THE CALCULUS OF FINITE DIFFERENCES.
The different fundamental considerations indicated in the five preceding
chapters constitute, in reality, all the essential bases of a complete
exposition of mathematical analysis, regarded in the philosophical point
of view. Nevertheless, in order not to neglect any truly important
general conception relating to this analysis, I think that I should here
very summarily explain the veritable character of a kind of calculus
which is very extended, and which, though at bottom it really belongs to
ordinary analysis, is still regarded as being of an essentially distinct
nature. I refer to the _Calculus of Finite Differences_, which will be
the special subject of this chapter.
_Its general Character._ This calculus, created by Taylor, in his
celebrated work entitled _Methodus Incrementorum_, consists essentially
in the consideration of the finite increments which functions receive as
a consequence of analogous increments on the part of the corresponding
variables. These increments or _differences_, which take the
characteristic Δ, to distinguish them from _differentials_, or
infinitely small increments, may be in their turn regarded as new
functions, and become the subject of a second similar consideration, and
so on; from which results the notion of differences of various
successive orders, analogous, at least in appearance, to the consecutive
orders of differentials. Such a calculus evidently presents, like the
calculus of indirect functions, two general classes of questions:
1°. To determine the successive differences of all the various
analytical functions of one or more variables, as the result of a
definite manner of increase of the independent variables, which are
generally supposed to augment in arithmetical progression.
2°. Reciprocally, to start from these differences, or, more generally,
from any equations established between them, and go back to the
primitive functions themselves, or to their corresponding relations.
Hence follows the decomposition of this calculus into two distinct ones,
to which are usually given the names of the _Direct_, and the _Inverse
Calculus of Finite Differences_, the latter being also sometimes called
the _Integral Calculus of Finite Differences_. Each of these would,
also, evidently admit of a logical distribution similar to that given in
the fourth chapter for the differential and the integral calculus.
_Its true Nature._ There is no doubt that Taylor thought that by such a
conception he had founded a calculus of an entirely new nature,
absolutely distinct from ordinary analysis, and more general than the
calculus of Leibnitz, although resting on an analogous consideration. It
is in this way, also, that almost all geometers have viewed the analysis
of Taylor; but Lagrange, with his usual profundity, clearly perceived
that these properties belonged much more to the forms and to the
notations employed by Taylor than to the substance of his theory. In
fact, that which constitutes the peculiar character of the analysis of
Leibnitz, and makes of it a truly distinct and superior calculus, is the
circumstance that the derived functions are in general of an entirely
different nature from the primitive functions, so that they may give
rise to more simple and more easily formed relations: whence result the
admirable fundamental properties of the transcendental analysis, which
have been already explained. But it is not so with the _differences_
considered by Taylor; for these differences are, by their nature,
functions essentially similar to those which have produced them, a
circumstance which renders them unsuitable to facilitate the
establishment of equations, and prevents their leading to more general
relations. Every equation of finite differences is truly, at bottom, an
equation directly relating to the very magnitudes whose successive
states are compared. The scaffolding of new signs, which produce an
illusion respecting the true character of these equations, disguises it,
however, in a very imperfect manner, since it could always be easily
made apparent by replacing the _differences_ by the equivalent
combinations of the primitive magnitudes, of which they are really only
the abridged designations. Thus the calculus of Taylor never has
offered, and never can offer, in any question of geometry or of
mechanics, that powerful general aid which we have seen to result
necessarily from the analysis of Leibnitz. Lagrange has, moreover, very
clearly proven that the pretended analogy observed between the calculus
of differences and the infinitesimal calculus was radically vicious, in
this way, that the formulas belonging to the former calculus can never
furnish, as particular cases, those which belong to the latter, the
nature of which is essentially distinct.
From these considerations I am led to think that the calculus of finite
differences is, in general, improperly classed with the transcendental
analysis proper, that is, with the calculus of indirect functions. I
consider it, on the contrary, in accordance with the views of Lagrange,
to be only a very extensive and very important branch of ordinary
analysis, that is to say, of that which I have named the calculus of
direct functions, the equations which it considers being always, in
spite of the notation, simple _direct_ equations.
GENERAL THEORY OF SERIES.
To sum up as briefly as possible the preceding explanation, the calculus
of Taylor ought to be regarded as having constantly for its true object
the general theory of _Series_, the most simple cases of which had alone
been considered before that illustrious geometer. I ought, properly, to
have mentioned this important theory in treating, in the second chapter,
of Algebra proper, of which it is such an extensive branch. But, in
order to avoid a double reference to it, I have preferred to notice it
only in the consideration of the calculus of finite differences, which,
reduced to its most simple general expression, is nothing but a complete
logical study of questions relating to _series_.
Every _Series_, or succession of numbers deduced from one another
according to any constant law, necessarily gives rise to these two
fundamental questions:
1°. The law of the series being supposed known, to find the expression
for its general term, so as to be able to calculate immediately any term
whatever without being obliged to form successively all the preceding
terms.
2°. In the same circumstances, to determine the _sum_ of any number of
terms of the series by means of their places, so that it can be known
without the necessity of continually adding these terms together.
These two fundamental questions being considered to be resolved, it may
be proposed, reciprocally, to find the law of a series from the form of
its general term, or the expression of the sum. Each of these different
problems has so much the more extent and difficulty, as there can be
conceived a greater number of different _laws_ for the series, according
to the number of preceding terms on which each term directly depends,
and according to the function which expresses that dependence. We may
even consider series with several variable indices, as Laplace has done
in his "Analytical Theory of Probabilities," by the analysis to which he
has given the name of _Theory of Generating Functions_, although it is
really only a new and higher branch of the calculus of finite
differences or of the general theory of series.
These general views which I have indicated give only an imperfect idea
of the truly infinite extent and variety of the questions to which
geometers have risen by means of this single consideration of series, so
simple in appearance and so limited in its origin. It necessarily
presents as many different cases as the algebraic resolution of
equations, considered in its whole extent; and it is, by its nature,
much more complicated, so much, indeed, that it always needs this last
to conduct it to a complete solution. We may, therefore, anticipate what
must still be its extreme imperfection, in spite of the successive
labours of several geometers of the first order. We do not, indeed,
possess as yet the complete and logical solution of any but the most
simple questions of this nature.
_Its identity with this Calculus._ It is now easy to conceive the
necessary and perfect identity, which has been already announced,
between the calculus of finite differences and the theory of series
considered in all its bearings. In fact, every differentiation after the
manner of Taylor evidently amounts to finding the _law_ of formation of
a series with one or with several variable indices, from the expression
of its general term; in the same way, every analogous integration may be
regarded as having for its object the summation of a series, the general
term of which would be expressed by the proposed difference. In this
point of view, the various problems of the calculus of differences,
direct or inverse, resolved by Taylor and his successors, have really a
very great value, as treating of important questions relating to series.
