, all
geometrical phenomena relating to surfaces will admit of being
translated by certain equivalent analytical conditions appropriate to
equations of three variables; and in the establishment and
interpretation of this general and necessary harmony will essentially
consist the science of analytical geometry of three dimensions.


_Expression of Equations by Surfaces._ Considering next this fundamental
conception in the inverse point of view, we see in the same manner that
every equation of three variables may, in general, be represented
geometrically by a determinate surface, primitively defined by the very
characteristic property, that the co-ordinates of all its points always
retain the mutual relation enunciated in this equation. This geometrical
locus will evidently change, for the same equation, according to the
system of co-ordinates which may serve for the construction of this
representation. In adopting, for example, the rectilinear system, it is
clear that in the equation between the three variables, _x_, _y_, _z_,
every particular value attributed to _z_ will give an equation between
at _x_ and _y_, the geometrical locus of which will be a certain line
situated in a plane parallel to the plane of _x_ and _y_, and at a
distance from this last equal to the value of _z_; so that the complete
geometrical locus will present itself as composed of an infinite series
of lines superimposed in a series of parallel planes (excepting the
interruptions which may exist), and will consequently form a veritable
surface. It would be the same in considering any other system of
co-ordinates, although the geometrical construction of the equation
becomes more difficult to follow.

Such is the elementary conception, the complement of the original idea
of Descartes, on which is founded general geometry relative to surfaces.
It would be useless to take up here directly the other considerations
which have been above indicated, with respect to lines, and which any
one can easily extend to surfaces; whether to show that every definition
of a surface by any method of generation whatever is really a direct
equation of that surface in a certain system of co-ordinates, or to
determine among all the different systems of possible co-ordinates that
one which is generally the most convenient. I will only add, on this
last point, that the necessary superiority of the ordinary rectilinear
system, as to the representation of equations, is evidently still more
marked in analytical geometry of three dimensions than in that of two,
because of the incomparably greater geometrical complication which would
result from the choice of any other system. This can be verified in the
most striking manner by considering the polar system in particular,
which is the most employed after the ordinary rectilinear system, for
surfaces as well as for plane curves, and for the same reasons.

In order to complete the general exposition of the fundamental
conception relative to the analytical study of surfaces, a philosophical
examination should be made of a final improvement of the highest
importance, which Monge has introduced into the very elements of this
theory, for the classification of surfaces in natural families,
established according to the mode of generation, and expressed
algebraically by common differential equations, or by finite equations
containing arbitrary functions.




CURVES OF DOUBLE CURVATURE.


Let us now consider the last elementary point of view of analytical
geometry of three dimensions; that relating to the algebraic
representation of curves considered in space, in the most general
manner. In continuing to follow the principle which has been constantly
employed, that of the degree of indetermination of the geometrical
locus, corresponding to the degree of independence of the variables, it
is evident, as a general principle, that when a point is required to be
situated upon some certain curve, a single co-ordinate is enough for
completely determining its position, by the intersection of this curve
with the surface which results from this co-ordinate. Thus, in this
case, the two other co-ordinates of the point must be conceived as
functions necessarily determinate and distinct from the first. It
follows that every line, considered in space, is then represented
analytically, no longer by a single equation, but by the system of two
equations between the three co-ordinates of any one of its points. It is
clear, indeed, from another point of view, that since each of these
equations, considered separately, expresses a certain surface, their
combination presents the proposed line as the intersection of two
determinate surfaces. Such is the most general manner of conceiving the
algebraic representation of a line in analytical geometry of three
dimensions. This conception is commonly considered in too restricted a
manner, when we confine ourselves to considering a line as determined by
the system of its two _projections_ upon two of the co-ordinate planes;
a system characterized, analytically, by this peculiarity, that each of
the two equations of the line then contains only two of the three
co-ordinates, instead of simultaneously including the three variables.
This consideration, which consists in regarding the line as the
intersection of two cylindrical surfaces parallel to two of the three
axes of the co-ordinates, besides the inconvenience of being confined to
the ordinary rectilinear system, has the fault, if we strictly confine
ourselves to it, of introducing useless difficulties into the analytical
representation of lines, since the combination of these two cylinders
would evidently not be always the most suitable for forming the
equations of a line. Thus, considering this fundamental notion in its
entire generality, it will be necessary in each case to choose, from
among the infinite number of couples of surfaces, the intersection of
which might produce the proposed curve, that one which will lend itself
the best to the establishment of equations, as being composed of the
best known surfaces. Thus, if the problem is to express analytically a
circle in space, it will evidently be preferable to consider it as the
intersection of a sphere and a plane, rather than as proceeding from any
other combination of surfaces which could equally produce it.

