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DIFFERENCE, ASCENSIONAL. 



DIFFERENTIAL CALCULUS. 



823 



of diTerenoea ; and our limits will not allow us to go further into the 

 subject. We shill only observe, that it is most desirable that this very 

 elementary branch of pure mathematics should be taught aa a part of 

 common algebra, or at least with the first rudiments of the differential 

 calculus. The almost universal practice of deferring this subject until 

 the student is master of the integral calculus, is entirely subversive of 

 all natural order, and is perhaps one of the reasons why the differential 

 calculus is proverbially difficult. Various developments and applications 

 are to be found in every work on the subject. 



The term difference is Continental ; the older English term was 



,i?nt. 



DIFFERENCE, ASCENSIONAL. [ASCENSION.] 

 DIFFERENCE, EQUATIONS OF. [EQUATIONS OF DIFFERENCES.] 

 DIFFERENTIAL CALCULUS, the name given by Leibnitz to the 

 science which was digested nearly about the same time by himself and 

 Newton, independently of each other [FLUXIONS ; COMMERCIUM EPI- 

 STOLICCM], and which has of late years almost exclusively prevailed in 

 this country, to the exclusion of the name, notation, and (so far as they 

 differ) methods of Newton's fluxions. 



It is impossible, in the smallest degree, to exhibit the present state 

 and uses of a science into which all others merge as the student ap- 

 proaches the higher applications of mathematics. The article DIFFER- 

 ENTIAL COEFFICIENT will, so far as it goes, give some idea of the nature 

 of its first step; but the following remarks must be considered as 

 intended for the student who has made some progress in a modern 

 elementary work. 



The history of the differential calculus, at its first rise, is so con- 

 nected with that of the Newtonian FLUXIONS, in consequence of the 

 celebrated dispute as to the right of invention, that we have thought 

 it best to refer the whole point to the last-named article. On the 

 history of the science since the time of Newton, there is no work from 

 which we can trace out a connected account of the various steps by 

 which the present system has been formed. In fact, most of the new 

 investigations have been made with reference to some particular points 

 of physical science. It would be very difficult to write the history of 

 this calculus without entering at the game time into that of mechanics, 

 optics, astronomy, Ac. &.C., and of every subject to which it has ever 

 been applied. An attempt at the former without the latter would be 

 an account of the progress of language without mention of literature, 

 oratory, or the drama. 



In the meanwhile, seeing that notions as to the most proper and 

 useful basis on which to build this science are far from being fixed, 

 the most advantageous course which we can here adopt is to give a 

 short account of the various systems which have been proposed, 

 referring to such articles and treatises as will enable the student to 

 obtain further information. These different systems all produce the 

 name results, expressed hi very similar manners ; there is no question 

 between them as to the truth or falsehood of any one deduction, and a 

 practised intellect can always see how the principles of any one, 

 (usitiiied <w granted, may be made to furnish demonstrations of those 

 of any other. It is therefore, with some exception, a psychological 

 rather than a mathematical difference which agitates (or rather which 

 did agitate) the mathematical world. We do not mean to say that 

 opinions are now agreed ; but it seems that the question is left open, 

 it being admitted that the manner in which a student arrives at his 

 knowledge of the subject in the first instance U not of the greatest 

 importance, provided that, when it has been obtained, he give his 

 attention to the comparison of the various methods by which he might 

 have attained the same end. 



The precursors of Newton and Leibnitz, namely, Archimedes, 

 C'avalieri, Wallis, Barrow, Fermat, Roberval, and others, touched so 

 near upon the differential calculus, that it is obvious any one of them 

 might have taken the place of either of the first, if he had possessed 

 more powerful means of algebraical development. After Vieta, Des- 

 cartes, W-illis, and Newton (considered only as the discoverer of the 

 binomial theorem), the step to a formal calculus was comparatively 

 small. The essential part of the difficulty had been removed, and by 

 much the greater part of the distance between Archimedes and Leibnitz 

 had been gained. This point once attained, methodi sprung up with 

 rapidity, and in little more than a century we find the introduction of 

 the various schemes which it will be necessary to mention, namely 

 Leibnitz's method of infinitesimals ; Newton's method of prime and 

 ultimate ratios ; Newton's method of fluxions ; Landen's method ol 

 vanishing fractions, or residual analysis ; D'Aleinbert's method of limits ; 

 Lagrange's method of derivation. In saying, however, that the step 

 to a formal calculus made by Newton and Leibnitz was comparatively 

 small, we only mean that it appears small to those who have been 

 taught how to make it. For one man who can make this lint simple 

 ttep, which put* the many into the position formerly held by the few 

 there are scores who can att but do it, and who have claims put in foi 

 them when the thing is done, but not before. A proverb, it is said, is 

 the wisdom of many, but the wit of one ; a great method in science is 

 always within the perception of many, before it is within the grasp 

 of one. 