But it is very doubtful if the form and the notation introduced by
Taylor really give any essential facility in the solution of questions
of this kind. It would be, perhaps, more advantageous for most cases,
and certainly more logical, to replace the _differences_ by the terms
themselves, certain combinations of which they represent. As the
calculus of Taylor does not rest on a truly distinct fundamental idea,
and has nothing peculiar to it but its system of signs, there could
never really be any important advantage in considering it as detached
from ordinary analysis, of which it is, in reality, only an immense
branch. This consideration of _differences_, most generally useless,
even if it does not cause complication, seems to me to retain the
character of an epoch in which, analytical ideas not being sufficiently
familiar to geometers, they were naturally led to prefer the special
forms suitable for simple numerical comparisons.
PERIODIC OR DISCONTINUOUS FUNCTIONS.
However that may be, I must not finish this general appreciation of the
calculus of finite differences without noticing a new conception to
which it has given birth, and which has since acquired a great
importance. It is the consideration of those periodic or discontinuous
functions which preserve the same value for an infinite series of values
of the corresponding variables, subjected to a certain law, and which
must be necessarily added to the integrals of the equations of finite
differences in order to render them sufficiently general, as simple
arbitrary constants are added to all quadratures in order to complete
their generality. This idea, primitively introduced by Euler, has since
been the subject of extended investigation by M. Fourier, who has made
new and important applications of it in his mathematical theory of heat.
APPLICATIONS OF THIS CALCULUS.
_Series._ Among the principal general applications which have been made
of the calculus of finite differences, it would be proper to place in
the first rank, as the most extended and the most important, the
solution of questions relating to series; if, as has been shown, the
general theory of series ought not to be considered as constituting, by
its nature, the actual foundation of the calculus of Taylor.
_Interpolations._ This great class of problems being then set aside, the
most essential of the veritable applications of the analysis of Taylor
is, undoubtedly, thus far, the general method of _interpolations_, so
frequently and so usefully employed in the investigation of the
empirical laws of natural phenomena. The question consists, as is well
known, in intercalating between certain given numbers other intermediate
numbers, subjected to the same law which we suppose to exist between the
first. We can abundantly verify, in this principal application of the
calculus of Taylor, how truly foreign and often inconvenient is the
consideration of _differences_ with respect to the questions which
depend on that analysis. Indeed, Lagrange has replaced the formulas of
interpolation, deduced from the ordinary algorithm of the calculus of
finite differences, by much simpler general formulas, which are now
almost always preferred, and which have been found directly, without
making any use of the notion of _differences_, which only complicates
the question.
_Approximate Rectification, &c._ A last important class of applications
of the calculus of finite differences, which deserves to be
distinguished from the preceding, consists in the eminently useful
employment made of it in geometry for determining by approximation the
length and the area of any curve, and in the same way the cubature of a
body of any form whatever. This procedure (which may besides be
conceived abstractly as depending on the same analytical investigation
as the question of interpolation) frequently offers a valuable
supplement to the entirely logical geometrical methods which often lead
to integrations, which we do not yet know how to effect, or to
calculations of very complicated execution.
* * * * *
Such are the various principal considerations to be noticed with respect
to the calculus of finite differences. This examination completes the
proposed philosophical outline of ABSTRACT MATHEMATICS.
CONCRETE MATHEMATICS will now be the subject of a similar labour. In it
we shall particularly devote ourselves to examining how it has been
possible (supposing the general science of the calculus to be perfect),
by invariable procedures, to reduce to pure questions of analysis all
the problems which can be presented by _Geometry_ and _Mechanics_, and
thus to impress on these two fundamental bases of natural philosophy a
degree of precision and especially of unity; in a word, a character of
high perfection, which could be communicated to them by such a course
alone.
BOOK II.
GEOMETRY.
BOOK II.
GEOMETRY.
CHAPTER I.
GENERAL VIEW OF GEOMETRY.
_Its true Nature._ After the general exposition of the philosophical
character of concrete mathematics, compared with that of abstract
mathematics, given in the introductory chapter, it need not here be
shown in a special manner that geometry must be considered as a true
natural science, only much more simple, and therefore much more perfect,
than any other. This necessary perfection of geometry, obtained
essentially by the application of mathematical analysis, which it so
eminently admits, is apt to produce erroneous views of the real nature
of this fundamental science, which most minds at present conceive to be
a purely logical science quite independent of observation. It is
nevertheless evident, to any one who examines with attention the
character of geometrical reasonings, even in the present state of
abstract geometry, that, although the facts which are considered in it
are much more closely united than those relating to any other science,
still there always exists, with respect to every body studied by
geometers, a certain number of primitive phenomena, which, since they
are not established by any reasoning, must be founded on observation
alone, and which form the necessary basis of all the deductions.
The scientific superiority of geometry arises from the phenomena which
it considers being necessarily the most universal and the most simple of
all. Not only may all the bodies of nature give rise to geometrical
inquiries, as well as mechanical ones, but still farther, geometrical
phenomena would still exist, even though all the parts of the universe
should be considered as immovable. Geometry is then, by its nature, more
general than mechanics. At the same time, its phenomena are more simple,
for they are evidently independent of mechanical phenomena, while these
latter are always complicated with the former. The same relations hold
good in comparing geometry with abstract thermology.
For these reasons, in our classification we have made geometry the first
part of concrete mathematics; that part the study of which, in addition
to its own importance, serves as the indispensable basis of all the
rest.
Before considering directly the philosophical study of the different
orders of inquiries which constitute our present geometry, we should
obtain a clear and exact idea of the general destination of that
science, viewed in all its bearings. Such is the object of this chapter.
_Definition._ Geometry is commonly defined in a very vague and entirely
improper manner, as being _the science of extension_. An improvement on
this would be to say that geometry has for its object the _measurement_
of extension; but such an explanation would be very insufficient,
although at bottom correct, and would be far from giving any idea of the
true general character of geometrical science.
To do this, I think that I should first explain _two fundamental ideas_,
which, very simple in themselves, have been singularly obscured by the
employment of metaphysical considerations.
_The Idea of Space._ The first is that of _Space_. This conception
properly consists simply in this, that, instead of considering extension
in the bodies themselves, we view it in an indefinite medium, which we
regard as containing all the bodies of the universe. This notion is
naturally suggested to us by observation, when we think of the
_impression_ which a body would leave in a fluid in which it had been
placed. It is clear, in fact, that, as regards its geometrical
relations, such an _impression_ may be substituted for the body itself,
without altering the reasonings respecting it. As to the physical nature
of this indefinite _space_, we are spontaneously led to represent it to
ourselves, as being entirely analogous to the actual medium in which we
live; so that if this medium was liquid instead of gaseous, our
geometrical _space_ would undoubtedly be conceived as liquid also. This
circumstance is, moreover, only very secondary, the essential object of
such a conception being only to make us view extension separately from
the bodies which manifest it to us. We can easily understand in advance
the importance of this fundamental image, since it permits us to study
geometrical phenomena in themselves, abstraction being made of all the
other phenomena which constantly accompany them in real bodies, without,
however, exerting any influence over them. The regular establishment of
this general abstraction must be regarded as the first step which has
been made in the rational study of geometry, which would have been
impossible if it had been necessary to consider, together with the form
and the magnitude of bodies, all their other physical properties. The
use of such an hypothesis, which is perhaps the most ancient
philosophical conception created by the human mind, has now become so
familiar to us, that we have difficulty in exactly estimating its
importance, by trying to appreciate the consequences which would result
from its suppression.
