THE DIFFERENTIAL AND INTEGRAL CALCULUS.
ITS TWO FUNDAMENTAL DIVISIONS.
The _calculus of indirect functions_, in accordance with the
considerations explained in the preceding chapter, is necessarily
divided into two parts (or, more properly, is decomposed into two
different _calculi_ entirely distinct, although intimately connected by
their nature), according as it is proposed to find the relations between
the auxiliary magnitudes (the introduction of which constitutes the
general spirit of this calculus) by means of the relations between the
corresponding primitive magnitudes; or, conversely, to try to discover
these direct equations by means of the indirect equations originally
established. Such is, in fact, constantly the double object of the
transcendental analysis.
These two systems have received different names, according to the point
of view under which this analysis has been regarded. The infinitesimal
method, properly so called, having been the most generally employed for
the reasons which have been given, almost all geometers employ
habitually the denominations of _Differential Calculus_ and of _Integral
Calculus_, established by Leibnitz, and which are, in fact, very
rational consequences of his conception. Newton, in accordance with his
method, named the first the _Calculus of Fluxions_, and the second the
_Calculus of Fluents_, expressions which were commonly employed in
England. Finally, following the eminently philosophical theory founded
by Lagrange, one would be called the _Calculus of Derived Functions_,
and the other the _Calculus of Primitive Functions_. I will continue to
make use of the terms of Leibnitz, as being more convenient for the
formation of secondary expressions, although I ought, in accordance with
the suggestions made in the preceding chapter, to employ concurrently
all the different conceptions, approaching as nearly as possible to that
of Lagrange.
THEIR RELATIONS TO EACH OTHER.
The differential calculus is evidently the logical basis of the integral
calculus; for we do not and cannot know how to integrate directly any
other differential expressions than those produced by the
differentiation of the ten simple functions which constitute the general
elements of our analysis. The art of integration consists, then,
essentially in bringing all the other cases, as far as is possible, to
finally depend on only this small number of fundamental integrations.
In considering the whole body of the transcendental analysis, as I have
characterized it in the preceding chapter, it is not at first apparent
what can be the peculiar utility of the differential calculus,
independently of this necessary relation with the integral calculus,
which seems as if it must be, by itself, the only one directly
indispensable. In fact, the elimination of the _infinitesimals_ or of
the _derivatives_, introduced as auxiliaries to facilitate the
establishment of equations, constituting, as we have seen, the final and
invariable object of the calculus of indirect functions, it is natural
to think that the calculus which teaches how to deduce from the
equations between these auxiliary magnitudes, those which exist between
the primitive magnitudes themselves, ought strictly to suffice for the
general wants of the transcendental analysis without our perceiving, at
the first glance, what special and constant part the solution of the
inverse question can have in such an analysis. It would be a real error,
though a common one, to assign to the differential calculus, in order to
explain its peculiar, direct, and necessary influence, the destination
of forming the differential equations, from which the integral calculus
then enables us to arrive at the finite equations; for the primitive
formation of differential equations is not and cannot be, properly
speaking, the object of any calculus, since, on the contrary, it forms
by its nature the indispensable starting point of any calculus whatever.
How, in particular, could the differential calculus, which in itself is
reduced to teaching the means of _differentiating_ the different
equations, be a general procedure for establishing them? That which in
every application of the transcendental analysis really facilitates the
formation of equations, is the infinitesimal _method_, and not the
infinitesimal _calculus_, which is perfectly distinct from it, although
it is its indispensable complement. Such a consideration would, then,
give a false idea of the special destination which characterizes the
differential calculus in the general system of the transcendental
analysis.
But we should nevertheless very imperfectly conceive the real peculiar
importance of this first branch of the calculus of indirect functions,
if we saw in it only a simple preliminary labour, having no other
general and essential object than to prepare indispensable foundations
for the integral calculus. As the ideas on this matter are generally
confused, I think that I ought here to explain in a summary manner this
important relation as I view it, and to show that in every application
of the transcendental analysis a primary, direct, and necessary part is
constantly assigned to the differential calculus.
1. _Use of the Differential Calculus as preparatory to that of the
Integral._ In forming the differential equations of any phenomenon
whatever, it is very seldom that we limit ourselves to introduce
differentially only those magnitudes whose relations are sought. To
impose that condition would be to uselessly diminish the resources
presented by the transcendental analysis for the expression of the
mathematical laws of phenomena. Most frequently we introduce into the
primitive equations, through their differentials, other magnitudes whose
relations are already known or supposed to be so, and without the
consideration of which it would be frequently impossible to establish
equations. Thus, for example, in the general problem of the
rectification of curves, the differential equation,
_ds_² = _dy_² + _dx_², or _ds_² = _dx_² + _dy_² + _dz_²,
is not only established between the desired function s and the
independent variable _x_, to which it is referred, but, at the same
time, there have been introduced, as indispensable intermediaries, the
differentials of one or two other functions, _y_ and _z_, which are
among the data of the problem; it would not have been possible to form
directly the equation between _ds_ and _dx_, which would, besides, be
peculiar to each curve considered. It is the same for most questions.
