TRANSCENDENTAL ANALYSIS:
DIFFERENT MODES OF VIEWING IT.
We determined, in the second chapter, the philosophical character of the
transcendental analysis, in whatever manner it may be conceived,
considering only the general nature of its actual destination as a part
of mathematical science. This analysis has been presented by geometers
under several points of view, really distinct, although necessarily
equivalent, and leading always to identical results. They may be reduced
to three principal ones; those of LEIBNITZ, of NEWTON, and of LAGRANGE,
of which all the others are only secondary modifications. In the present
state of science, each of these three general conceptions offers
essential advantages which pertain to it exclusively, without our having
yet succeeded in constructing a single method uniting all these
different characteristic qualities. This combination will probably be
hereafter effected by some method founded upon the conception of
Lagrange when that important philosophical labour shall have been
accomplished, the study of the other conceptions will have only a
historic interest; but, until then, the science must be considered as in
only a provisional state, which requires the simultaneous consideration
of all the various modes of viewing this calculus. Illogical as may
appear this multiplicity of conceptions of one identical subject, still,
without them all, we could form but a very insufficient idea of this
analysis, whether in itself, or more especially in relation to its
applications. This want of system in the most important part of
mathematical analysis will not appear strange if we consider, on the one
hand, its great extent and its superior difficulty, and, on the other,
its recent formation.
ITS EARLY HISTORY.
If we had to trace here the systematic history of the successive
formation of the transcendental analysis, it would be necessary
previously to distinguish carefully from the _calculus of indirect
functions_, properly so called, the original idea of the _infinitesimal
method_, which can be conceived by itself, independently of any
_calculus_. We should see that the first germ of this idea is found in
the procedure constantly employed by the Greek geometers, under the name
of the _Method of Exhaustions_, as a means of passing from the
properties of straight lines to those of curves, and consisting
essentially in substituting for the curve the auxiliary consideration of
an inscribed or circumscribed polygon, by means of which they rose to
the curve itself, taking in a suitable manner the limits of the
primitive ratios. Incontestable as is this filiation of ideas, it would
be giving it a greatly exaggerated importance to see in this method of
exhaustions the real equivalent of our modern methods, as some geometers
have done; for the ancients had no logical and general means for the
determination of these limits, and this was commonly the greatest
difficulty of the question; so that their solutions were not subjected
to abstract and invariable rules, the uniform application of which would
lead with certainty to the knowledge sought; which is, on the contrary,
the principal characteristic of our transcendental analysis. In a word,
there still remained the task of generalizing the conceptions used by
the ancients, and, more especially, by considering it in a manner purely
abstract, of reducing it to a complete system of calculation, which to
them was impossible.
The first idea which was produced in this new direction goes back to the
great geometer Fermat, whom Lagrange has justly presented as having
blocked out the direct formation of the transcendental analysis by his
method for the determination of _maxima_ and _minima_, and for the
finding of _tangents_, which consisted essentially in introducing the
auxiliary consideration of the correlative increments of the proposed
variables, increments afterward suppressed as equal to zero when the
equations had undergone certain suitable transformations. But, although
Fermat was the first to conceive this analysis in a truly abstract
manner, it was yet far from being regularly formed into a general and
distinct calculus having its own notation, and especially freed from the
superfluous consideration of terms which, in the analysis of Fermat,
were finally not taken into the account, after having nevertheless
greatly complicated all the operations by their presence. This is what
Leibnitz so happily executed, half a century later, after some
intermediate modifications of the ideas of Fermat introduced by Wallis,
and still more by Barrow; and he has thus been the true creator of the
transcendental analysis, such as we now employ it. This admirable
discovery was so ripe (like all the great conceptions of the human
intellect at the moment of their manifestation), that Newton, on his
side, had arrived, at the same time, or a little earlier, at a method
exactly equivalent, by considering this analysis under a very different
point of view, which, although more logical in itself, is really less
adapted to give to the common fundamental method all the extent and the
facility which have been imparted to it by the ideas of Leibnitz.
Finally, Lagrange, putting aside the heterogeneous considerations which
had guided Leibnitz and Newton, has succeeded in reducing the
transcendental analysis, in its greatest perfection, to a purely
algebraic system, which only wants more aptitude for its practical
applications.
After this summary glance at the general history of the transcendental
analysis, we will proceed to the dogmatic exposition of the three
principal conceptions, in order to appreciate exactly their
characteristic properties, and to show the necessary identity of the
methods which are thence derived. Let us begin with that of Leibnitz.
METHOD OF LEIBNITZ.
_Infinitely small Elements._ This consists in introducing into the
calculus, in order to facilitate the establishment of equations, the
infinitely small elements of which all the quantities, the relations
between which are sought, are considered to be composed. These elements
or _differentials_ will have certain relations to one another, which are
constantly and necessarily more simple and easy to discover than those
of the primitive quantities, and by means of which we will be enabled
(by a special calculus having for its peculiar object the elimination of
these auxiliary infinitesimals) to go back to the desired equations,
which it would have been most frequently impossible to obtain directly.