_Different Kinds of Extension._ The second preliminary geometrical
conception which we have to examine is that of the different kinds of
extension, designated by the words _volume_, _surface_, _line_, and even
_point_, and of which the ordinary explanation is so unsatisfactory.[13]
[Footnote 13: Lacroix has justly criticised the expression of
_solid_, commonly used by geometers to designate a _volume_. It is
certain, in fact, that when we wish to consider separately a
certain portion of indefinite space, conceived as gaseous, we
mentally solidify its exterior envelope, so that a _line_ and a
_surface_ are habitually, to our minds, just as _solid_ as a
_volume_. It may also be remarked that most generally, in order
that bodies may penetrate one another with more facility, we are
obliged to imagine the interior of the _volumes_ to be hollow,
which renders still more sensible the impropriety of the word
_solid_.]
Although it is evidently impossible to conceive any extension absolutely
deprived of any one of the three fundamental dimensions, it is no less
incontestable that, in a great number of occasions, even of immediate
utility, geometrical questions depend on only two dimensions, considered
separately from the third, or on a single dimension, considered
separately from the two others. Again, independently of this direct
motive, the study of extension with a single dimension, and afterwards
with two, clearly presents itself as an indispensable preliminary for
facilitating the study of complete bodies of three dimensions, the
immediate theory of which would be too complicated. Such are the two
general motives which oblige geometers to consider separately extension
with regard to one or to two dimensions, as well as relatively to all
three together.
The general notions of _surface_ and of _line_ have been formed by the
human mind, in order that it may be able to think, in a permanent
manner, of extension in two directions, or in one only. The hyperbolical
expressions habitually employed by geometers to define these notions
tend to convey false ideas of them; but, examined in themselves, they
have no other object than to permit us to reason with facility
respecting these two kinds of extension, making complete abstraction of
that which ought not to be taken into consideration. Now for this it is
sufficient to conceive the dimension which we wish to eliminate as
becoming gradually smaller and smaller, the two others remaining the
same, until it arrives at such a degree of tenuity that it can no longer
fix the attention. It is thus that we naturally acquire the real idea of
a _surface_, and, by a second analogous operation, the idea of a _line_,
by repeating for breadth what we had at first done for thickness.
Finally, if we again repeat the same operation, we arrive at the idea of
a _point_, or of an extension considered only with reference to its
place, abstraction being made of all magnitude, and designed
consequently to determine positions.
_Surfaces_ evidently have, moreover, the general property of exactly
circumscribing volumes; and in the same way, _lines_, in their turn,
circumscribe _surfaces_ and are limited by _points_. But this
consideration, to which too much importance is often given, is only a
secondary one.
Surfaces and lines are, then, in reality, always conceived with three
dimensions; it would be, in fact, impossible to represent to one's self
a surface otherwise than as an extremely thin plate, and a line
otherwise than as an infinitely fine thread. It is even plain that the
degree of tenuity attributed by each individual to the dimensions of
which he wishes to make abstraction is not constantly identical, for it
must depend on the degree of subtilty of his habitual geometrical
observations. This want of uniformity has, besides, no real
inconvenience, since it is sufficient, in order that the ideas of
surface and of line should satisfy the essential condition of their
destination, for each one to represent to himself the dimensions which
are to be neglected as being smaller than all those whose magnitude his
daily experience gives him occasion to appreciate.
We hence see how devoid of all meaning are the fantastic discussions of
metaphysicians upon the foundations of geometry. It should also be
remarked that these primordial ideas are habitually presented by
geometers in an unphilosophical manner, since, for example, they explain
the notions of the different sorts of extent in an order absolutely the
inverse of their natural dependence, which often produces the most
serious inconveniences in elementary instruction.
THE FINAL OBJECT OF GEOMETRY.
These preliminaries being established, we can proceed directly to the
general definition of geometry, continuing to conceive this science as
having for its final object the _measurement_ of extension.
It is necessary in this matter to go into a thorough explanation,
founded on the distinction of the three kinds of extension, since the
notion of _measurement_ is not exactly the same with reference to
surfaces and volumes as to lines.
_Nature of Geometrical Measurement._ If we take the word _measurement_
in its direct and general mathematical acceptation, which signifies
simply the determination of the value of the _ratios_ between any
homogeneous magnitudes, we must consider, in geometry, that the
_measurement_ of surfaces and of volumes, unlike that of lines, is never
conceived, even in the most simple and the most favourable cases, as
being effected directly. The comparison of two lines is regarded as
direct; that of two surfaces or of two volumes is, on the contrary,
always indirect. Thus we conceive that two lines may be superposed; but
the superposition of two surfaces, or, still more so, of two volumes, is
evidently impossible in most cases; and, even when it becomes rigorously
practicable, such a comparison is never either convenient or exact. It
is, then, very necessary to explain wherein properly consists the truly
geometrical measurement of a surface or of a volume.
_Measurement of Surfaces and of Volumes._ For this we must consider
that, whatever may be the form of a body, there always exists a certain
number of lines, more or less easy to be assigned, the length of which
is sufficient to define exactly the magnitude of its surface or of its
volume. Geometry, regarding these lines as alone susceptible of being
directly measured, proposes to deduce, from the simple determination of
them, the ratio of the surface or of the volume sought, to the unity of
surface, or to the unity of volume. Thus the general object of
geometry, with respect to surfaces and to volumes, is properly to reduce
all comparisons of surfaces or of volumes to simple comparisons of
lines.
Besides the very great facility which such a transformation evidently
offers for the measurement of volumes and of surfaces, there results
from it, in considering it in a more extended and more scientific
manner, the general possibility of reducing to questions of lines all
questions relating to volumes and to surfaces, considered with reference
to their magnitude. Such is often the most important use of the
geometrical expressions which determine surfaces and volumes in
functions of the corresponding lines.
It is true that direct comparisons between surfaces or between volumes
are sometimes employed; but such measurements are not regarded as
geometrical, but only as a supplement sometimes necessary, although too
rarely applicable, to the insufficiency or to the difficulty of truly
rational methods. It is thus that we often determine the volume of a
body, and in certain cases its surface, by means of its weight. In the
same way, on other occasions, when we can substitute for the proposed
volume an equivalent liquid volume, we establish directly the comparison
of the two volumes, by profiting by the property possessed by liquid
masses, of assuming any desired form. But all means of this nature are
purely mechanical, and rational geometry necessarily rejects them.
