The philosophy of mathematics · Auguste Comte
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.
Text from Project Gutenberg, public domain in the United States.