This indirect analysis may have different degrees of indirectness; for,
when there is too much difficulty in forming immediately the equation
between the differentials of the magnitudes under consideration, a
second application of the same general artifice will have to be made,
and these differentials be treated, in their turn, as new primitive
quantities, and a relation be sought between their infinitely small
elements (which, with reference to the final objects of the question,
will be _second differentials_), and so on; the same transformation
admitting of being repeated any number of times, on the condition of
finally eliminating the constantly increasing number of infinitesimal
quantities introduced as auxiliaries.
A person not yet familiar with these considerations does not perceive at
once how the employment of these auxiliary quantities can facilitate the
discovery of the analytical laws of phenomena; for the infinitely small
increments of the proposed magnitudes being of the same species with
them, it would seem that their relations should not be obtained with
more ease, inasmuch as the greater or less value of a quantity cannot,
in fact, exercise any influence on an inquiry which is necessarily
independent, by its nature, of every idea of value. But it is easy,
nevertheless, to explain very clearly, and in a quite general manner,
how far the question must be simplified by such an artifice. For this
purpose, it is necessary to begin by distinguishing _different orders_
of infinitely small quantities, a very precise idea of which may be
obtained by considering them as being either the successive powers of
the same primitive infinitely small quantity, or as being quantities
which may be regarded as having finite ratios with these powers; so
that, to take an example, the second, third, &c., differentials of any
one variable are classed as infinitely small quantities of the second
order, the third, &c., because it is easy to discover in them finite
multiples of the second, third, &c., powers of a certain first
differential. These preliminary ideas being established, the spirit of
the infinitesimal analysis consists in constantly neglecting the
infinitely small quantities in comparison with finite quantities, and
generally the infinitely small quantities of any order whatever in
comparison with all those of an inferior order. It is at once apparent
how much such a liberty must facilitate the formation of equations
between the differentials of quantities, since, in the place of these
differentials, we can substitute such other elements as we may choose,
and as will be more simple to consider, only taking care to conform to
this single condition, that the new elements differ from the preceding
ones only by quantities infinitely small in comparison with them. It is
thus that it will be possible, in geometry, to treat curved lines as
composed of an infinity of rectilinear elements, curved surfaces as
formed of plane elements, and, in mechanics, variable motions as an
infinite series of uniform motions, succeeding one another at infinitely
small intervals of time.
EXAMPLES. Considering the importance of this admirable conception, I
think that I ought here to complete the illustration of its fundamental
character by the summary indication of some leading examples.
1. _Tangents._ Let it be required to determine, for each point of a
plane curve, the equation of which is given, the direction of its
tangent; a question whose general solution was the primitive object of
the inventors of the transcendental analysis. We will consider the
tangent as a secant joining two points infinitely near to each other;
and then, designating by _dy_ and _dx_ the infinitely small differences
of the co-ordinates of those two points, the elementary principles of
geometry will immediately give the equation _t_ = _dy_/_dx_ for the
trigonometrical tangent of the angle which is made with the axis of the
abscissas by the desired tangent, this being the most simple way of
fixing its position in a system of rectilinear co-ordinates. This
equation, common to all curves, being established, the question is
reduced to a simple analytical problem, which will consist in
eliminating the infinitesimals _dx_ and _dy_, which were introduced as
auxiliaries, by determining in each particular case, by means of the
equation of the proposed curve, the ratio of _dy_ to _dx_, which will be
constantly done by uniform and very simple methods.
2. _Rectification of an Arc._ In the second place, suppose that we wish
to know the length of the arc of any curve, considered as a function of
the co-ordinates of its extremities. It would be impossible to establish
directly the equation between this arc s and these co-ordinates, while
it is easy to find the corresponding relation between the differentials
of these different magnitudes. The most simple theorems of elementary
geometry will in fact give at once, considering the infinitely small arc
_ds_ as a right line, the equations
_ds²_ = _dy²_ + _dx²_, or _ds²_ = _dx²_ + _dy²_ + _dz²_,
according as the curve is of single or double curvature. In either case,
the question is now entirely within the domain of analysis, which, by
the elimination of the differentials (which is the peculiar object of
the calculus of indirect functions), will carry us back from this
relation to that which exists between the finite quantities themselves
under examination.
3. _Quadrature of a Curve._ It would be the same with the quadrature of
curvilinear areas. If the curve is a plane one, and referred to
rectilinear co-ordinates, we will conceive the area A comprised between
this curve, the axis of the abscissas, and two extreme co-ordinates, to
increase by an infinitely small quantity _d_A, as the result of a
corresponding increment of the abscissa. The relation between these two
differentials can be immediately obtained with the greatest facility by
substituting for the curvilinear element of the proposed area the
rectangle formed by the extreme ordinate and the element of the
abscissa, from which it evidently differs only by an infinitely small
quantity of the second order. This will at once give, whatever may be
the curve, the very simple differential equation
_d_A = _ydx_,
from which, when the curve is defined, the calculus of indirect
functions will show how to deduce the finite equation, which is the
immediate object of the problem.
4. _Velocity in Variable Motion._ In like manner, in Dynamics, when we
desire to know the expression for the velocity acquired at each instant
by a body impressed with a motion varying according to any law, we will
consider the motion as being uniform during an infinitely small element
of the time _t_, and we will thus immediately form the differential
equation _de_ = _vdt_, in which _v_ designates the velocity acquired
when the body has passed over the space _e_; and thence it will be easy
to deduce, by simple and invariable analytical procedures, the formula
which would give the velocity in each particular motion, in accordance
with the corresponding relation between the time and the space; or,
reciprocally, what this relation would be if the mode of variation of
the velocity was supposed to be known, whether with respect to the space
or to the time.