Now in these cases it is evident that the differential equation is not
immediately suitable for integration. It is previously necessary that
the differentials of the functions supposed to be known, which have
been employed as intermediaries, should be entirely eliminated, in order
that equations may be obtained between the differentials of the
functions which alone are sought and those of the really independent
variables, after which the question depends on only the integral
calculus. Now this preparatory elimination of certain differentials, in
order to reduce the infinitesimals to the smallest number possible,
belongs simply to the differential calculus; for it must evidently be
done by determining, by means of the equations between the functions
supposed to be known, taken as intermediaries, the relations of their
differentials, which is merely a question of differentiation. Thus, for
example, in the case of rectifications, it will be first necessary to
calculate _dy_, or _dy_ and _dz_, by differentiating the equation or the
equations of each curve proposed; after eliminating these expressions,
the general differential formula above enunciated will then contain only
_ds_ and _dx_; having arrived at this point, the elimination of the
infinitesimals can be completed only by the integral calculus.
Such is, then, the general office necessarily belonging to the
differential calculus in the complete solution of the questions which
exact the employment of the transcendental analysis; to produce, as far
as is possible, the elimination of the infinitesimals, that is, to
reduce in each case the primitive differential equations so that they
shall contain only the differentials of the really independent
variables, and those of the functions sought, by causing to disappear,
by elimination, the differentials of all the other known functions which
may have been taken as intermediaries at the time of the formation of
the differential equations of the problem which is under consideration.
2. _Employment of the Differential Calculus alone._ For certain
questions, which, although few in number, have none the less, as we
shall see hereafter, a very great importance, the magnitudes which are
sought enter directly, and not by their differentials, into the
primitive differential equations, which then contain differentially only
the different known functions employed as intermediaries, in accordance
with the preceding explanation. These cases are the most favourable of
all; for it is evident that the differential calculus is then entirely
sufficient for the complete elimination of the infinitesimals, without
the question giving rise to any integration. This is what occurs, for
example, in the problem of _tangents_ in geometry; in that of
_velocities_ in mechanics, &c.
3. _Employment of the Integral Calculus alone._ Finally, some other
questions, the number of which is also very small, but the importance of
which is no less great, present a second exceptional case, which is in
its nature exactly the converse of the preceding. They are those in
which the differential equations are found to be immediately ready for
integration, because they contain, at their first formation, only the
infinitesimals which relate to the functions sought, or to the really
independent variables, without its being necessary to introduce,
differentially, other functions as intermediaries. If in these new cases
we introduce these last functions, since, by hypothesis, they will enter
directly and not by their differentials, ordinary algebra will suffice
to eliminate them, and to bring the question to depend on only the
integral calculus. The differential calculus will then have no special
part in the complete solution of the problem, which will depend entirely
upon the integral calculus. The general question of _quadratures_ offers
an important example of this, for the differential equation being then
_dA = ydx_, will become immediately fit for integration as soon as we
shall have eliminated, by means of the equation of the proposed curve,
the intermediary function _y_, which does not enter into it
differentially. The same circumstances exist in the problem of
_cubatures_, and in some others equally important.
_Three classes of Questions hence resulting._ As a general result of the
previous considerations, it is then necessary to divide into three
classes the mathematical questions which require the use of the
transcendental analysis; the _first_ class comprises the problems
susceptible of being entirely resolved by means of the differential
calculus alone, without any need of the integral calculus; the _second_,
those which are, on the contrary, entirely dependent upon the integral
calculus, without the differential calculus having any part in their
solution; lastly, in the _third_ and the most extensive, which
constitutes the normal case, the two others being only exceptional, the
differential and the integral calculus have each in their turn a
distinct and necessary part in the complete solution of the problem, the
former making the primitive differential equations undergo a preparation
which is indispensable for the application of the latter. Such are
exactly their general relations, of which too indefinite and inexact
ideas are generally formed.
* * * * *
Let us now take a general survey of the logical composition of each
calculus, beginning with the differential.
THE DIFFERENTIAL CALCULUS.
In the exposition of the transcendental analysis, it is customary to
intermingle with the purely analytical part (which reduces itself to the
treatment of the abstract principles of differentiation and integration)
the study of its different principal applications, especially those
which concern geometry. This confusion of ideas, which is a consequence
of the actual manner in which the science has been developed, presents,
in the dogmatic point of view, serious inconveniences in this respect,
that it makes it difficult properly to conceive either analysis or
geometry. Having to consider here the most rational co-ordination which
is possible, I shall include, in the following sketch, only the calculus
of indirect functions properly so called, reserving for the portion of
this volume which relates to the philosophical study of _concrete_
mathematics the general examination of its great geometrical and
mechanical applications.
