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Title: The philosophy of mathematics

Author: Auguste Comte

Translator: W. M. Gillespie

Release date: May 15, 2012 [eBook #39702]
Most recently updated: September 1, 2020

Language: English

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*** START OF THE PROJECT GUTENBERG EBOOK THE
PHILOSOPHY OF MATHEMATICS ***

THE
PHILOSOPHY
OF
MATHEMATICS.

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: 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.

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.

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.

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
1⁰. Different Questions with respect to the
same Figure 204
2⁰. 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 212
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 232
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."

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.
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.
1° y = a + x Sum.
1st couple
2° y = a - x Difference.
1° y = ax Product.
2d couple
2° y = a/x Quotient.
1° y = x^a Power.
3d couple
2° y = [aroot]x Root.
1° y = a^x Exponential.
4th couple
2° y = [log a]x Logarithmic.
1° y = sin. x Direct Circular.
5th couple
2° y = arc(sin. = x). Inverse Circular.[3]

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.
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]
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.
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 . . . . . + pzm,
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 ax, 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, 1o, infinitely

small elements, the differentials (of different orders) of those quantities, if
we regard this analysis in the manner of Leibnitz; or, 2o, 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, 3o, 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.

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.
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
mth degree would necessarily include radicals of the mth 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]

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 dA, 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
dA = 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 = 2axdx + 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 = 2ax(dx),
being sufficient to eliminate entirely the infinitesimals, the finite result, t =
2ax, 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 Δy and
Δx, we would have at each instant, for the trigonometrical tangent of the
angle which the secant makes with the axis of abscissas,
t = Δy/Δx;
from which, taking the limits, we will obtain, relatively to the tangent itself,
this general formula of transcendental analysis,

t = L(Δy/Δx),
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
Δy = 2axΔx + a(Δx)2,
from which we shall obtain
Δy/Δx = 2ax + aΔx.
Now it is clear that the limit towards which the second number tends, in
proportion as Δx diminishes, is 2ax. We shall therefore find, by this method,
t = 2ax, 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/Δx)² = 1 + (LΔy/Δx)²,
or (LΔs/Δx)2 = 1 + (LΔy/Δx)2 + (LΔz/Δx)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(Δy/Δx) 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(Δy/Δx), 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,
ds2 = dy2 + dx2, or ds2 = dx2 + dy2 + dz2,
is not only established between the desired function s and the independent
variable x, to which it is referred, but, at the same time, there have been
introduced, as indispensable intermediaries, the differentials of one or two
other functions, y and z, which are among the data of the problem; it would
not have been possible to form directly the equation between ds and dx,
which would, besides, be peculiar to each curve considered. It is the same
for most questions. Now in these cases it is evident that the differential
equation is not immediately suitable for integration. It is previously
necessary that the differentials of the functions supposed to be known,
which have been employed as intermediaries, should be entirely eliminated,
in order that equations may be obtained between the differentials of the
functions which alone are sought and those of the really independent
variables, after which the question depends on only the integral calculus.
Now this preparatory elimination of certain differentials, in order to reduce
the infinitesimals to the smallest number possible, belongs simply to the

differential calculus; for it must evidently be done by determining, by
means of the equations between the functions supposed to be known, taken
as intermediaries, the relations of their differentials, which is merely a
question of differentiation. Thus, for example, in the case of rectifications,
it will be first necessary to calculate dy, or dy and dz, by differentiating the
equation or the equations of each curve proposed; after eliminating these
expressions, the general differential formula above enunciated will then
contain only ds and dx; having arrived at this point, the elimination of the
infinitesimals can be completed only by the integral calculus.
Such is, then, the general office necessarily belonging to the differential
calculus in the complete solution of the questions which exact the
employment of the transcendental analysis; to produce, as far as is possible,
the elimination of the infinitesimals, that is, to reduce in each case the
primitive differential equations so that they shall contain only the
differentials of the really independent variables, and those of the functions
sought, by causing to disappear, by elimination, the differentials of all the
other known functions which may have been taken as intermediaries at the
time of the formation of the differential equations of the problem which is
under consideration.
2. Employment of the Differential Calculus alone. For certain questions,
which, although few in number, have none the less, as we shall see
hereafter, a very great importance, the magnitudes which are sought enter
directly, and not by their differentials, into the primitive differential
equations, which then contain differentially only the different known
functions employed as intermediaries, in accordance with the preceding
explanation. These cases are the most favourable of all; for it is evident that
the differential calculus is then entirely sufficient for the complete
elimination of the infinitesimals, without the question giving rise to any
integration. This is what occurs, for example, in the problem of tangents in
geometry; in that of velocities in mechanics, &c.
3. Employment of the Integral Calculus alone. Finally, some other
questions, the number of which is also very small, but the importance of
which is no less great, present a second exceptional case, which is in its
nature exactly the converse of the preceding. They are those in which the
differential equations are found to be immediately ready for integration,
because they contain, at their first formation, only the infinitesimals which

relate to the functions sought, or to the really independent variables, without
its being necessary to introduce, differentially, other functions as
intermediaries. If in these new cases we introduce these last functions,
since, by hypothesis, they will enter directly and not by their differentials,
ordinary algebra will suffice to eliminate them, and to bring the question to
depend on only the integral calculus. The differential calculus will then
have no special part in the complete solution of the problem, which will
depend entirely upon the integral calculus. The general question of
quadratures offers an important example of this, for the differential
equation being then dA = ydx, will become immediately fit for integration
as soon as we shall have eliminated, by means of the equation of the
proposed curve, the intermediary function y, which does not enter into it
differentially. The same circumstances exist in the problem of cubatures,
and in some others equally important.
Three classes of Questions hence resulting. As a general result of the
previous considerations, it is then necessary to divide into three classes the
mathematical questions which require the use of the transcendental
analysis; the first class comprises the problems susceptible of being entirely
resolved by means of the differential calculus alone, without any need of
the integral calculus; the second, those which are, on the contrary, entirely
dependent upon the integral calculus, without the differential calculus
having any part in their solution; lastly, in the third and the most extensive,
which constitutes the normal case, the two others being only exceptional,
the differential and the integral calculus have each in their turn a distinct
and necessary part in the complete solution of the problem, the former
making the primitive differential equations undergo a preparation which is
indispensable for the application of the latter. Such are exactly their general
relations, of which too indefinite and inexact ideas are generally formed.

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

THE DIFFERENTIAL CALCULUS.

In the exposition of the transcendental analysis, it is customary to
intermingle with the purely analytical part (which reduces itself to the
treatment of the abstract principles of differentiation and integration) the
study of its different principal applications, especially those which concern
geometry. This confusion of ideas, which is a consequence of the actual
manner in which the science has been developed, presents, in the dogmatic
point of view, serious inconveniences in this respect, that it makes it
difficult properly to conceive either analysis or geometry. Having to
consider here the most rational co-ordination which is possible, I shall
include, in the following sketch, only the calculus of indirect functions
properly so called, reserving for the portion of this volume which relates to
the philosophical study of concrete mathematics the general examination of
its great geometrical and mechanical applications.
Two Cases: explicit and implicit Functions. The fundamental division of the
differential calculus, or of the general subject of differentiation, consists in
distinguishing two cases, according as the analytical functions which are to
be differentiated are explicit or implicit; from which flow two parts
ordinarily designated by the names of differentiation of formulas and
differentiation of equations. It is easy to understand, à priori, the
importance of this classification. In fact, such a distinction would be
illusory if the ordinary analysis was perfect; that is, if we knew how to
resolve all equations algebraically, for then it would be possible to render
every implicit function explicit; and, by differentiating it in that state alone,
the second part of the differential calculus would be immediately comprised
in the first, without giving rise to any new difficulty. But the algebraical
resolution of equations being, as we have seen, still almost in its infancy,
and as yet impossible for most cases, it is plain that the case is very
different, since we have, properly speaking, to differentiate a function
without knowing it, although it is determinate. The differentiation of
implicit functions constitutes then, by its nature, a question truly distinct
from that presented by explicit functions, and necessarily more
complicated. It is thus evident that we must commence with the
differentiation of formulas, and reduce the differentiation of equations to
this primary case by certain invariable analytical considerations, which
need not be here mentioned.

These two general cases of differentiation are also distinct in another point
of view equally necessary, and too important to be left unnoticed. The
relation which is obtained between the differentials is constantly more
indirect, in comparison with that of the finite quantities, in the
differentiation of implicit functions than in that of explicit functions. We
know, in fact, from the considerations presented by Lagrange on the general
formation of differential equations, that, on the one hand, the same
primitive equation may give rise to a greater or less number of derived
equations of very different forms, although at bottom equivalent, depending
upon which of the arbitrary constants is eliminated, which is not the case in
the differentiation of explicit formulas; and that, on the other hand, the
unlimited system of the different primitive equations, which correspond to
the same derived equation, presents a much more profound analytical
variety than that of the different functions, which admit of one same explicit
differential, and which are distinguished from each other only by a constant
term. Implicit functions must therefore be regarded as being in reality still
more modified by differentiation than explicit functions. We shall again
meet with this consideration relatively to the integral calculus, where it
acquires a preponderant importance.
Two Sub-cases: A single Variable or several Variables. Each of the two
fundamental parts of the Differential Calculus is subdivided into two very
distinct theories, according as we are required to differentiate functions of a
single variable or functions of several independent variables. This second
case is, by its nature, quite distinct from the first, and evidently presents
more complication, even in considering only explicit functions, and still
more those which are implicit. As to the rest, one of these cases is deduced
from the other in a general manner, by the aid of an invariable and very
simple principle, which consists in regarding the total differential of a
function which is produced by the simultaneous increments of the different
independent variables which it contains, as the sum of the partial
differentials which would be produced by the separate increment of each
variable in turn, if all the others were constant. It is necessary, besides,
carefully to remark, in connection with this subject, a new idea which is
introduced by the distinction of functions into those of one variable and of
several; it is the consideration of these different special derived functions,
relating to each variable separately, and the number of which increases
more and more in proportion as the order of the derivation becomes higher,

and also when the variables become more numerous. It results from this that
the differential relations belonging to functions of several variables are, by
their nature, both much more indirect, and especially much more
indeterminate, than those relating to functions of a single variable. This is
most apparent in the case of implicit functions, in which, in the place of the
simple arbitrary constants which elimination causes to disappear when we
form the proper differential equations for functions of a single variable, it is
the arbitrary functions of the proposed variables which are then eliminated;
whence must result special difficulties when these equations come to be
integrated.
Finally, to complete this summary sketch of the different essential parts of
the differential calculus proper, I should add, that in the differentiation of
implicit functions, whether of a single variable or of several, it is necessary
to make another distinction; that of the case in which it is required to
differentiate at once different functions of this kind, combined in certain
primitive equations, from that in which all these functions are separate.
The functions are evidently, in fact, still more implicit in the first case than
in the second, if we consider that the same imperfection of ordinary
analysis, which forbids our converting every implicit function into an
equivalent explicit function, in like manner renders us unable to separate
the functions which enter simultaneously into any system of equations. It is
then necessary to differentiate, not only without knowing how to resolve the
primitive equations, but even without being able to effect the proper
eliminations among them, thus producing a new difficulty.
Reduction of the whole to the Differentiation of the ten elementary
Functions. Such, then, are the natural connection and the logical
distribution of the different principal theories which compose the general
system of differentiation. Since the differentiation of implicit functions is
deduced from that of explicit functions by a single constant principle, and
the differentiation of functions of several variables is reduced by another
fixed principle to that of functions of a single variable, the whole of the
differential calculus is finally found to rest upon the differentiation of
explicit functions with a single variable, the only one which is ever
executed directly. Now it is easy to understand that this first theory, the
necessary basis of the entire system, consists simply in the differentiation of
the ten simple functions, which are the uniform elements of all our

analytical combinations, and the list of which has been given in the first
chapter, on page 51; for the differentiation of compound functions is
evidently deduced, in an immediate and necessary manner, from that of the
simple functions which compose them. It is, then, to the knowledge of these
ten fundamental differentials, and to that of the two general principles just
mentioned, which bring under it all the other possible cases, that the whole
system of differentiation is properly reduced. We see, by the combination of
these different considerations, how simple and how perfect is the entire
system of the differential calculus. It certainly constitutes, in its logical
relations, the most interesting spectacle which mathematical analysis can
present to our understanding.
Transformation of derived Functions for new Variables. The general sketch
which I have just summarily drawn would nevertheless present an
important deficiency, if I did not here distinctly indicate a final theory,
which forms, by its nature, the indispensable complement of the system of
differentiation. It is that which has for its object the constant transformation
of derived functions, as a result of determinate changes in the independent
variables, whence results the possibility of referring to new variables all the
general differential formulas primitively established for others. This
question is now resolved in the most complete and the most simple manner,
as are all those of which the differential calculus is composed. It is easy to
conceive the general importance which it must have in any of the
applications of the transcendental analysis, the fundamental resources of
which it may be considered as augmenting, by permitting us to choose (in
order to form the differential equations, in the first place, with more ease)
that system of independent variables which may appear to be the most
advantageous, although it is not to be finally retained. It is thus, for
example, that most of the principal questions of geometry are resolved
much more easily by referring the lines and surfaces to rectilinear co-
ordinates, and that we may, nevertheless, have occasion to express these
lines, etc., analytically by the aid of polar co-ordinates, or in any other
manner. We will then be able to commence the differential solution of the
problem by employing the rectilinear system, but only as an intermediate
step, from which, by the general theory here referred to, we can pass to the
final system, which sometimes could not have been considered directly.

Different Orders of Differentiation. In the logical classification of the
differential calculus which has just been given, some may be inclined to
suggest a serious omission, since I have not subdivided each of its four
essential parts according to another general consideration, which seems at
first view very important; namely, that of the higher or lower order of
differentiation. But it is easy to understand that this distinction has no real
influence in the differential calculus, inasmuch as it does not give rise to
any new difficulty. If, indeed, the differential calculus was not rigorously
complete, that is, if we did not know how to differentiate at will any
function whatever, the differentiation to the second or higher order of each
determinate function might engender special difficulties. But the perfect
universality of the differential calculus plainly gives us the assurance of
being able to differentiate, to any order whatever, all known functions
whatever, the question reducing itself to a constantly repeated
differentiation of the first order. This distinction, unimportant as it is for the
differential calculus, acquires, however, a very great importance in the
integral calculus, on account of the extreme imperfection of the latter.
Analytical Applications. Finally, though this is not the place to consider the
various applications of the differential calculus, yet an exception may be
made for those which consist in the solution of questions which are purely
analytical, which ought, indeed, to be logically treated in continuation of a
system of differentiation, because of the evident homogeneity of the
considerations involved. These questions may be reduced to three essential
ones.
Firstly, the development into series of functions of one or more variables,
or, more generally, the transformation of functions, which constitutes the
most beautiful and the most important application of the differential
calculus to general analysis, and which comprises, besides the fundamental
series discovered by Taylor, the remarkable series discovered by Maclaurin,
John Bernouilli, Lagrange, &c.:
Secondly, the general theory of maxima and minima values for any
functions whatever, of one or more variables; one of the most interesting
problems which analysis can present, however elementary it may now have
become, and to the complete solution of which the differential calculus
naturally applies:

Thirdly, the general determination of the true value of functions which
present themselves under an indeterminate appearance for certain
hypotheses made on the values of the corresponding variables; which is the
least extensive and the least important of the three.
The first question is certainly the principal one in all points of view; it is
also the most susceptible of receiving a new extension hereafter, especially
by conceiving, in a broader manner than has yet been done, the employment
of the differential calculus in the transformation of functions, on which
subject Lagrange has left some valuable hints.

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

THE INTEGRAL CALCULUS.