5. _Distribution of Heat._ Lastly, to indicate another kind of
questions, it is by similar steps that we are able, in the study of
thermological phenomena, according to the happy conception of M.
Fourier, to form in a very simple manner the general differential
equation which expresses the variable distribution of heat in any body
whatever, subjected to any influences, by means of the single and
easily-obtained relation, which represents the uniform distribution of
heat in a right-angled parallelopipedon, considering (geometrically)
every other body as decomposed into infinitely small elements of a
similar form, and (thermologically) the flow of heat as constant during
an infinitely small element of time. Henceforth, all the questions which
can be presented by abstract thermology will be reduced, as in geometry
and mechanics, to mere difficulties of analysis, which will always
consist in the elimination of the differentials introduced as
auxiliaries to facilitate the establishment of the equations.
Examples of such different natures are more than sufficient to give a
clear general idea of the immense scope of the fundamental conception of
the transcendental analysis as formed by Leibnitz, constituting, as it
undoubtedly does, the most lofty thought to which the human mind has as
yet attained.
It is evident that this conception was indispensable to complete the
foundation of mathematical science, by enabling us to establish, in a
broad and fruitful manner, the relation of the concrete to the abstract.
In this respect it must be regarded as the necessary complement of the
great fundamental idea of Descartes on the general analytical
representation of natural phenomena: an idea which did not begin to be
worthily appreciated and suitably employed till after the formation of
the infinitesimal analysis, without which it could not produce, even in
geometry, very important results.
_Generality of the Formulas._ Besides the admirable facility which is
given by the transcendental analysis for the investigation of the
mathematical laws of all phenomena, a second fundamental and inherent
property, perhaps as important as the first, is the extreme generality
of the differential formulas, which express in a single equation each
determinate phenomenon, however varied the subjects in relation to which
it is considered. Thus we see, in the preceding examples, that a single
differential equation gives the tangents of all curves, another their
rectifications, a third their quadratures; and in the same way, one
invariable formula expresses the mathematical law of every variable
motion; and, finally, a single equation constantly represents the
distribution of heat in any body and for any case. This generality,
which is so exceedingly remarkable, and which is for geometers the basis
of the most elevated considerations, is a fortunate and necessary
consequence of the very spirit of the transcendental analysis,
especially in the conception of Leibnitz. Thus the infinitesimal
analysis has not only furnished a general method for indirectly forming
equations which it would have been impossible to discover in a direct
manner, but it has also permitted us to consider, for the mathematical
study of natural phenomena, a new order of more general laws, which
nevertheless present a clear and precise signification to every mind
habituated to their interpretation. By virtue of this second
characteristic property, the entire system of an immense science, such
as geometry or mechanics, has been condensed into a small number of
analytical formulas, from which the human mind can deduce, by certain
and invariable rules, the solution of all particular problems.
_Demonstration of the Method._ To complete the general exposition of the
conception of Leibnitz, there remains to be considered the demonstration
of the logical procedure to which it leads, and this, unfortunately, is
the most imperfect part of this beautiful method.
In the beginning of the infinitesimal analysis, the most celebrated
geometers rightly attached more importance to extending the immortal
discovery of Leibnitz and multiplying its applications than to
rigorously establishing the logical bases of its operations. They
contented themselves for a long time by answering the objections of
second-rate geometers by the unhoped-for solution of the most difficult
problems; doubtless persuaded that in mathematical science, much more
than in any other, we may boldly welcome new methods, even when their
rational explanation is imperfect, provided they are fruitful in
results, inasmuch as its much easier and more numerous verifications
would not permit any error to remain long undiscovered. But this state
of things could not long exist, and it was necessary to go back to the
very foundations of the analysis of Leibnitz in order to prove, in a
perfectly general manner, the rigorous exactitude of the procedures
employed in this method, in spite of the apparent infractions of the
ordinary rules of reasoning which it permitted.
Leibnitz, urged to answer, had presented an explanation entirely
erroneous, saying that he treated infinitely small quantities as
_incomparables_, and that he neglected them in comparison with finite
quantities, "like grains of sand in comparison with the sea:" a view
which would have completely changed the nature of his analysis, by
reducing it to a mere approximative calculus, which, under this point of
view, would be radically vicious, since it would be impossible to
foresee, in general, to what degree the successive operations might
increase these first errors, which could thus evidently attain any
amount. Leibnitz, then, did not see, except in a very confused manner,
the true logical foundations of the analysis which he had created. His
earliest successors limited themselves, at first, to verifying its
exactitude by showing the conformity of its results, in particular
applications, to those obtained by ordinary algebra or the geometry of
the ancients; reproducing, according to the ancient methods, so far as
they were able, the solutions of some problems after they had been once
obtained by the new method, which alone was capable of discovering them
in the first place.
When this great question was considered in a more general manner,
geometers, instead of directly attacking the difficulty, preferred to
elude it in some way, as Euler and D'Alembert, for example, have done,
by demonstrating the necessary and constant conformity of the conception
of Leibnitz, viewed in all its applications, with other fundamental
conceptions of the transcendental analysis, that of Newton especially,
the exactitude of which was free from any objection. Such a general
verification is undoubtedly strictly sufficient to dissipate any
uncertainty as to the legitimate employment of the analysis of Leibnitz.