_Two Cases: explicit and implicit Functions._ The fundamental division
of the differential calculus, or of the general subject of
differentiation, consists in distinguishing two cases, according as the
analytical functions which are to be differentiated are _explicit_ or
_implicit_; from which flow two parts ordinarily designated by the names
of differentiation _of formulas_ and differentiation _of equations_. It
is easy to understand, _à priori_, the importance of this
classification. In fact, such a distinction would be illusory if the
ordinary analysis was perfect; that is, if we knew how to resolve all
equations algebraically, for then it would be possible to render every
_implicit_ function _explicit_; and, by differentiating it in that
state alone, the second part of the differential calculus would be
immediately comprised in the first, without giving rise to any new
difficulty. But the algebraical resolution of equations being, as we
have seen, still almost in its infancy, and as yet impossible for most
cases, it is plain that the case is very different, since we have,
properly speaking, to differentiate a function without knowing it,
although it is determinate. The differentiation of implicit functions
constitutes then, by its nature, a question truly distinct from that
presented by explicit functions, and necessarily more complicated. It is
thus evident that we must commence with the differentiation of formulas,
and reduce the differentiation of equations to this primary case by
certain invariable analytical considerations, which need not be here
mentioned.
These two general cases of differentiation are also distinct in another
point of view equally necessary, and too important to be left unnoticed.
The relation which is obtained between the differentials is constantly
more indirect, in comparison with that of the finite quantities, in the
differentiation of implicit functions than in that of explicit
functions. We know, in fact, from the considerations presented by
Lagrange on the general formation of differential equations, that, on
the one hand, the same primitive equation may give rise to a greater or
less number of derived equations of very different forms, although at
bottom equivalent, depending upon which of the arbitrary constants is
eliminated, which is not the case in the differentiation of explicit
formulas; and that, on the other hand, the unlimited system of the
different primitive equations, which correspond to the same derived
equation, presents a much more profound analytical variety than that of
the different functions, which admit of one same explicit differential,
and which are distinguished from each other only by a constant term.
Implicit functions must therefore be regarded as being in reality still
more modified by differentiation than explicit functions. We shall again
meet with this consideration relatively to the integral calculus, where
it acquires a preponderant importance.
_Two Sub-cases: A single Variable or several Variables._ Each of the two
fundamental parts of the Differential Calculus is subdivided into two
very distinct theories, according as we are required to differentiate
functions of a single variable or functions of several independent
variables. This second case is, by its nature, quite distinct from the
first, and evidently presents more complication, even in considering
only explicit functions, and still more those which are implicit. As to
the rest, one of these cases is deduced from the other in a general
manner, by the aid of an invariable and very simple principle, which
consists in regarding the total differential of a function which is
produced by the simultaneous increments of the different independent
variables which it contains, as the sum of the partial differentials
which would be produced by the separate increment of each variable in
turn, if all the others were constant. It is necessary, besides,
carefully to remark, in connection with this subject, a new idea which
is introduced by the distinction of functions into those of one variable
and of several; it is the consideration of these different special
derived functions, relating to each variable separately, and the number
of which increases more and more in proportion as the order of the
derivation becomes higher, and also when the variables become more
numerous. It results from this that the differential relations belonging
to functions of several variables are, by their nature, both much more
indirect, and especially much more indeterminate, than those relating to
functions of a single variable. This is most apparent in the case of
implicit functions, in which, in the place of the simple arbitrary
constants which elimination causes to disappear when we form the proper
differential equations for functions of a single variable, it is the
arbitrary functions of the proposed variables which are then eliminated;
whence must result special difficulties when these equations come to be
integrated.
Finally, to complete this summary sketch of the different essential
parts of the differential calculus proper, I should add, that in the
differentiation of implicit functions, whether of a single variable or
of several, it is necessary to make another distinction; that of the
case in which it is required to differentiate at once different
functions of this kind, _combined_ in certain primitive equations, from
that in which all these functions are _separate_.
The functions are evidently, in fact, still more implicit in the first
case than in the second, if we consider that the same imperfection of
ordinary analysis, which forbids our converting every implicit function
into an equivalent explicit function, in like manner renders us unable
to separate the functions which enter simultaneously into any system of
equations. It is then necessary to differentiate, not only without
knowing how to resolve the primitive equations, but even without being
able to effect the proper eliminations among them, thus producing a new
difficulty.
_Reduction of the whole to the Differentiation of the ten elementary
Functions._ Such, then, are the natural connection and the logical
distribution of the different principal theories which compose the
general system of differentiation. Since the differentiation of implicit
functions is deduced from that of explicit functions by a single
constant principle, and the differentiation of functions of several
variables is reduced by another fixed principle to that of functions of
a single variable, the whole of the differential calculus is finally
found to rest upon the differentiation of explicit functions with a
single variable, the only one which is ever executed directly. Now it is
easy to understand that this first theory, the necessary basis of the
entire system, consists simply in the differentiation of the ten simple
functions, which are the uniform elements of all our analytical
combinations, and the list of which has been given in the first chapter,
on page 51; for the differentiation of compound functions is evidently
deduced, in an immediate and necessary manner, from that of the simple
functions which compose them. It is, then, to the knowledge of these ten
fundamental differentials, and to that of the two general principles
just mentioned, which bring under it all the other possible cases, that
the whole system of differentiation is properly reduced. We see, by the
combination of these different considerations, how simple and how
perfect is the entire system of the differential calculus. It certainly
constitutes, in its logical relations, the most interesting spectacle
which mathematical analysis can present to our understanding.
