The extreme imperfection of algebra, with respect to the resolution of equations, has led analysts to occupy themselves with a new class of questions, whose true character should be here noted. They have busied themselves in filling up the immense gap in the resolution of algebraic equations of the higher degrees, by what they have named thenumerical resolutionof equations. Not being able to obtain, in general, theformulawhich expresses what explicit function of the given quantities the unknown one is, they have sought (in the absence of this kind of resolution, the only one reallyalgebraic) to determine, independently of that formula, at least thevalueof each unknown quantity, for various designated systems of particular values attributed to the given quantities. By the successive labours of analysts, this incomplete and illegitimate operation, which presents an intimate mixture of truly algebraic questions with others which are purely arithmetical, has been rendered possible in all cases for equations of any degree and even of any form. The methods for this which we now possess are sufficiently general, although the calculations to which they lead are often so complicated as to render it almost impossible to execute them. We have nothing else to do, then, in this part of algebra, but to simplify the methods sufficiently to render them regularly applicable, which we may hope hereafter to effect. In this condition of the calculus of direct functions, we endeavour, in its application,so to dispose the proposed questions as finally to require only this numerical resolution of the equations.
Its limited Usefulness.Valuable as is such a resource in the absence of the veritable solution, it is essential not to misconceive the true character of these methods, which analysts rightly regard as a very imperfect algebra. In fact, we are far from being always able to reduce our mathematical questions to depend finally upon only thenumericalresolution of equations; that can be done only for questions quite isolated or truly final, that is, for the smallest number. Most questions, in fact, are only preparatory, and intended to serve as an indispensable preparation for the solution of other questions. Now, for such an object, it is evident that it is not the actualvalueof the unknown quantity which it is important to discover, but theformula, which shows how it is derived from the other quantities under consideration. It is this which happens, for example, in a very extensive class of cases, whenever a certain question includes at the same time several unknown quantities. We have then, first of all, to separate them. By suitably employing the simple and general method so happily invented by analysts, and which consists in referring all the other unknown quantities to one of them, the difficulty would always disappear if we knew how to obtain the algebraic resolution of the equations under consideration, while thenumericalsolution would then be perfectly useless. It is only for want of knowing thealgebraicresolution of equations with a single unknown quantity, that we are obliged to treatEliminationas a distinct question, which forms one of the greatest special difficulties of common algebra. Laborious as are themethods by the aid of which we overcome this difficulty, they are not even applicable, in an entirely general manner, to the elimination of one unknown quantity between two equations of any form whatever.
In the most simple questions, and when we have really to resolve only a single equation with a single unknown quantity, thisnumericalresolution is none the less a very imperfect method, even when it is strictly sufficient. It presents, in fact, this serious inconvenience of obliging us to repeat the whole series of operations for the slightest change which may take place in a single one of the quantities considered, although their relations to one another remain unchanged; the calculations made for one case not enabling us to dispense with any of those which relate to a case very slightly different. This happens because of our inability to abstract and treat separately that purely algebraic part of the question which is common to all the cases which result from the mere variation of the given numbers.
According to the preceding considerations, the calculus of direct functions, viewed in its present state, divides into two very distinct branches, according as its subject is thealgebraicresolution of equations or theirnumericalresolution. The first department, the only one truly satisfactory, is unhappily very limited, and will probably always remain so; the second, too often insufficient, has, at least, the advantage of a much greater generality. The necessity of clearly distinguishing these two parts is evident, because of the essentially different object proposed in each, and consequently the peculiar point of view under which quantities are therein considered.
Different Divisions of the two Methods of Resolution.If, moreover, we consider these parts with reference to the different methods of which each is composed, we find in their logical distribution an entirely different arrangement. In fact, the first part must be divided according to the nature of the equations which we are able to resolve, and independently of every consideration relative to thevaluesof the unknown quantities. In the second part, on the contrary, it is not according to thedegreesof the equations that the methods are naturally distinguished, since they are applicable to equations of any degree whatever; it is according to the numerical character of thevaluesof the unknown quantities; for, in calculating these numbers directly, without deducing them from general formulas, different means would evidently be employed when the numbers are not susceptible of having their values determined otherwise than by a series of approximations, always incomplete, or when they can be obtained with entire exactness. This distinction ofincommensurableand ofcommensurableroots, which require quite different principles for their determination, important as it is in the numerical resolution of equations, is entirely insignificant in the algebraic resolution, in which therationalorirrationalnature of the numbers which are obtained is a mere accident of the calculation, which cannot exercise any influence over the methods employed; it is, in a word, a simple arithmetical consideration. We may say as much, though in a less degree, of the division of the commensurable roots themselves intoentireandfractional. In fine, the case is the same, in a still greater degree, with the most general classification of roots, asrealandimaginary. Allthese different considerations, which are preponderant as to the numerical resolution of equations, and which are of no importance in their algebraic resolution, render more and more sensible the essentially distinct nature of these two principal parts of algebra.