But the infinitesimal method is so important--it offers still, in almost
all its applications, such a practical superiority over the other
general conceptions which have been successively proposed--that there
would be a real imperfection in the philosophical character of the
science if it could not justify itself, and needed to be logically
founded on considerations of another order, which would then cease to be
employed.
It was, then, of real importance to establish directly and in a general
manner the necessary rationality of the infinitesimal method. After
various attempts more or less imperfect, a distinguished geometer,
Carnot, presented at last the true direct logical explanation of the
method of Leibnitz, by showing it to be founded on the principle of the
necessary compensation of errors, this being, in fact, the precise and
luminous manifestation of what Leibnitz had vaguely and confusedly
perceived. Carnot has thus rendered the science an essential service,
although, as we shall see towards the end of this chapter, all this
logical scaffolding of the infinitesimal method, properly so called, is
very probably susceptible of only a provisional existence, inasmuch as
it is radically vicious in its nature. Still, we should not fail to
notice the general system of reasoning proposed by Carnot, in order to
directly legitimate the analysis of Leibnitz. Here is the substance of
it:
In establishing the differential equation of a phenomenon, we
substitute, for the immediate elements of the different quantities
considered, other simpler infinitesimals, which differ from them
infinitely little in comparison with them; and this substitution
constitutes the principal artifice of the method of Leibnitz, which
without it would possess no real facility for the formation of
equations. Carnot regards such an hypothesis as really producing an
error in the equation thus obtained, and which for this reason he calls
_imperfect_; only, it is clear that this error must be infinitely small.
Now, on the other hand, all the analytical operations, whether of
differentiation or of integration, which are performed upon these
differential equations, in order to raise them to finite equations by
eliminating all the infinitesimals which have been introduced as
auxiliaries, produce as constantly, by their nature, as is easily seen,
other analogous errors, so that an exact compensation takes place, and
the final equations, in the words of Carnot, become _perfect_. Carnot
views, as a certain and invariable indication of the actual
establishment of this necessary compensation, the complete elimination
of the various infinitely small quantities, which is always, in fact,
the final object of all the operations of the transcendental analysis;
for if we have committed no other infractions of the general rules of
reasoning than those thus exacted by the very nature of the
infinitesimal method, the infinitely small errors thus produced cannot
have engendered other than infinitely small errors in all the equations,
and the relations are necessarily of a rigorous exactitude as soon as
they exist between finite quantities alone, since the only errors then
possible must be finite ones, while none such can have entered. All this
general reasoning is founded on the conception of infinitesimal
quantities, regarded as indefinitely decreasing, while those from which
they are derived are regarded as fixed.
_Illustration by Tangents._ Thus, to illustrate this abstract exposition
by a single example, let us take up again the question of _tangents_,
which is the most easy to analyze completely. We will regard the
equation _t_ = _dy/dx_, obtained above, as being affected with an
infinitely small error, since it would be perfectly rigorous only for
the secant. Now let us complete the solution by seeking, according to
the equation of each curve, the ratio between the differentials of the
co-ordinates. If we suppose this equation to be _y_ = _ax²_, we shall
evidently have
_dy_ = 2_axdx_ + _adx²_.
In this formula we shall have to neglect the term _dx²_ as an infinitely
small quantity of the second order. Then the combination of the two
_imperfect_ equations.
_t_ = _dy/dx_, _dy_ = 2_ax(dx)_,
being sufficient to eliminate entirely the infinitesimals, the finite
result, _t_ = 2_ax_, will necessarily be rigorously correct, from the
effect of the exact compensation of the two errors committed; since, by
its finite nature, it cannot be affected by an infinitely small error,
and this is, nevertheless, the only one which it could have, according
to the spirit of the operations which have been executed.
It would be easy to reproduce in a uniform manner the same reasoning
with reference to all the other general applications of the analysis of
Leibnitz.
This ingenious theory is undoubtedly more subtile than solid, when we
examine it more profoundly; but it has really no other radical logical
fault than that of the infinitesimal method itself, of which it is, it
seems to me, the natural development and the general explanation, so
that it must be adopted for as long a time as it shall be thought proper
to employ this method directly.
* * * * *
I pass now to the general exposition of the two other fundamental
conceptions of the transcendental analysis, limiting myself in each to
its principal idea, the philosophical character of the analysis having
been sufficiently determined above in the examination of the conception
of Leibnitz, which I have specially dwelt upon because it admits of
being most easily grasped as a whole, and most rapidly described.
METHOD OF NEWTON.
Newton has successively presented his own method of conceiving the
transcendental analysis under several different forms. That which is at
present the most commonly adopted was designated by Newton, sometimes
under the name of the _Method of prime and ultimate Ratios_, sometimes
under that of the _Method of Limits_.
_Method of Limits._ The general spirit of the transcendental analysis,
from this point of view, consists in introducing as auxiliaries, in the
place of the primitive quantities, or concurrently with them, in order
to facilitate the establishment of equations, the _limits of the ratios_
of the simultaneous increments of these quantities; or, in other words,
the _final ratios_ of these increments; limits or final ratios which can
be easily shown to have a determinate and finite value. A special
calculus, which is the equivalent of the infinitesimal calculus, is then
employed to pass from the equations between these limits to the
corresponding equations between the primitive quantities themselves.
