science would be perpetually reproduced.
. The methods that we have are, the comknowlecb'-e " plete resolution of the equations of the first four degrees; of any binomial equations; of certain sjiecial equations of the superior degrees; and of a very small number of exponential, logarithmic, and circular equations. These elements are very limited; but geometers have succeeded in treating with them a great number of important questions in an admirable manner. The improvements introduced within a century into mathematical analysis have contributed more to render the little knowledge that we have immeasurably useful, than to increase it.
NUMERICAL RESOLUTION OF EQUATIONS. 61 To fill up the vast gap in the resolution of Numerical realgebraic equations of the higher degrees, solutions of analysts have had recourse to a new order of equations, questions, — to what they call the numerical resolution of equations. Not being able to obtain the real algebraic formula, they have sought to determine at least the value of each unknown quantity for such or such a designated system of particular values attributed to the given quantities.
This operation is a mixture of algebraic with arithmetical questions; and it has been so cultivated as to be rendered possible in all cases, for equations of any degree and even of any form. The methods for this are now sufficiently general; and what remains is to simplify them so as to fit them for regular application. While such is the state of algebra, we have to endeavour so to dispose the questions to be worked as to require finally only this numerical resolution of the equations. We must not forget however that this is very imperfect algebi'a; and it is only isolated, or truly final questions (which are very few), that can be brought finally to depend upon only the numerical resolution of equations. Most questions are only preparatory,— a first stage of the solution of other questions; and in these cases it is evidently not the value of the unknown quantity that we want to discover, but the formula which exhibits its derivation. Even in the most simple questions, when this numei'ical resolution is strictly sufficient, it is not the less a veiw imperfect method. Because we cannot abstract and treat separately the algebraic part of the question, which is common to all the cases which result from the mere variation of the given numbers, we are obliged to go over again the whole series of operations for the slightest change that may take place in any one of the quantities concerned.
Thus is the calculus of direct functions at present divided into two parts, as it is employed for the algebraic or the numerical resolution of equations. The first, the only satisfactory one, is unfortunately very restricted, and there is little hope that it will ever be otherwise: the second, usually insufficient, has at least the advantage of a much greater generality. They mvist be carefully distinguislied in our minds, on account of their different objects, and 62 POSIllYK PHILOSOPHY.
therefore of the different ways in which quantities are considered by them. Moreover, there is, in i-egard to their methods, an entirely different procedure in their rational distribution. In the first part, we have nothing to do with the values of the unknown quantities, and the division must take place according to the nature of the equations which we are able to resolve; whereas in the second, we have nothing to do with the degrees of the equations, as the methods are applicable to equations of any degree whatever; but the concern is with the numerical character of the values of the unknown quantities.
These two parts, which constitute the ime, nations!"^' mediate object of the Calculus of direct functions, are subordinated to a third, purely speculative, from Avhich both derive their most eiiectual resources, and which has been very exactly designated by the general name of Theory of Equations, though it relates, as yet, only to algebraic equations. The numerical resolution of equations has, on account of its generality, special need of this rational foundation.
Two orders of questions divide this important department of algebra between them; first, those which relate to the composition of eqiiations, and then those that relate to tlieir transformation; the business of these last being to modify the roots of an ec{uation without knowing them, according to any given law, provided this law is uniform in relation to all these roots.
One more theory remains to be noticed, to complete our rapid exhibition of the different essential parts of the cal-IVlethod of cuius of direct functions. This theory, which indeterminate relates to the transformation of functions Coefticients. j^^q series by the aid of what is called the IMethod of indeterminate Coefficients, is one of the most fertile and important in algebra. This eminently analytical method is one of the most remarkable discoveries of Descartes. The invention and development of the infinitesimal calculus, for which it might be very happily substituted in some respects, has undoubtedly deprived it of some of its importance; but the growing extension of the transcendental analysis has, while lessening its necessity, multiplied its applications and enlarged its resources; so CALCULUS OF INDIRECT FUNXTIONS. 63 that, by the useful combiuatiou of the two theories, the employmeut of the method of indetei-miuate coefticieuts has become much more extensive than it was even before the formation of the calculus of indirect functions.
I have now completed my sketch of the Calculus of Direct Functions. We must next pass on to the more important and extensive branch of our science, the Calculus of Indirect Functions.
TRANSCENDENTAL ANALYSIS, OR CALCULUS OF INDIRECT FUNCTIONS.