These two departments, which constitute the immediate object of the calculus of direct functions, are subordinate to a third one, purely speculative, from which both of them borrow their most powerful resources, and which has been very exactly designated by the general name ofTheory of Equations, although it as yet relates only toAlgebraicequations. The numerical resolution of equations, because of its generality, has special need of this rational foundation.
This last and important branch of algebra is naturally divided into two orders of questions, viz., those which refer to thecompositionof equations, and those which concern theirtransformation; these latter having for their object to modify the roots of an equation without knowing them, in accordance with any given law, providing that this law is uniform in relation to all the parts.[9]
To complete this rapid general enumeration of the different essential parts of the calculus of direct functions, I must, lastly, mention expressly one of the most fruitful and important theories of algebra proper, that relating to the transformation of functions into series by the aid of what is called theMethod of indeterminate Coefficients. This method, so eminently analytical, and which must be regarded as one of the most remarkable discoveries of Descartes, has undoubtedly lost some of its importance since the invention and the development of the infinitesimal calculus, the place of which it might so happily take in some particular respects. But the increasing extension of the transcendental analysis, although it has rendered this method much less necessary, has, on the other hand, multiplied its applications and enlarged its resources; so that by the useful combination between the two theories, which has finally been effected, the use of the method of indeterminate coefficients has become at present much more extensive than it was even before the formation of the calculus of indirect functions.
Having thus sketched the general outlines of algebra proper, I have now to offer some considerations on several leading points in the calculus of direct functions, our ideas of which may be advantageously made more clear by a philosophical examination.
The difficulties connected with several peculiar symbols to which algebraic calculations sometimes lead, and especially to the expressions calledimaginary, have been, I think, much exaggerated through purely metaphysical considerations, which have been forced upon them, in the place of regarding these abnormal results in their true point of view as simple analytical facts. Viewing them thus, we readily see that, since the spirit of mathematical analysis consists in considering magnitudes in reference to their relations only, and without any regard to their determinate value, analysts are obliged to admit indifferently every kind of expression which can be engendered by algebraic combinations. The interdiction of even one expression because of its apparent singularity would destroy the generality of their conceptions. The common embarrassment on this subject seems to me to proceed essentially from an unconscious confusion between the idea offunctionand the idea ofvalue, or, what comes to the same thing, between thealgebraicand thearithmeticalpoint of view. A thorough examination would show mathematical analysis to be much more clear in its nature than even mathematicians commonly suppose.
As to negative quantities, which have given rise to so many misplaced discussions, as irrational as useless, we must distinguish between theirabstractsignification and theirconcreteinterpretation, which have been almost always confounded up to the present day. Under the firstpoint of view, the theory of negative quantities can be established in a complete manner by a single algebraical consideration. The necessity of admitting such expressions is the same as for imaginary quantities, as above indicated; and their employment as an analytical artifice, to render the formulas more comprehensive, is a mechanism of calculation which cannot really give rise to any serious difficulty. We may therefore regard the abstract theory of negative quantities as leaving nothing essential to desire; it presents no obstacles but those inappropriately introduced by sophistical considerations.