Its Fundamental Division. The fundamental division of the Integral
Calculus is founded on the same principle as that of the Differential
Calculus, in distinguishing the integration of explicit differential formulas,
and the integration of implicit differentials or of differential equations. The
separation of these two cases is even much more profound in relation to
integration than to differentiation. In the differential calculus, in fact, this
distinction rests, as we have seen, only on the extreme imperfection of
ordinary analysis. But, on the other hand, it is easy to see that, even though
all equations could be algebraically resolved, differential equations would
none the less constitute a case of integration quite distinct from that
presented by the explicit differential formulas; for, limiting ourselves, for
the sake of simplicity, to the first order, and to a single function y of a single
variable x, if we suppose any differential equation between x, y, and dy/dx,
to be resolved with reference to dy/dx, the expression of the derived
function being then generally found to contain the primitive function itself,
which is the object of the inquiry, the question of integration will not have
at all changed its nature, and the solution will not really have made any

other progress than that of having brought the proposed differential
equation to be of only the first degree relatively to the derived function,
which is in itself of little importance. The differential would not then be
determined in a manner much less implicit than before, as regards the
integration, which would continue to present essentially the same
characteristic difficulty. The algebraic resolution of equations could not
make the case which we are considering come within the simple integration
of explicit differentials, except in the special cases in which the proposed
differential equation did not contain the primitive function itself, which
would consequently permit us, by resolving it, to find dy/dx in terms of x
only, and thus to reduce the question to the class of quadratures. Still greater
difficulties would evidently be found in differential equations of higher
orders, or containing simultaneously different functions of several
independent variables.
The integration of differential equations is then necessarily more
complicated than that of explicit differentials, by the elaboration of which
last the integral calculus has been created, and upon which the others have
been made to depend as far as it has been possible. All the various
analytical methods which have been proposed for integrating differential
equations, whether it be the separation of the variables, the method of
multipliers, &c., have in fact for their object to reduce these integrations to
those of differential formulas, the only one which, by its nature, can be
undertaken directly. Unfortunately, imperfect as is still this necessary base
of the whole integral calculus, the art of reducing to it the integration of
differential equations is still less advanced.
Subdivisions: one variable or several. Each of these two fundamental
branches of the integral calculus is next subdivided into two others (as in
the differential calculus, and for precisely analogous reasons), according as
we consider functions with a single variable, or functions with several
independent variables.
This distinction is, like the preceding one, still more important for
integration than for differentiation. This is especially remarkable in
reference to differential equations. Indeed, those which depend on several
independent variables may evidently present this characteristic and much
more serious difficulty, that the desired function may be differentially
defined by a simple relation between its different special derivatives relative

to the different variables taken separately. Hence results the most difficult
and also the most extensive branch of the integral calculus, which is
commonly named the Integral Calculus of partial differences, created by
D'Alembert, and in which, according to the just appreciation of Lagrange,
geometers ought to have seen a really new calculus, the philosophical
character of which has not yet been determined with sufficient exactness. A
very striking difference between this case and that of equations with a
single independent variable consists, as has been already observed, in the
arbitrary functions which take the place of the simple arbitrary constants, in
order to give to the corresponding integrals all the proper generality.
It is scarcely necessary to say that this higher branch of transcendental
analysis is still entirely in its infancy, since, even in the most simple case,
that of an equation of the first order between the partial derivatives of a
single function with two independent variables, we are not yet completely
able to reduce the integration to that of the ordinary differential equations.
The integration of functions of several variables is much farther advanced
in the case (infinitely more simple indeed) in which it has to do with only
explicit differential formulas. We can then, in fact, when these formulas
fulfil the necessary conditions of integrability, always reduce their
integration to quadratures.
Other Subdivisions: different Orders of Differentiation. A new general
distinction, applicable as a subdivision to the integration of explicit or
implicit differentials, with one variable or several, is drawn from the higher
or lower order of the differentials: a distinction which, as we have above
remarked, does not give rise to any special question in the differential
calculus.
Relatively to explicit differentials, whether of one variable or of several, the
necessity of distinguishing their different orders belongs only to the extreme
imperfection of the integral calculus. In fact, if we could always integrate
every differential formula of the first order, the integration of a formula of
the second order, or of any other, would evidently not form a new question,
since, by integrating it at first in the first degree, we would arrive at the
differential expression of the immediately preceding order, from which, by
a suitable series of analogous integrations, we would be certain of finally
arriving at the primitive function, the final object of these operations. But
the little knowledge which we possess on integration of even the first order

causes quite another state of affairs, so that a higher order of differentials
produces new difficulties; for, having differential formulas of any order
above the first, it may happen that we may be able to integrate them, either
once, or several times in succession, and that we may still be unable to go
back to the primitive functions, if these preliminary labours have produced,
for the differentials of a lower order, expressions whose integrals are not
known. This circumstance must occur so much the oftener (the number of
known integrals being still very small), seeing that these successive
integrals are generally very different functions from the derivatives which
have produced them.
With reference to implicit differentials, the distinction of orders is still more
important; for, besides the preceding reason, the influence of which is
evidently analogous in this case, and is even greater, it is easy to perceive
that the higher order of the differential equations necessarily gives rise to
questions of a new nature. In fact, even if we could integrate every equation
of the first order relating to a single function, that would not be sufficient
for obtaining the final integral of an equation of any order whatever,
inasmuch as every differential equation is not reducible to that of an
immediately inferior order. Thus, for example, if we have given any relation
between x, y, dx/dy, and d2y/dx2, to determine a function y of a variable x,
we shall not be able to deduce from it at once, after effecting a first
integration, the corresponding differential relation between x, y, and dy/dx,
from which, by a second integration, we could ascend to the primitive
equations. This would not necessarily take place, at least without
introducing new auxiliary functions, unless the proposed equation of the
second order did not contain the required function y, together with its
derivatives. As a general principle, differential equations will have to be
regarded as presenting cases which are more and more implicit, as they are
of a higher order, and which cannot be made to depend on one another
except by special methods, the investigation of which consequently forms a
new class of questions, with respect to which we as yet know scarcely any
thing, even for functions of a single variable.[10]
Another equivalent distinction. Still farther, when we examine more
profoundly this distinction of different orders of differential equations, we
find that it can be always made to come under a final general distinction,
relative to differential equations, which remains to be noticed. Differential

equations with one or more independent variables may contain simply a
single function, or (in a case evidently more complicated and more implicit,
which corresponds to the differentiation of simultaneous implicit functions)
we may have to determine at the same time several functions from the
differential equations in which they are found united, together with their
different derivatives. It is clear that such a state of the question necessarily
presents a new special difficulty, that of separating the different functions
desired, by forming for each, from the proposed differential equations, an
isolated differential equation which does not contain the other functions or
their derivatives. This preliminary labour, which is analogous to the
elimination of algebra, is evidently indispensable before attempting any
direct integration, since we cannot undertake generally (except by special
artifices which are very rarely applicable) to determine directly several
distinct functions at once.
Now it is easy to establish the exact and necessary coincidence of this new
distinction with the preceding one respecting the order of differential
equations. We know, in fact, that the general method for isolating functions
in simultaneous differential equations consists essentially in forming
differential equations, separately in relation to each function, and of an
order equal to the sum of all those of the different proposed equations. This
transformation can always be effected. On the other hand, every differential
equation of any order in relation to a single function might evidently always
be reduced to the first order, by introducing a suitable number of auxiliary
differential equations, containing at the same time the different anterior
derivatives regarded as new functions to be determined. This method has,
indeed, sometimes been actually employed with success, though it is not the
natural one.
Here, then, are two necessarily equivalent orders of conditions in the
general theory of differential equations; the simultaneousness of a greater or
smaller number of functions, and the higher or lower order of differentiation
of a single function. By augmenting the order of the differential equations,
we can isolate all the functions; and, by artificially multiplying the number
of the functions, we can reduce all the equations to the first order. There is,
consequently, in both cases, only one and the same difficulty from two
different points of sight. But, however we may conceive it, this new
difficulty is none the less real, and constitutes none the less, by its nature, a

marked separation between the integration of equations of the first order
and that of equations of a higher order. I prefer to indicate the distinction
under this last form as being more simple, more general, and more logical.
Quadratures. From the different considerations which have been indicated
respecting the logical dependence of the various principal parts of the
integral calculus, we see that the integration of explicit differential formulas
of the first order and of a single variable is the necessary basis of all other
integrations, which we never succeed in effecting but so far as we reduce
them to this elementary case, evidently the only one which, by its nature, is
capable of being treated directly. This simple fundamental integration is
often designated by the convenient expression of quadratures, seeing that
every integral of this kind, Sf(x)dx, may, in fact, be regarded as representing
the area of a curve, the equation of which in rectilinear co-ordinates would
be y = f(x). Such a class of questions corresponds, in the differential
calculus, to the elementary case of the differentiation of explicit functions
of a single variable. But the integral question is, by its nature, very
differently complicated, and especially much more extensive than the
differential question. This latter is, in fact, necessarily reduced, as we have
seen, to the differentiation of the ten simple functions, the elements of all
which are considered in analysis. On the other hand, the integration of
compound functions does not necessarily follow from that of the simple
functions, each combination of which may present special difficulties with
respect to the integral calculus. Hence results the naturally indefinite extent,
and the so varied complication of the question of quadratures, upon which,
in spite of all the efforts of analysts, we still possess so little complete
knowledge.
In decomposing this question, as is natural, according to the different forms
which may be assumed by the derivative function, we distinguish the case
of algebraic functions and that of transcendental functions.
Integration of Transcendental Functions. The truly analytical integration of
transcendental functions is as yet very little advanced, whether for
exponential, or for logarithmic, or for circular functions. But a very small
number of cases of these three different kinds have as yet been treated, and
those chosen from among the simplest; and still the necessary calculations
are in most cases extremely laborious. A circumstance which we ought
particularly to remark in its philosophical connection is, that the different

procedures of quadrature have no relation to any general view of
integration, and consist of simple artifices very incoherent with each other,
and very numerous, because of the very limited extent of each.
One of these artifices should, however, here be noticed, which, without
being really a method of integration, is nevertheless remarkable for its
generality; it is the procedure invented by John Bernouilli, and known
under the name of integration by parts, by means of which every integral
may be reduced to another which is sometimes found to be more easy to be
obtained. This ingenious relation deserves to be noticed for another reason,
as having suggested the first idea of that transformation of integrals yet
unknown, which has lately received a greater extension, and of which M.
Fourier especially has made so new and important a use in the analytical
questions produced by the theory of heat.
Integration of Algebraic Functions. As to the integration of algebraic
functions, it is farther advanced. However, we know scarcely any thing in
relation to irrational functions, the integrals of which have been obtained
only in extremely limited cases, and particularly by rendering them rational.
The integration of rational functions is thus far the only theory of the
integral calculus which has admitted of being treated in a truly complete
manner; in a logical point of view, it forms, then, its most satisfactory part,
but perhaps also the least important. It is even essential to remark, in order
to have a just idea of the extreme imperfection of the integral calculus, that
this case, limited as it is, is not entirely resolved except for what properly
concerns integration viewed in an abstract manner; for, in the execution, the
theory finds its progress most frequently quite stopped, independently of the
complication of the calculations, by the imperfection of ordinary analysis,
seeing that it makes the integration finally depend upon the algebraic
resolution of equations, which greatly limits its use.
To grasp in a general manner the spirit of the different procedures which are
employed in quadratures, we must observe that, by their nature, they can be
primitively founded only on the differentiation of the ten simple functions.
The results of this, conversely considered, establish as many direct
theorems of the integral calculus, the only ones which can be directly
known. All the art of integration afterwards consists, as has been said in the
beginning of this chapter, in reducing all the other quadratures, so far as is

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

represent the required curve; the general integral, which thenceforth it
becomes unnecessary to know, designating only the system of the tangents,
or of the osculating circles of this curve. We may hence easily understand
all the importance of this theory, which seems to me to be not as yet
sufficiently appreciated by most geometers.
Definite Integrals. Finally, to complete our review of the vast collection of
analytical researches of which is composed the integral calculus, properly
so called, there remains to be mentioned one theory, very important in all
the applications of the transcendental analysis, which I have had to leave
outside of the system, as not being really destined for veritable integration,
and proposing, on the contrary, to supply the place of the knowledge of
truly analytical integrals, which are most generally unknown. I refer to the
determination of definite integrals.
The expression, always possible, of integrals in infinite series, may at first
be viewed as a happy general means of compensating for the extreme
imperfection of the integral calculus. But the employment of such series,
because of their complication, and of the difficulty of discovering the law of
their terms, is commonly of only moderate utility in the algebraic point of
view, although sometimes very essential relations have been thence
deduced. It is particularly in the arithmetical point of view that this
procedure acquires a great importance, as a means of calculating what are
called definite integrals, that is, the values of the required functions for
certain determinate values of the corresponding variables.
An inquiry of this nature exactly corresponds, in transcendental analysis, to
the numerical resolution of equations in ordinary analysis. Being generally
unable to obtain the veritable integral—named by opposition the general or
indefinite integral; that is, the function which, differentiated, has produced
the proposed differential formula—analysts have been obliged to employ
themselves in determining at least, without knowing this function, the
particular numerical values which it would take on assigning certain
designated values to the variables. This is evidently resolving the
arithmetical question without having previously resolved the corresponding
algebraic one, which most generally is the most important one. Such an
analysis is, then, by its nature, as imperfect as we have seen the numerical
resolution of equations to be. It presents, like this last, a vicious confusion
of arithmetical and algebraic considerations, whence result analogous

inconveniences both in the purely logical point of view and in the
applications. We need not here repeat the considerations suggested in our
third chapter. But it will be understood that, unable as we almost always are
to obtain the true integrals, it is of the highest importance to have been able
to obtain this solution, incomplete and necessarily insufficient as it is. Now
this has been fortunately attained at the present day for all cases, the
determination of the value of definite integrals having been reduced to
entirely general methods, which leave nothing to desire, in a great number
of cases, but less complication in the calculations, an object towards which
are at present directed all the special transformations of analysts. Regarding
now this sort of transcendental arithmetic as perfect, the difficulty in the
applications is essentially reduced to making the proposed research depend,
finally, on a simple determination of definite integrals, which evidently
cannot always be possible, whatever analytical skill may be employed in
effecting such a transformation.
Prospects of the Integral Calculus. From the considerations indicated in this
chapter, we see that, while the differential calculus constitutes by its nature
a limited and perfect system, to which nothing essential remains to be
added, the integral calculus, or the simple system of integration, presents
necessarily an inexhaustible field for the activity of the human mind,
independently of the indefinite applications of which the transcendental
analysis is evidently susceptible. The general argument by which I have
endeavoured, in the second chapter, to make apparent the impossibility of
ever discovering the algebraic solution of equations of any degree and form
whatsoever, has undoubtedly infinitely more force with regard to the search
for a single method of integration, invariably applicable to all cases. "It is,"
says Lagrange, "one of those problems whose general solution we cannot
hope for." The more we meditate on this subject, the more we will be
convinced that such a research is utterly chimerical, as being far above the
feeble reach of our intelligence; although the labours of geometers must
certainly augment hereafter the amount of our knowledge respecting
integration, and thus create methods of greater generality. The
transcendental analysis is still too near its origin—there is especially too
little time since it has been conceived in a truly rational manner—for us
now to be able to have a correct idea of what it will hereafter become. But,
whatever should be our legitimate hopes, let us not forget to consider,
before all, the limits which are imposed by our intellectual constitution, and

which, though not susceptible of a precise determination, have none the less
an incontestable reality.
I am induced to think that, when geometers shall have exhausted the most
important applications of our present transcendental analysis, instead of
striving to impress upon it, as now conceived, a chimerical perfection, they
will rather create new resources by changing the mode of derivation of the
auxiliary quantities introduced in order to facilitate the establishment of
equations, and the formation of which might follow an infinity of other laws
besides the very simple relation which has been chosen, according to the
conception suggested in the first chapter. The resources of this nature
appear to me susceptible of a much greater fecundity than those which
would consist of merely pushing farther our present calculus of indirect
functions. It is a suggestion which I submit to the geometers who have
turned their thoughts towards the general philosophy of analysis.
Finally, although, in the summary exposition which was the object of this
chapter, I have had to exhibit the condition of extreme imperfection which
still belongs to the integral calculus, the student would have a false idea of
the general resources of the transcendental analysis if he gave that
consideration too great an importance. It is with it, indeed, as with ordinary
analysis, in which a very small amount of fundamental knowledge
respecting the resolution of equations has been employed with an immense
degree of utility. Little advanced as geometers really are as yet in the
science of integrations, they have nevertheless obtained, from their scanty
abstract conceptions, the solution of a multitude of questions of the first
importance in geometry, in mechanics, in thermology, &c. The
philosophical explanation of this double general fact results from the
necessarily preponderating importance and grasp of abstract branches of
knowledge, the least of which is naturally found to correspond to a crowd
of concrete researches, man having no other resource for the successive
extension of his intellectual means than in the consideration of ideas more
and more abstract, and still positive.

In order to finish the complete exposition of the philosophical character of
the transcendental analysis, there remains to be considered a final

conception, by which the immortal Lagrange has rendered this analysis still
better adapted to facilitate the establishment of equations in the most
difficult problems, by considering a class of equations still more indirect
than the ordinary differential equations. It is the Calculus, or, rather, the
Method of Variations; the general appreciation of which will be our next
subject.

CHAPTER V.
THE CALCULUS OF VARIATIONS.

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

PROBLEMS GIVING RISE TO IT.

The mathematical questions which have given birth to the Calculus of
Variations consist generally in the investigation of the maxima and minima
of certain indeterminate integral formulas, which express the analytical law
of such or such a phenomenon of geometry or mechanics, considered
independently of any particular subject. Geometers for a long time
designated all the questions of this character by the common name of
Isoperimetrical Problems, which, however, is really suitable to only the
smallest number of them.
Ordinary Questions of Maxima and Minima. In the common theory of
maxima and minima, it is proposed to discover, with reference to a given
function of one or more variables, what particular values must be assigned
to these variables, in order that the corresponding value of the proposed
function may be a maximum or a minimum with respect to those values
which immediately precede and follow it; that is, properly speaking, we
seek to know at what instant the function ceases to increase and commences
to decrease, or reciprocally. The differential calculus is perfectly sufficient,
as we know, for the general resolution of this class of questions, by showing
that the values of the different variables, which suit either the maximum or
minimum, must always reduce to zero the different first derivatives of the
given function, taken separately with reference to each independent

variable, and by indicating, moreover, a suitable characteristic for
distinguishing the maximum from the minimum; consisting, in the case of a
function of a single variable, for example, in the derived function of the
second order taking a negative value for the maximum, and a positive value
for the minimum. Such are the well-known fundamental conditions
belonging to the greatest number of cases.
A new Class of Questions. The construction of this general theory having
necessarily destroyed the chief interest which questions of this kind had for
geometers, they almost immediately rose to the consideration of a new
order of problems, at once much more important and of much greater
difficulty—those of isoperimeters. It is, then, no longer the values of the
variables belonging to the maximum or the minimum of a given function
that it is required to determine. It is the form of the function itself which is
required to be discovered, from the condition of the maximum or of the
minimum of a certain definite integral, merely indicated, which depends
upon that function.
Solid of least Resistance. The oldest question of this nature is that of the
solid of least resistance, treated by Newton in the second book of the
Principia, in which he determines what ought to be the meridian curve of a
solid of revolution, in order that the resistance experienced by that body in
the direction of its axis may be the least possible. But the course pursued by
Newton, from the nature of his special method of transcendental analysis,
had not a character sufficiently simple, sufficiently general, and especially
sufficiently analytical, to attract geometers to this new order of problems.
To effect this, the application of the infinitesimal method was needed; and
this was done, in 1695, by John Bernouilli, in proposing the celebrated
problem of the Brachystochrone.
This problem, which afterwards suggested such a long series of analogous
questions, consists in determining the curve which a heavy body must
follow in order to descend from one point to another in the shortest possible
time. Limiting the conditions to the simple fall in a vacuum, the only case
which was at first considered, it is easily found that the required curve must
be a reversed cycloid with a horizontal base, and with its origin at the
highest point. But the question may become singularly complicated, either
by taking into account the resistance of the medium, or the change in the
intensity of gravity.