To render more sensible the difference between these modes of
determination and true geometrical measurements, I will cite a single
very remarkable example; the manner in which Galileo determined the
ratio of the ordinary cycloid to that of the generating circle. The
geometry of his time was as yet insufficient for the rational solution
of such a problem. Galileo conceived the idea of discovering that ratio
by a direct experiment. Having weighed as exactly as possible two plates
of the same material and of equal thickness, one of them having the form
of a circle and the other that of the generated cycloid, he found the
weight of the latter always triple that of the former; whence he
inferred that the area of the cycloid is triple that of the generating
circle, a result agreeing with the veritable solution subsequently
obtained by Pascal and Wallis. Such a success evidently depends on the
extreme simplicity of the ratio sought; and we can understand the
necessary insufficiency of such expedients, even when they are actually
practicable.
We see clearly, from what precedes, the nature of that part of geometry
relating to _volumes_ and that relating to _surfaces_. But the character
of the geometry of _lines_ is not so apparent, since, in order to
simplify the exposition, we have considered the measurement of lines as
being made directly. There is, therefore, needed a complementary
explanation with respect to them.
_Measurement of curved Lines._ For this purpose, it is sufficient to
distinguish between the right line and curved lines, the measurement of
the first being alone regarded as direct, and that of the other as
always indirect. Although superposition is sometimes strictly
practicable for curved lines, it is nevertheless evident that truly
rational geometry must necessarily reject it, as not admitting of any
precision, even when it is possible. The geometry of lines has, then,
for its general object, to reduce in every case the measurement of
curved lines to that of right lines; and consequently, in the most
extended point of view, to reduce to simple questions of right lines all
questions relating to the magnitude of any curves whatever. To
understand the possibility of such a transformation, we must remark,
that in every curve there always exist certain right lines, the length
of which must be sufficient to determine that of the curve. Thus, in a
circle, it is evident that from the length of the radius we must be able
to deduce that of the circumference; in the same way, the length of an
ellipse depends on that of its two axes; the length of a cycloid upon
the diameter of the generating circle, &c.; and if, instead of
considering the whole of each curve, we demand, more generally, the
length of any arc, it will be sufficient to add to the different
rectilinear parameters, which determine the whole curve, the chord of
the proposed arc, or the co-ordinates of its extremities. To discover
the relation which exists between the length of a curved line and that
of similar right lines, is the general problem of the part of geometry
which relates to the study of lines.
Combining this consideration with those previously suggested with
respect to volumes and to surfaces, we may form a very clear idea of the
science of geometry, conceived in all its parts, by assigning to it, for
its general object, the final reduction of the comparisons of all kinds
of extent, volumes, surfaces, or lines, to simple comparisons of right
lines, the only comparisons regarded as capable of being made directly,
and which indeed could not be reduced to any others more easy to effect.
Such a conception, at the same time, indicates clearly the veritable
character of geometry, and seems suited to show at a single glance its
utility and its perfection.
_Measurement of right Lines._ In order to complete this fundamental
explanation, I have yet to show how there can be, in geometry, a special
section relating to the right line, which seems at first incompatible
with the principle that the measurement of this class of lines must
always be regarded as direct.
It is so, in fact, as compared with that of curved lines, and of all the
other objects which geometry considers. But it is evident that the
estimation of a right line cannot be viewed as direct except so far as
the linear unit can be applied to it. Now this often presents
insurmountable difficulties, as I had occasion to show, for another
reason, in the introductory chapter. We must, then, make the measurement
of the proposed right line depend on other analogous measurements
capable of being effected directly. There is, then, necessarily a
primary distinct branch of geometry, exclusively devoted to the right
line; its object is to determine certain right lines from others by
means of the relations belonging to the figures resulting from their
assemblage. This preliminary part of geometry, which is almost
imperceptible in viewing the whole of the science, is nevertheless
susceptible of a great development. It is evidently of especial
importance, since all other geometrical measurements are referred to
those of right lines, and if they could not be determined, the solution
of every question would remain unfinished.
Such, then, are the various fundamental parts of rational geometry,
arranged according to their natural dependence; the geometry of _lines_
being first considered, beginning with the right line; then the geometry
of _surfaces_, and, finally, that of _solids_.
INFINITE EXTENT OF ITS FIELD.
Having determined with precision the general and final object of
geometrical inquiries, the science must now be considered with respect
to the field embraced by each of its three fundamental sections.
Thus considered, geometry is evidently susceptible, by its nature, of an
extension which is rigorously infinite; for the measurement of lines, of
surfaces, or of volumes presents necessarily as many distinct questions
as we can conceive different figures subjected to exact definitions; and
their number is evidently infinite.
Geometers limited themselves at first to consider the most simple
figures which were directly furnished them by nature, or which were
deduced from these primitive elements by the least complicated
combinations. But they have perceived, since Descartes, that, in order
to constitute the science in the most philosophical manner, it was
necessary to make it apply to all imaginable figures. This abstract
geometry will then inevitably comprehend as particular cases all the
different real figures which the exterior world could present. It is
then a fundamental principle in truly rational geometry to consider, as
far as possible, all figures which can be rigorously conceived.
The most superficial examination is enough to convince us that these
figures present a variety which is quite infinite.
_Infinity of Lines._ With respect to curved _lines_, regarding them as
generated by the motion of a point governed by a certain law, it is
plain that we shall have, in general, as many different curves as we
conceive different laws for this motion, which may evidently be
determined by an infinity of distinct conditions; although it may
sometimes accidentally happen that new generations produce curves which
have been already obtained. Thus, among plane curves, if a point moves
so as to remain constantly at the same distance from a fixed point, it
will generate a _circle_; if it is the sum or the difference of its
distances from two fixed points which remains constant, the curve
described will be an _ellipse_ or an _hyperbola_; if it is their
product, we shall have an entirely different curve; if the point departs
equally from a fixed point and from a fixed line, it will describe a
_parabola_; if it revolves on a circle at the same time that this circle
rolls along a straight line, we shall have a _cycloid_; if it advances
along a straight line, while this line, fixed at one of its extremities,
turns in any manner whatever, there will result what in general terms
are called _spirals_, which of themselves evidently present as many
perfectly distinct curves as we can suppose different relations between
these two motions of translation and of rotation, &c. Each of these
different curves may then furnish new ones, by the different general
constructions which geometers have imagined, and which give rise to
evolutes, to epicycloids, to caustics, &c. Finally, there exists a still
greater variety among curves of double curvature.
_Infinity of Surfaces._ As to _surfaces_, the figures are necessarily
more different still, considering them as generated by the motion of
lines. Indeed, the figure may then vary, not only in considering, as in
curves, the different infinitely numerous laws to which the motion of
the generating line may be subjected, but also in supposing that this
line itself may change its nature; a circumstance which has nothing
analogous in curves, since the points which describe them cannot have
any distinct figure. Two classes of very different conditions may then
cause the figures of surfaces to vary, while there exists only one for
lines. It is useless to cite examples of this doubly infinite
multiplicity of surfaces. It would be sufficient to consider the extreme
variety of the single group of surfaces which may be generated by a
right line, and which comprehends the whole family of cylindrical
surfaces, that of conical surfaces, the most general class of
developable surfaces, &c.