We referred (p. 53) in a former section to „..• • i the views of the transcendental analysis pre- view^^^'"^^^^'*^ sented by Leibnitz, ISewton, and Lagrange. We shall see that each concejjtion has advantages of its own, that all are finally equivalent, and that uo method has yet been found which unites their respective characteristics. Whenever the combination takes place, it will 2)robably be by some method founded on the conception of Lagrange. The other two will then offer only an historical interest; and meanwhile, the science must be regarded as in a merely provisional state, which requires the use of all the three conceptions at the same time; for it is only by the use of them all that an adequate idea of the analysis and its applications can be formed. The vast extent and difficulty of this part of mathematics, and its recent formation, should prevent our being at all surprised at the existing want of system. The conception which will doubtless give a fixed and uniform character to the science has come into the hands of only one new generation of geometers since its creation; and the intellectual habits requisite to perfect it have not been sufficiently formed.
The first germ of the infinitesimal method tt-^^ (which can be conceived of independently of the Calculus) may be recognized in the old Greek Method of Exhaustions, employed to pass from the properties of straight lines to those of curves. The method consisted in substituting for the curve the auxihary consideration of a jiolygou, inscribed or circumscribed, by means of which the curve itself was reached, the limits of the primitive ratios being suitably taken. There is no doubt of the filiation of ideas in this case; but there was in it no equivalent for our modern methods; for the ancients had no logical and general means for the determination of these limits, which was the chief difficulty of the question. The task remaining for modern geometers was to generalize the conception of the ancients, and, considering it in an abstract manner, to reduce it to a system of calculation, Avhich was impossible to them.
Lagrange justly ascribes to the great geometer Fermat the first idea in this new direction. Permat may be regarded as having initiated the direct formation of transcendental analysis by his method for the determination of maxima and minima, and for the finding of tangents, in which process he introduced auxiliaries which he afterwards suppressed as null when the equations obtained had imdergone certain suitable transformations. After some modifications of the ideas of Format in the intermediate time, Leibnitz stripped the process of some complications, and formed the analysis into a general and distinct calculus, having his own notation: and Leibnitz is thus the creator of transcendental analysis, as we employ it now. This pre-eminent discovery was so ripe, as all great conceptions are at the hour of their advent, that Newton had at the same time, or rather earlier, discovered a method exactly equivalent, regarding the analysis from a different point of view, much moi'e logical in itself, but less adapted than that of Leibnitz to give all practicable extent and facility to the fundamental method. Lagrange afterwards, discarding the heterogeneous considerations which had guided Leibnitz and Newton, reduced the analysis to a purely algebraic system, which only wants more aptitude for application.
We will notice the three methods in their order.
The method of Leibnitz consists in intro- L^BNiTZ °^ ducing into the calculus, in order to facilitate the establishment of equations, the infinitely small elements or differentials which are supposed to con- METHOD OF LEIBNITZ. 65 stitute tlie quantities whose relations we are seeking. There are relations between these differentials which are simpler and more discoverable than those of the [ivimitive quantities; and hy these we maj afterwards (through a special calculus employed to eliminate these auxdiary infinitesimals) recur to the equations sought, which it would usually have been im])Ossible to obtain directly. This indirect analysis may have various degrees of indirectness; for, when there is too miich difiiculty in forming the equation between the diiferentials of the magnitudes under notice, a second application of the method is required, the differentials being now treated as new primitive quantities. and a relation being sought between their infinitely small elements, or second differentials, and so on; the same transformation beiug repeated any numlier of times, provided the whole number of auxiliaries be fiually eliminated.
It may be asked by novices in these studies, how these auxiliary quantities can be of use while they are of the same species with the magnitudes to be treated, seeing that the greater or less value of any quantity cannot affect any inquiry which has nothing to do with value at all. The explanation is this. We must begin by distinguishing the different orders of infinitely small quantities, obtaining a precise idea of this by considering them as l»eing either the successive powers of the same primitive infinitely small quantity, or as being quantities which may be regarded as having finite ratios \v'itli these powers; so that, for instance, the second or third or other differentials of the same variable are classed as infinitely small quantities of the second, third or other order, because it is easy to exhibit in them finite multiples of the second, third, or other powers of a certain first differential. These preliminary ideas being laid down, 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. We see at once how such a power must facilitate the formation of equations between the differentials of quantities, since we can substitute for these differentials such other elements as we may choose, and as will be more simjjle to treat, only observing the con- I. F dition that the new elements shall differ from the precediniij only by quantities infinitely small in relation to them. It is thus that it becomes possible in geometry to treat curved lines as composed of an infinity of rectilinear elements, and curved surfaces as formed of plane elements; and, in mechanics, varied motions as an infinite series of uniform motion'', succeeding each other at infinitely small intervals of time. Such a mere hint as this of the varied application of this method may give some idea of the vast scope of the conception of transcendental analysis, as formed by Leibnitz. It is, beyond all question, the loftiest idea ever yet attained by the human mind.