It is far from being so, however, with their concrete theory. This consists essentially in that admirable property of the signs + and-, of representing analytically the oppositions of directions of which certain magnitudes are susceptible. Thisgeneral theoremon the relation of the concrete to the abstract in mathematics is one of the most beautiful discoveries which we owe to the genius of Descartes, who obtained it as a simple result of properly directed philosophical observation. A great number of geometers have since striven to establish directly its general demonstration, but thus far their efforts have been illusory. Their vain metaphysical considerations and heterogeneous minglings of the abstract and the concrete have so confused the subject, that it becomes necessary to here distinctly enunciate the general fact. It consists in this: if, in any equation whatever, expressing the relation of certain quantities which are susceptible of opposition of directions, one or more of those quantities come to be reckoned in a direction contrary to that which belonged to them when the equation was first established, it will not be necessary to form directly a newequation for this second state of the phenomena; it will suffice to change, in the first equation, the sign of each of the quantities which shall have changed its direction; and the equation, thus modified, will always rigorously coincide with that which we would have arrived at in recommencing to investigate, for this new case, the analytical law of the phenomenon. The general theorem consists in this constant and necessary coincidence. Now, as yet, no one has succeeded in directly proving this; we have assured ourselves of it only by a great number of geometrical and mechanical verifications, which are, it is true, sufficiently multiplied, and especially sufficiently varied, to prevent any clear mind from having the least doubt of the exactitude and the generality of this essential property, but which, in a philosophical point of view, do not at all dispense with the research for so important an explanation. The extreme extent of the theorem must make us comprehend both the fundamental difficulties of this research and the high utility for the perfecting of mathematical science which would belong to the general conception of this great truth. This imperfection of theory, however, has not prevented geometers from making the most extensive and the most important use of this property in all parts of concrete mathematics.
It follows from the above general enunciation of the fact, independently of any demonstration, that the property of which we speak must never be applied to magnitudes whose directions are continually varying, without giving rise to a simple opposition of direction; in that case, the sign with which every result of calculation is necessarily affected is not susceptible of any concrete interpretation, and the attempts sometimes made to establishone are erroneous. This circumstance occurs, among other occasions, in the case of a radius vector in geometry, and diverging forces in mechanics.
A second general theorem on the relation of the concrete to the abstract is that which is ordinarily designated under the name ofPrinciple of Homogeneity. It is undoubtedly much less important in its applications than the preceding, but it particularly merits our attention as having, by its nature, a still greater extent, since it is applicable to all phenomena without distinction, and because of the real utility which it often possesses for the verification of their analytical laws. I can, moreover, exhibit a direct and general demonstration of it which seems to me very simple. It is founded on this single observation, which is self-evident, that the exactitude of every relation between any concrete magnitudes whatsoever is independent of the value of theunitsto which they are referred for the purpose of expressing them in numbers. For example, the relation which exists between the three sides of a right-angled triangle is the same, whether they are measured by yards, or by miles, or by inches.
It follows from this general consideration, that every equation which expresses the analytical law of any phenomenon must possess this property of being in no way altered, when all the quantities which are found in it are made to undergo simultaneously the change corresponding to that which their respective units would experience. Now this change evidently consists in all the quantities of each sort becoming at oncemtimessmaller, if the unit which corresponds to them becomesmtimes greater, or reciprocally. Thus every equation which represents any concrete relation whatever must possess this characteristic of remaining the same, when we makemtimes greater all the quantities which it contains, and which express the magnitudes between which the relation exists; excepting always the numbers which designate simply the mutualratiosof these different magnitudes, and which therefore remain invariable during the change of the units. It is this property which constitutes the law of Homogeneity in its most extended signification, that is, of whatever analytical functions the equations may be composed.
But most frequently we consider only the cases in which the functions are such as are calledalgebraic, and to which the idea ofdegreeis applicable. In this case we can give more precision to the general proposition by determining the analytical character which must be necessarily presented by the equation, in order that this property may be verified. It is easy to see, then, that, by the modification just explained, all thetermsof the first degree, whatever may be their form, rational or irrational, entire or fractional, will becomemtimes greater; all those of the second degree,m2times; those of the third,m3times, &c. Thus the terms of the same degree, however different may be their composition, varying in the same manner, and the terms of different degrees varying in an unequal proportion, whatever similarity there may be in their composition, it will be necessary, to prevent the equation from being disturbed, that all the terms which it contains should be of the same degree. It is in this that properly consists the ordinarytheorem ofHomogeneity, and it is from this circumstance that the general law has derived its name, which, however, ceases to be exactly proper for all other functions.
