CALCULUS OF INDIRECT FUNCTIONS. 63 that, by the useful combination of the two theories, the employment of the method of indeterminate coefficients 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. SECTION II. TRANSCENDENTAL ANALYSIS, OR CALCULUS OF INDIRECT FUNCTIONS. We referred (p. 53) in a former section to the views of the transcendental analysis pre- views. Three principal sented by Leibnitz, Newton, and Lagrange. We shall see that each conception has advantages of its own, that all are finally equivalent, and that no method has yet been found which unites their respective characteristics. Whenever the combination takes place, it will probably 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 (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 History. substituting for the curve the auxiliary consideration of a polygon, 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, which was impossible to them. Lagrange justly ascribes to the great geometer Fermat the first idea in this new direction. Fermat 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 undergone certain suitable transformations. After some modifications of the ideas of Fermat 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 more 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. METHOD OF We will notice the three methods in their order. The method of Leibnitz consists in introducing 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 the quantities whose relations we are seeking. There are relations between these differentials which are simpler and more discoverable than those of the primitive quantities; and by these we may afterwards (through a special calculus employed to eliminate these auxiliary infinitesimals) recur to the equations sought, which it would usually have been impossible to obtain directly. This indirect analysis may have various degrees of indirectness; for, when there is too much difficulty in forming the equation between the differentials 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 being repeated any number of times, provided the whole number of auxiliaries be finally 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 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, 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 simple to treat, only observing the con It dition that the new elements shall differ from the preceding only by quantities infinitely small in relation to them. 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 motions, 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 facili tating the study of the mathematical laws of all phenomena, and perhaps not less important than that. The differential formulas exhibit an extreme generality, exGenerality of the formulas. pressing in 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. 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 positive approximations between classes of wholly different phenomena, through the This 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. Justification of This beautiful method is, however, imperfect in its logical basis. At first, geometers the 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 difficult problems. It became necessary, however, to recur to the basis of the new analysis, to establish the rigorous exactness of the processes 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 operations 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 Carnot who furnished this at last, by showing that the method was founded on the principle 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 reason |