In truth, this manner of conceiving the representation of lines by
equations, in analytical geometry of three dimensions, produces, by its
nature, a necessary inconvenience, that of a certain analytical
confusion, consisting in this: that the same line may thus be expressed,
with the same system of co-ordinates, by an infinite number of different
couples of equations, on account of the infinite number of couples of
surfaces which can form it; a circumstance which may cause some
difficulties in recognizing this line under all the algebraical
disguises of which it admits. But there exists a very simple method for
causing this inconvenience to disappear; it consists in giving up the
facilities which result from this variety of geometrical constructions.
It suffices, in fact, whatever may be the analytical system primitively
established for a certain line, to be able to deduce from it the system
corresponding to a single couple of surfaces uniformly generated; as,
for example, to that of the two cylindrical surfaces which _project_ the
proposed line upon two of the co-ordinate planes; surfaces which will
evidently be always identical, in whatever manner the line may have been
obtained, and which will not vary except when that line itself shall
change. Now, in choosing this fixed system, which is actually the most
simple, we shall generally be able to deduce from the primitive
equations those which correspond to them in this special construction,
by transforming them, by two successive eliminations, into two
equations, each containing only two of the variable co-ordinates, and
thereby corresponding to the two surfaces of projection. Such is really
the principal destination of this sort of geometrical combination, which
thus offers to us an invariable and certain means of recognizing the
identity of lines in spite of the diversity of their equations, which is
sometimes very great.




IMPERFECTIONS OF ANALYTICAL GEOMETRY.


Having now considered the fundamental conception of analytical geometry
under its principal elementary aspects, it is proper, in order to make
the sketch complete, to notice here the general imperfections yet
presented by this conception with respect to both geometry and to
analysis.

_Relatively to geometry_, we must remark that the equations are as yet
adapted to represent only entire geometrical loci, and not at all
determinate portions of those loci. It would, however, be necessary, in
some circumstances, to be able to express analytically a part of a line
or of a surface, or even a _discontinuous_ line or surface, composed of
a series of sections belonging to distinct geometrical figures, such as
the contour of a polygon, or the surface of a polyhedron. Thermology,
especially, often gives rise to such considerations, to which our
present analytical geometry is necessarily inapplicable. The labours of
M. Fourier on discontinuous functions have, however, begun to fill up
this great gap, and have thereby introduced a new and essential
improvement into the fundamental conception of Descartes. But this
manner of representing heterogeneous or partial figures, being founded
on the employment of trigonometrical series proceeding according to the
sines of an infinite series of multiple arcs, or on the use of certain
definite integrals equivalent to those series, and the general integral
of which is unknown, presents as yet too much complication to admit of
being immediately introduced into the system of analytical geometry.

_Relatively to analysis_, we must begin by observing that our inability
to conceive a geometrical representation of equations containing four,
five, or more variables, analogous to those representations which all
equations of two or of three variables admit, must not be viewed as an
imperfection of our system of analytical geometry, for it evidently
belongs to the very nature of the subject. Analysis being necessarily
more general than geometry, since it relates to all possible phenomena,
it would be very unphilosophical to desire always to find among
geometrical phenomena alone a concrete representation of all the laws
which analysis can express.

There exists, however, another imperfection of less importance, which
must really be viewed as proceeding from the manner in which we conceive
analytical geometry. It consists in the evident incompleteness of our
present representation of equations of two or of three variables by
lines or surfaces, inasmuch as in the construction of the geometric
locus we pay regard only to the _real_ solutions of equations, without
at all noticing any _imaginary_ solutions. The general course of these
last should, however, by its nature, be quite as susceptible as that of
the others of a geometrical representation. It follows from this
omission that the graphic picture of the equation is constantly
imperfect, and sometimes even so much so that there is no geometric
representation at all when the equation admits of only imaginary
solutions. But, even in this last case, we evidently ought to be able to
distinguish between equations as different in themselves as these, for
example,

  _x²_ + _y²_ + 1 = 0, _x⁶_ + _y⁴_ + 1 = 0, _y²_ + _e^x_ = 0.

We know, moreover, that this principal imperfection often brings with
it, in analytical geometry of two or of three dimensions, a number of
secondary inconveniences, arising from several analytical modifications
not corresponding to any geometrical phenomena.

       *       *       *       *       *

Our philosophical exposition of the fundamental conception of analytical
geometry shows us clearly that this science consists essentially in
determining what is the general analytical expression of such or such a
geometrical phenomenon belonging to lines or to surfaces; and,
reciprocally, in discovering the geometrical interpretation of such or
such an analytical consideration. A detailed examination of the most
important general questions would show us how geometers have succeeded
in actually establishing this beautiful harmony, and in thus imprinting
on geometrical science, regarded as a whole, its present eminently
perfect character of rationality and of simplicity.

     _Note._--The author devotes the two following chapters of his
     course to the more detailed examination of Analytical Geometry of
     two and of three dimensions; but his subsequent publication of a
     separate work upon this branch of mathematics has been thought to
     render unnecessary the reproduction of these two chapters in the
     present volume.


THE END.





End of Project Gutenberg's The philosophy of mathematics, by Auguste Comte