Isoperimeters. Although this new class of problems was in the first place
furnished by mechanics, it is in geometry that the principal investigations of
this character were subsequently made. Thus it was proposed to discover
which, among all the curves of the same contour traced between two given
points, is that whose area is a maximum or minimum, whence has come the
name of Problem of Isoperimeters; or it was required that the maximum or
minimum should belong to the surface produced by the revolution of the
required curve about an axis, or to the corresponding volume; in other
cases, it was the vertical height of the center of gravity of the unknown
curve, or of the surface and of the volume which it might generate, which
was to become a maximum or minimum, &c. Finally, these problems were
varied and complicated almost to infinity by the Bernouillis, by Taylor, and
especially by Euler, before Lagrange reduced their solution to an abstract
and entirely general method, the discovery of which has put a stop to the
enthusiasm of geometers for such an order of inquiries. This is not the place
for tracing the history of this subject. I have only enumerated some of the
simplest principal questions, in order to render apparent the original general
object of the method of variations.
Analytical Nature of these Problems. We see that all these problems,
considered in an analytical point of view, consist, by their nature, in
determining what form a certain unknown function of one or more variables
ought to have, in order that such or such an integral, dependent upon that
function, shall have, within assigned limits, a value which is a maximum or
a minimum with respect to all those which it would take if the required
function had any other form whatever.
Thus, for example, in the problem of the brachystochrone, it is well known
that if y = f(z), x = π(z), are the rectilinear equations of the required curve,
supposing the axes of x and of y to be horizontal, and the axis of z to be
vertical, the time of the fall of a heavy body in that curve from the point
whose ordinate is z1, to that whose ordinate is z2, is expressed in general
terms by the definite integral
∫_{z_{2}}z_{1}√(1 + (f'(z))2 + (π'(z))2/(2gz))dz.
It is, then, necessary to find what the two unknown functions f and π must
be, in order that this integral may be a minimum.

In the same way, to demand what is the curve among all plane
isoperimetrical curves, which includes the greatest area, is the same thing as
to propose to find, among all the functions f(x) which can give a certain
constant value to the integral
∫dx√(1 + (f'(x) )2),
that one which renders the integral ∫f(x)dx, taken between the same limits, a
maximum. It is evidently always so in other questions of this class.
Methods of the older Geometers. In the solutions which geometers before
Lagrange gave of these problems, they proposed, in substance, to reduce
them to the ordinary theory of maxima and minima. But the means
employed to effect this transformation consisted in special simple artifices
peculiar to each case, and the discovery of which did not admit of
invariable and certain rules, so that every really new question constantly
reproduced analogous difficulties, without the solutions previously obtained
being really of any essential aid, otherwise than by their discipline and
training of the mind. In a word, this branch of mathematics presented, then,
the necessary imperfection which always exists when the part common to
all questions of the same class has not yet been distinctly grasped in order
to be treated in an abstract and thenceforth general manner.

METHOD OF LAGRANGE.

Lagrange, in endeavouring to bring all the different problems of
isoperimeters to depend upon a common analysis, organized into a distinct
calculus, was led to conceive a new kind of differentiation, to which he has
applied the characteristic δ, reserving the characteristic d for the common
differentials. These differentials of a new species, which he has designated
under the name of Variations, consist of the infinitely small increments
which the integrals receive, not by virtue of analogous increments on the
part of the corresponding variables, as in the ordinary transcendental
analysis, but by supposing that the form of the function placed under the
sign of integration undergoes an infinitely small change. This distinction is
easily conceived with reference to curves, in which we see the ordinate, or
any other variable of the curve, admit of two sorts of differentials, evidently
very different, according as we pass from one point to another infinitely

near it on the same curve, or to the corresponding point of the infinitely
near curve produced by a certain determinate modification of the first
curve.[11] It is moreover clear, that the relative variations of different
magnitudes connected with each other by any laws whatever are calculated,
all but the characteristic, almost exactly in the same manner as the
differentials. Finally, from the general notion of variations are in like
manner deduced the fundamental principles of the algorithm proper to this
method, consisting simply in the evidently permissible liberty of
transposing at will the characteristics specially appropriated to variations,
before or after those which correspond to the ordinary differentials.
This abstract conception having been once formed, Lagrange was able to
reduce with ease, and in the most general manner, all the problems of
Isoperimeters to the simple ordinary theory of maxima and minima. To
obtain a clear idea of this great and happy transformation, we must
previously consider an essential distinction which arises in the different
questions of isoperimeters.
Two Classes of Questions. These investigations must, in fact, be divided
into two general classes, according as the maxima and minima demanded
are absolute or relative, to employ the abridged expressions of geometers.
Questions of the first Class. The first case is that in which the indeterminate
definite integrals, the maximum or minimum of which is sought, are not
subjected, by the nature of the problem, to any condition; as happens, for
example, in the problem of the brachystochrone, in which the choice is to
be made between all imaginable curves. The second case takes place when,
on the contrary, the variable integrals can vary only according to certain
conditions, which usually consist in other definite integrals (which depend,
in like manner, upon the required functions) always retaining the same
given value; as, for example, in all the geometrical questions relating to real
isoperimetrical figures, and in which, by the nature of the problem, the
integral relating to the length of the curve, or to the area of the surface, must
remain constant during the variation of that integral which is the object of
the proposed investigation.
The Calculus of Variations gives immediately the general solution of
questions of the former class; for it evidently follows, from the ordinary
theory of maxima and minima, that the required relation must reduce to

zero the variation of the proposed integral with reference to each
independent variable; which gives the condition common to both the
maximum and the minimum: and, as a characteristic for distinguishing the
one from the other, that the variation of the second order of the same
integral must be negative for the maximum and positive for the minimum.
Thus, for example, in the problem of the brachystochrone, we will have, in
order to determine the nature of the curve sought, the equation of condition
δ∫_{z_{2}}z_{1}√([1 + (f'(z))2 + (π'(z))2]/(2gz))dz = 0,
which, being decomposed into two, with respect to the two unknown
functions f and π, which are independent of each other, will completely
express the analytical definition of the required curve. The only difficulty
peculiar to this new analysis consists in the elimination of the characteristic
δ, for which the calculus of variations furnishes invariable and complete
rules, founded, in general, on the method of "integration by parts," from
which Lagrange has thus derived immense advantage. The constant object
of this first analytical elaboration (which this is not the place for treating in
detail) is to arrive at real differential equations, which can always be done;
and thereby the question comes under the ordinary transcendental analysis,
which furnishes the solution, at least so far as to reduce it to pure algebra if
the integration can be effected. The general object of the method of
variations is to effect this transformation, for which Lagrange has
established rules, which are simple, invariable, and certain of success.
Equations of Limits. Among the greatest special advantages of the method
of variations, compared with the previous isolated solutions of
isoperimetrical problems, is the important consideration of what Lagrange
calls Equations of Limits, which were entirely neglected before him, though
without them the greater part of the particular solutions remained
necessarily incomplete. When the limits of the proposed integrals are to be
fixed, their variations being zero, there is no occasion for noticing them.
But it is no longer so when these limits, instead of being rigorously
invariable, are only subjected to certain conditions; as, for example, if the
two points between which the required curve is to be traced are not fixed,
and have only to remain upon given lines or surfaces. Then it is necessary
to pay attention to the variation of their co-ordinates, and to establish
between them the relations which correspond to the equations of these lines
or of these surfaces.

A more general consideration. This essential consideration is only the final
complement of a more general and more important consideration relative to
the variations of different independent variables. If these variables are really
independent of one another, as when we compare together all the
imaginable curves susceptible of being traced between two points, it will be
the same with their variations, and, consequently, the terms relating to each
of these variations will have to be separately equal to zero in the general
equation which expresses the maximum or the minimum. But if, on the
contrary, we suppose the variables to be subjected to any fixed conditions, it
will be necessary to take notice of the resulting relation between their
variations, so that the number of the equations into which this general
equation is then decomposed is always equal to only the number of the
variables which remain truly independent. It is thus, for example, that
instead of seeking for the shortest path between any two points, in choosing
it from among all possible ones, it may be proposed to find only what is the
shortest among all those which may be taken on any given surface; a
question the general solution of which forms certainly one of the most
beautiful applications of the method of variations.
Questions of the second Class. Problems in which such modifying
conditions are considered approach very nearly, in their nature, to the
second general class of applications of the method of variations,
characterized above as consisting in the investigation of relative maxima
and minima. There is, however, this essential difference between the two
cases, that in this last the modification is expressed by an integral which
depends upon the function sought, while in the other it is designated by a
finite equation which is immediately given. It is hence apparent that the
investigation of relative maxima and minima is constantly and necessarily
more complicated than that of absolute maxima and minima. Luckily, a
very important general theory, discovered by the genius of the great Euler
before the invention of the Calculus of Variations, gives a uniform and very
simple means of making one of these two classes of questions dependent on
the other. It consists in this, that if we add to the integral which is to be a
maximum or a minimum, a constant and indeterminate multiple of that one
which, by the nature of the problem, is to remain constant, it will be
sufficient to seek, by the general method of Lagrange above indicated, the
absolute maximum or minimum of this whole expression. It can be easily
conceived, indeed, that the part of the complete variation which would

proceed from the last integral must be equal to zero (because of the constant
character of this last) as well as the portion due to the first integral, which
disappears by virtue of the maximum or minimum state. These two
conditions evidently unite to produce, in that respect, effects exactly alike.
Such is a sketch of the general manner in which the method of variation is
applied to all the different questions which compose what is called the
Theory of Isoperimeters. It will undoubtedly have been remarked in this
summary exposition how much use has been made in this new analysis of
the second fundamental property of the transcendental analysis noticed in
the third chapter, namely, the generality of the infinitesimal expressions for
the representation of the same geometrical or mechanical phenomenon, in
whatever body it may be considered. Upon this generality, indeed, are
founded, by their nature, all the solutions due to the method of variations. If
a single formula could not express the length or the area of any curve
whatever; if another fixed formula could not designate the time of the fall of
a heavy body, according to whatever line it may descend, &c., how would it
have been possible to resolve questions which unavoidably require, by their
nature, the simultaneous consideration of all the cases which can be
determined in each phenomenon by the different subjects which exhibit it.
Other Applications of this Method. Notwithstanding the extreme importance
of the theory of isoperimeters, and though the method of variations had at
first no other object than the logical and general solution of this order of
problems, we should still have but an incomplete idea of this beautiful
analysis if we limited its destination to this. In fact, the abstract conception
of two distinct natures of differentiation is evidently applicable not only to
the cases for which it was created, but also to all those which present, for
any reason whatever, two different manners of making the same magnitudes
vary. It is in this way that Lagrange himself has made, in his "Méchanique
Analytique," an extensive and important application of his calculus of
variations, by employing it to distinguish the two sorts of changes which are
naturally presented by the questions of rational mechanics for the different
points which are considered, according as we compare the successive
positions which are occupied, in virtue of its motion, by the same point of
each body in two consecutive instants, or as we pass from one point of the
body to another in the same instant. One of these comparisons produces
ordinary differentials; the other gives rise to variations, which, there as

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

ITS RELATIONS TO THE ORDINARY CALCULUS.

In order to make as clear as possible the philosophical character of the
Calculus of Variations, I think that I should, in conclusion, briefly indicate a
consideration which seems to me important, and by which I can approach it
to the ordinary transcendental analysis in a higher degree than Lagrange
seems to me to have done.[12]
We noticed in the preceding chapter the formation of the calculus of partial
differences, created by D'Alembert, as having introduced into the
transcendental analysis a new elementary idea; the notion of two kinds of
increments, distinct and independent of one another, which a function of
two variables may receive by virtue of the change of each variable

separately. It is thus that the vertical ordinate of a surface, or any other
magnitude which is referred to it, varies in two manners which are quite
distinct, and which may follow the most different laws, according as we
increase either the one or the other of the two horizontal co-ordinates. Now
such a consideration seems to me very nearly allied, by its nature, to that
which serves as the general basis of the method of variations. This last,
indeed, has in reality done nothing but transfer to the independent variables
themselves the peculiar conception which had been already adopted for the
functions of these variables; a modification which has remarkably enlarged
its use. I think, therefore, that so far as regards merely the fundamental
conceptions, we may consider the calculus created by D'Alembert as having
established a natural and necessary transition between the ordinary
infinitesimal calculus and the calculus of variations; such a derivation of
which seems to be adapted to make the general notion more clear and
simple.
According to the different considerations indicated in this chapter, the
method of variations presents itself as the highest degree of perfection
which the analysis of indirect functions has yet attained. In its primitive
state, this last analysis presented itself as a powerful general means of
facilitating the mathematical study of natural phenomena, by introducing,
for the expression of their laws, the consideration of auxiliary magnitudes,
chosen in such a manner that their relations are necessarily more simple and
more easy to obtain than those of the direct magnitudes. But the formation
of these differential equations was not supposed to admit of any general and
abstract rules. Now the Analysis of Variations, considered in the most
philosophical point of view, may be regarded as essentially destined, by its
nature, to bring within the reach of the calculus the actual establishment of
the differential equations; for, in a great number of important and difficult
questions, such is the general effect of the varied equations, which, still
more indirect than the simple differential equations with respect to the
special objects of the investigation, are also much more easy to form, and
from which we may then, by invariable and complete analytical methods,
the object of which is to eliminate the new order of auxiliary infinitesimals
which have been introduced, deduce those ordinary differential equations
which it would often have been impossible to establish directly. The method
of variations forms, then, the most sublime part of that vast system of
mathematical analysis, which, setting out from the most simple elements of

algebra, organizes, by an uninterrupted succession of ideas, general
methods more and more powerful, for the study of natural philosophy, and
which, in its whole, presents the most incomparably imposing and
unequivocal monument of the power of the human intellect.
We must, however, also admit that the conceptions which are habitually
considered in the method of variations being, by their nature, more indirect,
more general, and especially more abstract than all others, the employment
of such a method exacts necessarily and continuously the highest known
degree of intellectual exertion, in order never to lose sight of the precise
object of the investigation, in following reasonings which offer to the mind
such uncertain resting-places, and in which signs are of scarcely any
assistance. We must undoubtedly attribute in a great degree to this difficulty
the little real use which geometers, with the exception of Lagrange, have as
yet made of such an admirable conception.

CHAPTER VI.
THE CALCULUS OF FINITE DIFFERENCES.

The different fundamental considerations indicated in the five preceding
chapters constitute, in reality, all the essential bases of a complete
exposition of mathematical analysis, regarded in the philosophical point of
view. Nevertheless, in order not to neglect any truly important general
conception relating to this analysis, I think that I should here very
summarily explain the veritable character of a kind of calculus which is
very extended, and which, though at bottom it really belongs to ordinary
analysis, is still regarded as being of an essentially distinct nature. I refer to
the Calculus of Finite Differences, which will be the special subject of this
chapter.
Its general Character. This calculus, created by Taylor, in his celebrated
work entitled Methodus Incrementorum, consists essentially in the
consideration of the finite increments which functions receive as a
consequence of analogous increments on the part of the corresponding
variables. These increments or differences, which take the characteristic Δ,
to distinguish them from differentials, or infinitely small increments, may
be in their turn regarded as new functions, and become the subject of a
second similar consideration, and so on; from which results the notion of
differences of various successive orders, analogous, at least in appearance,
to the consecutive orders of differentials. Such a calculus evidently
presents, like the calculus of indirect functions, two general classes of
questions:
1°. To determine the successive differences of all the various analytical
functions of one or more variables, as the result of a definite manner of
increase of the independent variables, which are generally supposed to
augment in arithmetical progression.
2°. Reciprocally, to start from these differences, or, more generally, from
any equations established between them, and go back to the primitive
functions themselves, or to their corresponding relations.

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

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

GENERAL THEORY OF SERIES.