It is clear that this conception was necessary to complete the basis of mathematical science, by enabling us to establish, in a broad and practical manner, the relation of the concrete to the abstract. In this respect, we must regard it as the necessary complement of the great fundamental idea of Descartes on the general analytical representation of natural phenomena; an idea which could not be duly estimated or put to use till after the formation of the infinitesimal analysis.
This analysis has another property, besides that of facilitating the study of the mathematical laws of all phenomena, and perhaps not less important than that. The differential ^,.. formulas exhibit an extreme generality, exthe formulas pressing m a single equation each determinate phenomenon, however varied may be the subjects to which it belongs. Thus, one such 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 all variable motion; and one single equation represents the distribution of heat in any body, and for any case. This remarkable generality is the basis of the loftiest views of the geometers. Thus this analysis has not only furnished a general method for forming equations indirectly which could not have been directly discovered, but it has introduced a new order of more natural laws for our use in the mathematical study of natural phenomena, enabling us to rise at times to a perception of ])ositi\ e approximations between classes of wholly different phenomena, through the JUSTIFICATION OF THE LEIBNITZIAN METHOD. 67 analogies presented by the differential expressions of their mathematical laws. In virtue of this second property of the analysis, the entire system of an immense science, like geometry or mechanics, has submitted to a condensation into a small number of analytical formulas, from which the solution of all particular problems can be deduced, by invariable rules.
This beautiful method is, however, iniper-.,.
feet in its logical basis. At first, geometers ^.j^^ Method were naturally more intent upon extending the discovery and multiplying its applications than upon establishing the logical foundation of its processes. It was enough for some time to be able to produce, in answer to objections, unhoped-for solutions of the most difiicult problems. It became necessary, however, to recur to the basis of the new analysis, to establish the rigorous exactness of the jirocesses employed, notwithstanding their apparent breaches of the ordinary laws of reasoning, Leibnitz himself failed to justify his conception, giving, when urged, an answer which represented it as a mere approximative calculus, the successive opei'ations of which might, it is evident, admit an augmenting amount of error.
Some of his successors were satisfied with showing that its results accorded with those obtained by ordinary algebra, or the geometry of the ancients, reproducing by these last some solutions which could be at first obtained only by the new method. Some, again, demonstrated the conformity of the new conception with others; that of Newton especially, which was unquestionably exact. This afforded a practical justification: but, in a case of such unequalled importance, a logical justification is also required, — a direct proof of the necessary rationality of the infinitesimal method. It was Caruot who furnished this at last, by showing that the method was founded on the pi'inciple of the necessary compensation of errors. We cannot say that all the logical scaffolding of the infinitesimal method may not have a merely provisional existence, vicious as it is in its nature: but, in the present state of our knowledge, Carnot's principle of the necessary compensation of errors is of more importance, in legitimating the analysis of Leibnitz, than is even yet commonly supposed. His reasoning is founded on the conception of infinitesimal quantities indefinitely decreasing, while those from which they are derived are fixed. The infinitely small errors introduced with the auxiliaries cannot have occasioned other than infinitely small errors in all the equations; and when the relations of finite c^uantities are reached, these relations must be rigorously exact, since the only errors then possible must be finite ones, which cannot have entered: and thus the final equations become perfect. Caruot's theory is doubtless more subtle than solid; but it has no other radical logical vice than that of the infiaitesimal method itself, of which it is, as it seems to me, the natural development and general explanation; so that it must be adopted as long as that method is directly employed.
The philosophical character of the transcendental analysis has now been sufficiently exhil>ited to allow of my giving only the principal idea of tlie other two methods.
^,^ Newton offered his conception under several Method different forms in succession. That which is now most commonly adopted, at least on the continent, was called by himself, sometimes the Mdhod of prime and tdtimate Ratios, sometimes the Method of Limits, by which last term it is now usually known.
Under this Method, the auxiliaries introsimultaneous increments of the primitive quantities; or, in other words, the final ratios of these increments; limits or final ratios which we can easily show to have a determinate and finite value. A special calculus, which is the equivalent of the infinitesimal calculus, is afterwards employed, to rise from the equations between these limits to the corresponding equations between the primitive quantities themselves.