_Transformation of derived Functions for new Variables._ The general
sketch which I have just summarily drawn would nevertheless present an
important deficiency, if I did not here distinctly indicate a final
theory, which forms, by its nature, the indispensable complement of the
system of differentiation. It is that which has for its object the
constant transformation of derived functions, as a result of determinate
changes in the independent variables, whence results the possibility of
referring to new variables all the general differential formulas
primitively established for others. This question is now resolved in the
most complete and the most simple manner, as are all those of which the
differential calculus is composed. It is easy to conceive the general
importance which it must have in any of the applications of the
transcendental analysis, the fundamental resources of which it may be
considered as augmenting, by permitting us to choose (in order to form
the differential equations, in the first place, with more ease) that
system of independent variables which may appear to be the most
advantageous, although it is not to be finally retained. It is thus, for
example, that most of the principal questions of geometry are resolved
much more easily by referring the lines and surfaces to _rectilinear_
co-ordinates, and that we may, nevertheless, have occasion to express
these lines, etc., analytically by the aid of _polar_ co-ordinates, or
in any other manner. We will then be able to commence the differential
solution of the problem by employing the rectilinear system, but only as
an intermediate step, from which, by the general theory here referred
to, we can pass to the final system, which sometimes could not have been
considered directly.
_Different Orders of Differentiation._ In the logical classification of
the differential calculus which has just been given, some may be
inclined to suggest a serious omission, since I have not subdivided each
of its four essential parts according to another general consideration,
which seems at first view very important; namely, that of the higher or
lower order of differentiation. But it is easy to understand that this
distinction has no real influence in the differential calculus, inasmuch
as it does not give rise to any new difficulty. If, indeed, the
differential calculus was not rigorously complete, that is, if we did
not know how to differentiate at will any function whatever, the
differentiation to the second or higher order of each determinate
function might engender special difficulties. But the perfect
universality of the differential calculus plainly gives us the assurance
of being able to differentiate, to any order whatever, all known
functions whatever, the question reducing itself to a constantly
repeated differentiation of the first order. This distinction,
unimportant as it is for the differential calculus, acquires, however, a
very great importance in the integral calculus, on account of the
extreme imperfection of the latter.
_Analytical Applications._ Finally, though this is not the place to
consider the various applications of the differential calculus, yet an
exception may be made for those which consist in the solution of
questions which are purely analytical, which ought, indeed, to be
logically treated in continuation of a system of differentiation,
because of the evident homogeneity of the considerations involved. These
questions may be reduced to three essential ones.
Firstly, the _development into series_ of functions of one or more
variables, or, more generally, the transformation of functions, which
constitutes the most beautiful and the most important application of the
differential calculus to general analysis, and which comprises, besides
the fundamental series discovered by Taylor, the remarkable series
discovered by Maclaurin, John Bernouilli, Lagrange, &c.:
Secondly, the general _theory of maxima and minima_ values for any
functions whatever, of one or more variables; one of the most
interesting problems which analysis can present, however elementary it
may now have become, and to the complete solution of which the
differential calculus naturally applies:
Thirdly, the general determination of the true value of functions which
present themselves under an _indeterminate_ appearance for certain
hypotheses made on the values of the corresponding variables; which is
the least extensive and the least important of the three.
The first question is certainly the principal one in all points of view;
it is also the most susceptible of receiving a new extension hereafter,
especially by conceiving, in a broader manner than has yet been done,
the employment of the differential calculus in the transformation of
functions, on which subject Lagrange has left some valuable hints.
* * * * *
Having thus summarily, though perhaps too briefly, considered the chief
points in the differential calculus, I now proceed to an equally rapid
exposition of a systematic outline of the Integral Calculus, properly so
called, that is, the abstract subject of integration.
THE INTEGRAL CALCULUS.
