The philosophy of mathematics
by Auguste Comte · English · Gutenberg #39702 · 440,173 chars · public domain (US)
The Project Gutenberg eBook of The philosophy of mathematics
This eBook is for the use of anyone anywhere in the United States and
most other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms
of the Project Gutenberg License included with this eBook or online
at www.gutenberg.org. If you are not located in the United States,
you will have to check the laws of the country where you are located
before using this eBook.
Title: The philosophy of mathematics
Author: Auguste Comte
Translator: W. M. Gillespie
Release date: May 15, 2012 [eBook #39702]
Language: English
Other information and formats: www.gutenberg.org/ebooks/39702
Credits: Produced by Anna Hall, Albert László and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive)
*** START OF THE PROJECT GUTENBERG EBOOK THE PHILOSOPHY OF MATHEMATICS ***
Produced by Anna Hall, Albert László and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive)
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, i
Showing first 200,000 characters. Full text: gutenberg.org/ebooks/39702