To sum up as briefly as possible the preceding explanation, the calculus of
Taylor ought to be regarded as having constantly for its true object the
general theory of Series, the most simple cases of which had alone been
considered before that illustrious geometer. I ought, properly, to have
mentioned this important theory in treating, in the second chapter, of
Algebra proper, of which it is such an extensive branch. But, in order to
avoid a double reference to it, I have preferred to notice it only in the
consideration of the calculus of finite differences, which, reduced to its
most simple general expression, is nothing but a complete logical study of
questions relating to series.
Every Series, or succession of numbers deduced from one another
according to any constant law, necessarily gives rise to these two
fundamental questions:
1°. The law of the series being supposed known, to find the expression for
its general term, so as to be able to calculate immediately any term
whatever without being obliged to form successively all the preceding
terms.
2°. In the same circumstances, to determine the sum of any number of terms
of the series by means of their places, so that it can be known without the
necessity of continually adding these terms together.
These two fundamental questions being considered to be resolved, it may
be proposed, reciprocally, to find the law of a series from the form of its

general term, or the expression of the sum. Each of these different problems
has so much the more extent and difficulty, as there can be conceived a
greater number of different laws for the series, according to the number of
preceding terms on which each term directly depends, and according to the
function which expresses that dependence. We may even consider series
with several variable indices, as Laplace has done in his "Analytical Theory
of Probabilities," by the analysis to which he has given the name of Theory
of Generating Functions, although it is really only a new and higher branch
of the calculus of finite differences or of the general theory of series.
These general views which I have indicated give only an imperfect idea of
the truly infinite extent and variety of the questions to which geometers
have risen by means of this single consideration of series, so simple in
appearance and so limited in its origin. It necessarily presents as many
different cases as the algebraic resolution of equations, considered in its
whole extent; and it is, by its nature, much more complicated, so much,
indeed, that it always needs this last to conduct it to a complete solution. We
may, therefore, anticipate what must still be its extreme imperfection, in
spite of the successive labours of several geometers of the first order. We do
not, indeed, possess as yet the complete and logical solution of any but the
most simple questions of this nature.
Its identity with this Calculus. It is now easy to conceive the necessary and
perfect identity, which has been already announced, between the calculus of
finite differences and the theory of series considered in all its bearings. In
fact, every differentiation after the manner of Taylor evidently amounts to
finding the law of formation of a series with one or with several variable
indices, from the expression of its general term; in the same way, every
analogous integration may be regarded as having for its object the
summation of a series, the general term of which would be expressed by the
proposed difference. In this point of view, the various problems of the
calculus of differences, direct or inverse, resolved by Taylor and his
successors, have really a very great value, as treating of important questions
relating to series. But it is very doubtful if the form and the notation
introduced by Taylor really give any essential facility in the solution of
questions of this kind. It would be, perhaps, more advantageous for most
cases, and certainly more logical, to replace the differences by the terms
themselves, certain combinations of which they represent. As the calculus

of Taylor does not rest on a truly distinct fundamental idea, and has nothing
peculiar to it but its system of signs, there could never really be any
important advantage in considering it as detached from ordinary analysis, of
which it is, in reality, only an immense branch. This consideration of
differences, most generally useless, even if it does not cause complication,
seems to me to retain the character of an epoch in which, analytical ideas
not being sufficiently familiar to geometers, they were naturally led to
prefer the special forms suitable for simple numerical comparisons.

PERIODIC OR DISCONTINUOUS FUNCTIONS.

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

APPLICATIONS OF THIS CALCULUS.

Series. Among the principal general applications which have been made of
the calculus of finite differences, it would be proper to place in the first
rank, as the most extended and the most important, the solution of questions
relating to series; if, as has been shown, the general theory of series ought
not to be considered as constituting, by its nature, the actual foundation of
the calculus of Taylor.
Interpolations. This great class of problems being then set aside, the most
essential of the veritable applications of the analysis of Taylor is,

undoubtedly, thus far, the general method of interpolations, so frequently
and so usefully employed in the investigation of the empirical laws of
natural phenomena. The question consists, as is well known, in intercalating
between certain given numbers other intermediate numbers, subjected to the
same law which we suppose to exist between the first. We can abundantly
verify, in this principal application of the calculus of Taylor, how truly
foreign and often inconvenient is the consideration of differences with
respect to the questions which depend on that analysis. Indeed, Lagrange
has replaced the formulas of interpolation, deduced from the ordinary
algorithm of the calculus of finite differences, by much simpler general
formulas, which are now almost always preferred, and which have been
found directly, without making any use of the notion of differences, which
only complicates the question.
Approximate Rectification, &c. A last important class of applications of the
calculus of finite differences, which deserves to be distinguished from the
preceding, consists in the eminently useful employment made of it in
geometry for determining by approximation the length and the area of any
curve, and in the same way the cubature of a body of any form whatever.
This procedure (which may besides be conceived abstractly as depending
on the same analytical investigation as the question of interpolation)
frequently offers a valuable supplement to the entirely logical geometrical
methods which often lead to integrations, which we do not yet know how to
effect, or to calculations of very complicated execution.

Such are the various principal considerations to be noticed with respect to
the calculus of finite differences. This examination completes the proposed
philosophical outline of abstract Mathematics.
Concrete Mathematics will now be the subject of a similar labour. In it we
shall particularly devote ourselves to examining how it has been possible
(supposing the general science of the calculus to be perfect), by invariable
procedures, to reduce to pure questions of analysis all the problems which
can be presented by Geometry and Mechanics, and thus to impress on these
two fundamental bases of natural philosophy a degree of precision and

especially of unity; in a word, a character of high perfection, which could
be communicated to them by such a course alone.

BOOK II.
GEOMETRY.

BOOK II.

GEOMETRY.

CHAPTER I.
GENERAL VIEW OF GEOMETRY.

Its true Nature. After the general exposition of the philosophical character
of concrete mathematics, compared with that of abstract mathematics, given
in the introductory chapter, it need not here be shown in a special manner
that geometry must be considered as a true natural science, only much more
simple, and therefore much more perfect, than any other. This necessary
perfection of geometry, obtained essentially by the application of
mathematical analysis, which it so eminently admits, is apt to produce
erroneous views of the real nature of this fundamental science, which most
minds at present conceive to be a purely logical science quite independent
of observation. It is nevertheless evident, to any one who examines with
attention the character of geometrical reasonings, even in the present state
of abstract geometry, that, although the facts which are considered in it are
much more closely united than those relating to any other science, still there
always exists, with respect to every body studied by geometers, a certain
number of primitive phenomena, which, since they are not established by
any reasoning, must be founded on observation alone, and which form the
necessary basis of all the deductions.
The scientific superiority of geometry arises from the phenomena which it
considers being necessarily the most universal and the most simple of all.
Not only may all the bodies of nature give rise to geometrical inquiries, as
well as mechanical ones, but still farther, geometrical phenomena would
still exist, even though all the parts of the universe should be considered as
immovable. Geometry is then, by its nature, more general than mechanics.
At the same time, its phenomena are more simple, for they are evidently
independent of mechanical phenomena, while these latter are always
complicated with the former. The same relations hold good in comparing
geometry with abstract thermology.
For these reasons, in our classification we have made geometry the first part
of concrete mathematics; that part the study of which, in addition to its own

importance, serves as the indispensable basis of all the rest.
Before considering directly the philosophical study of the different orders of
inquiries which constitute our present geometry, we should obtain a clear
and exact idea of the general destination of that science, viewed in all its
bearings. Such is the object of this chapter.
Definition. Geometry is commonly defined in a very vague and entirely
improper manner, as being the science of extension. An improvement on
this would be to say that geometry has for its object the measurement of
extension; but such an explanation would be very insufficient, although at
bottom correct, and would be far from giving any idea of the true general
character of geometrical science.
To do this, I think that I should first explain two fundamental ideas, which,
very simple in themselves, have been singularly obscured by the
employment of metaphysical considerations.
The Idea of Space. The first is that of Space. This conception properly
consists simply in this, that, instead of considering extension in the bodies
themselves, we view it in an indefinite medium, which we regard as
containing all the bodies of the universe. This notion is naturally suggested
to us by observation, when we think of the impression which a body would
leave in a fluid in which it had been placed. It is clear, in fact, that, as
regards its geometrical relations, such an impression may be substituted for
the body itself, without altering the reasonings respecting it. As to the
physical nature of this indefinite space, we are spontaneously led to
represent it to ourselves, as being entirely analogous to the actual medium
in which we live; so that if this medium was liquid instead of gaseous, our
geometrical space would undoubtedly be conceived as liquid also. This
circumstance is, moreover, only very secondary, the essential object of such
a conception being only to make us view extension separately from the
bodies which manifest it to us. We can easily understand in advance the
importance of this fundamental image, since it permits us to study
geometrical phenomena in themselves, abstraction being made of all the
other phenomena which constantly accompany them in real bodies, without,
however, exerting any influence over them. The regular establishment of
this general abstraction must be regarded as the first step which has been
made in the rational study of geometry, which would have been impossible

if it had been necessary to consider, together with the form and the
magnitude of bodies, all their other physical properties. The use of such an
hypothesis, which is perhaps the most ancient philosophical conception
created by the human mind, has now become so familiar to us, that we have
difficulty in exactly estimating its importance, by trying to appreciate the
consequences which would result from its suppression.
Different Kinds of Extension. The second preliminary geometrical
conception which we have to examine is that of the different kinds of
extension, designated by the words volume, surface, line, and even point,
and of which the ordinary explanation is so unsatisfactory.[13]
Although it is evidently impossible to conceive any extension absolutely
deprived of any one of the three fundamental dimensions, it is no less
incontestable that, in a great number of occasions, even of immediate
utility, geometrical questions depend on only two dimensions, considered
separately from the third, or on a single dimension, considered separately
from the two others. Again, independently of this direct motive, the study of
extension with a single dimension, and afterwards with two, clearly
presents itself as an indispensable preliminary for facilitating the study of
complete bodies of three dimensions, the immediate theory of which would
be too complicated. Such are the two general motives which oblige
geometers to consider separately extension with regard to one or to two
dimensions, as well as relatively to all three together.
The general notions of surface and of line have been formed by the human
mind, in order that it may be able to think, in a permanent manner, of
extension in two directions, or in one only. The hyperbolical expressions
habitually employed by geometers to define these notions tend to convey
false ideas of them; but, examined in themselves, they have no other object
than to permit us to reason with facility respecting these two kinds of
extension, making complete abstraction of that which ought not to be taken
into consideration. Now for this it is sufficient to conceive the dimension
which we wish to eliminate as becoming gradually smaller and smaller, the
two others remaining the same, until it arrives at such a degree of tenuity
that it can no longer fix the attention. It is thus that we naturally acquire the
real idea of a surface, and, by a second analogous operation, the idea of a
line, by repeating for breadth what we had at first done for thickness.
Finally, if we again repeat the same operation, we arrive at the idea of a

point, or of an extension considered only with reference to its place,
abstraction being made of all magnitude, and designed consequently to
determine positions.
Surfaces evidently have, moreover, the general property of exactly
circumscribing volumes; and in the same way, lines, in their turn,
circumscribe surfaces and are limited by points. But this consideration, to
which too much importance is often given, is only a secondary one.
Surfaces and lines are, then, in reality, always conceived with three
dimensions; it would be, in fact, impossible to represent to one's self a
surface otherwise than as an extremely thin plate, and a line otherwise than
as an infinitely fine thread. It is even plain that the degree of tenuity
attributed by each individual to the dimensions of which he wishes to make
abstraction is not constantly identical, for it must depend on the degree of
subtilty of his habitual geometrical observations. This want of uniformity
has, besides, no real inconvenience, since it is sufficient, in order that the
ideas of surface and of line should satisfy the essential condition of their
destination, for each one to represent to himself the dimensions which are to
be neglected as being smaller than all those whose magnitude his daily
experience gives him occasion to appreciate.
We hence see how devoid of all meaning are the fantastic discussions of
metaphysicians upon the foundations of geometry. It should also be
remarked that these primordial ideas are habitually presented by geometers
in an unphilosophical manner, since, for example, they explain the notions
of the different sorts of extent in an order absolutely the inverse of their
natural dependence, which often produces the most serious inconveniences
in elementary instruction.

THE FINAL OBJECT OF GEOMETRY.

These preliminaries being established, we can proceed directly to the
general definition of geometry, continuing to conceive this science as
having for its final object the measurement of extension.
It is necessary in this matter to go into a thorough explanation, founded on
the distinction of the three kinds of extension, since the notion of

measurement is not exactly the same with reference to surfaces and
volumes as to lines.
Nature of Geometrical Measurement. If we take the word measurement in
its direct and general mathematical acceptation, which signifies simply the
determination of the value of the ratios between any homogeneous
magnitudes, we must consider, in geometry, that the measurement of
surfaces and of volumes, unlike that of lines, is never conceived, even in the
most simple and the most favourable cases, as being effected directly. The
comparison of two lines is regarded as direct; that of two surfaces or of two
volumes is, on the contrary, always indirect. Thus we conceive that two
lines may be superposed; but the superposition of two surfaces, or, still
more so, of two volumes, is evidently impossible in most cases; and, even
when it becomes rigorously practicable, such a comparison is never either
convenient or exact. It is, then, very necessary to explain wherein properly
consists the truly geometrical measurement of a surface or of a volume.
Measurement of Surfaces and of Volumes. For this we must consider that,
whatever may be the form of a body, there always exists a certain number
of lines, more or less easy to be assigned, the length of which is sufficient to
define exactly the magnitude of its surface or of its volume. Geometry,
regarding these lines as alone susceptible of being directly measured,
proposes to deduce, from the simple determination of them, the ratio of the
surface or of the volume sought, to the unity of surface, or to the unity of
volume. Thus the general object of geometry, with respect to surfaces and to
volumes, is properly to reduce all comparisons of surfaces or of volumes to
simple comparisons of lines.
Besides the very great facility which such a transformation evidently offers
for the measurement of volumes and of surfaces, there results from it, in
considering it in a more extended and more scientific manner, the general
possibility of reducing to questions of lines all questions relating to volumes
and to surfaces, considered with reference to their magnitude. Such is often
the most important use of the geometrical expressions which determine
surfaces and volumes in functions of the corresponding lines.
It is true that direct comparisons between surfaces or between volumes are
sometimes employed; but such measurements are not regarded as
geometrical, but only as a supplement sometimes necessary, although too

rarely applicable, to the insufficiency or to the difficulty of truly rational
methods. It is thus that we often determine the volume of a body, and in
certain cases its surface, by means of its weight. In the same way, on other
occasions, when we can substitute for the proposed volume an equivalent
liquid volume, we establish directly the comparison of the two volumes, by
profiting by the property possessed by liquid masses, of assuming any
desired form. But all means of this nature are purely mechanical, and
rational geometry necessarily rejects them.
To render more sensible the difference between these modes of
determination and true geometrical measurements, I will cite a single very
remarkable example; the manner in which Galileo determined the ratio of
the ordinary cycloid to that of the generating circle. The geometry of his
time was as yet insufficient for the rational solution of such a problem.
Galileo conceived the idea of discovering that ratio by a direct experiment.
Having weighed as exactly as possible two plates of the same material and
of equal thickness, one of them having the form of a circle and the other
that of the generated cycloid, he found the weight of the latter always triple
that of the former; whence he inferred that the area of the cycloid is triple
that of the generating circle, a result agreeing with the veritable solution
subsequently obtained by Pascal and Wallis. Such a success evidently
depends on the extreme simplicity of the ratio sought; and we can
understand the necessary insufficiency of such expedients, even when they
are actually practicable.
We see clearly, from what precedes, the nature of that part of geometry
relating to volumes and that relating to surfaces. But the character of the
geometry of lines is not so apparent, since, in order to simplify the
exposition, we have considered the measurement of lines as being made
directly. There is, therefore, needed a complementary explanation with
respect to them.
Measurement of curved Lines. For this purpose, it is sufficient to distinguish
between the right line and curved lines, the measurement of the first being
alone regarded as direct, and that of the other as always indirect. Although
superposition is sometimes strictly practicable for curved lines, it is
nevertheless evident that truly rational geometry must necessarily reject it,
as not admitting of any precision, even when it is possible. The geometry of
lines has, then, for its general object, to reduce in every case the

measurement of curved lines to that of right lines; and consequently, in the
most extended point of view, to reduce to simple questions of right lines all
questions relating to the magnitude of any curves whatever. To understand
the possibility of such a transformation, we must remark, that in every
curve there always exist certain right lines, the length of which must be
sufficient to determine that of the curve. Thus, in a circle, it is evident that
from the length of the radius we must be able to deduce that of the
circumference; in the same way, the length of an ellipse depends on that of
its two axes; the length of a cycloid upon the diameter of the generating
circle, &c.; and if, instead of considering the whole of each curve, we
demand, more generally, the length of any arc, it will be sufficient to add to
the different rectilinear parameters, which determine the whole curve, the
chord of the proposed arc, or the co-ordinates of its extremities. To discover
the relation which exists between the length of a curved line and that of
similar right lines, is the general problem of the part of geometry which
relates to the study of lines.
Combining this consideration with those previously suggested with respect
to volumes and to surfaces, we may form a very clear idea of the science of
geometry, conceived in all its parts, by assigning to it, for its general object,
the final reduction of the comparisons of all kinds of extent, volumes,
surfaces, or lines, to simple comparisons of right lines, the only
comparisons regarded as capable of being made directly, and which indeed
could not be reduced to any others more easy to effect. Such a conception,
at the same time, indicates clearly the veritable character of geometry, and
seems suited to show at a single glance its utility and its perfection.
Measurement of right Lines. In order to complete this fundamental
explanation, I have yet to show how there can be, in geometry, a special
section relating to the right line, which seems at first incompatible with the
principle that the measurement of this class of lines must always be
regarded as direct.
It is so, in fact, as compared with that of curved lines, and of all the other
objects which geometry considers. But it is evident that the estimation of a
right line cannot be viewed as direct except so far as the linear unit can be
applied to it. Now this often presents insurmountable difficulties, as I had
occasion to show, for another reason, in the introductory chapter. We must,
then, make the measurement of the proposed right line depend on other

analogous measurements capable of being effected directly. There is, then,
necessarily a primary distinct branch of geometry, exclusively devoted to
the right line; its object is to determine certain right lines from others by
means of the relations belonging to the figures resulting from their
assemblage. This preliminary part of geometry, which is almost
imperceptible in viewing the whole of the science, is nevertheless
susceptible of a great development. It is evidently of especial importance,
since all other geometrical measurements are referred to those of right lines,
and if they could not be determined, the solution of every question would
remain unfinished.
Such, then, are the various fundamental parts of rational geometry, arranged
according to their natural dependence; the geometry of lines being first
considered, beginning with the right line; then the geometry of surfaces,
and, finally, that of solids.

INFINITE EXTENT OF ITS FIELD.