_Its Fundamental Division._ The fundamental division of the Integral
Calculus is founded on the same principle as that of the Differential
Calculus, in distinguishing the integration of _explicit_ differential
formulas, and the integration of _implicit_ differentials or of
differential equations. The separation of these two cases is even much
more profound in relation to integration than to differentiation. In the
differential calculus, in fact, this distinction rests, as we have seen,
only on the extreme imperfection of ordinary analysis. But, on the other
hand, it is easy to see that, even though all equations could be
algebraically resolved, differential equations would none the less
constitute a case of integration quite distinct from that presented by
the explicit differential formulas; for, limiting ourselves, for the
sake of simplicity, to the first order, and to a single function _y_ of
a single variable _x_, if we suppose any differential equation between
_x_, _y_, and _dy/dx_, to be resolved with reference to _dy/dx_, the
expression of the derived function being then generally found to contain
the primitive function itself, which is the object of the inquiry, the
question of integration will not have at all changed its nature, and the
solution will not really have made any other progress than that of
having brought the proposed differential equation to be of only the
first degree relatively to the derived function, which is in itself of
little importance. The differential would not then be determined in a
manner much less _implicit_ than before, as regards the integration,
which would continue to present essentially the same characteristic
difficulty. The algebraic resolution of equations could not make the
case which we are considering come within the simple integration of
explicit differentials, except in the special cases in which the
proposed differential equation did not contain the primitive function
itself, which would consequently permit us, by resolving it, to find
_dy/dx_ in terms of _x_ only, and thus to reduce the question to the
class of quadratures. Still greater difficulties would evidently be
found in differential equations of higher orders, or containing
simultaneously different functions of several independent variables.
The integration of differential equations is then necessarily more
complicated than that of explicit differentials, by the elaboration of
which last the integral calculus has been created, and upon which the
others have been made to depend as far as it has been possible. All the
various analytical methods which have been proposed for integrating
differential equations, whether it be the separation of the variables,
the method of multipliers, &c., have in fact for their object to reduce
these integrations to those of differential formulas, the only one
which, by its nature, can be undertaken directly. Unfortunately,
imperfect as is still this necessary base of the whole integral
calculus, the art of reducing to it the integration of differential
equations is still less advanced.
_Subdivisions: one variable or several._ Each of these two fundamental
branches of the integral calculus is next subdivided into two others (as
in the differential calculus, and for precisely analogous reasons),
according as we consider functions with a _single variable_, or
functions with _several independent variables_.
This distinction is, like the preceding one, still more important for
integration than for differentiation. This is especially remarkable in
reference to differential equations. Indeed, those which depend on
several independent variables may evidently present this characteristic
and much more serious difficulty, that the desired function may be
differentially defined by a simple relation between its different
special derivatives relative to the different variables taken
separately. Hence results the most difficult and also the most extensive
branch of the integral calculus, which is commonly named the _Integral
Calculus of partial differences_, created by D'Alembert, and in which,
according to the just appreciation of Lagrange, geometers ought to have
seen a really new calculus, the philosophical character of which has not
yet been determined with sufficient exactness. A very striking
difference between this case and that of equations with a single
independent variable consists, as has been already observed, in the
arbitrary functions which take the place of the simple arbitrary
constants, in order to give to the corresponding integrals all the
proper generality.
It is scarcely necessary to say that this higher branch of
transcendental analysis is still entirely in its infancy, since, even in
the most simple case, that of an equation of the first order between the
partial derivatives of a single function with two independent variables,
we are not yet completely able to reduce the integration to that of the
ordinary differential equations. The integration of functions of several
variables is much farther advanced in the case (infinitely more simple
indeed) in which it has to do with only explicit differential formulas.
We can then, in fact, when these formulas fulfil the necessary
conditions of integrability, always reduce their integration to
quadratures.
_Other Subdivisions: different Orders of Differentiation._ A new general
distinction, applicable as a subdivision to the integration of explicit
or implicit differentials, with one variable or several, is drawn from
the _higher or lower order of the differentials_: a distinction which,
as we have above remarked, does not give rise to any special question in
the differential calculus.
Relatively to _explicit differentials_, whether of one variable or of
several, the necessity of distinguishing their different orders belongs
only to the extreme imperfection of the integral calculus. In fact, if
we could always integrate every differential formula of the first order,
the integration of a formula of the second order, or of any other, would
evidently not form a new question, since, by integrating it at first in
the first degree, we would arrive at the differential expression of the
immediately preceding order, from which, by a suitable series of
analogous integrations, we would be certain of finally arriving at the
primitive function, the final object of these operations. But the little
knowledge which we possess on integration of even the first order causes
quite another state of affairs, so that a higher order of differentials
produces new difficulties; for, having differential formulas of any
order above the first, it may happen that we may be able to integrate
them, either once, or several times in succession, and that we may still
be unable to go back to the primitive functions, if these preliminary
labours have produced, for the differentials of a lower order,
expressions whose integrals are not known. This circumstance must occur
so much the oftener (the number of known integrals being still very
small), seeing that these successive integrals are generally very
different functions from the derivatives which have produced them.
With reference to _implicit differentials_, the distinction of orders is
still more important; for, besides the preceding reason, the influence
of which is evidently analogous in this case, and is even greater, it is
easy to perceive that the higher order of the differential equations
necessarily gives rise to questions of a new nature. In fact, even if we
could integrate every equation of the first order relating to a single
function, that would not be sufficient for obtaining the final integral
of an equation of any order whatever, inasmuch as every differential
equation is not reducible to that of an immediately inferior order.