_Infinity of Volumes._ With respect to _volumes_, there is no occasion
for any special consideration, since they are distinguished from each
other only by the surfaces which bound them.
In order to complete this sketch, it should be added that surfaces
themselves furnish a new general means of conceiving new curves, since
every curve may be regarded as produced by the intersection of two
surfaces. It is in this way, indeed, that the first lines which we may
regard as having been truly invented by geometers were obtained, since
nature gave directly the straight line and the circle. We know that the
ellipse, the parabola, and the hyperbola, the only curves completely
studied by the ancients, were in their origin conceived only as
resulting from the intersection of a cone with circular base by a plane
in different positions. It is evident that, by the combined employment
of these different general means for the formation of lines and of
surfaces, we could produce a rigorously infinitely series of distinct
forms in starting from only a very small number of figures directly
furnished by observation.
_Analytical invention of Curves, &c._ Finally, all the various direct
means for the invention of figures have scarcely any farther importance,
since rational geometry has assumed its final character in the hands of
Descartes. Indeed, as we shall see more fully in chapter iii., the
invention of figures is now reduced to the invention of equations, so
that nothing is more easy than to conceive new lines and new surfaces,
by changing at will the functions introduced into the equations. This
simple abstract procedure is, in this respect, infinitely more fruitful
than all the direct resources of geometry, developed by the most
powerful imagination, which should devote itself exclusively to that
order of conceptions. It also explains, in the most general and the most
striking manner, the necessarily infinite variety of geometrical forms,
which thus corresponds to the diversity of analytical functions. Lastly,
it shows no less clearly that the different forms of surfaces must be
still more numerous than those of lines, since lines are represented
analytically by equations with two variables, while surfaces give rise
to equations with three variables, which necessarily present a greater
diversity.
The preceding considerations are sufficient to show clearly the
rigorously infinite extent of each of the three general sections of
geometry.
EXPANSION OF ORIGINAL DEFINITION.
To complete the formation of an exact and sufficiently extended idea of
the nature of geometrical inquiries, it is now indispensable to return
to the general definition above given, in order to present it under a
new point of view, without which the complete science would be only very
imperfectly conceived.
When we assign as the object of geometry the _measurement_ of all sorts
of lines, surfaces, and volumes, that is, as has been explained, the
reduction of all geometrical comparisons to simple comparisons of right
lines, we have evidently the advantage of indicating a general
destination very precise and very easy to comprehend. But if we set
aside every definition, and examine the actual composition of the
science of geometry, we will at first be induced to regard the preceding
definition as much too narrow; for it is certain that the greater part
of the investigations which constitute our present geometry do not at
all appear to have for their object the _measurement_ of extension. In
spite of this fundamental objection, I will persist in retaining this
definition; for, in fact, if, instead of confining ourselves to
considering the different questions of geometry isolatedly, we endeavour
to grasp the leading questions, in comparison with which all others,
however important they may be, must be regarded as only secondary, we
will finally recognize that the measurement of lines, of surfaces, and
of volumes, is the invariable object, sometimes _direct_, though most
often _indirect_, of all geometrical labours.
This general proposition being fundamental, since it can alone give our
definition all its value, it is indispensable to enter into some
developments upon this subject.
PROPERTIES OF LINES AND SURFACES.
When we examine with attention the geometrical investigations which do
not seem to relate to the _measurement_ of extent, we find that they
consist essentially in the study of the different _properties_ of each
line or of each surface; that is, in the knowledge of the different
modes of generation, or at least of definition, peculiar to each figure
considered. Now we can easily establish in the most general manner the
necessary relation of such a study to the question of _measurement_, for
which the most complete knowledge of the properties of each form is an
indispensable preliminary. This is concurrently proven by two
considerations, equally fundamental, although quite distinct in their
nature.
NECESSITY OF THEIR STUDY: 1. _To find the most suitable Property._ The
_first_, purely scientific, consists in remarking that, if we did not
know any other characteristic property of each line or surface than that
one according to which geometers had first conceived it, in most cases
it would be impossible to succeed in the solution of questions relating
to its _measurement_. In fact, it is easy to understand that the
different definitions which each figure admits of are not all equally
suitable for such an object, and that they even present the most
complete oppositions in that respect. Besides, since the primitive
definition of each figure was evidently not chosen with this condition
in view, it is clear that we must not expect, in general, to find it the
most suitable; whence results the necessity of discovering others, that
is, of studying as far as is possible the _properties_ of the proposed
figure. Let us suppose, for example, that the circle is defined to be
"the curve which, with the same contour, contains the greatest area."
This is certainly a very characteristic property, but we would evidently
find insurmountable difficulties in trying to deduce from such a
starting point the solution of the fundamental questions relating to the
rectification or to the quadrature of this curve. It is clear, in
advance, that the property of having all its points equally distant from
a fixed point must evidently be much better adapted to inquiries of this
nature, even though it be not precisely the most suitable. In like
manner, would Archimedes ever have been able to discover the quadrature
of the parabola if he had known no other property of that curve than
that it was the section of a cone with a circular base, by a plane
parallel to its generatrix? The purely speculative labours of preceding
geometers, in transforming this first definition, were evidently
indispensable preliminaries to the direct solution of such a question.
The same is true, in a still greater degree, with respect to surfaces.
To form a just idea of this, we need only compare, as to the question of
cubature or quadrature, the common definition of the sphere with that
one, no less characteristic certainly, which would consist in regarding
a spherical body, as that one which, with the same area, contains the
greatest volume.
No more examples are needed to show the necessity of knowing, so far as
is possible, all the properties of each line or of each surface, in
order to facilitate the investigation of rectifications, of quadratures,
and of cubatures, which constitutes the final object of geometry. We may
even say that the principal difficulty of questions of this kind
consists in employing in each case the property which is best adapted
to the nature of the proposed problem. Thus, while we continue to
indicate, for more precision, the measurement of extension as the
general destination of geometry, this first consideration, which goes to
the very bottom of the subject, shows clearly the necessity of including
in it the study, as thorough as possible, of the different generations
or definitions belonging to the same form.
2. _To pass from the Concrete to the Abstract._ A second consideration,
of at least equal importance, consists in such a study being
indispensable for organizing in a rational manner the relation of the
abstract to the concrete in geometry.
The science of geometry having to consider all imaginable figures which
admit of an exact definition, it necessarily results from this, as we
have remarked, that questions relating to any figures presented by
nature are always implicitly comprised in this abstract geometry,
supposed to have attained its perfection. But when it is necessary to
actually pass to concrete geometry, we constantly meet with a
fundamental difficulty, that of knowing to which of the different
abstract types we are to refer, with sufficient approximation, the real
lines or surfaces which we have to study. Now it is for the purpose of
establishing such a relation that it is particularly indispensable to
know the greatest possible number of properties of each figure
considered in geometry.