Having determined with precision the general and final object of
geometrical inquiries, the science must now be considered with respect to
the field embraced by each of its three fundamental sections.
Thus considered, geometry is evidently susceptible, by its nature, of an
extension which is rigorously infinite; for the measurement of lines, of
surfaces, or of volumes presents necessarily as many distinct questions as
we can conceive different figures subjected to exact definitions; and their
number is evidently infinite.
Geometers limited themselves at first to consider the most simple figures
which were directly furnished them by nature, or which were deduced from
these primitive elements by the least complicated combinations. But they
have perceived, since Descartes, that, in order to constitute the science in
the most philosophical manner, it was necessary to make it apply to all
imaginable figures. This abstract geometry will then inevitably comprehend
as particular cases all the different real figures which the exterior world
could present. It is then a fundamental principle in truly rational geometry

to consider, as far as possible, all figures which can be rigorously
conceived.
The most superficial examination is enough to convince us that these
figures present a variety which is quite infinite.
Infinity of Lines. With respect to curved lines, regarding them as generated
by the motion of a point governed by a certain law, it is plain that we shall
have, in general, as many different curves as we conceive different laws for
this motion, which may evidently be determined by an infinity of distinct
conditions; although it may sometimes accidentally happen that new
generations produce curves which have been already obtained. Thus, among
plane curves, if a point moves so as to remain constantly at the same
distance from a fixed point, it will generate a circle; if it is the sum or the
difference of its distances from two fixed points which remains constant,
the curve described will be an ellipse or an hyperbola; if it is their product,
we shall have an entirely different curve; if the point departs equally from a
fixed point and from a fixed line, it will describe a parabola; if it revolves
on a circle at the same time that this circle rolls along a straight line, we
shall have a cycloid; if it advances along a straight line, while this line,
fixed at one of its extremities, turns in any manner whatever, there will
result what in general terms are called spirals, which of themselves
evidently present as many perfectly distinct curves as we can suppose
different relations between these two motions of translation and of rotation,
&c. Each of these different curves may then furnish new ones, by the
different general constructions which geometers have imagined, and which
give rise to evolutes, to epicycloids, to caustics, &c. Finally, there exists a
still greater variety among curves of double curvature.
Infinity of Surfaces. As to surfaces, the figures are necessarily more
different still, considering them as generated by the motion of lines. Indeed,
the figure may then vary, not only in considering, as in curves, the different
infinitely numerous laws to which the motion of the generating line may be
subjected, but also in supposing that this line itself may change its nature; a
circumstance which has nothing analogous in curves, since the points which
describe them cannot have any distinct figure. Two classes of very different
conditions may then cause the figures of surfaces to vary, while there exists
only one for lines. It is useless to cite examples of this doubly infinite
multiplicity of surfaces. It would be sufficient to consider the extreme

variety of the single group of surfaces which may be generated by a right
line, and which comprehends the whole family of cylindrical surfaces, that
of conical surfaces, the most general class of developable surfaces, &c.
Infinity of Volumes. With respect to volumes, there is no occasion for any
special consideration, since they are distinguished from each other only by
the surfaces which bound them.
In order to complete this sketch, it should be added that surfaces themselves
furnish a new general means of conceiving new curves, since every curve
may be regarded as produced by the intersection of two surfaces. It is in this
way, indeed, that the first lines which we may regard as having been truly
invented by geometers were obtained, since nature gave directly the straight
line and the circle. We know that the ellipse, the parabola, and the
hyperbola, the only curves completely studied by the ancients, were in their
origin conceived only as resulting from the intersection of a cone with
circular base by a plane in different positions. It is evident that, by the
combined employment of these different general means for the formation of
lines and of surfaces, we could produce a rigorously infinitely series of
distinct forms in starting from only a very small number of figures directly
furnished by observation.
Analytical invention of Curves, &c. Finally, all the various direct means for
the invention of figures have scarcely any farther importance, since rational
geometry has assumed its final character in the hands of Descartes. Indeed,
as we shall see more fully in chapter iii., the invention of figures is now
reduced to the invention of equations, so that nothing is more easy than to
conceive new lines and new surfaces, by changing at will the functions
introduced into the equations. This simple abstract procedure is, in this
respect, infinitely more fruitful than all the direct resources of geometry,
developed by the most powerful imagination, which should devote itself
exclusively to that order of conceptions. It also explains, in the most general
and the most striking manner, the necessarily infinite variety of geometrical
forms, which thus corresponds to the diversity of analytical functions.
Lastly, it shows no less clearly that the different forms of surfaces must be
still more numerous than those of lines, since lines are represented
analytically by equations with two variables, while surfaces give rise to
equations with three variables, which necessarily present a greater diversity.

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

EXPANSION OF ORIGINAL DEFINITION.

To complete the formation of an exact and sufficiently extended idea of the
nature of geometrical inquiries, it is now indispensable to return to the
general definition above given, in order to present it under a new point of
view, without which the complete science would be only very imperfectly
conceived.
When we assign as the object of geometry the measurement of all sorts of
lines, surfaces, and volumes, that is, as has been explained, the reduction of
all geometrical comparisons to simple comparisons of right lines, we have
evidently the advantage of indicating a general destination very precise and
very easy to comprehend. But if we set aside every definition, and examine
the actual composition of the science of geometry, we will at first be
induced to regard the preceding definition as much too narrow; for it is
certain that the greater part of the investigations which constitute our
present geometry do not at all appear to have for their object the
measurement of extension. In spite of this fundamental objection, I will
persist in retaining this definition; for, in fact, if, instead of confining
ourselves to considering the different questions of geometry isolatedly, we
endeavour to grasp the leading questions, in comparison with which all
others, however important they may be, must be regarded as only
secondary, we will finally recognize that the measurement of lines, of
surfaces, and of volumes, is the invariable object, sometimes direct, though
most often indirect, of all geometrical labours.
This general proposition being fundamental, since it can alone give our
definition all its value, it is indispensable to enter into some developments
upon this subject.

PROPERTIES OF LINES AND SURFACES.

When we examine with attention the geometrical investigations which do
not seem to relate to the measurement of extent, we find that they consist
essentially in the study of the different properties of each line or of each
surface; that is, in the knowledge of the different modes of generation, or at
least of definition, peculiar to each figure considered. Now we can easily
establish in the most general manner the necessary relation of such a study
to the question of measurement, for which the most complete knowledge of
the properties of each form is an indispensable preliminary. This is
concurrently proven by two considerations, equally fundamental, although
quite distinct in their nature.
Necessity of their Study: 1. To find the most suitable Property. The first,
purely scientific, consists in remarking that, if we did not know any other
characteristic property of each line or surface than that one according to
which geometers had first conceived it, in most cases it would be
impossible to succeed in the solution of questions relating to its
measurement. In fact, it is easy to understand that the different definitions
which each figure admits of are not all equally suitable for such an object,
and that they even present the most complete oppositions in that respect.
Besides, since the primitive definition of each figure was evidently not
chosen with this condition in view, it is clear that we must not expect, in
general, to find it the most suitable; whence results the necessity of
discovering others, that is, of studying as far as is possible the properties of
the proposed figure. Let us suppose, for example, that the circle is defined
to be "the curve which, with the same contour, contains the greatest area."
This is certainly a very characteristic property, but we would evidently find
insurmountable difficulties in trying to deduce from such a starting point
the solution of the fundamental questions relating to the rectification or to
the quadrature of this curve. It is clear, in advance, that the property of
having all its points equally distant from a fixed point must evidently be
much better adapted to inquiries of this nature, even though it be not
precisely the most suitable. In like manner, would Archimedes ever have
been able to discover the quadrature of the parabola if he had known no
other property of that curve than that it was the section of a cone with a
circular base, by a plane parallel to its generatrix? The purely speculative
labours of preceding geometers, in transforming this first definition, were
evidently indispensable preliminaries to the direct solution of such a
question. The same is true, in a still greater degree, with respect to surfaces.

To form a just idea of this, we need only compare, as to the question of
cubature or quadrature, the common definition of the sphere with that one,
no less characteristic certainly, which would consist in regarding a spherical
body, as that one which, with the same area, contains the greatest volume.
No more examples are needed to show the necessity of knowing, so far as is
possible, all the properties of each line or of each surface, in order to
facilitate the investigation of rectifications, of quadratures, and of
cubatures, which constitutes the final object of geometry. We may even say
that the principal difficulty of questions of this kind consists in employing
in each case the property which is best adapted to the nature of the proposed
problem. Thus, while we continue to indicate, for more precision, the
measurement of extension as the general destination of geometry, this first
consideration, which goes to the very bottom of the subject, shows clearly
the necessity of including in it the study, as thorough as possible, of the
different generations or definitions belonging to the same form.
2. To pass from the Concrete to the Abstract. A second consideration, of at
least equal importance, consists in such a study being indispensable for
organizing in a rational manner the relation of the abstract to the concrete in
geometry.
The science of geometry having to consider all imaginable figures which
admit of an exact definition, it necessarily results from this, as we have
remarked, that questions relating to any figures presented by nature are
always implicitly comprised in this abstract geometry, supposed to have
attained its perfection. But when it is necessary to actually pass to concrete
geometry, we constantly meet with a fundamental difficulty, that of
knowing to which of the different abstract types we are to refer, with
sufficient approximation, the real lines or surfaces which we have to study.
Now it is for the purpose of establishing such a relation that it is particularly
indispensable to know the greatest possible number of properties of each
figure considered in geometry.
In fact, if we always confined ourselves to the single primitive definition of
a line or of a surface, supposing even that we could then measure it (which,
according to the first order of considerations, would generally be
impossible), this knowledge would remain almost necessarily barren in the
application, since we should not ordinarily know how to recognize that

figure in nature when it presented itself there; to ensure that, it would be
necessary that the single characteristic, according to which geometers had
conceived it, should be precisely that one whose verification external
circumstances would admit: a coincidence which would be purely
fortuitous, and on which we could not count, although it might sometimes
take place. It is, then, only by multiplying as much as possible the
characteristic properties of each abstract figure, that we can be assured, in
advance, of recognizing it in the concrete state, and of thus turning to
account all our rational labours, by verifying in each case the definition
which is susceptible of being directly proven. This definition is almost
always the only one in given circumstances, and varies, on the other hand,
for the same figure, with different circumstances; a double reason for its
previous determination.
Illustration: Orbits of the Planets. The geometry of the heavens furnishes us
with a very memorable example in this matter, well suited to show the
general necessity of such a study. We know that the ellipse was discovered
by Kepler to be the curve which the planets describe about the sun, and the
satellites about their planets. Now would this fundamental discovery, which
re-created astronomy, ever have been possible, if geometers had been
always confined to conceiving the ellipse only as the oblique section of a
circular cone by a plane? No such definition, it is evident, would admit of
such a verification. The most general property of the ellipse, that the sum of
the distances from any of its points to two fixed points is a constant
quantity, is undoubtedly much more susceptible, by its nature, of causing
the curve to be recognized in this case, but still is not directly suitable. The
only characteristic which can here be immediately verified is that which is
derived from the relation which exists in the ellipse between the length of
the focal distances and their direction; the only relation which admits of an
astronomical interpretation, as expressing the law which connects the
distance from the planet to the sun, with the time elapsed since the
beginning of its revolution. It was, then, necessary that the purely
speculative labours of the Greek geometers on the properties of the conic
sections should have previously presented their generation under a
multitude of different points of view, before Kepler could thus pass from
the abstract to the concrete, in choosing from among all these different
characteristics that one which could be most easily proven for the planetary
orbits.

Illustration: Figure of the Earth. Another example of the same order, but
relating to surfaces, occurs in considering the important question of the
figure of the earth. If we had never known any other property of the sphere
than its primitive character of having all its points equally distant from an
interior point, how would we ever have been able to discover that the
surface of the earth was spherical? For this, it was necessary previously to
deduce from this definition of the sphere some properties capable of being
verified by observations made upon the surface alone, such as the constant
ratio which exists between the length of the path traversed in the direction
of any meridian of a sphere going towards a pole, and the angular height of
this pole above the horizon at each point. Another example, but involving a
much longer series of preliminary speculations, is the subsequent proof that
the earth is not rigorously spherical, but that its form is that of an ellipsoid
of revolution.
After such examples, it would be needless to give any others, which any
one besides may easily multiply. All of them prove that, without a very
extended knowledge of the different properties of each figure, the relation
of the abstract to the concrete, in geometry, would be purely accidental, and
that the science would consequently want one of its most essential
foundations.
Such, then, are two general considerations which fully demonstrate the
necessity of introducing into geometry a great number of investigations
which have not the measurement of extension for their direct object; while
we continue, however, to conceive such a measurement as being the final
destination of all geometrical science. In this way we can retain the
philosophical advantages of the clearness and precision of this definition,
and still include in it, in a very logical though indirect manner, all known
geometrical researches, in considering those which do not seem to relate to
the measurement of extension, as intended either to prepare for the solution
of the final questions, or to render possible the application of the solutions
obtained.
Having thus recognized, as a general principle, the close and necessary
connexion of the study of the properties of lines and surfaces with those
researches which constitute the final object of geometry, it is evident that
geometers, in the progress of their labours, must by no means constrain
themselves to keep such a connexion always in view. Knowing, once for all,

how important it is to vary as much as possible the manner of conceiving
each figure, they should pursue that study, without considering of what
immediate use such or such a special property may be for rectifications,
quadratures, and cubatures. They would uselessly fetter their inquiries by
attaching a puerile importance to the continued establishment of that co-
ordination.
This general exposition of the general object of geometry is so much the
more indispensable, since, by the very nature of the subject, this study of
the different properties of each line and of each surface necessarily
composes by far the greater part of the whole body of geometrical
researches. Indeed, the questions directly relating to rectifications, to
quadratures, and to cubatures, are evidently, by themselves, very few in
number for each figure considered. On the other hand, the study of the
properties of the same figure presents an unlimited field to the activity of
the human mind, in which it may always hope to make new discoveries.
Thus, although geometers have occupied themselves for twenty centuries,
with more or less activity undoubtedly, but without any real interruption, in
the study of the conic sections, they are far from regarding that so simple
subject as being exhausted; and it is certain, indeed, that in continuing to
devote themselves to it, they would not fail to find still unknown properties
of those different curves. If labours of this kind have slackened
considerably for a century past, it is not because they are completed, but
only, as will be presently explained, because the philosophical revolution in
geometry, brought about by Descartes, has singularly diminished the
importance of such researches.
It results from the preceding considerations that not only is the field of
geometry necessarily infinite because of the variety of figures to be
considered, but also in virtue of the diversity of the points of view under the
same figure may be regarded. This last conception is, indeed, that which
gives the broadest and most complete idea of the whole body of geometrical
researches. We see that studies of this kind consist essentially, for each line
or for each surface, in connecting all the geometrical phenomena which it
can present, with a single fundamental phenomenon, regarded as the
primitive definition.

THE TWO GENERAL METHODS OF GEOMETRY.

Having now explained in a general and yet precise manner the final object
of geometry, and shown how the science, thus defined, comprehends a very
extensive class of researches which did not at first appear necessarily to
belong to it, there remains to be considered the method to be followed for
the formation of this science. This discussion is indispensable to complete
this first sketch of the philosophical character of geometry. I shall here
confine myself to indicating the most general consideration in this matter,
developing and summing up this important fundamental idea in the
following chapters.
Geometrical questions may be treated according to two methods so
different, that there result from them two sorts of geometry, so to say, the
philosophical character of which does not seem to me to have yet been
properly apprehended. The expressions of Synthetical Geometry and
Analytical Geometry, habitually employed to designate them, give a very
false idea of them. I would much prefer the purely historical denominations
of Geometry of the Ancients and Geometry of the Moderns, which have at
least the advantage of not causing their true character to be misunderstood.
But I propose to employ henceforth the regular expressions of Special
Geometry and General Geometry, which seem to me suited to characterize
with precision the veritable nature of the two methods.
Their fundamental Difference. The fundamental difference between the
manner in which we conceive Geometry since Descartes, and the manner in
which the geometers of antiquity treated geometrical questions, is not the
use of the Calculus (or Algebra), as is commonly thought to be the case. On
the one hand, it is certain that the use of the calculus was not entirely
unknown to the ancient geometers, since they used to make continual and
very extensive applications of the theory of proportions, which was for
them, as a means of deduction, a sort of real, though very imperfect and
especially extremely limited equivalent for our present algebra. The
calculus may even be employed in a much more complete manner than they
have used it, in order to obtain certain geometrical solutions, which will still
retain all the essential character of the ancient geometry; this occurs very
frequently with respect to those problems of geometry of two or of three
dimensions, which are commonly designated under the name of
determinate. On the other hand, important as is the influence of the calculus
in our modern geometry, various solutions obtained without algebra may

sometimes manifest the peculiar character which distinguishes it from the
ancient geometry, although analysis is generally indispensable. I will cite,
as an example, the method of Roberval for tangents, the nature of which is
essentially modern, and which, however, leads in certain cases to complete
solutions, without any aid from the calculus. It is not, then, the instrument
of deduction employed which is the principal distinction between the two
courses which the human mind can take in geometry.
The real fundamental difference, as yet imperfectly apprehended, seems to
me to consist in the very nature of the questions considered. In truth,
geometry, viewed as a whole, and supposed to have attained entire
perfection, must, as we have seen on the one hand, embrace all imaginable
figures, and, on the other, discover all the properties of each figure. It
admits, from this double consideration, of being treated according to two
essentially distinct plans; either, 1°, by grouping together all the questions,
however different they may be, which relate to the same figure, and
isolating those relating to different bodies, whatever analogy there may
exist between them; or, 2°, on the contrary, by uniting under one point of
view all similar inquiries, to whatever different figures they may relate, and
separating the questions relating to the really different properties of the
same body. In a word, the whole body of geometry may be essentially
arranged either with reference to the bodies studied or to the phenomena to
be considered. The first plan, which is the most natural, was that of the
ancients; the second, infinitely more rational, is that of the moderns since
Descartes.
Geometry of the Ancients. Indeed, the principal characteristics of the ancient
geometry is that they studied, one by one, the different lines and the
different surfaces, not passing to the examination of a new figure till they
thought they had exhausted all that there was interesting in the figures
already known. In this way of proceeding, when they undertook the study
of a new curve, the whole of the labour bestowed on the preceding ones
could not offer directly any essential assistance, otherwise than by the
geometrical practice to which it had trained the mind. Whatever might be
the real similarity of the questions proposed as to two different figures, the
complete knowledge acquired for the one could not at all dispense with
taking up again the whole of the investigation for the other. Thus the
progress of the mind was never assured; so that they could not be certain, in