Thus, for example, if we have given any relation between _x_, _y_,
_dx/dy_, and _d_²_y_/_dx_², to determine a function _y_ of a variable
_x_, we shall not be able to deduce from it at once, after effecting a
first integration, the corresponding differential relation between _x_,
_y_, and _dy/dx_, from which, by a second integration, we could ascend
to the primitive equations. This would not necessarily take place, at
least without introducing new auxiliary functions, unless the proposed
equation of the second order did not contain the required function _y_,
together with its derivatives. As a general principle, differential
equations will have to be regarded as presenting cases which are more
and more _implicit_, as they are of a higher order, and which cannot be
made to depend on one another except by special methods, the
investigation of which consequently forms a new class of questions, with
respect to which we as yet know scarcely any thing, even for functions
of a single variable.[10]
[Footnote 10: The only important case of this class which has thus
far been completely treated is the general integration of _linear_
equations of any order whatever, with constant coefficients. Even
this case finally depends on the algebraic resolution of equations
of a degree equal to the order of differentiation.]
_Another equivalent distinction._ Still farther, when we examine more
profoundly this distinction of different orders of differential
equations, we find that it can be always made to come under a final
general distinction, relative to differential equations, which remains
to be noticed. Differential equations with one or more independent
variables may contain simply a single function, or (in a case evidently
more complicated and more implicit, which corresponds to the
differentiation of simultaneous implicit functions) we may have to
determine at the same time several functions from the differential
equations in which they are found united, together with their different
derivatives. It is clear that such a state of the question necessarily
presents a new special difficulty, that of separating the different
functions desired, by forming for each, from the proposed differential
equations, an isolated differential equation which does not contain the
other functions or their derivatives. This preliminary labour, which is
analogous to the elimination of algebra, is evidently indispensable
before attempting any direct integration, since we cannot undertake
generally (except by special artifices which are very rarely applicable)
to determine directly several distinct functions at once.
Now it is easy to establish the exact and necessary coincidence of this
new distinction with the preceding one respecting the order of
differential equations. We know, in fact, that the general method for
isolating functions in simultaneous differential equations consists
essentially in forming differential equations, separately in relation to
each function, and of an order equal to the sum of all those of the
different proposed equations. This transformation can always be
effected. On the other hand, every differential equation of any order in
relation to a single function might evidently always be reduced to the
first order, by introducing a suitable number of auxiliary differential
equations, containing at the same time the different anterior
derivatives regarded as new functions to be determined. This method has,
indeed, sometimes been actually employed with success, though it is not
the natural one.
Here, then, are two necessarily equivalent orders of conditions in the
general theory of differential equations; the simultaneousness of a
greater or smaller number of functions, and the higher or lower order of
differentiation of a single function. By augmenting the order of the
differential equations, we can isolate all the functions; and, by
artificially multiplying the number of the functions, we can reduce all
the equations to the first order. There is, consequently, in both cases,
only one and the same difficulty from two different points of sight.
But, however we may conceive it, this new difficulty is none the less
real, and constitutes none the less, by its nature, a marked separation
between the integration of equations of the first order and that of
equations of a higher order. I prefer to indicate the distinction under
this last form as being more simple, more general, and more logical.
_Quadratures._ From the different considerations which have been
indicated respecting the logical dependence of the various principal
parts of the integral calculus, we see that the integration of explicit
differential formulas of the first order and of a single variable is the
necessary basis of all other integrations, which we never succeed in
effecting but so far as we reduce them to this elementary case,
evidently the only one which, by its nature, is capable of being treated
directly. This simple fundamental integration is often designated by the
convenient expression of _quadratures_, seeing that every integral of
this kind, S_f_(_x_)_dx_, may, in fact, be regarded as representing the
area of a curve, the equation of which in rectilinear co-ordinates would
be _y_ = _f_(_x_). Such a class of questions corresponds, in the
differential calculus, to the elementary case of the differentiation of
explicit functions of a single variable. But the integral question is,
by its nature, very differently complicated, and especially much more
extensive than the differential question. This latter is, in fact,
necessarily reduced, as we have seen, to the differentiation of the ten
simple functions, the elements of all which are considered in analysis.
On the other hand, the integration of compound functions does not
necessarily follow from that of the simple functions, each combination
of which may present special difficulties with respect to the integral
calculus. Hence results the naturally indefinite extent, and the so
varied complication of the question of _quadratures_, upon which, in
spite of all the efforts of analysts, we still possess so little
complete knowledge.
In decomposing this question, as is natural, according to the different
forms which may be assumed by the derivative function, we distinguish
the case of _algebraic_ functions and that of _transcendental_
functions.
_Integration of Transcendental Functions._ The truly analytical
integration of transcendental functions is as yet very little advanced,
whether for _exponential_, or for _logarithmic_, or for _circular_
functions. But a very small number of cases of these three different
kinds have as yet been treated, and those chosen from among the
simplest; and still the necessary calculations are in most cases
extremely laborious. A circumstance which we ought particularly to
remark in its philosophical connection is, that the different procedures
of quadrature have no relation to any general view of integration, and
consist of simple artifices very incoherent with each other, and very
numerous, because of the very limited extent of each.
