The Positive Philosophy of Auguste Comte

"[T]he auxiliary quantities... might be derived, according to any law whatever, from the immediate elements of the question."

"[O]ur future improved analytical resources may perhaps be found in a new mode of derivation. But, at present, the only auxiliary quantities habitually substituted for the primitive quantities in transcendental analysis are what are called— 1st, infinitely small elements, the differentials of different orders of those quantities, if we conceive of this analysis in the manner of Leibnitz: or 2nd, the s, the limits of the ratios of the simultaneous increments of the primitive quantities, compared with one another; or, more briefly, the prime and ultimate ratios of these increments, if we adopt the conception of Newton: or 3rd, the derivatives... of these quantities; that is, the coefficients of the different terms of their respective increments, according to the conception of Lagrange. These conceptions, and all others that have been proposed, are by their nature identical."

"We now see that the Calculus of functions, or Algebra, must consist of two distinct branches."

"To ordinary analysis I... give the name of Calculus of Direct Functions. To transcendental analysis, (...Infinitesimal Calculus, Calculus of fluxions and of fluents, Calculus of Vanishing quantities, the Differential and Integral Calculus, etc...) I shall give the title of Calculus of Indirect Functions.... by generalizing and giving precision to the ideas of Lagrange, and employ them to indicate the exact character of the two forms of analysis."

"[A]nalysts first divide equations... into two principal classes, according as they contain functions of only the first three of the [five] couples, or as they include also either exponential or circular functions. Though the names of algebraic and s given to these principal groups are inapt, the division between the corresponding equations is real enough, insofar as that the resolution of equations containing the transcendental functions is more difficult than that of s. Hence the study of the first is extremely imperfect, and our analytical methods relate almost exclusively to the elaboration of the second."

"[W]e must observe that, though [Algebraic equations] may often contain irrational functions of the unknown quantities, as well as rational functions, the first case can always be brought under the second, by transformations more or less easy..."

"[T]he resolution of algebraic equations is as yet known to us only in the four first degrees. In this respect, algebra has advanced but little since the labours of Descartes and the Italian analysts of the sixteenth century..."

"The only question... of eminent importance... in its logical relations, would be the general resolution of algebraic equations of any degree whatever. But... we are led to suppose, with Lagrange, that it exceeds the scope of our understandings."

"[I]f we had obtained the resolution of algebraic equations of any degree whatever, we should still have treated only a very small part of algebra... that is, of the calculus of direct functions, comprehending the resolution of all the equations that can be formed by the s known to us..."

"[B]y a law of our nature, we shall always remain below the difficulty of science, our means of conceiving of new questions being always more powerful than our resources for resolving them... [i.e.,] the human mind being more apt at imagining than at reasoning."

"Thus, if we... resolved all the analytical equations now known, and if, to do this, we had found new analytical elements, these again would introduce classes of equations of which we now know nothing: and so, however great might be the increase of our knowledge, the imperfection of our algebraic science would be perpetually reproduced."

"The methods that we have are, the complete resolution of the equations of the first four degrees; of any binomial equations; of certain special 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."

"To fill up the vast gap in the resolution of algebraic equations of the higher degrees, analysts have had recourse to a new order of questions,—to... 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."

"[T]his is very imperfect algebra; 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 numerical resolution is strictly sufficient, it is... a very 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... 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 must be carefully distinguished in our minds, on account of their different objects, and therefore of the different ways in which quantities are considered by them. Moreover, there is, in regard 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 immediate object of the Calculus of direct functions, are subordinated to a third, purely speculative, from which both derive their most effectual resources, ...designated by the general name of ', 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 equations, and then those that relate to their transformation; the business of these last being to modify the roots of an equation without knowing them..."

"One more theory... to complete our rapid exhibition of the different essential parts of the calculus of direct functions... relates to the transformation of functions into series by the aid of the Method of indeterminate Coefficients... one of the most fertile and important in algebra. This... is one of the most remarkable discoveries of Descartes."

"[I]nfinitesimal calculus, for which it might be... substituted in some respects, has... deprived it of some... importance; but the growing extension of the transcendental analysis has, while lessening its necessity, multiplied its applications and enlarged its resources; so by the useful combination of the two theories, the employment of the method of indeterminate coefficients has become much more extensive than... even before the formation of the calculus of indirect functions."

"We must next pass on to the more important and extensive branch... the Calculus of Indirect Functions."

"[T]he views of the transcendental analysis... by Leibnitz, Newton, and Lagrange... each... has advantages... all are finally equivalent, and... no method has yet been found which unites their respective characteristics."

"[I]t is only by the use of them all that an adequate idea of the analysis and its applications can be 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 substituting for the curve the auxiliary consideration of a polygon, inscribed or circumscribed, by means of which the curve itself was reached... but there was in it no equivalent for our modern methods; for the ancients had no logical and general means for the determination of... 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..."

"Lagrange justly ascribes to... Fermat the first idea in this new direction. Fermat... initiated the direct formation of transcendental analysis by his method for the determination of ', and for the finding of s, in which process he introduced auxiliaries which he afterwards suppressed as null when the equations obtained had undergone... 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... 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... earlier, discovered a method exactly equivalent, regarding the analysis from a different point of view, much more logical... but less adapted than that of Leibnitz to give all practicable extent and facility to the fundamental method."

"Lagrange... 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."

"The method of Leibnitz consists in introducing... in order to facilitate the establishment of equations, the infinitely small elements or differentials which are supposed to constitute 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."

"[W]hen 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... repeated any number of times..."

"[P]reliminary 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."

"[I]t 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."

"[T]he conception of transcendental analysis, as formed by Leibnitz... is... the loftiest idea ever yet attained by the human mind."

"[T]his conception was necessary to complete the basis of mathematical science, by enabling us to establish... 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."

"The differential formulas exhibit an extreme generality, expressing in a single equation each determinate phenomenon, however varied may be the subjects to which it belongs."

"Thus, one... 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 positive approximations between classes of wholly different phenomena, through the 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, imperfect in its logical basis."

"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... 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... some solutions..."

"Some... demonstrated the conformity of the new conception with others; that of Newton especially, which was unquestionably exact. This afforded a practical justification: but... a logical justification is also required,—a direct proof of the necessary rationality of the infinitesimal method."

"Carnot... 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... may not have a merely provisional existence... but, in the present state of our knowledge, Carnot's principle... is of... importance, in legitimating the analysis of Leibnitz... 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... Carnot's theory is doubtless more subtle than solid; but it has no other radical logical vice than that of the infinitesimal method itself..."