In fact, if we always confined ourselves to the single primitive
definition of a line or of a surface, supposing even that we could then
_measure_ it (which, according to the first order of considerations,
would generally be impossible), this knowledge would remain almost
necessarily barren in the application, since we should not ordinarily
know how to recognize that figure in nature when it presented itself
there; to ensure that, it would be necessary that the single
characteristic, according to which geometers had conceived it, should be
precisely that one whose verification external circumstances would
admit: a coincidence which would be purely fortuitous, and on which we
could not count, although it might sometimes take place. It is, then,
only by multiplying as much as possible the characteristic properties of
each abstract figure, that we can be assured, in advance, of recognizing
it in the concrete state, and of thus turning to account all our
rational labours, by verifying in each case the definition which is
susceptible of being directly proven. This definition is almost always
the only one in given circumstances, and varies, on the other hand, for
the same figure, with different circumstances; a double reason for its
previous determination.
_Illustration: Orbits of the Planets._ The geometry of the heavens
furnishes us with a very memorable example in this matter, well suited
to show the general necessity of such a study. We know that the ellipse
was discovered by Kepler to be the curve which the planets describe
about the sun, and the satellites about their planets. Now would this
fundamental discovery, which re-created astronomy, ever have been
possible, if geometers had been always confined to conceiving the
ellipse only as the oblique section of a circular cone by a plane? No
such definition, it is evident, would admit of such a verification. The
most general property of the ellipse, that the sum of the distances from
any of its points to two fixed points is a constant quantity, is
undoubtedly much more susceptible, by its nature, of causing the curve
to be recognized in this case, but still is not directly suitable. The
only characteristic which can here be immediately verified is that which
is derived from the relation which exists in the ellipse between the
length of the focal distances and their direction; the only relation
which admits of an astronomical interpretation, as expressing the law
which connects the distance from the planet to the sun, with the time
elapsed since the beginning of its revolution. It was, then, necessary
that the purely speculative labours of the Greek geometers on the
properties of the conic sections should have previously presented their
generation under a multitude of different points of view, before Kepler
could thus pass from the abstract to the concrete, in choosing from
among all these different characteristics that one which could be most
easily proven for the planetary orbits.
_Illustration: Figure of the Earth._ Another example of the same order,
but relating to surfaces, occurs in considering the important question
of the figure of the earth. If we had never known any other property of
the sphere than its primitive character of having all its points equally
distant from an interior point, how would we ever have been able to
discover that the surface of the earth was spherical? For this, it was
necessary previously to deduce from this definition of the sphere some
properties capable of being verified by observations made upon the
surface alone, such as the constant ratio which exists between the
length of the path traversed in the direction of any meridian of a
sphere going towards a pole, and the angular height of this pole above
the horizon at each point. Another example, but involving a much longer
series of preliminary speculations, is the subsequent proof that the
earth is not rigorously spherical, but that its form is that of an
ellipsoid of revolution.
After such examples, it would be needless to give any others, which any
one besides may easily multiply. All of them prove that, without a very
extended knowledge of the different properties of each figure, the
relation of the abstract to the concrete, in geometry, would be purely
accidental, and that the science would consequently want one of its most
essential foundations.
Such, then, are two general considerations which fully demonstrate the
necessity of introducing into geometry a great number of investigations
which have not the _measurement_ of extension for their direct object;
while we continue, however, to conceive such a measurement as being the
final destination of all geometrical science. In this way we can retain
the philosophical advantages of the clearness and precision of this
definition, and still include in it, in a very logical though indirect
manner, all known geometrical researches, in considering those which do
not seem to relate to the measurement of extension, as intended either
to prepare for the solution of the final questions, or to render
possible the application of the solutions obtained.
Having thus recognized, as a general principle, the close and necessary
connexion of the study of the properties of lines and surfaces with
those researches which constitute the final object of geometry, it is
evident that geometers, in the progress of their labours, must by no
means constrain themselves to keep such a connexion always in view.
Knowing, once for all, how important it is to vary as much as possible
the manner of conceiving each figure, they should pursue that study,
without considering of what immediate use such or such a special
property may be for rectifications, quadratures, and cubatures. They
would uselessly fetter their inquiries by attaching a puerile importance
to the continued establishment of that co-ordination.
This general exposition of the general object of geometry is so much the
more indispensable, since, by the very nature of the subject, this study
of the different properties of each line and of each surface necessarily
composes by far the greater part of the whole body of geometrical
researches. Indeed, the questions directly relating to rectifications,
to quadratures, and to cubatures, are evidently, by themselves, very few
in number for each figure considered. On the other hand, the study of
the properties of the same figure presents an unlimited field to the
activity of the human mind, in which it may always hope to make new
discoveries. Thus, although geometers have occupied themselves for
twenty centuries, with more or less activity undoubtedly, but without
any real interruption, in the study of the conic sections, they are far
from regarding that so simple subject as being exhausted; and it is
certain, indeed, that in continuing to devote themselves to it, they
would not fail to find still unknown properties of those different
curves. If labours of this kind have slackened considerably for a
century past, it is not because they are completed, but only, as will be
presently explained, because the philosophical revolution in geometry,
brought about by Descartes, has singularly diminished the importance of
such researches.
It results from the preceding considerations that not only is the field
of geometry necessarily infinite because of the variety of figures to
be considered, but also in virtue of the diversity of the points of view
under the same figure may be regarded. This last conception is, indeed,
that which gives the broadest and most complete idea of the whole body
of geometrical researches. We see that studies of this kind consist
essentially, for each line or for each surface, in connecting all the
geometrical phenomena which it can present, with a single fundamental
phenomenon, regarded as the primitive definition.
THE TWO GENERAL METHODS OF GEOMETRY.
Having now explained in a general and yet precise manner the final
object of geometry, and shown how the science, thus defined, comprehends
a very extensive class of researches which did not at first appear
necessarily to belong to it, there remains to be considered the _method_
to be followed for the formation of this science. This discussion is
indispensable to complete this first sketch of the philosophical
character of geometry. I shall here confine myself to indicating the
most general consideration in this matter, developing and summing up
this important fundamental idea in the following chapters.
Geometrical questions may be treated according to _two methods_ so
different, that there result from them two sorts of geometry, so to say,
the philosophical character of which does not seem to me to have yet
been properly apprehended. The expressions of _Synthetical Geometry_ and
_Analytical Geometry_, habitually employed to designate them, give a
very false idea of them. I would much prefer the purely historical
denominations of _Geometry of the Ancients_ and _Geometry of the
Moderns_, which have at least the advantage of not causing their true
character to be misunderstood. But I propose to employ henceforth the
regular expressions of _Special Geometry_ and _General Geometry_, which
seem to me suited to characterize with precision the veritable nature of
the two methods.