The power of easy expression of the mathematical laws of phenomena given by this analysis arises from the calculus applying, not to the increments themselves of the proposed quantities, but to the limits of the ratios of those increments; and from our being therefore able always to substitute for each increment any other magnitude more easy to treat, provided their final ratio is the ratio of equality; or, in other words, that the limit of their ratio is unity. It Newton's method of limits. 69 is clear, in fact, that the calculus of limits can he in no way affected by this substitution. Starting from this principle, we find nearly the equivalent of the facilities offei'ed by the analysis of Leibnitz, which are merely considered from another point of view. Thus, curves will be regarded as the limits of a series of rectilinear polygons, and variable motions as the limits of an aggregate of uniform motions of continually nearer approximation, etc., etc. Such is, in substance, Newton's conception; or rather, that which Maclaurin and d'Alembert have offered as the most rational basis of the transcendental analysis, in the endeavour to fix and arrange Newton's ideas on the subject.
Newton had another view, however, which ought to be presented here, because it is still fj^p^Ig"^ ^"' the special form of the calculus of indirect functions commonly adopted by English geometers; and also, on account of its ingenious clearness in some cases, and of its having furnished the notation best adapted to this manner of regarding the ti'anscendental analysis. I mean the Calculus oi fluxions and oifltients, founded on the general notion of velocities.
To facilitate the conception of the fundamental idea, let us conceive of every curve as generated by a point affected by a motion varying according to any law whatever. The different quantities presented by the (turve, the abscissa, the ordinate, the arc, tlie area, etc., will be regarded as simiiltaneously produced by successive degrees during this motion. The velocity with which each one will have been described will be called the fluxion of that quantity, which inversely would have been called its fluted Henceforth, the transcendental analysis will, accoi'ding to this conception, consist in forming directly the equations between the fluxions of the proposed quantities, to deduce from them afterwards, by a special Calculus, the equations between the fluents themselves. Wliat has just been stated respecting curves may evidently be transferred to any magnitudes whatever, regarded, by the help of a suitable image, as some being produced by the motion of others. This method is evidently the same with that of limits complicated with the foreign idea of motion. It is, in fact, only a way of representing, by a comparison derived from mechanics, the luetliod of prime aud ultimate ratios, which alone is reducible to a calculus. It therefore necessarily admits of the same genei'al advantages in the various principal applications of the transcendental analysis, without its being requisite for us to offer special proofs of this.
Lagrange's conception consists, in its ad- Method mirable simplicity, in considering the transcendental analysis to be a great algebraic artifice, by which, to facilitate the establishment of equations, we must introduce, in the place of or with the primitive functions, their derived functions; that is, according to the definition of Lagrange, the coefficient of the first term of the increment of each function, arranged according to the ascending powers of the increment of its variable. The Calculus of indirect functions, properly so called, is destined here, as well as in the conceptions of Leibnitz and Newton, to eliminate these derivatives, employed as auxiliaries, to deduce from their relations the corresponding equations 1>etween the primitive magnitudes. The transcendental analysis is then only a simple, but very considerable extension of ordinary analysis. It has long been a common practice with geometers to introduce, in analytical investigations, in the place of the magnitudes in question, their different powers, or their logarithms, or their sines, etc., in order to simjjlify the equations, and even to obtain them more easily. Successive derivation is a general artifice of the same nature, only of greater exteutT and commanding, in consequence, much more important resources for this common object.
But, though we may easily conceive, a priori, that the auxiliary use of these derivatives may facilitate the study of equations, it is not easy to explain why it must be so under this method of derivation, rather than any other transformation. This is the weak side of Lagrange's great idea. We liave not yet become able to lay hold of its precise advantages, in an abstract manner, and without recurrence to the other conceptions of the transcendental analysis. These advantages can be established only in the separate consideration of each principal question; and this verification becomes laborious, in the treatment of a complex problem.
COxMPAEISON OF THE THREE METHODS. 71 Other theories have been proposed, such as Euler's Calculus of vanishing quantities: but they are merely modifications of the thi'ee just exhibited. We must next compare and estimate these methods; and in the first j)lace observe their perfect and necessary conformity.