One of these artifices should, however, here be noticed, which, without
being really a method of integration, is nevertheless remarkable for its
generality; it is the procedure invented by John Bernouilli, and known
under the name of _integration by parts_, by means of which every
integral may be reduced to another which is sometimes found to be more
easy to be obtained. This ingenious relation deserves to be noticed for
another reason, as having suggested the first idea of that
transformation of integrals yet unknown, which has lately received a
greater extension, and of which M. Fourier especially has made so new
and important a use in the analytical questions produced by the theory
of heat.
_Integration of Algebraic Functions._ As to the integration of algebraic
functions, it is farther advanced. However, we know scarcely any thing
in relation to irrational functions, the integrals of which have been
obtained only in extremely limited cases, and particularly by rendering
them rational. The integration of rational functions is thus far the
only theory of the integral calculus which has admitted of being treated
in a truly complete manner; in a logical point of view, it forms, then,
its most satisfactory part, but perhaps also the least important. It is
even essential to remark, in order to have a just idea of the extreme
imperfection of the integral calculus, that this case, limited as it is,
is not entirely resolved except for what properly concerns integration
viewed in an abstract manner; for, in the execution, the theory finds
its progress most frequently quite stopped, independently of the
complication of the calculations, by the imperfection of ordinary
analysis, seeing that it makes the integration finally depend upon the
algebraic resolution of equations, which greatly limits its use.
To grasp in a general manner the spirit of the different procedures
which are employed in quadratures, we must observe that, by their
nature, they can be primitively founded only on the differentiation of
the ten simple functions. The results of this, conversely considered,
establish as many direct theorems of the integral calculus, the only
ones which can be directly known. All the art of integration afterwards
consists, as has been said in the beginning of this chapter, in reducing
all the other quadratures, so far as is possible, to this small number
of elementary ones, which unhappily we are in most cases unable to
effect.
_Singular Solutions._ In this systematic enumeration of the various
essential parts of the integral calculus, considered in their logical
relations, I have designedly neglected (in order not to break the chain
of sequence) to consider a very important theory, which forms implicitly
a portion of the general theory of the integration of differential
equations, but which I ought here to notice separately, as being, so to
speak, outside of the integral calculus, and being nevertheless of the
greatest interest, both by its logical perfection and by the extent of
its applications. I refer to what are called _Singular Solutions_ of
differential equations, called sometimes, but improperly, _particular_
solutions, which have been the subject of very remarkable investigations
by Euler and Laplace, and of which Lagrange especially has presented
such a beautiful and simple general theory. Clairaut, who first had
occasion to remark their existence, saw in them a paradox of the
integral calculus, since these solutions have the peculiarity of
satisfying the differential equations without being comprised in the
corresponding general integrals. Lagrange has since explained this
paradox in the most ingenious and most satisfactory manner, by showing
how such solutions are always derived from the general integral by the
variation of the arbitrary constants. He was also the first to suitably
appreciate the importance of this theory, and it is with good reason
that he devoted to it so full a development in his "Calculus of
Functions." In a logical point of view, this theory deserves all our
attention by the character of perfect generality which it admits of,
since Lagrange has given invariable and very simple procedures for
finding the _singular_ solution of any differential equation which is
susceptible of it; and, what is no less remarkable, these procedures
require no integration, consisting only of differentiations, and are
therefore always applicable. Differentiation has thus become, by a
happy artifice, a means of compensating, in certain circumstances, for
the imperfection of the integral calculus. Indeed, certain problems
especially require, by their nature, the knowledge of these _singular_
solutions; such, for example, in geometry, are all the questions in
which a curve is to be determined from any property of its tangent or
its osculating circle. In all cases of this kind, after having expressed
this property by a differential equation, it will be, in its analytical
relations, the _singular_ equation which will form the most important
object of the inquiry, since it alone will represent the required curve;
the general integral, which thenceforth it becomes unnecessary to know,
designating only the system of the tangents, or of the osculating
circles of this curve. We may hence easily understand all the importance
of this theory, which seems to me to be not as yet sufficiently
appreciated by most geometers.
_Definite Integrals._ Finally, to complete our review of the vast
collection of analytical researches of which is composed the integral
calculus, properly so called, there remains to be mentioned one theory,
very important in all the applications of the transcendental analysis,
which I have had to leave outside of the system, as not being really
destined for veritable integration, and proposing, on the contrary, to
supply the place of the knowledge of truly analytical integrals, which
are most generally unknown. I refer to the determination of _definite
integrals_.
The expression, always possible, of integrals in infinite series, may at
first be viewed as a happy general means of compensating for the extreme
imperfection of the integral calculus. But the employment of such
series, because of their complication, and of the difficulty of
discovering the law of their terms, is commonly of only moderate utility
in the algebraic point of view, although sometimes very essential
relations have been thence deduced. It is particularly in the
arithmetical point of view that this procedure acquires a great
importance, as a means of calculating what are called _definite
integrals_, that is, the values of the required functions for certain
determinate values of the corresponding variables.