_Their fundamental Difference._ The fundamental difference between the
manner in which we conceive Geometry since Descartes, and the manner in
which the geometers of antiquity treated geometrical questions, is not
the use of the Calculus (or Algebra), as is commonly thought to be the
case. On the one hand, it is certain that the use of the calculus was
not entirely unknown to the ancient geometers, since they used to make
continual and very extensive applications of the theory of proportions,
which was for them, as a means of deduction, a sort of real, though very
imperfect and especially extremely limited equivalent for our present
algebra. The calculus may even be employed in a much more complete
manner than they have used it, in order to obtain certain geometrical
solutions, which will still retain all the essential character of the
ancient geometry; this occurs very frequently with respect to those
problems of geometry of two or of three dimensions, which are commonly
designated under the name of _determinate_. On the other hand, important
as is the influence of the calculus in our modern geometry, various
solutions obtained without algebra may sometimes manifest the peculiar
character which distinguishes it from the ancient geometry, although
analysis is generally indispensable. I will cite, as an example, the
method of Roberval for tangents, the nature of which is essentially
modern, and which, however, leads in certain cases to complete
solutions, without any aid from the calculus. It is not, then, the
instrument of deduction employed which is the principal distinction
between the two courses which the human mind can take in geometry.
The real fundamental difference, as yet imperfectly apprehended, seems
to me to consist in the very nature of the questions considered. In
truth, geometry, viewed as a whole, and supposed to have attained entire
perfection, must, as we have seen on the one hand, embrace all
imaginable figures, and, on the other, discover all the properties of
each figure. It admits, from this double consideration, of being treated
according to two essentially distinct plans; either, 1°, by grouping
together all the questions, however different they may be, which relate
to the same figure, and isolating those relating to different bodies,
whatever analogy there may exist between them; or, 2°, on the contrary,
by uniting under one point of view all similar inquiries, to whatever
different figures they may relate, and separating the questions relating
to the really different properties of the same body. In a word, the
whole body of geometry may be essentially arranged either with reference
to the _bodies_ studied or to the _phenomena_ to be considered. The
first plan, which is the most natural, was that of the ancients; the
second, infinitely more rational, is that of the moderns since
Descartes.
_Geometry of the Ancients._ Indeed, the principal characteristics of the
ancient geometry is that they studied, one by one, the different lines
and the different surfaces, not passing to the examination of a new
figure till they thought they had exhausted all that there was
interesting in the figures already known. In this way of proceeding,
when they undertook the study of a new curve, the whole of the labour
bestowed on the preceding ones could not offer directly any essential
assistance, otherwise than by the geometrical practice to which it had
trained the mind. Whatever might be the real similarity of the questions
proposed as to two different figures, the complete knowledge acquired
for the one could not at all dispense with taking up again the whole of
the investigation for the other. Thus the progress of the mind was never
assured; so that they could not be certain, in advance, of obtaining any
solution whatever, however analogous the proposed problem might be to
questions which had been already resolved. Thus, for example, the
determination of the tangents to the three conic sections did not
furnish any rational assistance for drawing the tangent to any other new
curve, such as the conchoid, the cissoid, &c. In a word, the geometry of
the ancients was, according to the expression proposed above,
essentially special.
_Geometry of the Moderns._ In the system of the moderns, geometry is, on
the contrary, eminently _general_, that is to say, relating to any
figures whatever. It is easy to understand, in the first place, that all
geometrical expressions of any interest may be proposed with reference
to all imaginable figures. This is seen directly in the fundamental
problems--of rectifications, quadratures, and cubatures--which
constitute, as has been shown, the final object of geometry. But this
remark is no less incontestable, even for investigations which relate to
the different _properties_ of lines and of surfaces, and of which the
most essential, such as the question of tangents or of tangent planes,
the theory of curvatures, &c., are evidently common to all figures
whatever. The very few investigations which are truly peculiar to
particular figures have only an extremely secondary importance. This
being understood, modern geometry consists essentially in abstracting,
in order to treat it by itself, in an entirely general manner, every
question relating to the same geometrical phenomenon, in whatever bodies
it may be considered. The application of the universal theories thus
constructed to the special determination of the phenomenon which is
treated of in each particular body, is now regarded as only a subaltern
labour, to be executed according to invariable rules, and the success of
which is certain in advance. This labour is, in a word, of the same
character as the numerical calculation of an analytical formula. There
can be no other merit in it than that of presenting in each case the
solution which is necessarily furnished by the general method, with all
the simplicity and elegance which the line or the surface considered can
admit of. But no real importance is attached to any thing but the
conception and the complete solution of a new question belonging to any
figure whatever. Labours of this kind are alone regarded as producing
any real advance in science. The attention of geometers, thus relieved
from the examination of the peculiarities of different figures, and
wholly directed towards general questions, has been thereby able to
elevate itself to the consideration of new geometrical conceptions,
which, applied to the curves studied by the ancients, have led to the
discovery of important properties which they had not before even
suspected. Such is geometry, since the radical revolution produced by
Descartes in the general system of the science.
_The Superiority of the modern Geometry._ The mere indication of the
fundamental character of each of the two geometries is undoubtedly
sufficient to make apparent the immense necessary superiority of modern
geometry. We may even say that, before the great conception of
Descartes, rational geometry was not truly constituted upon definitive
bases, whether in its abstract or concrete relations. In fact, as
regards science, considered speculatively, it is clear that, in
continuing indefinitely to follow the course of the ancients, as did the
moderns before Descartes, and even for a little while afterwards, by
adding some new curves to the small number of those which they had
studied, the progress thus made, however rapid it might have been, would
still be found, after a long series of ages, to be very inconsiderable
in comparison with the general system of geometry, seeing the infinite
variety of the forms which would still have remained to be studied. On
the contrary, at each question resolved according to the method of the
moderns, the number of geometrical problems to be resolved is then, once
for all, diminished by so much with respect to all possible bodies.
Another consideration is, that it resulted, from their complete want of
general methods, that the ancient geometers, in all their
investigations, were entirely abandoned to their own strength, without
ever having the certainty of obtaining, sooner or later, any solution
whatever. Though this imperfection of the science was eminently suited
to call forth all their admirable sagacity, it necessarily rendered
their progress extremely slow; we can form some idea of this by the
considerable time which they employed in the study of the conic
sections. Modern geometry, making the progress of our mind certain,
permits us, on the contrary, to make the greatest possible use of the
forces of our intelligence, which the ancients were often obliged to
waste on very unimportant questions.
A no less important difference between the two systems appears when we
come to consider geometry in the concrete point of view. Indeed, we have
already remarked that the relation of the abstract to the concrete in
geometry can be founded upon rational bases only so far as the
investigations are made to bear directly upon all imaginable figures. In
studying lines, only one by one, whatever may be the number, always
necessarily very small, of those which we shall have considered, the
application of such theories to figures really existing in nature will
never have any other than an essentially accidental character, since
there is nothing to assure us that these figures can really be brought
under the abstract types considered by geometers.