In order to treat this subject in its whole extent, it is important to observe an essential condition, to which attention must be paid in applying this property when the phenomenon expressed by the equation presents magnitudes of different natures. Thus it may happen that the respective units are completely independent of each other, and then the theorem of Homogeneity will hold good, either with reference to all the corresponding classes of quantities, or with regard to only a single one or more of them. But it will happen on other occasions that the different units will have fixed relations to one another, determined by the nature of the question; then it will be necessary to pay attention to this subordination of the units in verifying the homogeneity, which will not exist any longer in a purely algebraic sense, and the precise form of which will vary according to the nature of the phenomena. Thus, for example, to fix our ideas, when, in the analytical expression of geometrical phenomena, we are considering at once lines, areas, and volumes, it will be necessary to observe that the three corresponding units are necessarily so connected with each other that, according to the subordination generally established in that respect, when the first becomesmtimes greater, the second becomesm2times, and the thirdm3times. It is with such a modification that homogeneity will exist in the equations, in which, if they arealgebraic, we will have to estimate the degree of each term by doubling the exponents of the factors which correspondto areas, and tripling those of the factors relating to volumes.
Such are the principal general considerations relating to theCalculus of Direct Functions. We have now to pass to the philosophical examination of theCalculus of Indirect Functions, the much superior importance and extent of which claim a fuller development.
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 ofLeibnitz, ofNewton, and ofLagrange, 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 insufficientidea 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.
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 thecalculus of indirect functions, properly so called, the original idea of theinfinitesimal method, which can be conceived by itself, independently of anycalculus. 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 theMethod 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 ofmaximaandminima, and for the finding oftangents, 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.
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 ordifferentialswill 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 havedifferent 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 besecond 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 distinguishingdifferent ordersof 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, totake 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 inventorsof the transcendental analysis. We will consider the tangent as a secant joining two points infinitely near to each other; and then, designating bydyanddxthe infinitely small differences of the co-ordinates of those two points, the elementary principles of geometry will immediately give the equationt=dy/dxfor 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 infinitesimalsdxanddy, which were introduced as auxiliaries, by determining in each particular case, by means of the equation of the proposed curve, the ratio ofdytodx, 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 arcdsas a right line, the equations
ds2=dy2+dx2, ords2=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 ofindirect 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 quantitydA, as the result of a corresponding increment of the abscissa. The relation between these two differentials can be immediately obtained with the greatest facility by substituting for the curvilinear element of the proposed area the rectangle formed by the extreme ordinate and the element of the abscissa, from which it evidently differs only by an infinitely small quantity of the second order. This will at once give, whatever may be the curve, the very simple differential equation
dA =ydx,
from which, when the curve is defined, the calculus of indirect functions will show how to deduce the finite equation, which is the immediate object of the problem.
4.Velocity in Variable Motion.In like manner, in Dynamics, when we desire to know the expression for the velocity acquired at each instant by a body impressed with a motion varying according to any law, we will consider the motion as being uniform during an infinitely small element of the timet, and we will thus immediately form the differential equationde=vdt, in whichvdesignates the velocity acquired when the body has passed over the spacee; 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 enablingus 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, forthe 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 spiteof 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 asincomparables, 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 verificationis 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 comparisonwith 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 callsimperfect; 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, becomeperfect. 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 oftangents, which is the most easy to analyze completely. We will regard the equationt=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 bey=ax2, we shall evidently have
dy= 2axdx+adx2.
In this formula we shall have to neglect the termdx2as an infinitely small quantity of the second order. Then the combination of the twoimperfectequations.
t=dy/dx,dy= 2ax(dx),
being sufficient to eliminate entirely the infinitesimals, the finite result,t= 2ax, will necessarily be rigorously correct, from the effect of the exact compensation of the two errors committed; since, by its finite nature, it cannot be affected by an infinitely small error, and this is, nevertheless, the only one which it could have, according to the spirit of the operations which have been executed.
It would be easy to reproduce in a uniform manner the same reasoning with reference to all the other general applications of the analysis of Leibnitz.
This ingenious theory is undoubtedly more subtile than solid, when we examine it more profoundly; but it has really no other radical logical fault than that of the infinitesimal method itself, of which it is, it seems to me, the natural development and the general explanation, sothat 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.