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

studied by the ancients, have led to the discovery of important properties
which they had not before even suspected. Such is geometry, since the
radical revolution produced by Descartes in the general system of the
science.
The Superiority of the modern Geometry. The mere indication of the
fundamental character of each of the two geometries is undoubtedly
sufficient to make apparent the immense necessary superiority of modern
geometry. We may even say that, before the great conception of Descartes,
rational geometry was not truly constituted upon definitive bases, whether
in its abstract or concrete relations. In fact, as regards science, considered
speculatively, it is clear that, in continuing indefinitely to follow the course
of the ancients, as did the moderns before Descartes, and even for a little
while afterwards, by adding some new curves to the small number of those
which they had studied, the progress thus made, however rapid it might
have been, would still be found, after a long series of ages, to be very
inconsiderable in comparison with the general system of geometry, seeing
the infinite variety of the forms which would still have remained to be
studied. On the contrary, at each question resolved according to the method
of the moderns, the number of geometrical problems to be resolved is then,
once for all, diminished by so much with respect to all possible bodies.
Another consideration is, that it resulted, from their complete want of
general methods, that the ancient geometers, in all their investigations, were
entirely abandoned to their own strength, without ever having the certainty
of obtaining, sooner or later, any solution whatever. Though this
imperfection of the science was eminently suited to call forth all their
admirable sagacity, it necessarily rendered their progress extremely slow;
we can form some idea of this by the considerable time which they
employed in the study of the conic sections. Modern geometry, making the
progress of our mind certain, permits us, on the contrary, to make the
greatest possible use of the forces of our intelligence, which the ancients
were often obliged to waste on very unimportant questions.
A no less important difference between the two systems appears when we
come to consider geometry in the concrete point of view. Indeed, we have
already remarked that the relation of the abstract to the concrete in
geometry can be founded upon rational bases only so far as the
investigations are made to bear directly upon all imaginable figures. In

studying lines, only one by one, whatever may be the number, always
necessarily very small, of those which we shall have considered, the
application of such theories to figures really existing in nature will never
have any other than an essentially accidental character, since there is
nothing to assure us that these figures can really be brought under the
abstract types considered by geometers.
Thus, for example, there is certainly something fortuitous in the happy
relation established between the speculations of the Greek geometers upon
the conic sections and the determination of the true planetary orbits. In
continuing geometrical researches upon the same plan, there was no good
reason for hoping for similar coincidences; and it would have been possible,
in these special studies, that the researches of geometers should have been
directed to abstract figures entirely incapable of any application, while they
neglected others, susceptible perhaps of an important and immediate
application. It is clear, at least, that nothing positively guaranteed the
necessary applicability of geometrical speculations. It is quite another thing
in the modern geometry. From the single circumstance that in it we proceed
by general questions relating to any figures whatever, we have in advance
the evident certainty that the figures really existing in the external world
could in no case escape the appropriate theory if the geometrical
phenomenon which it considers presents itself in them.
From these different considerations, we see that the ancient system of
geometry wears essentially the character of the infancy of the science,
which did not begin to become completely rational till after the
philosophical resolution produced by Descartes. But it is evident, on the
other hand, that geometry could not be at first conceived except in this
special manner. General geometry would not have been possible, and its
necessity could not even have been felt, if a long series of special labours
on the most simple figures had not previously furnished bases for the
conception of Descartes, and rendered apparent the impossibility of
persisting indefinitely in the primitive geometrical philosophy.
The Ancient the Base of the Modern. From this last consideration we must
infer that, although the geometry which I have called general must be now
regarded as the only true dogmatical geometry, and that to which we shall
chiefly confine ourselves, the other having no longer much more than an
historical interest, nevertheless it is not possible to entirely dispense with

special geometry in a rational exposition of the science. We undoubtedly
need not borrow directly from ancient geometry all the results which it has
furnished; but, from the very nature of the subject, it is necessarily
impossible entirely to dispense with the ancient method, which will always
serve as the preliminary basis of the science, dogmatically as well as
historically. The reason of this is easy to understand. In fact, general
geometry being essentially founded, as we shall soon establish, upon the
employment of the calculus in the transformation of geometrical into
analytical considerations, such a manner of proceeding could not take
possession of the subject immediately at its origin. We know that the
application of mathematical analysis, from its nature, can never commence
any science whatever, since evidently it cannot be employed until the
science has already been sufficiently cultivated to establish, with respect to
the phenomena considered, some equations which can serve as starting
points for the analytical operations. These fundamental equations being
once discovered, analysis will enable us to deduce from them a multitude of
consequences which it would have been previously impossible even to
suspect; it will perfect the science to an immense degree, both with respect
to the generality of its conceptions and to the complete co-ordination
established between them. But mere mathematical analysis could never be
sufficient to form the bases of any natural science, not even to demonstrate
them anew when they have once been established. Nothing can dispense
with the direct study of the subject, pursued up to the point of the discovery
of precise relations.
We thus see that the geometry of the ancients will always have, by its
nature, a primary part, absolutely necessary and more or less extensive, in
the complete system of geometrical knowledge. It forms a rigorously
indispensable introduction to general geometry. But it is to this that it must
be limited in a completely dogmatic exposition. I will consider, then,
directly, in the following chapter, this special or preliminary geometry
restricted to exactly its necessary limits, in order to occupy myself
thenceforth only with the philosophical examination of general or definitive
geometry, the only one which is truly rational, and which at present
essentially composes the science.

CHAPTER II.
ANCIENT OR SYNTHETIC GEOMETRY.

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

ITS PROPER EXTENT.

Lines; Polygons; Polyhedrons. In considering it under this point of view, it
is easy to recognize that we might restrict it to the study of the right line
alone for what concerns the geometry of lines; to the quadrature of
rectilinear plane areas; and, lastly, to the cubature of bodies terminated by
plane faces. The elementary propositions relating to these three fundamental
questions form, in fact, the necessary starting point of all geometrical
inquiries; they alone cannot be obtained except by a direct study of the
subject; while, on the contrary, the complete theory of all other figures,
even that of the circle, and of the surfaces and volumes which are connected
with it, may at the present day be completely comprehended in the domain
of general or analytical geometry; these primitive elements at once
furnishing equations which are sufficient to allow of the application of the
calculus to geometrical questions, which would not have been possible
without this previous condition.
It results from this consideration that, in common practice, we give to
elementary geometry more extent than would be rigorously necessary to it;
since, besides the right line, polygons, and polyhedrons, we also include in
it the circle and the "round" bodies; the study of which might, however, be
as purely analytical as that, for example, of the conic sections. An
unreflecting veneration for antiquity contributes to maintain this defect in
method; but the best reason which can be given for it is the serious
inconvenience for ordinary instruction which there would be in postponing,

to so distant an epoch of mathematical education, the solution of several
essential questions, which are susceptible of a direct and continual
application to a great number of important uses. In fact, to proceed in the
most rational manner, we should employ the integral calculus in obtaining
the interesting results relating to the length or the area of the circle, or to the
quadrature of the sphere, &c., which have been determined by the ancients
from extremely simple considerations. This inconvenience would be of little
importance with regard to the persons destined to study the whole of
mathematical science, and the advantage of proceeding in a perfectly
logical order would have a much greater comparative value. But the
contrary case being the more frequent, theories so essential have necessarily
been retained in elementary geometry. Perhaps the conic sections, the
cycloid, &c., might be advantageously added in such cases.
Not to be farther restricted. While this preliminary portion of geometry,
which cannot be founded on the application of the calculus, is reduced by
its nature to a very limited series of fundamental researches, relating to the
right line, polygonal areas, and polyhedrons, it is certain, on the other hand,
that we cannot restrict it any more; although, by a veritable abuse of the
spirit of analysis, it has been recently attempted to present the establishment
of the principal theorems of elementary geometry under an algebraical point
of view. Thus some have pretended to demonstrate, by simple abstract
considerations of mathematical analysis, the constant relation which exists
between the three angles of a rectilinear triangle, the fundamental
proposition of the theory of similar triangles, that of parallelopipedons, &c.;
in a word, precisely the only geometrical propositions which cannot be
obtained except by a direct study of the subject, without the calculus being
susceptible of having any part in it. Such aberrations are the unreflecting
exaggerations of that natural and philosophical tendency which leads us to
extend farther and farther the influence of analysis in mathematical studies.
In mechanics, the pretended analytical demonstrations of the parallelogram
of forces are of similar character.
The viciousness of such a manner of proceeding follows from the principles
previously presented. We have already, in fact, recognized that, since the
calculus is not, and cannot be, any thing but a means of deduction, it would
indicate a radically false idea of it to wish to employ it in establishing the
elementary foundations of any science whatever; for on what would the

analytical reasonings in such an operation repose? A labour of this nature,
very far from really perfecting the philosophical character of a science,
would constitute a return towards the metaphysical age, in presenting real
facts as mere logical abstractions.
When we examine in themselves these pretended analytical demonstrations
of the fundamental propositions of elementary geometry, we easily verify
their necessary want of meaning. They are all founded on a vicious manner
of conceiving the principle of homogeneity, the true general idea of which
was explained in the second chapter of the preceding book. These
demonstrations suppose that this principle does not allow us to admit the
coexistence in the same equation of numbers obtained by different concrete
comparisons, which is evidently false, and contrary to the constant practice
of geometers. Thus it is easy to recognize that, by employing the law of
homogeneity in this arbitrary and illegitimate acceptation, we could succeed
in "demonstrating," with quite as much apparent rigour, propositions whose
absurdity is manifest at the first glance. In examining attentively, for
example, the procedure by the aid of which it has been attempted to prove
analytically that the sum of the three angles of any rectilinear triangle is
constantly equal to two right angles, we see that it is founded on this
preliminary principle that, if two triangles have two of their angles
respectively equal, the third angle of the one will necessarily be equal to the
third angle of the other. This first point being granted, the proposed relation
is immediately deduced from it in a very exact and simple manner. Now the
analytical consideration by which this previous proposition has been
attempted to be established, is of such a nature that, if it could be correct,
we could rigorously deduce from it, in reproducing it conversely, this
palpable absurdity, that two sides of a triangle are sufficient, without any
angle, for the entire determination of the third side. We may make
analogous remarks on all the demonstrations of this sort, the sophisms of
which will be thus verified in a perfectly apparent manner.
The more reason that we have here to consider geometry as being at the
present day essentially analytical, the more necessary was it to guard
against this abusive exaggeration of mathematical analysis, according to
which all geometrical observation would be dispensed with, in establishing
upon pure algebraical abstractions the very foundations of this natural
science.

Attempted Demonstrations of Axioms, &c. Another indication that
geometers have too much overlooked the character of a natural science
which is necessarily inherent in geometry, appears from their vain attempts,
so long made, to demonstrate rigorously, not by the aid of the calculus, but
by means of certain constructions, several fundamental propositions of
elementary geometry. Whatever may be effected, it will evidently be
impossible to avoid sometimes recurring to simple and direct observation in
geometry as a means of establishing various results. While, in this science,
the phenomena which are considered are, by virtue of their extreme
simplicity, much more closely connected with one another than those
relating to any other physical science, some must still be found which
cannot be deduced, and which, on the contrary, serve as starting points. It
may be admitted that the greatest logical perfection of the science is to
reduce these to the smallest number possible, but it would be absurd to
pretend to make them completely disappear. I avow, moreover, that I find
fewer real inconveniences in extending, a little beyond what would be
strictly necessary, the number of these geometrical notions thus established
by direct observation, provided they are sufficiently simple, than in making
them the subjects of complicated and indirect demonstrations, even when
these demonstrations may be logically irreproachable.
The true dogmatic destination of the geometry of the ancients, reduced to its
least possible indispensable developments, having thus been characterized
as exactly as possible, it is proper to consider summarily each of the
principal parts of which it must be composed. I think that I may here limit
myself to considering the first and the most extensive of these parts, that
which has for its object the study of the right line; the two other sections,
namely, the quadrature of polygons and the cubature of polyhedrons, from
their limited extent, not being capable of giving rise to any philosophical
consideration of any importance, distinct from those indicated in the
preceding chapter with respect to the measure of areas and of volumes in
general.

GEOMETRY OF THE RIGHT LINE.

The final question which we always have in view in the study of the right
line, properly consists in determining, by means of one another, the

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

GRAPHICAL SOLUTIONS.

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

This exact reproduction of any figure whatever on a different scale cannot
present any great theoretical difficulty when all the parts of the proposed
figure lie in the same plane. But if we suppose, as most frequently happens,
that they are situated in different planes, we see, then, a new order of
geometrical considerations arise. The artificial figure, which is constantly
plane, not being capable, in that case, of being a perfectly faithful image of
the real figure, it is necessary previously to fix with precision the mode of
representation, which gives rise to different systems of Projection.
It then remains to be determined according to what laws the geometrical
phenomena correspond in the two figures. This consideration generates a
new series of geometrical investigations, the final object of which is
properly to discover how we can replace constructions in relief by plane
constructions. The ancients had to resolve several elementary questions of
this kind for various cases in which we now employ spherical trigonometry,
principally for different problems relating to the celestial sphere. Such was
the object of their analemmas, and of the other plane figures which for a
long time supplied the place of the calculus. We see by this that the ancients
really knew the elements of what we now name Descriptive Geometry,
although they did not conceive it in a distinct and general manner.
I think it proper briefly to indicate in this place the true philosophical
character of this "Descriptive Geometry;" although, being essentially a
science of application, it ought not to be included within the proper domain
of this work.

DESCRIPTIVE GEOMETRY.

All questions of geometry of three dimensions necessarily give rise, when
we consider their graphical solution, to a common difficulty which is
peculiar to them; that of substituting for the different constructions in relief,
which are necessary to resolve them directly, and which it is almost always
impossible to execute, simple equivalent plane constructions, by means of
which we finally obtain the same results. Without this indispensable
transformation, every solution of this kind would be evidently incomplete
and really inapplicable in practice, although theoretically the constructions
in space are usually preferable as being more direct. It was in order to
furnish general means for always effecting such a transformation that

Descriptive Geometry was created, and formed into a distinct and
homogeneous system, by the illustrious Monge. He invented, in the first
place, a uniform method of representing bodies by figures traced on a single
plane, by the aid of projections on two different planes, usually
perpendicular to each other, and one of which is supposed to turn about
their common intersection so as to coincide with the other produced; in this
system, or in any other equivalent to it, it is sufficient to regard points and
lines as being determined by their projections, and surfaces by the
projections of their generating lines. This being established, Monge—
analyzing with profound sagacity the various partial labours of this kind
which had before been executed by a number of incongruous procedures,
and considering also, in a general and direct manner, in what any questions
of that nature must consist—found that they could always be reduced to a
very small number of invariable abstract problems, capable of being
resolved separately, once for all, by uniform operations, relating essentially
some to the contacts and others to the intersections of surfaces. Simple and
entirely general methods for the graphical solution of these two orders of
problems having been formed, all the geometrical questions which may
arise in any of the various arts of construction—stone-cutting, carpentry,
perspective, dialling, fortification, &c.—can henceforth be treated as simple
particular cases of a single theory, the invariable application of which will
always necessarily lead to an exact solution, which may be facilitated in
practice by profiting by the peculiar circumstances of each case.

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

the arts better than any one else, endeavoured to sketch out a corresponding
system for the mechanical arts.
Essential as the conception of descriptive geometry really is, it is very
important not to deceive ourselves with respect to its true destination, as did
those who, in the excitement of its first discovery, saw in it a means of
enlarging the general and abstract domain of rational geometry. The result
has in no way answered to these mistaken hopes. And, indeed, is it not
evident that descriptive geometry has no special value except as a science
of application, and as forming the true special theory of the geometrical
arts? Considered in its abstract relations, it could not introduce any truly
distinct order of geometrical speculations. We must not forget that, in order
that a geometrical question should fall within the peculiar domain of
descriptive geometry, it must necessarily have been previously resolved by
speculative geometry, the solutions of which then, as we have seen, always
need to be prepared for practice in such a way as to supply the place of
constructions in relief by plane constructions; a substitution which really
constitutes the only characteristic function of descriptive geometry.
It is proper, however, to remark here, that, with regard to intellectual
education, the study of descriptive geometry possesses an important
philosophical peculiarity, quite independent of its high industrial utility.
This is the advantage which it so pre-eminently offers—in habituating the
mind to consider very complicated geometrical combinations in space, and
to follow with precision their continual correspondence with the figures
which are actually traced—of thus exercising to the utmost, in the most
certain and precise manner, that important faculty of the human mind which
is properly called "imagination," and which consists, in its elementary and
positive acceptation, in representing to ourselves, clearly and easily, a vast
and variable collection of ideal objects, as if they were really before us.
Finally, to complete the indication of the general nature of descriptive
geometry by determining its logical character, we have to observe that,
while it belongs to the geometry of the ancients by the character of its
solutions, on the other hand it approaches the geometry of the moderns by
the nature of the questions which compose it. These questions are in fact
eminently remarkable for that generality which, as we saw in the preceding
chapter, constitutes the true fundamental character of modern geometry; for
the methods used are always conceived as applicable to any figures

whatever, the peculiarity of each having only a purely secondary influence.
The solutions of descriptive geometry are then graphical, like most of those
of the ancients, and at the same time general, like those of the moderns.