The power which is given by such an analysis, of expressing with more
ease the mathematical laws of phenomena, depends in general on this,
that since the calculus applies, not to the increments themselves of the
proposed quantities, but to the limits of the ratios of those
increments, we can always substitute for each increment any other
magnitude more easy to consider, provided that their final ratio is the
ratio of equality, or, in other words, that the limit of their ratio is
unity. It is clear, indeed, that the calculus of limits would be in no
way affected by this substitution. Starting from this principle, we find
nearly the equivalent of the facilities offered by the analysis of
Leibnitz, which are then merely conceived under another point of view.
Thus curves will be regarded as the _limits_ of a series of rectilinear
polygons, variable motions as the _limits_ of a collection of uniform
motions of constantly diminishing durations, and so on.
EXAMPLES. 1. _Tangents._ Suppose, for example, that we wish to determine
the direction of the tangent to a curve; we will regard it as the limit
towards which would tend a secant, which should turn about the given
point so that its second point of intersection should indefinitely
approach the first. Representing the differences of the co-ordinates of
the two points by Δ_y_ and Δ_x_, we would have at each instant, for the
trigonometrical tangent of the angle which the secant makes with the
axis of abscissas,
_t_ = Δ_y_/Δ_x_;
from which, taking the limits, we will obtain, relatively to the tangent
itself, this general formula of transcendental analysis,
_t_ = _L_(Δ_y_/Δ_x_),
the characteristic _L_ being employed to designate the limit. The
calculus of indirect functions will show how to deduce from this formula
in each particular case, when the equation of the curve is given, the
relation between _t_ and _x_, by eliminating the auxiliary quantities
which have been introduced. If we suppose, in order to complete the
solution, that the equation of the proposed curve is _y_ = _ax²_, we
shall evidently have
Δ_y_ = 2_ax_Δ_x_ + _a_(Δ_x_)²,
from which we shall obtain
Δ_y_/Δ_x_ = 2_ax_ + _a_Δ_x_.
Now it is clear that the _limit_ towards which the second number tends,
in proportion as Δ_x_ diminishes, is 2_ax_. We shall therefore find, by
this method, _t_ = 2_ax_, as we obtained it for the same case by the
method of Leibnitz.
2. _Rectifications._ In like manner, when the rectification of a curve
is desired, we must substitute for the increment of the arc s the chord
of this increment, which evidently has such a connexion with it that the
limit of their ratio is unity; and then we find (pursuing in other
respects the same plan as with the method of Leibnitz) this general
equation of rectifications:
(_LΔs_/Δ_x_)² = 1 + (_LΔy_/Δ_x_)²,
or (_LΔs_/Δ_x_)² = 1 + (_LΔy_/Δ_x_)² + (_LΔz_/Δ_x_)²,
according as the curve is plane or of double curvature. It will now be
necessary, for each particular curve, to pass from this equation to that
between the arc and the abscissa, which depends on the transcendental
calculus properly so called.
We could take up, with the same facility, by the method of limits, all
the other general questions, the solution of which has been already
indicated according to the infinitesimal method.
Such is, in substance, the conception which Newton formed for the
transcendental analysis, or, more precisely, that which Maclaurin and
D'Alembert have presented as the most rational basis of that analysis,
in seeking to fix and to arrange the ideas of Newton upon that subject.
_Fluxions and Fluents._ Another distinct form under which Newton has
presented this same method should be here noticed, and deserves
particularly to fix our attention, as much by its ingenious clearness in
some cases as by its having furnished the notation best suited to this
manner of viewing the transcendental analysis, and, moreover, as having
been till lately the special form of the calculus of indirect functions
commonly adopted by the English geometers. I refer to the calculus of
_fluxions_ and of _fluents_, founded on the general idea of
_velocities_.
To facilitate the conception of the fundamental idea, let us consider
every curve as generated by a point impressed with a motion varying
according to any law whatever. The different quantities which the curve
can present, the abscissa, the ordinate, the arc, the area, &c., will be
regarded as simultaneously produced by successive degrees during this
motion. The _velocity_ with which each shall have been described will be
called the _fluxion_ of that quantity, which will be inversely named its
_fluent_. Henceforth the transcendental analysis will consist, according
to this conception, in forming directly the equations between the
fluxions of the proposed quantities, in order to deduce therefrom, by a
special calculus, the equations between the fluents themselves. What
has been stated respecting curves may, moreover, evidently be applied to
any magnitudes whatever, regarded, by the aid of suitable images, as
produced by motion.
It is easy to understand the general and necessary identity of this
method with that of limits complicated with the foreign idea of motion.
In fact, resuming the case of the curve, if we suppose, as we evidently
always may, that the motion of the describing point is uniform in a
certain direction, that of the abscissa, for example, then the fluxion
of the abscissa will be constant, like the element of the time; for all
the other quantities generated, the motion cannot be conceived to be
uniform, except for an infinitely small time. Now the velocity being in
general according to its mechanical conception, the ratio of each space
to the time employed in traversing it, and this time being here
proportional to the increment of the abscissa, it follows that the
fluxions of the ordinate, of the arc, of the area, &c., are really
nothing else (rejecting the intermediate consideration of time) than the
final ratios of the increments of these different quantities to the
increment of the abscissa. This method of fluxions and fluents is, then,
in reality, only a manner of representing, by a comparison borrowed from
mechanics, the method of prime and ultimate ratios, which alone can be
reduced to a calculus. It evidently, then, offers the same general
advantages in the various principal applications of the transcendental
analysis, without its being necessary to present special proofs of
this.