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 theMethod of prime and ultimate Ratios, sometimes under that of theMethod 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, thelimits of the ratiosof the simultaneous increments of these quantities; or, in other words, thefinal ratiosof 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 thelimitsof a series of rectilinear polygons, variable motions as thelimitsof 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 Δyand Δx, we would have at each instant, for the trigonometrical tangent of the angle which the secant makes with the axis of abscissas,
t= Δy/Δx;
from which, taking the limits, we will obtain, relatively to the tangent itself, this general formula of transcendental analysis,
t=L(Δy/Δx),
the characteristicLbeing 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 betweentandx, 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 isy=ax2, we shall evidently have
Δy= 2axΔx+a(Δx)2,
from which we shall obtain
Δy/Δx= 2ax+aΔx.
Now it is clear that thelimittowards which the second number tends, in proportion as Δxdiminishes, is 2ax. We shall therefore find, by this method,t= 2ax, as we obtained it for the same case by the method of Leibnitz.
2.Rectifications.In like manner, when the rectification of a curve is desired, we must substitute for the increment of the arc s the chord of this increment, which evidently has such a connexion with it that the limit of their ratio is unity; and then we find (pursuing in other respects the same plan as with the method of Leibnitz) this general equation of rectifications:
(LΔs/Δx)² = 1 + (LΔy/Δx)²,or (LΔs/Δx)2= 1 + (LΔy/Δx)2+ (LΔz/Δx)2,
according as the curve is plane or of double curvature. It will now be necessary, for each particular curve, to pass from this equation to that between the arc and the abscissa, which depends on the transcendental calculus properly so called.
We could take up, with the same facility, by the method of limits, all the other general questions, the solution of which has been already indicated according to the infinitesimal method.
Such is, in substance, the conception which Newton formed for the transcendental analysis, or, more precisely, that which Maclaurin and D'Alembert have presented as the most rational basis of that analysis, in seeking to fix and to arrange the ideas of Newton upon that subject.
Fluxions and Fluents.Another distinct form under which Newton has presented this same method should be here noticed, and deserves particularly to fix our attention, as much by its ingenious clearness in some cases as by its having furnished the notation best suited to this manner of viewing the transcendental analysis, and, moreover, as having been till lately the special form of the calculus of indirect functions commonly adopted by the English geometers. I refer to the calculus offluxionsand offluents, founded on the general idea ofvelocities.
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. Thevelocitywith which each shall have been described will be called thefluxionof that quantity, which will be inversely named itsfluent. 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.
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, theirderivedfunctions; 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 thesederivativeswhich 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 successivederivationis 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 derivativesmayfacilitatethe establishment of equations, it is not easy to explain why thismustnecessarily 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 abscissax+h, will be represented by the formula
(f'(x) -t)h+qh2+rh3+ etc.,
in whichtdesignates, as above, the unknown trigonometricaltangent of the angle which the required line makes with the axis of abscissas, andf'(x) the derived function of the ordinatef(x). This being understood, it is easy to see that, by disposing oftso 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 whichtdid 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 expressiont=f'(x), a result exactly equivalent to those furnished by the Infinitesimal Method and the Method of Limits. We have yet to findf'(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'sCalculus 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. Theirperfect and necessary conformityis first to be proven in a general manner.
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, theintroduction 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 asthe 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 thecalculus, 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 aredifferentialequations, according to the idea of Leibnitz, or equations oflimits, conformably to the conception of Newton, or, lastly,derivedequations, 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 ofderived functions, that they are nothing else than what Leibnitz callsdifferential 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 necessarylimittowards 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 bydx/dyin the method of Leibnitz; that which ought to be noted asL(Δy/Δx) in that of Newton; and that which Lagrange has indicated byf'(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.
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 ofequations 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 hisMéchanique 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 afalse 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 oflimitsis, 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 consideringthe 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 bydy/dxthat which logically ought, in the theory of limits, to be denoted byL(Δy/Δx), and in extending to all the other analytical conceptions this displacement of signs, they intended, undoubtedly, to combine the special advantages of the two methods; but, in reality, they have only succeeded in causing a vicious confusion between them, a familiarity with which hinders the formation of clear and exact ideas of either. It would certainly be singular, considering this usage in itself, that, by the mere means of signs, it could be possible to effecta 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 oflimits, 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.