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

ALGEBRAIC SOLUTIONS.

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

means of one another, which reduces polygonometry to simple
trigonometry.

TRIGONOMETRY.

The difficulty in resolving algebraically such a question as the above,
consists essentially in forming, between the angles and the sides of a
triangle, three distinct equations; which, when once obtained, will evidently
reduce all trigonometrical problems to mere questions of analysis.
How to introduce Angles. In considering the establishment of these
equations in the most general manner, we immediately meet with a
fundamental distinction with respect to the manner of introducing the
angles into the calculation, according as they are made to enter directly, by
themselves or by the circular arcs which are proportional to them; or
indirectly, by the chords of these arcs, which are hence called their
trigonometrical lines. Of these two systems of trigonometry the second was
of necessity the only one originally adopted, as being the only practicable
one, since the condition of geometry made it easy enough to find exact
relations between the sides of the triangles and the trigonometrical lines
which represent the angles, while it would have been absolutely impossible
at that epoch to establish equations between the sides and the angles
themselves.
Advantages of introducing Trigonometrical Lines. At the present day, since
the solution can be obtained by either system indifferently, that motive for
preference no longer exists; but geometers have none the less persisted in
following from choice the system primitively admitted from necessity; for,
the same reason which enabled these trigonometrical equations to be
obtained with much more facility, must, in like manner, as it is still more
easy to conceive à priori, render these equations much more simple, since
they then exist only between right lines, instead of being established
between right lines and arcs of circles. Such a consideration has so much
the more importance, as the question relates to formulas which are

eminently elementary, and destined to be continually employed in all parts
of mathematical science, as well as in all its various applications.
It may be objected, however, that when an angle is given, it is, in reality,
always given by itself, and not by its trigonometrical lines; and that when it
is unknown, it is its angular value which is properly to be determined, and
not that of any of its trigonometrical lines. It seems, according to this, that
such lines are only useless intermediaries between the sides and the angles,
which have to be finally eliminated, and the introduction of which does not
appear capable of simplifying the proposed research. It is indeed important
to explain, with more generality and precision than is customary, the great
real utility of this manner of proceeding.
Division of Trigonometry into two Parts. It consists in the fact that the
introduction of these auxiliary magnitudes divides the entire question of
trigonometry into two others essentially distinct, one of which has for its
object to pass from the angles to their trigonometrical lines, or the converse,
and the other of which proposes to determine the sides of the triangles by
the trigonometrical lines of their angles, or the converse. Now the first of
these two fundamental questions is evidently susceptible, by its nature, of
being entirely treated and reduced to numerical tables once for all, in
considering all possible angles, since it depends only upon those angles, and
not at all upon the particular triangles in which they may enter in each case;
while the solution of the second question must necessarily be renewed, at
least in its arithmetical relations, for each new triangle which it is necessary
to resolve. This is the reason why the first portion of the complete work,
which would be precisely the most laborious, is no longer taken into the
account, being always done in advance; while, if such a decomposition had
not been performed, we would evidently have found ourselves under the
obligation of recommencing the entire calculation in each particular case.
Such is the essential property of the present trigonometrical system, which
in fact would really present no actual advantage, if it was necessary to
calculate continually the trigonometrical line of each angle to be
considered, or the converse; the intermediate agency introduced would then
be more troublesome than convenient.
In order to clearly comprehend the true nature of this conception, it will be
useful to compare it with a still more important one, designed to produce an
analogous effect either in its algebraic, or, still more, in its arithmetical

relations—the admirable theory of logarithms. In examining in a
philosophical manner the influence of this theory, we see in fact that its
general result is to decompose all imaginable arithmetical operations into
two distinct parts. The first and most complicated of these is capable of
being executed in advance once for all (since it depends only upon the
numbers to be considered, and not at all upon the infinitely different
combinations into which they can enter), and consists in considering all
numbers as assignable powers of a constant number. The second part of the
calculation, which must of necessity be recommenced for each new formula
which is to have its value determined, is thenceforth reduced to executing
upon these exponents correlative operations which are infinitely more
simple. I confine myself here to merely indicating this resemblance, which
any one can carry out for himself.
We must besides observe, as a property (secondary at the present day, but
all-important at its origin) of the trigonometrical system adopted, the very
remarkable circumstance that the determination of angles by their
trigonometrical lines, or the converse, admits of an arithmetical solution
(the only one which is directly indispensable for the special destination of
trigonometry) without the previous resolution of the corresponding
algebraic question. It is doubtless to such a peculiarity that the ancients
owed the possibility of knowing trigonometry. The investigation conceived
in this way was so much the more easy, inasmuch as tables of chords
(which the ancients naturally took as the trigonometrical lines) had been
previously constructed for quite a different object, in the course of the
labours of Archimedes on the rectification of the circle, from which resulted
the actual determination of a certain series of chords; so that when
Hipparchus subsequently invented trigonometry, he could confine himself
to completing that operation by suitable intercalations; which shows clearly
the connexion of ideas in that matter.
The Increase of such Trigonometrical Lines. To complete this philosophical
sketch of trigonometry, it is proper now to observe that the extension of the
same considerations which lead us to replace angles or arcs of circles by
straight lines, with the view of simplifying our equations, must also lead us
to employ concurrently several trigonometrical lines, instead of confining
ourselves to one only (as did the ancients), so as to perfect this system by
choosing that one which will be algebraically the most convenient on each

occasion. In this point of view, it is clear that the number of these lines is in
itself no ways limited; provided that they are determined by the arc, and
that they determine it, whatever may be the law according to which they are
derived from it, they are suitable to be substituted for it in the equations.
The Arabians, and subsequently the moderns, in confining themselves to the
most simple constructions, have carried to four or five the number of direct
trigonometrical lines, which might be extended much farther.
But instead of recurring to geometrical formations, which would finally
become very complicated, we conceive with the utmost facility as many
new trigonometrical lines as the analytical transformations may require, by
means of a remarkable artifice, which is not usually apprehended in a
sufficiently general manner. It consists in not directly multiplying the
trigonometrical lines appropriate to each arc considered, but in introducing
new ones, by considering this arc as indirectly determined by all lines
relating to an arc which is a very simple function of the first. It is thus, for
example, that, in order to calculate an angle with more facility, we will
determine, instead of its sine, the sine of its half, or of its double, &c. Such
a creation of indirect trigonometrical lines is evidently much more fruitful
than all the direct geometrical methods for obtaining new ones. We may
accordingly say that the number of trigonometrical lines actually employed
at the present day by geometers is in reality unlimited, since at every
instant, so to say, the transformations of analysis may lead us to augment it
by the method which I have just indicated. Special names, however, have
been given to those only of these indirect lines which refer to the
complement of the primitive arc, the others not occurring sufficiently often
to render such denominations necessary; a circumstance which has caused a
common misconception of the true extent of the system of trigonometry.
Study of their Mutual Relations. This multiplicity of trigonometrical lines
evidently gives rise to a third fundamental question in trigonometry, the
study of the relations which exist between these different lines; since,
without such a knowledge, we could not make use, for our analytical
necessities, of this variety of auxiliary magnitudes, which, however, have
no other destination. It is clear, besides, from the consideration just
indicated, that this essential part of trigonometry, although simply
preparatory, is, by its nature, susceptible of an indefinite extension when we

view it in its entire generality, while the two others are circumscribed within
rigorously defined limits.
It is needless to add that these three principal parts of trigonometry have to
be studied in precisely the inverse order from that in which we have seen
them necessarily derived from the general nature of the subject; for the third
is evidently independent of the two others, and the second, of that which
was first presented—the resolution of triangles, properly so called—which
must for that reason be treated in the last place; which rendered so much the
more important the consideration of their natural succession and logical
relations to one another.
It is useless to consider here separately spherical trigonometry, which
cannot give rise to any special philosophical consideration; since, essential
as it is by the importance and the multiplicity of its uses, it can be treated at
the present day only as a simple application of rectilinear trigonometry,
which furnishes directly its fundamental equations, by substituting for the
spherical triangle the corresponding trihedral angle.
This summary exposition of the philosophy of trigonometry has been here
given in order to render apparent, by an important example, that rigorous
dependence and those successive ramifications which are presented by what
are apparently the most simple questions of elementary geometry.

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

CHAPTER III.
MODERN OR ANALYTICAL GEOMETRY.

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

ANALYTICAL REPRESENTATION OF FIGURES.

In order to succeed in expressing all imaginable geometrical phenomena by
simple analytical relations, we must evidently, in the first place, establish a
general method for representing analytically the subjects themselves in
which these phenomena are found, that is, the lines or the surfaces to be
considered. The subject being thus habitually considered in a purely
analytical point of view, we see how it is thenceforth possible to conceive in
the same manner the various accidents of which it is susceptible.
In order to organize the representation of geometrical figures by analytical
equations, we must previously surmount a fundamental difficulty; that of
reducing the general elements of the various conceptions of geometry to
simply numerical ideas; in a word, that of substituting in geometry pure
considerations of quantity for all considerations of quality.

Reduction of Figure to Position. For this purpose let us observe, in the first
place, that all geometrical ideas relate necessarily to these three universal
categories: the magnitude, the figure, and the position of the extensions to
be considered. As to the first, there is evidently no difficulty; it enters at
once into the ideas of numbers. With relation to the second, it must be
remarked that it will always admit of being reduced to the third. For the
figure of a body evidently results from the mutual position of the different
points of which it is composed, so that the idea of position necessarily
comprehends that of figure, and every circumstance of figure can be
translated by a circumstance of position. It is in this way, in fact, that the
human mind has proceeded in order to arrive at the analytical representation
of geometrical figures, their conception relating directly only to positions.
All the elementary difficulty is then properly reduced to that of referring
ideas of situation to ideas of magnitude. Such is the direct destination of the
preliminary conception upon which Descartes has established the general
system of analytical geometry.
His philosophical labour, in this relation, has consisted simply in the entire
generalization of an elementary operation, which we may regard as natural
to the human mind, since it is performed spontaneously, so to say, in all
minds, even the most uncultivated. Thus, when we have to indicate the
situation of an object without directly pointing it out, the method which we
always adopt, and evidently the only one which can be employed, consists
in referring that object to others which are known, by assigning the
magnitude of the various geometrical elements, by which we conceive it
connected with the known objects. These elements constitute what
Descartes, and after him all geometers, have called the co-ordinates of each
point considered. They are necessarily two in number, if it is known in
advance in what plane the point is situated; and three, if it may be found
indifferently in any region of space. As many different constructions as can
be imagined for determining the position of a point, whether on a plane or
in space, so many distinct systems of co-ordinates may be conceived; they
are consequently susceptible of being multiplied to infinity. But, whatever
may be the system adopted, we shall always have reduced the ideas of
situation to simple ideas of magnitude, so that we will consider the change
in the position of a point as produced by mere numerical variations in the
values of its co-ordinates.

Determination of the Position of a Point. Considering at first only the least
complicated case, that of plane geometry, it is in this way that we usually
determine the position of a point on a plane, by its distances from two fixed
right lines considered as known, which are called axes, and which are
commonly supposed to be perpendicular to each other. This system is that
most frequently adopted, because of its simplicity; but geometers employ
occasionally an infinity of others. Thus the position of a point on a plane
may be determined, 1°, by its distances from two fixed points; or, 2°, by its
distance from a single fixed point, and the direction of that distance,
estimated by the greater or less angle which it makes with a fixed right line,
which constitutes the system of what are called polar co-ordinates, the most
frequently used after the system first mentioned; or, 3°, by the angles which
the right lines drawn from the variable point to two fixed points make with
the right line which joins these last; or, 4°, by the distances from that point
to a fixed right line and a fixed point, &c. In a word, there is no geometrical
figure whatever from which it is not possible to deduce a certain system of
co-ordinates more or less susceptible of being employed.
A general observation, which it is important to make in this connexion, is,
that every system of co-ordinates is equivalent to determining a point, in
plane geometry, by the intersection of two lines, each of which is subjected
to certain fixed conditions of determination; a single one of these conditions
remaining variable, sometimes the one, sometimes the other, according to
the system considered. We could not, indeed, conceive any other means of
constructing a point than to mark it by the meeting of two lines. Thus, in the
most common system, that of rectilinear co-ordinates, properly so called,
the point is determined by the intersection of two right lines, each of which
remains constantly parallel to a fixed axis, at a greater or less distance from
it; in the polar system, the position of the point is marked by the meeting of
a circle, of variable radius and fixed centre, with a movable right line
compelled to turn about this centre: in other systems, the required point
might be designated by the intersection of two circles, or of any other two
lines, &c. In a word, to assign the value of one of the co-ordinates of a point
in any system whatever, is always necessarily equivalent to determining a
certain line on which that point must be situated. The geometers of antiquity
had already made this essential remark, which served as the base of their
method of geometrical loci, of which they made so happy a use to direct
their researches in the resolution of determinate problems, in considering

separately the influence of each of the two conditions by which was defined
each point constituting the object, direct or indirect, of the proposed
question. It was the general systematization of this method which was the
immediate motive of the labours of Descartes, which led him to create
analytical geometry.
After having clearly established this preliminary conception—by means of
which ideas of position, and thence, implicitly, all elementary geometrical
conceptions are capable of being reduced to simple numerical
considerations—it is easy to form a direct conception, in its entire
generality, of the great original idea of Descartes, relative to the analytical
representation of geometrical figures: it is this which forms the special
object of this chapter. I will continue to consider at first, for more facility,
only geometry of two dimensions, which alone was treated by Descartes;
and will afterwards examine separately, under the same point of view, the
theory of surfaces and curves of double curvature.

PLANE CURVES.

Expression of Lines by Equations. In accordance with the manner of
expressing analytically the position of a point on a plane, it can be easily
established that, by whatever property any line may be defined, that
definition always admits of being replaced by a corresponding equation
between the two variable co-ordinates of the point which describes this line;
an equation which will be thenceforth the analytical representation of the
proposed line, every phenomenon of which will be translated by a certain
algebraic modification of its equation. Thus, if we suppose that a point
moves on a plane without its course being in any manner determined, we
shall evidently have to regard its co-ordinates, to whatever system they may
belong, as two variables entirely independent of one another. But if, on the
contrary, this point is compelled to describe a certain line, we shall
necessarily be compelled to conceive that its co-ordinates, in all the
positions which it can take, retain a certain permanent and precise relation
to each other, which is consequently susceptible of being expressed by a
suitable equation; which will become the very clear and very rigorous

analytical definition of the line under consideration, since it will express an
algebraical property belonging exclusively to the co-ordinates of all the
points of this line. It is clear, indeed, that when a point is not subjected to
any condition, its situation is not determined except in giving at once its
two co-ordinates, independently of each other; while, when the point must
continue upon a defined line, a single co-ordinate is sufficient for
completely fixing its position. The second co-ordinate is then a determinate
function of the first; or, in other words, there must exist between them a
certain equation, of a nature corresponding to that of the line on which the
point is compelled to remain. In a word, each of the co-ordinates of a point
requiring it to be situated on a certain line, we conceive reciprocally that the
condition, on the part of a point, of having to belong to a line defined in any
manner whatever, is equivalent to assigning the value of one of the two co-
ordinates; which is found in that case to be entirely dependent on the other.
The analytical relation which expresses this dependence may be more or
less difficult to discover, but it must evidently be always conceived to exist,
even in the cases in which our present means may be insufficient to make it
known. It is by this simple consideration that we may demonstrate, in an
entirely general manner—independently of the particular verifications on
which this fundamental conception is ordinarily established for each special
definition of a line—the necessity of the analytical representation of lines
by equations.
Expression of Equations by Lines. Taking up again the same reflections in
the inverse direction, we could show as easily the geometrical necessity of
the representation of every equation of two variables, in a determinate
system of co-ordinates, by a certain line; of which such a relation would be,
in the absence of any other known property, a very characteristic definition,
the scientific destination of which will be to fix the attention directly upon
the general course of the solutions of the equation, which will thus be noted
in the most striking and the most simple manner. This picturing of equations
is one of the most important fundamental advantages of analytical
geometry, which has thereby reacted in the highest degree upon the general
perfecting of analysis itself; not only by assigning to purely abstract
researches a clearly determined object and an inexhaustible career, but, in a
still more direct relation, by furnishing a new philosophical medium for
analytical meditation which could not be replaced by any other. In fact, the
purely algebraic discussion of an equation undoubtedly makes known its

solutions in the most precise manner, but in considering them only one by
one, so that in this way no general view of them could be obtained, except
as the final result of a long and laborious series of numerical comparisons.
On the other hand, the geometrical locus of the equation, being only
designed to represent distinctly and with perfect clearness the summing up
of all these comparisons, permits it to be directly considered, without
paying any attention to the details which have furnished it. It can thereby
suggest to our mind general analytical views, which we should have arrived
at with much difficulty in any other manner, for want of a means of clearly
characterizing their object. It is evident, for example, that the simple
inspection of the logarithmic curve, or of the curve y = sin. x, makes us
perceive much more distinctly the general manner of the variations of
logarithms with respect to their numbers, or of sines with respect to their
arcs, than could the most attentive study of a table of logarithms or of
natural sines. It is well known that this method has become entirely
elementary at the present day, and that it is employed whenever it is desired
to get a clear idea of the general character of the law which reigns in a
series of precise observations of any kind whatever.
Any Change in the Line causes a Change in the Equation. Returning to the
representation of lines by equations, which is our principal object, we see
that this representation is, by its nature, so faithful, that the line could not
experience any modification, however slight it might be, without causing a
corresponding change in the equation. This perfect exactitude even gives
rise oftentimes to special difficulties; for since, in our system of analytical
geometry, the mere displacements of lines affect the equations, as well as
their real variations in magnitude or form, we should be liable to confound
them with one another in our analytical expressions, if geometers had not
discovered an ingenious method designed expressly to always distinguish
them. This method is founded on this principle, that although it is
impossible to change analytically at will the position of a line with respect
to the axes of the co-ordinates, we can change in any manner whatever the
situation of the axes themselves, which evidently amounts to the same;
then, by the aid of the very simple general formula by which this
transformation of the axes is produced, it becomes easy to discover whether
two different equations are the analytical expressions of only the same line
differently situated, or refer to truly distinct geometrical loci; since, in the
former case, one of them will pass into the other by suitably changing the

axes or the other constants of the system of co-ordinates employed. It must,
moreover, be remarked on this subject, that general inconveniences of this
nature seem to be absolutely inevitable in analytical geometry; for, since the
ideas of position are, as we have seen, the only geometrical ideas
immediately reducible to numerical considerations, and the conceptions of
figure cannot be thus reduced, except by seeing in them relations of
situation, it is impossible for analysis to escape confounding, at first, the
phenomena of figure with simple phenomena of position, which alone are
directly expressed by the equations.
Every Definition of a Line is an Equation. In order to complete the
philosophical explanation of the fundamental conception which serves as
the base of analytical geometry, I think that I should here indicate a new
general consideration, which seems to me particularly well adapted for
putting in the clearest point of view this necessary representation of lines by
equations with two variables. It consists in this, that not only, as we have
shown, must every defined line necessarily give rise to a certain equation
between the two co-ordinates of any one of its points, but, still farther,
every definition of a line may be regarded as being already of itself an
equation of that line in a suitable system of co-ordinates.
It is easy to establish this principle, first making a preliminary logical
distinction with respect to different kinds of definitions. The rigorously
indispensable condition of every definition is that of distinguishing the
object defined from all others, by assigning to it a property which belongs
to it exclusively. But this end may be generally attained in two very
different ways; either by a definition which is simply characteristic, that is,
indicative of a property which, although truly exclusive, does not make
known the mode of generation of the object; or by a definition which is
really explanatory, that is, which characterizes the object by a property
which expresses one of its modes of generation. For example, in
considering the circle as the line, which, under the same contour, contains
the greatest area, we have evidently a definition of the first kind; while in
choosing the property of its having all its points equally distant from a fixed
point, we have a definition of the second kind. It is, besides, evident, as a
general principle, that even when any object whatever is known at first only
by a characteristic definition, we ought, nevertheless, to regard it as

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

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

say that we shall have so many more resources in this matter as we shall
know more of true analytical geometry, that is, as we shall know the
algebraical expression of a greater number of different algebraical
phenomena.