METHOD OF LAGRANGE.
_Derived Functions._ The conception of Lagrange, in its admirable
simplicity, consists in representing the transcendental analysis as a
great algebraic artifice, by which, in order to facilitate the
establishment of equations, we introduce, in the place of the primitive
functions, or concurrently with them, their _derived_ functions; that
is, according to the definition of Lagrange, the coefficient of the
first term of the increment of each function, arranged according to the
ascending powers of the increment of its variable. The special calculus
of indirect functions has for its constant object, here as well as in
the conceptions of Leibnitz and of Newton, to eliminate these
_derivatives_ which have been thus employed as auxiliaries, in order to
deduce from their relations the corresponding equations between the
primitive magnitudes.
_An Extension of ordinary Analysis._ The transcendental analysis is,
then, nothing but a simple though very considerable extension of
ordinary analysis. Geometers have long been accustomed to introduce in
analytical investigations, in the place of the magnitudes themselves
which they wished to study, their different powers, or their logarithms,
or their sines, &c., in order to simplify the equations, and even to
obtain them more easily. This successive _derivation_ is an artifice of
the same nature, only of greater extent, and procuring, in consequence,
much more important resources for this common object.
But, although we can readily conceive, _à priori_, that the auxiliary
consideration of these derivatives _may_ facilitate the establishment
of equations, it is not easy to explain why this _must_ necessarily
follow from this mode of derivation rather than from any other
transformation. Such is the weak point of the great idea of Lagrange.
The precise advantages of this analysis cannot as yet be grasped in an
abstract manner, but only shown by considering separately each principal
question, so that the verification is often exceedingly laborious.
EXAMPLE. _Tangents._ This manner of conceiving the transcendental
analysis may be best illustrated by its application to the most simple
of the problems above examined--that of tangents.
Instead of conceiving the tangent as the prolongation of the infinitely
small element of the curve, according to the notion of Leibnitz--or as
the limit of the secants, according to the ideas of Newton--Lagrange
considers it, according to its simple geometrical character, analogous
to the definitions of the ancients, to be a right line such that no
other right line can pass through the point of contact between it and
the curve. Then, to determine its direction, we must seek the general
expression of its distance from the curve, measured in any direction
whatever--in that of the ordinate, for example--and dispose of the
arbitrary constant relating to the inclination of the right line, which
will necessarily enter into that expression, in such a way as to
diminish that separation as much as possible. Now this distance, being
evidently equal to the difference of the two ordinates of the curve and
of the right line, which correspond to the same new abscissa _x_ + _h_,
will be represented by the formula
(_f'_(_x_) - _t_)_h_ + _qh²_ + _rh³_ + etc.,
in which _t_ designates, as above, the unknown trigonometrical tangent
of the angle which the required line makes with the axis of abscissas,
and _f'_(_x_) the derived function of the ordinate _f_(_x_). This being
understood, it is easy to see that, by disposing of _t_ so as to make
the first term of the preceding formula equal to zero, we will render
the interval between the two lines the least possible, so that any other
line for which _t_ did not have the value thus determined would
necessarily depart farther from the proposed curve. We have, then, for
the direction of the tangent sought, the general expression _t_ =
_f'_(_x_), a result exactly equivalent to those furnished by the
Infinitesimal Method and the Method of Limits. We have yet to find
_f'_(_x_) in each particular curve, which is a mere question of
analysis, quite identical with those which are presented, at this stage
of the operations, by the other methods.
After these considerations upon the principal general conceptions, we
need not stop to examine some other theories proposed, such as Euler's
_Calculus of Vanishing Quantities_, which are really modifications--more
or less important, and, moreover, no longer used--of the preceding
methods.
I have now to establish the comparison and the appreciation of these
three fundamental methods. Their _perfect and necessary conformity_ is
first to be proven in a general manner.
FUNDAMENTAL IDENTITY OF THE THREE METHODS.