Thus, for example, there is certainly something fortuitous in the happy
relation established between the speculations of the Greek geometers
upon the conic sections and the determination of the true planetary
orbits. In continuing geometrical researches upon the same plan, there
was no good reason for hoping for similar coincidences; and it would
have been possible, in these special studies, that the researches of
geometers should have been directed to abstract figures entirely
incapable of any application, while they neglected others, susceptible
perhaps of an important and immediate application. It is clear, at
least, that nothing positively guaranteed the necessary applicability of
geometrical speculations. It is quite another thing in the modern
geometry. From the single circumstance that in it we proceed by general
questions relating to any figures whatever, we have in advance the
evident certainty that the figures really existing in the external world
could in no case escape the appropriate theory if the geometrical
phenomenon which it considers presents itself in them.
From these different considerations, we see that the ancient system of
geometry wears essentially the character of the infancy of the science,
which did not begin to become completely rational till after the
philosophical resolution produced by Descartes. But it is evident, on
the other hand, that geometry could not be at first conceived except in
this _special_ manner. _General_ geometry would not have been possible,
and its necessity could not even have been felt, if a long series of
special labours on the most simple figures had not previously furnished
bases for the conception of Descartes, and rendered apparent the
impossibility of persisting indefinitely in the primitive geometrical
philosophy.
_The Ancient the Base of the Modern._ From this last consideration we
must infer that, although the geometry which I have called _general_
must be now regarded as the only true dogmatical geometry, and that to
which we shall chiefly confine ourselves, the other having no longer
much more than an historical interest, nevertheless it is not possible
to entirely dispense with _special_ geometry in a rational exposition of
the science. We undoubtedly need not borrow directly from ancient
geometry all the results which it has furnished; but, from the very
nature of the subject, it is necessarily impossible entirely to dispense
with the ancient method, which will always serve as the preliminary
basis of the science, dogmatically as well as historically. The reason
of this is easy to understand. In fact, _general_ geometry being
essentially founded, as we shall soon establish, upon the employment of
the calculus in the transformation of geometrical into analytical
considerations, such a manner of proceeding could not take possession of
the subject immediately at its origin. We know that the application of
mathematical analysis, from its nature, can never commence any science
whatever, since evidently it cannot be employed until the science has
already been sufficiently cultivated to establish, with respect to the
phenomena considered, some _equations_ which can serve as starting
points for the analytical operations. These fundamental equations being
once discovered, analysis will enable us to deduce from them a multitude
of consequences which it would have been previously impossible even to
suspect; it will perfect the science to an immense degree, both with
respect to the generality of its conceptions and to the complete
co-ordination established between them. But mere mathematical analysis
could never be sufficient to form the bases of any natural science, not
even to demonstrate them anew when they have once been established.
Nothing can dispense with the direct study of the subject, pursued up to
the point of the discovery of precise relations.
We thus see that the geometry of the ancients will always have, by its
nature, a primary part, absolutely necessary and more or less extensive,
in the complete system of geometrical knowledge. It forms a rigorously
indispensable introduction to _general_ geometry. But it is to this that
it must be limited in a completely dogmatic exposition. I will consider,
then, directly, in the following chapter, this _special_ or
_preliminary_ geometry restricted to exactly its necessary limits, in
order to occupy myself thenceforth only with the philosophical
examination of _general_ or _definitive_ geometry, the only one which is
truly rational, and which at present essentially composes the science.
CHAPTER II.
ANCIENT OR SYNTHETIC GEOMETRY.
The geometrical method of the ancients necessarily constituting a
preliminary department in the dogmatical system of geometry, designed to
furnish _general_ geometry with indispensable foundations, it is now
proper to begin with determining wherein strictly consists this
preliminary function of _special_ geometry, thus reduced to the
narrowest possible limits.
ITS PROPER EXTENT.
_Lines; Polygons; Polyhedrons._ In considering it under this point of
view, it is easy to recognize that we might restrict it to the study of
the right line alone for what concerns the geometry of _lines_; to the
_quadrature_ of rectilinear plane areas; and, lastly, to the _cubature_
of bodies terminated by plane faces. The elementary propositions
relating to these three fundamental questions form, in fact, the
necessary starting point of all geometrical inquiries; they alone cannot
be obtained except by a direct study of the subject; while, on the
contrary, the complete theory of all other figures, even that of the
circle, and of the surfaces and volumes which are connected with it, may
at the present day be completely comprehended in the domain of _general_
or _analytical_ geometry; these primitive elements at once furnishing
_equations_ which are sufficient to allow of the application of the
calculus to geometrical questions, which would not have been possible
without this previous condition.
It results from this consideration that, in common practice, we give to
_elementary_ geometry more extent than would be rigorously necessary to
it; since, besides the right line, polygons, and polyhedrons, we also
include in it the circle and the "round" bodies; the study of which
might, however, be as purely analytical as that, for example, of the
conic sections. An unreflecting veneration for antiquity contributes to
maintain this defect in method; but the best reason which can be given
for it is the serious inconvenience for ordinary instruction which there
would be in postponing, to so distant an epoch of mathematical
education, the solution of several essential questions, which are
susceptible of a direct and continual application to a great number of
important uses. In fact, to proceed in the most rational manner, we
should employ the integral calculus in obtaining the interesting results
relating to the length or the area of the circle, or to the quadrature
of the sphere, &c., which have been determined by the ancients from
extremely simple considerations. This inconvenience would be of little
importance with regard to the persons destined to study the whole of
mathematical science, and the advantage of proceeding in a perfectly
logical order would have a much greater comparative value. But the
contrary case being the more frequent, theories so essential have
necessarily been retained in elementary geometry. Perhaps the conic
sections, the cycloid, &c., might be advantageously added in such cases.
_Not to be farther restricted._ While this preliminary portion of
geometry, which cannot be founded on the application of the calculus,
is reduced by its nature to a very limited series of fundamental
researches, relating to the right line, polygonal areas, and
polyhedrons, it is certain, on the other hand, that we cannot restrict
it any more; although, by a veritable abuse of the spirit of analysis,
it has been recently attempted to present the establishment of the
principal theorems of elementary geometry under an algebraical point of
view. Thus some have pretended to demonstrate, by simple abstract
considerations of mathematical analysis, the constant relation which
exists between the three angles of a rectilinear triangle, the
fundamental proposition of the theory of similar triangles, that of
parallelopipedons, &c.; in a word, precisely the only geometrical
propositions which cannot be obtained except by a direct study of the
subject, without the calculus being susceptible of having any part in
it. Such aberrations are the unreflecting exaggerations of that natural
and philosophical tendency which leads us to extend farther and farther
the influence of analysis in mathematical studies. In mechanics, the
pretended analytical demonstrations of the parallelogram of forces are
of similar character.
The viciousness of such a manner of proceeding follows from the
principles previously presented. We have already, in fact, recognized
that, since the calculus is not, and cannot be, any thing but a means of
deduction, it would indicate a radically false idea of it to wish to
employ it in establishing the elementary foundations of any science
whatever; for on what would the analytical reasonings in such an
operation repose? A labour of this nature, very far from really
perfecting the philosophical character of a science, would constitute a
return towards the metaphysical age, in presenting real facts as mere
logical abstractions.