Considering the three methods in regard. •,, r, j to their destination, independently of pre- three m'etliod^ liminary ideas, it is clear that they all consist in the same general logical artifice; that is, the introduction of a certain system of auxiliary magnitudes uniformly correlative Avith those under investigation; the auxiliaries being substituted for the express object of facilitating the analytical expression of the mathematical laws of phenomena, though they must be finally eliminated by the help of a special calculus. It was this which determined me to define the transcendental analysis as the Calculus of indirect functions, in order to mark its true philosophical character, while excluding all discussion about the best manner of conceiving and applying it. Whatever may be the method employed, the general effect of this analysis is to bring every mathematical question more speedily into the domain of the calculus, and thus to lessen considerably the grand difficulty of the passage from the concrete to the abstract. We cannot hope that the Calculus Avill ever lay hold of all questions of natural philosophy — geometrical, mechanical, thermological, etc. — from their birth. That would be a contradiction. In every problem there must be a certain ])reliminai"y operation before the calculus can be of any use, and one which could not by its nature be subjected to abstract and invariable rules: — it is that which has for its object the establishment of equations, which are the indispensable point of departure for all analytical investigations. But this preliminary elaboration has been remarkably simplified by the creation of the transcendental analysis, which has thus hastened the moment at which general and abstract processes may be uniformly and exactly applied to the solution, by reducing the operation to fiudiug the equations between auxiliary magnitudes, whence the Calculus leads to equations directly relating to the proj^osed magnitudes, which had formerly to be established directly. Whether these indirect equations are differential equations, according to Leibnitz, or equations of ZimiVs, according to Newton, ov derived equations, according to Lagrange, the general procedure is evidently always the same. The coincidence is not only in the result but in the process; for the auxiliaries introduced are really identical, being only regarded from different points of view. The conceptions of Leibnitz and of Newton consist in making known in any case two general necessary properties of the derived function of Lagrange. The transcendental analysis, then, examined abstractly and in its principle, is always the same, whatever conception is adopted; and the processes of the Calculus of indirect functions ai^e necessarily identical in these different methods, which must therefore, under any aj^plication whatever, lead to rigorously uniform results.
„.. If we endeavour to estiinate their comparai\l^ r.r.1,',1^ '' tive value, we shall find in each of the three live Veil lit;,,, _ conceptions advantages and inconveniences which are peculiar to it, and which prevent geometers from adhering to any one of them, as exclusive and final.
The method of Leibnitz has eminently the advantage in the rapidity and ease with which it effects the formation of equations between auxiliary magnitiides. We owe to its use the high perfection attained by all the general theories of geometry and mechanics. Whatever may be the speculative opinions of geometers as to the infinitesimal method, they all employ it in the treatment of any new question. Lagrange himself, after having reconstructed tht^ analysis on a new basis, rendered a candid and decisive homage to the conception of Leibnitz, by employing it exclusively in the whole system of his "Analytical Mechanics." Such a fact needs no comment. Yet are we obliged to admit, with Lagrange, that the conception of Leibnitz is radically vicious in its logical relations. He himself declared the notion of infinitely small quantities to be a faJse idea: and it is in fact impossible to conceive of them clearly, though we may sometimes fancy that we do. This false idea bears, to my mind, the characteristic impress of the metaphysical age of its birth and tendencies of its originator. By the ingenious princij^le of the compensation of errors, we may, as we have already seen, explain the necessary exactness of COMPARISON OF THE THREE METHODS. 78 the processes which compose the method; but it is a radical inconvenience to be obhged to indicate, in Mathematics, two clashes of reasonings so unlike, as tliat the one order are perfectly rigoi'ons, while by the others we designedly commit errors which have to be afterwards compensated. There is nothing very logical in this; nor is anything obtained by pleading, as some do, that this method can be made to enter into that of limits, which is logically irrepr<.)achable. This is eluding the difficulty, and not resolving it; and besides, the advantages of this method, its ease and rapidity, are almost entirely lost under sucli a transformation. Finally, the infinitesimal method exhibits the very serious defect of breaking the unity of abstract mathematics by ci'eating a transcendental analysis founded upon principles widely different from those which serve as a basis to ordinary analysis. This division of analysis into two systems, almost wholly independent, tends to prevent the formation of general analytical conceptions. To estimate the consequences duly, we must recur in thought to the state of the science before Lagrange had established a general and complete harmony between these two great sections.
Newton's conception is free from the logical objections imputable to that of Leibnitz. The notion of limits is in fact remarkable for its distinctness and precision. The equations are, in this case, regarded as exact from their origin; and the general rules of reasoning are as constantly observed as in ordinary analysis. But it is weak in resources, and embarrassing in operation, compared with the infinitesimal method. In its applications, the relative inferiority of this theory is very strongly marked. It also separates the ordinary and transcendental analysis, though not so conspicuously as the theory of Leibnitz. As Lagrange remarked, the idea of liiidts, though clear and exact, is not the less a foreign idea, on which analytical theories ought not to be dependent.
This jierfect unity of analysis, and a ])\irely abstract character in the fundamental ideas, are found in the conception of Lagrange, and there alone. It is therefore the most philosophical of all. Discarding every heterogeneous consideration, Lagrange reduced the transcendental analysis