It is, in the first place, evident from what precedes, considering these
three methods as to their actual destination, independently of their
preliminary ideas, that they all consist in the same general logical
artifice, which has been characterized in the first chapter; to wit,
the introduction of a certain system of auxiliary magnitudes, having
uniform relations to those which are the special objects of the inquiry,
and substituted for them expressly to facilitate the analytical
expression of the mathematical laws of the phenomena, although they have
finally to be eliminated by the aid of a special calculus. It is this
which has determined me to regularly define the transcendental analysis
as _the calculus of indirect functions_, in order to mark its true
philosophical character, at the same time avoiding any discussion upon
the best manner of conceiving and applying it. The general effect of
this analysis, whatever the method employed, is, then, to bring every
mathematical question much more promptly within the power of the
_calculus_, and thus to diminish considerably the serious difficulty
which is usually presented by the passage from the concrete to the
abstract. Whatever progress we may make, we can never hope that the
calculus will ever be able to grasp every question of natural
philosophy, geometrical, or mechanical, or thermological, &c.,
immediately upon its birth, which would evidently involve a
contradiction. Every problem will constantly require a certain
preliminary labour to be performed, in which the calculus can be of no
assistance, and which, by its nature, cannot be subjected to abstract
and invariable rules; it is that which has for its special object the
establishment of equations, which form the indispensable starting point
of all analytical researches. But this preliminary labour has been
remarkably simplified by the creation of the transcendental analysis,
which has thus hastened the moment at which the solution admits of the
uniform and precise application of general and abstract methods; by
reducing, in each case, this special labour to the investigation of
equations between the auxiliary magnitudes; from which the calculus then
leads to equations directly referring to the proposed magnitudes, which,
before this admirable conception, it had been necessary to establish
directly and separately. Whether these indirect equations are
_differential_ equations, according to the idea of Leibnitz, or
equations of _limits_, conformably to the conception of Newton, or,
lastly, _derived_ equations, according to the theory of Lagrange, the
general procedure is evidently always the same.
But the coincidence of these three principal methods is not limited to
the common effect which they produce; it exists, besides, in the very
manner of obtaining it. In fact, not only do all three consider, in the
place of the primitive magnitudes, certain auxiliary ones, but, still
farther, the quantities thus introduced as subsidiary are exactly
identical in the three methods, which consequently differ only in the
manner of viewing them. This can be easily shown by taking for the
general term of comparison any one of the three conceptions, especially
that of Lagrange, which is the most suitable to serve as a type, as
being the freest from foreign considerations. Is it not evident, by the
very definition of _derived functions_, that they are nothing else than
what Leibnitz calls _differential coefficients_, or the ratios of the
differential of each function to that of the corresponding variable,
since, in determining the first differential, we will be obliged, by the
very nature of the infinitesimal method, to limit ourselves to taking
the only term of the increment of the function which contains the first
power of the infinitely small increment of the variable? In the same
way, is not the derived function, by its nature, likewise the necessary
_limit_ towards which tends the ratio between the increment of the
primitive function and that of its variable, in proportion as this last
indefinitely diminishes, since it evidently expresses what that ratio
becomes when we suppose the increment of the variable to equal zero?
That which is designated by _dx_/_dy_ in the method of Leibnitz; that
which ought to be noted as _L_(Δ_y_/Δ_x_) in that of Newton; and that
which Lagrange has indicated by _f'_(_x_), is constantly one same
function, seen from three different points of view, the considerations
of Leibnitz and Newton properly consisting in making known two general
necessary properties of the derived function. The transcendental
analysis, examined abstractedly and in its principle, is then always the
same, whatever may be the conception which is adopted, and the
procedures of the calculus of indirect functions are necessarily
identical in these different methods, which in like manner must, for any
application whatever, lead constantly to rigorously uniform results.
COMPARATIVE VALUE OF THE THREE METHODS.
If now we endeavour to estimate the comparative value of these three
equivalent conceptions, we shall find in each advantages and
inconveniences which are peculiar to it, and which still prevent
geometers from confining themselves to any one of them, considered as
final.
_That of Leibnitz._ The conception of Leibnitz presents incontestably,
in all its applications, a very marked superiority, by leading in a much
more rapid manner, and with much less mental effort, to the formation
of equations between the auxiliary magnitudes. It is to its use that we
owe the high perfection which has been acquired by all the general
theories of geometry and mechanics. Whatever may be the different
speculative opinions of geometers with respect to the infinitesimal
method, in an abstract point of view, all tacitly agree in employing it
by preference, as soon as they have to treat a new question, in order
not to complicate the necessary difficulty by this purely artificial
obstacle proceeding from a misplaced obstinacy in adopting a less
expeditious course. Lagrange himself, after having reconstructed the
transcendental analysis on new foundations, has (with that noble
frankness which so well suited his genius) rendered a striking and
decisive homage to the characteristic properties of the conception of
Leibnitz, by following it exclusively in the entire system of his
_Méchanique Analytique_. Such a fact renders any comments unnecessary.
But when we consider the conception of Leibnitz in itself and in its
logical relations, we cannot escape admitting, with Lagrange, that it is
radically vicious in this, that, adopting its own expressions, the
notion of infinitely small quantities is a _false idea_, of which it is
in fact impossible to obtain a clear conception, however we may deceive
ourselves in that matter. Even if we adopt the ingenious idea of the
compensation of errors, as above explained, this involves the radical
inconvenience of being obliged to distinguish in mathematics two classes
of reasonings, those which are perfectly rigorous, and those in which we
designedly commit errors which subsequently have to be compensated. A
conception which leads to such strange consequences is undoubtedly very
unsatisfactory in a logical point of view.
To say, as do some geometers, that it is possible in every case to
reduce the infinitesimal method to that of limits, the logical character
of which is irreproachable, would evidently be to elude the difficulty
rather than to remove it; besides, such a transformation almost entirely
strips the conception of Leibnitz of its essential advantages of
facility and rapidity.
Finally, even disregarding the preceding important considerations, the
infinitesimal method would no less evidently present by its nature the
very serious defect of breaking the unity of abstract mathematics, by
creating a transcendental analysis founded on principles so different
from those which form the basis of the ordinary analysis. This division
of analysis into two worlds almost entirely independent of each other,
tends to hinder the formation of truly general analytical conceptions.