An inquiry of this nature exactly corresponds, in transcendental
analysis, to the numerical resolution of equations in ordinary analysis.
Being generally unable to obtain the veritable integral--named by
opposition the _general_ or _indefinite_ integral; that is, the function
which, differentiated, has produced the proposed differential
formula--analysts have been obliged to employ themselves in determining
at least, without knowing this function, the particular numerical values
which it would take on assigning certain designated values to the
variables. This is evidently resolving the arithmetical question without
having previously resolved the corresponding algebraic one, which most
generally is the most important one. Such an analysis is, then, by its
nature, as imperfect as we have seen the numerical resolution of
equations to be. It presents, like this last, a vicious confusion of
arithmetical and algebraic considerations, whence result analogous
inconveniences both in the purely logical point of view and in the
applications. We need not here repeat the considerations suggested in
our third chapter. But it will be understood that, unable as we almost
always are to obtain the true integrals, it is of the highest importance
to have been able to obtain this solution, incomplete and necessarily
insufficient as it is. Now this has been fortunately attained at the
present day for all cases, the determination of the value of definite
integrals having been reduced to entirely general methods, which leave
nothing to desire, in a great number of cases, but less complication in
the calculations, an object towards which are at present directed all
the special transformations of analysts. Regarding now this sort of
_transcendental arithmetic_ as perfect, the difficulty in the
applications is essentially reduced to making the proposed research
depend, finally, on a simple determination of definite integrals, which
evidently cannot always be possible, whatever analytical skill may be
employed in effecting such a transformation.
_Prospects of the Integral Calculus._ From the considerations indicated
in this chapter, we see that, while the differential calculus
constitutes by its nature a limited and perfect system, to which nothing
essential remains to be added, the integral calculus, or the simple
system of integration, presents necessarily an inexhaustible field for
the activity of the human mind, independently of the indefinite
applications of which the transcendental analysis is evidently
susceptible. The general argument by which I have endeavoured, in the
second chapter, to make apparent the impossibility of ever discovering
the algebraic solution of equations of any degree and form whatsoever,
has undoubtedly infinitely more force with regard to the search for a
single method of integration, invariably applicable to all cases. "It
is," says Lagrange, "one of those problems whose general solution we
cannot hope for." The more we meditate on this subject, the more we
will be convinced that such a research is utterly chimerical, as being
far above the feeble reach of our intelligence; although the labours of
geometers must certainly augment hereafter the amount of our knowledge
respecting integration, and thus create methods of greater generality.
The transcendental analysis is still too near its origin--there is
especially too little time since it has been conceived in a truly
rational manner--for us now to be able to have a correct idea of what it
will hereafter become. But, whatever should be our legitimate hopes, let
us not forget to consider, before all, the limits which are imposed by
our intellectual constitution, and which, though not susceptible of a
precise determination, have none the less an incontestable reality.
I am induced to think that, when geometers shall have exhausted the most
important applications of our present transcendental analysis, instead
of striving to impress upon it, as now conceived, a chimerical
perfection, they will rather create new resources by changing the mode
of derivation of the auxiliary quantities introduced in order to
facilitate the establishment of equations, and the formation of which
might follow an infinity of other laws besides the very simple relation
which has been chosen, according to the conception suggested in the
first chapter. The resources of this nature appear to me susceptible of
a much greater fecundity than those which would consist of merely
pushing farther our present calculus of indirect functions. It is a
suggestion which I submit to the geometers who have turned their
thoughts towards the general philosophy of analysis.
Finally, although, in the summary exposition which was the object of
this chapter, I have had to exhibit the condition of extreme
imperfection which still belongs to the integral calculus, the student
would have a false idea of the general resources of the transcendental
analysis if he gave that consideration too great an importance. It is
with it, indeed, as with ordinary analysis, in which a very small amount
of fundamental knowledge respecting the resolution of equations has been
employed with an immense degree of utility. Little advanced as geometers
really are as yet in the science of integrations, they have nevertheless
obtained, from their scanty abstract conceptions, the solution of a
multitude of questions of the first importance in geometry, in
mechanics, in thermology, &c. The philosophical explanation of this
double general fact results from the necessarily preponderating
importance and grasp of _abstract_ branches of knowledge, the least of
which is naturally found to correspond to a crowd of _concrete_
researches, man having no other resource for the successive extension of
his intellectual means than in the consideration of ideas more and more
abstract, and still positive.
* * * * *
In order to finish the complete exposition of the philosophical
character of the transcendental analysis, there remains to be considered
a final conception, by which the immortal Lagrange has rendered this
analysis still better adapted to facilitate the establishment of
equations in the most difficult problems, by considering a class of
equations still more _indirect_ than the ordinary differential
equations. It is the _Calculus_, or, rather, the _Method of Variations_;
the general appreciation of which will be our next subject.