CHOICE OF CO-ORDINATES.

In order to complete the philosophical exposition of the conception which
serves as the base of analytical geometry, I have yet to notice the
considerations relating to the choice of the system of co-ordinates which is
in general the most suitable. They will give the rational explanation of the
preference unanimously accorded to the ordinary rectilinear system; a
preference which has hitherto been rather the effect of an empirical
sentiment of the superiority of this system, than the exact result of a direct
and thorough analysis.
Two different Points of View. In order to decide clearly between all the
different systems of co-ordinates, it is indispensable to distinguish with care
the two general points of view, the converse of one another, which belong to
analytical geometry; namely, the relation of algebra to geometry, founded
upon the representation of lines by equations; and, reciprocally, the relation
of geometry to algebra, founded on the representation of equations by lines.
It is evident that in every investigation of general geometry these two
fundamental points of view are of necessity always found combined, since
we have always to pass alternately, and at insensible intervals, so to say,
from geometrical to analytical considerations, and from analytical to
geometrical considerations. But the necessity of here temporarily separating
them is none the less real; for the answer to the question of method which
we are examining is, in fact, as we shall see presently, very far from being
the same in both these relations, so that without this distinction we could
not form any clear idea of it.
1. Representation of Lines by Equations. Under the first point of view—the
representation of lines by equations—the only reason which could lead us
to prefer one system of co-ordinates to another would be the greater
simplicity of the equation of each line, and greater facility in arriving at it.
Now it is easy to see that there does not exist, and could not be expected to
exist, any system of co-ordinates deserving in that respect a constant

preference over all others. In fact, we have above remarked that for each
geometrical definition proposed we can conceive a system of co-ordinates
in which the equation of the line is obtained at once, and is necessarily
found to be also very simple; and this system, moreover, inevitably varies
with the nature of the characteristic property under consideration. The
rectilinear system could not, therefore, be constantly the most advantageous
for this object, although it may often be very favourable; there is probably
no system which, in certain particular cases, should not be preferred to it, as
well as to every other.
2. Representation of Equations by Lines. It is by no means so, however,
under the second point of view. We can, indeed, easily establish, as a
general principle, that the ordinary rectilinear system must necessarily be
better adapted than any other to the representation of equations by the
corresponding geometrical loci; that is to say, that this representation is
constantly more simple and more faithful in it than in any other.
Let us consider, for this object, that, since every system of co-ordinates
consists in determining a point by the intersection of two lines, the system
adapted to furnish the most suitable geometrical loci must be that in which
these two lines are the simplest possible; a consideration which confines our
choice to the rectilinear system. In truth, there is evidently an infinite
number of systems which deserve that name, that is to say, which employ
only right lines to determine points, besides the ordinary system which
assigns the distances from two fixed lines as co-ordinates; such, for
example, would be that in which the co-ordinates of each point should be
the two angles which the right lines, which go from that point to two fixed
points, make with the right line, which joins these last points: so that this
first consideration is not rigorously sufficient to explain the preference
unanimously given to the common system. But in examining in a more
thorough manner the nature of every system of co-ordinates, we also
perceive that each of the two lines, whose meeting determines the point
considered, must necessarily offer at every instant, among its different
conditions of determination, a single variable condition, which gives rise to
the corresponding co-ordinate, all the rest being fixed, and constituting the
axes of the system, taking this term in its most extended mathematical
acceptation. The variation is indispensable, in order that we may be able to
consider all possible positions; and the fixity is no less so, in order that

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

secondary circumstance is no more universally proper than we have seen
the general nature of the system to be; since, according to the particular
occasion, any other inclination of the axes may deserve our preference in
that respect. But, in the inverse point of view, it is easy to see that
rectangular axes constantly permit us to represent equations in a more
simple and even more faithful manner; for, with oblique axes, space being
divided by them into regions which no longer have a perfect identity, it
follows that, if the geometrical locus of the equation extends into all these
regions at once, there will be presented, by reason merely of this inequality
of the angles, differences of figure which do not correspond to any
analytical diversity, and will necessarily alter the rigorous exactness of the
representation, by being confounded with the proper results of the algebraic
comparisons. For example, an equation like: xm + ym = c, which, by its
perfect symmetry, should evidently give a curve composed of four identical
quarters, will be represented, on the contrary, if we take axes not
rectangular, by a geometric locus, the four parts of which will be unequal. It
is plain that the only means of avoiding all inconveniences of this kind is to
suppose the angle of the two axes to be a right angle.
The preceding discussion clearly shows that, although the ordinary system
of rectilinear co-ordinates has no constant superiority over all others in one
of the two fundamental points of view which are continually combined in
analytical geometry, yet as, on the other hand, it is not constantly inferior,
its necessary and absolute greater aptitude for the representation of
equations must cause it to generally receive the preference; although it may
evidently happen, in some particular cases, that the necessity of simplifying
equations and of obtaining them more easily may determine geometers to
adopt a less perfect system. The rectilinear system is, therefore, the one by
means of which are ordinarily constructed the most essential theories of
general geometry, intended to express analytically the most important
geometrical phenomena. When it is thought necessary to choose some
other, the polar system is almost always the one which is fixed upon, this
system being of a nature sufficiently opposite to that of the rectilinear
system to cause the equations, which are too complicated with respect to the
latter, to become, in general, sufficiently simple with respect to the other.
Polar co-ordinates, moreover, have often the advantage of admitting of a
more direct and natural concrete signification; as is the case in mechanics,

for the geometrical questions to which the theory of circular movement
gives rise, and in almost all the cases of celestial geometry.

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

SURFACES.

Determination of a Point in Space. The complete analytical determination
of a point in space evidently requires the values of three co-ordinates to be
assigned; as, for example, in the system which is generally adopted, and
which corresponds to the rectilinear system of plane geometry, distances
from the point to three fixed planes, usually perpendicular to one another;
which presents the point as the intersection of three planes whose direction
is invariable. We might also employ the distances from the movable point to
three fixed points, which would determine it by the intersection of three
spheres with a common centre. In like manner, the position of a point would
be defined by giving its distance from a fixed point, and the direction of that
distance, by means of the two angles which this right line makes with two
invariable axes; this is the polar system of geometry of three dimensions;
the point is then constructed by the intersection of a sphere having a fixed
centre, with two right cones with circular bases, whose axes and common
summit do not change. In a word, there is evidently, in this case at least, the
same infinite variety among the various possible systems of co-ordinates
which we have already observed in geometry of two dimensions. In general,
we have to conceive a point as being always determined by the intersection

of any three surfaces whatever, as it was in the former case by that of two
lines: each of these three surfaces has, in like manner, all its conditions of
determination constant, excepting one, which gives rise to the
corresponding co-ordinates, whose peculiar geometrical influence is thus to
constrain the point to be situated upon that surface.
This being premised, it is clear that if the three co-ordinates of a point are
entirely independent of one another, that point can take successively all
possible positions in space. But if the point is compelled to remain upon a
certain surface defined in any manner whatever, then two co-ordinates are
evidently sufficient for determining its situation at each instant, since the
proposed surface will take the place of the condition imposed by the third
co-ordinate. We must then, in this case, under the analytical point of view,
necessarily conceive this last co-ordinate as a determinate function of the
two others, these latter remaining perfectly independent of each other. Thus
there will be a certain equation between the three variable co-ordinates,
which will be permanent, and which will be the only one, in order to
correspond to the precise degree of indetermination in the position of the
point.
Expression of Surfaces by Equations. This equation, more or less easy to be
discovered, but always possible, will be the analytical definition of the
proposed surface, since it must be verified for all the points of that surface,
and for them alone. If the surface undergoes any change whatever, even a
simple change of place, the equation must undergo a more or less serious
corresponding modification. In a word, all geometrical phenomena relating
to surfaces will admit of being translated by certain equivalent analytical
conditions appropriate to equations of three variables; and in the
establishment and interpretation of this general and necessary harmony will
essentially consist the science of analytical geometry of three dimensions.
Expression of Equations by Surfaces. Considering next this fundamental
conception in the inverse point of view, we see in the same manner that
every equation of three variables may, in general, be represented
geometrically by a determinate surface, primitively defined by the very
characteristic property, that the co-ordinates of all its points always retain
the mutual relation enunciated in this equation. This geometrical locus will
evidently change, for the same equation, according to the system of co-
ordinates which may serve for the construction of this representation. In

adopting, for example, the rectilinear system, it is clear that in the equation
between the three variables, x, y, z, every particular value attributed to z will
give an equation between at x and y, the geometrical locus of which will be
a certain line situated in a plane parallel to the plane of x and y, and at a
distance from this last equal to the value of z; so that the complete
geometrical locus will present itself as composed of an infinite series of
lines superimposed in a series of parallel planes (excepting the interruptions
which may exist), and will consequently form a veritable surface. It would
be the same in considering any other system of co-ordinates, although the
geometrical construction of the equation becomes more difficult to follow.
Such is the elementary conception, the complement of the original idea of
Descartes, on which is founded general geometry relative to surfaces. It
would be useless to take up here directly the other considerations which
have been above indicated, with respect to lines, and which any one can
easily extend to surfaces; whether to show that every definition of a surface
by any method of generation whatever is really a direct equation of that
surface in a certain system of co-ordinates, or to determine among all the
different systems of possible co-ordinates that one which is generally the
most convenient. I will only add, on this last point, that the necessary
superiority of the ordinary rectilinear system, as to the representation of
equations, is evidently still more marked in analytical geometry of three
dimensions than in that of two, because of the incomparably greater
geometrical complication which would result from the choice of any other
system. This can be verified in the most striking manner by considering the
polar system in particular, which is the most employed after the ordinary
rectilinear system, for surfaces as well as for plane curves, and for the same
reasons.
In order to complete the general exposition of the fundamental conception
relative to the analytical study of surfaces, a philosophical examination
should be made of a final improvement of the highest importance, which
Monge has introduced into the very elements of this theory, for the
classification of surfaces in natural families, established according to the
mode of generation, and expressed algebraically by common differential
equations, or by finite equations containing arbitrary functions.

CURVES OF DOUBLE CURVATURE.

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

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

IMPERFECTIONS OF ANALYTICAL GEOMETRY.

Having now considered the fundamental conception of analytical geometry
under its principal elementary aspects, it is proper, in order to make the

sketch complete, to notice here the general imperfections yet presented by
this conception with respect to both geometry and to analysis.
Relatively to geometry, we must remark that the equations are as yet
adapted to represent only entire geometrical loci, and not at all determinate
portions of those loci. It would, however, be necessary, in some
circumstances, to be able to express analytically a part of a line or of a
surface, or even a discontinuous line or surface, composed of a series of
sections belonging to distinct geometrical figures, such as the contour of a
polygon, or the surface of a polyhedron. Thermology, especially, often gives
rise to such considerations, to which our present analytical geometry is
necessarily inapplicable. The labours of M. Fourier on discontinuous
functions have, however, begun to fill up this great gap, and have thereby
introduced a new and essential improvement into the fundamental
conception of Descartes. But this manner of representing heterogeneous or
partial figures, being founded on the employment of trigonometrical series
proceeding according to the sines of an infinite series of multiple arcs, or on
the use of certain definite integrals equivalent to those series, and the
general integral of which is unknown, presents as yet too much
complication to admit of being immediately introduced into the system of
analytical geometry.
Relatively to analysis, we must begin by observing that our inability to
conceive a geometrical representation of equations containing four, five, or
more variables, analogous to those representations which all equations of
two or of three variables admit, must not be viewed as an imperfection of
our system of analytical geometry, for it evidently belongs to the very
nature of the subject. Analysis being necessarily more general than
geometry, since it relates to all possible phenomena, it would be very
unphilosophical to desire always to find among geometrical phenomena
alone a concrete representation of all the laws which analysis can express.
There exists, however, another imperfection of less importance, which must
really be viewed as proceeding from the manner in which we conceive
analytical geometry. It consists in the evident incompleteness of our present
representation of equations of two or of three variables by lines or surfaces,
inasmuch as in the construction of the geometric locus we pay regard only
to the real solutions of equations, without at all noticing any imaginary
solutions. The general course of these last should, however, by its nature, be

quite as susceptible as that of the others of a geometrical representation. It
follows from this omission that the graphic picture of the equation is
constantly imperfect, and sometimes even so much so that there is no
geometric representation at all when the equation admits of only imaginary
solutions. But, even in this last case, we evidently ought to be able to
distinguish between equations as different in themselves as these, for
example,
x2 + y2 + 1 = 0, x6 + y4 + 1 = 0, y2 + ex = 0.
We know, moreover, that this principal imperfection often brings with it, in
analytical geometry of two or of three dimensions, a number of secondary
inconveniences, arising from several analytical modifications not
corresponding to any geometrical phenomena.

Our philosophical exposition of the fundamental conception of analytical
geometry shows us clearly that this science consists essentially in
determining what is the general analytical expression of such or such a
geometrical phenomenon belonging to lines or to surfaces; and,
reciprocally, in discovering the geometrical interpretation of such or such an
analytical consideration. A detailed examination of the most important
general questions would show us how geometers have succeeded in actually
establishing this beautiful harmony, and in thus imprinting on geometrical
science, regarded as a whole, its present eminently perfect character of
rationality and of simplicity.
Note.—The author devotes the two following chapters of his
course to the more detailed examination of Analytical Geometry
of two and of three dimensions; but his subsequent publication
of a separate work upon this branch of mathematics has been
thought to render unnecessary the reproduction of these two
chapters in the present volume.
THE END.
FOOTNOTES:
[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.
[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.
[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.
[4] Suppose, for example, that a question gives the following equation between
an unknown magnitude x, and two known magnitudes, a and b,

x3 + 3ax = 2b,
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.
[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.
[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.
[7] Simple as may seem, for example, the equation

ax + bx = cx,
we do not yet know how to resolve it, which may give some idea of the extreme
imperfection of this part of algebra.
[8] The same error was afterward committed, in the infancy of the infinitesimal
calculus, in relation to the integration of differential equations.
[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.
[10] The only important case of this class which has thus far been completely
treated is the general integration of linear equations of any order whatever, with
constant coefficients. Even this case finally depends on the algebraic resolution
of equations of a degree equal to the order of differentiation.
[11] Leibnitz had already considered the comparison of one curve with an other
infinitely near to it, calling it "Differentiatio de curva in curvam." But this
comparison had no analogy with the conception of Lagrange, the curves of
Leibnitz being embraced in the same general equation, from which they were
deduced by the simple change of an arbitrary constant.
[12] I propose hereafter to develop this new consideration, in a special work
upon the Calculus of Variations, intended to present this hyper-transcendental
analysis in a new point of view, which I think adapted to extend its general
range.
[13] Lacroix has justly criticised the expression of solid, commonly used by
geometers to designate a volume. It is certain, in fact, that when we wish to
consider separately a certain portion of indefinite space, conceived as gaseous,
we mentally solidify its exterior envelope, so that a line and a surface are
habitually, to our minds, just as solid as a volume. It may also be remarked that
most generally, in order that bodies may penetrate one another with more facility,
we are obliged to imagine the interior of the volumes to be hollow, which renders
still more sensible the impropriety of the word solid.

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