To fully appreciate the consequences of this, we should have to go back
to the state of the science before Lagrange had established a general
and complete harmony between these two great sections.
_That of Newton._ Passing now to the conception of Newton, it is evident
that by its nature it is not exposed to the fundamental logical
objections which are called forth by the method of Leibnitz. The notion
of _limits_ is, in fact, remarkable for its simplicity and its
precision. In the transcendental analysis presented in this manner, the
equations are regarded as exact from their very origin, and the general
rules of reasoning are as constantly observed as in ordinary analysis.
But, on the other hand, it is very far from offering such powerful
resources for the solution of problems as the infinitesimal method. The
obligation which it imposes, of never considering the increments of
magnitudes separately and by themselves, nor even in their ratios, but
only in the limits of those ratios, retards considerably the operations
of the mind in the formation of auxiliary equations. We may even say
that it greatly embarrasses the purely analytical transformations. Thus
the transcendental analysis, considered separately from its
applications, is far from presenting in this method the extent and the
generality which have been imprinted upon it by the conception of
Leibnitz. It is very difficult, for example, to extend the theory of
Newton to functions of several independent variables. But it is
especially with reference to its applications that the relative
inferiority of this theory is most strongly marked.
Several Continental geometers, in adopting the method of Newton as the
more logical basis of the transcendental analysis, have partially
disguised this inferiority by a serious inconsistency, which consists in
applying to this method the notation invented by Leibnitz for the
infinitesimal method, and which is really appropriate to it alone. In
designating by _dy_/_dx_ that which logically ought, in the theory of
limits, to be denoted by _L_(Δ_y_/Δ_x_), and in extending to all the
other analytical conceptions this displacement of signs, they intended,
undoubtedly, to combine the special advantages of the two methods; but,
in reality, they have only succeeded in causing a vicious confusion
between them, a familiarity with which hinders the formation of clear
and exact ideas of either. It would certainly be singular, considering
this usage in itself, that, by the mere means of signs, it could be
possible to effect a veritable combination between two theories so
distinct as those under consideration.
Finally, the method of limits presents also, though in a less degree,
the greater inconvenience, which I have above noted in reference to the
infinitesimal method, of establishing a total separation between the
ordinary and the transcendental analysis; for the idea of _limits_,
though clear and rigorous, is none the less in itself, as Lagrange has
remarked, a foreign idea, upon which analytical theories ought not to be
dependent.
_That of Lagrange._ This perfect unity of analysis, and this purely
abstract character of its fundamental notions, are found in the highest
degree in the conception of Lagrange, and are found there alone; it is,
for this reason, the most rational and the most philosophical of all.
Carefully removing every heterogeneous consideration, Lagrange has
reduced the transcendental analysis to its true peculiar character, that
of presenting a very extensive class of analytical transformations,
which facilitate in a remarkable degree the expression of the conditions
of various problems. At the same time, this analysis is thus necessarily
presented as a simple extension of ordinary analysis; it is only a
higher algebra. All the different parts of abstract mathematics,
previously so incoherent, have from that moment admitted of being
conceived as forming a single system.
Unhappily, this conception, which possesses such fundamental properties,
independently of its so simple and so lucid notation, and which is
undoubtedly destined to become the final theory of transcendental
analysis, because of its high philosophical superiority over all the
other methods proposed, presents in its present state too many
difficulties in its applications, as compared with the conception of
Newton, and still more with that of Leibnitz, to be as yet exclusively
adopted. Lagrange himself has succeeded only with great difficulty in
rediscovering, by his method, the principal results already obtained by
the infinitesimal method for the solution of the general questions of
geometry and mechanics; we may judge from that what obstacles would be
found in treating in the same manner questions which were truly new and
important. It is true that Lagrange, on several occasions, has shown
that difficulties call forth, from men of genius, superior efforts,
capable of leading to the greatest results. It was thus that, in trying
to adapt his method to the examination of the curvature of lines, which
seemed so far from admitting its application, he arrived at that
beautiful theory of contacts which has so greatly perfected that
important part of geometry. But, in spite of such happy exceptions, the
conception of Lagrange has nevertheless remained, as a whole,
essentially unsuited to applications.
The final result of the general comparison which I have too briefly
sketched, is, then, as already suggested, that, in order to really
understand the transcendental analysis, we should not only consider it
in its principles according to the three fundamental conceptions of
Leibnitz, of Newton, and of Lagrange, but should besides accustom
ourselves to carry out almost indifferently, according to these three
principal methods, and especially according to the first and the last,
the solution of all important questions, whether of the pure calculus of
indirect functions or of its applications. This is a course which I
could not too strongly recommend to all those who desire to judge
philosophically of this admirable creation of the human mind, as well as
to those who wish to learn to make use of this powerful instrument with
success and with facility. In all the other parts of mathematical
science, the consideration of different methods for a single class of
questions may be useful, even independently of its historical interest,
but it is not indispensable; here, on the contrary, it is strictly
necessary.
Having determined with precision, in this chapter, the philosophical
character of the calculus of indirect functions, according to the
principal fundamental conceptions of which it admits, we have next to
consider, in the following chapter, the logical division and the general
composition of this calculus.