Many other forms have been proposed, which either coincide in 

 principle with one or other of the preceding, or are without any inde 

 pendent claim to notice. Several of the preceding, indeed, are more 

 distinguifhud from each other by historical notoriety than by esseutia 



.ifference of character. If we distinguish carefully between the first 

 irinciples of a method and the manner in which those principles are 

 ipplied to algebra, it would not be any great stretsh of assertion to 

 :ontend that all the methods except the last are different, ways of 

 ixpressing the same fundamental ideas ; and that the last (Lagrange's^ 

 s a proof that, so long as the preceding methods employed the usual 

 amount of algebraical assumption in the establishment of the connection 

 between themselves and algebra, that same quantity of assumption 

 vould have been sufficient for the basis of a purely algebraical science, 

 equivalent to the differential calculus. 



The method of Leibnitz assumes that quantities are made up of 

 nfinite numbers of infinitely small parts. [INFINITESIMAL CALCOLUS ; 

 .NFINITE.] It is a sort of atomic theory of pure magnitude, which is 

 most obviously either false or obscure ; for so far as infinitely small 

 quantities can be definitely explained as objective realities, it is obvious 

 that there are no such things ; and any obscurity left in their definition 

 extends itself throughout the whole science. But the falsehood of the 

 lupposition is not absolute ; for though magnitudes cannot be distinctly 

 aid down to be composed of an infinite number of infinitely small parts, 

 ret any magnitude can be divided into a number of parts greater than any 

 ce may happen to name, each of which parts shall be less than any magni- 

 tude we may happen to name. Thus it is perfectly obvious that a foot 

 may be divided into parts more than a million in number, each of 

 which shall be less than the hundred millionth part of an inch. If we 

 may use such a phrase, the falsehood of the assertion may be made of 

 as small an intensity as we please, and the consequence is, that its 

 results turn out absolutely correct. All the difficulties of the science 

 are concentrated into one single assertion ; and when this assertion is 

 once fairly understood and received in a correct sense, all that follows 

 is more easily understood and remembered, and far more easily applied, 

 than the results of any other method. Whichever of the systems a 

 student pursues, it is our decided opinion that he should accustom 

 tiimself to translate every result into the language of the infinitesimal 

 calculus, and endeavour to demonstrate it by the methods of the same. 

 It is usual to give a chapter on this method in elementary works ; in 

 addition to which we should strongly recommend to the student of 

 principles, Carnot, ' Reflexions sur la Metaphysique du Calcul Infini- 

 tesimal," last edition, Paris, 1839. But we must be understood not to 

 recommend the peculiar method of explaining the difficulties of infini- 

 tesimals adopted in this work, but only the manner of stating the 

 points of difficulty, and the comparison of the different systems. 



The system of Newton, known by the name of prime and ultimate 

 ratios, was set forth in the first section of the Principia, and is the 

 method pursued throughout that work. It is in reality a method of 

 limits, exhibited in a form which allows of a more ready application to 

 geometry than to algebra, and accordingly it is abandoned by Newton 

 himself in the method of fluxions. Instead of considering and com- 

 paring simultaneous increments of infinitely email magnitude, the 

 ratios of small but finite increments are taken ; and not these exactly, 

 but the limits towards which they approach when the increments are 

 diminished, which are called ultimate ratios, or nascent ratios, according 

 as the increments are supposed to be in the act of growing from or 

 diminishing towards nothing. The expression of Newton will justify 

 us in using the three words in italics : " Objectio est, quod quantitatum 

 evanescentium uulla sit ultima proportio : quippe quic, antequaui 



evanueruut, nou est ultima ; ubi evauueruut, nulla est Similiter 



per ultimaui rationem quautitatum evauescentium, intellegendam 

 ease rationem quantitatum, iton antequam evanescunt, non pustca, 

 ted qudcum ei-aneicunt. Pariter et ratio prima nascentium est ratio 

 quacum nascuutur ? " The student must seek for the account of this 

 method in the first section of the Principia already cited, and in the 

 article RATIOS, PRIME AND ULTIMATE. It may be observed that in 

 illustrating the preceding answer, Newton appeals to the fundamental 

 considerations on which his other method (if it be really another 

 method) is founded, to which we now come. 



The method of fluxions was also given by Newton, and with a pecu- 

 liar notation, which maintained its ground in this country until about 

 the year 1816. [FLUXIONS.] It will be seen in the articles DIRECTION, 

 VELOCITY, FORCE, &c., that there are many fundamental ideas, con- 

 nected with sensible objects, which lead to a practical differential cal- 

 culus, and might have happened to have been the means of suggesting 

 a strict and mathematical theory. Newton adopted one of these, that 

 of velocity, of which it may be said that its assumption as an answer 

 to objections is an evasion of all the metaphysical difficulties of the 

 subject. Since the proportions of all quantities may be represented by 

 those of- straight lines, the nature of the comparative changes which 

 take place in continuously increasing or decreasing quantities may be 

 referred to the velocities with which the terminal points of straight 

 lines change their places. Velocity once clearly defined, in cases where 

 it is variable, there is no further difficulty ; but unfortunately a distinct 

 conception of the measure of velocity is precisely equivalent to finding 

 a meaning for the differential or fluxional coefficient independently of 

 it. The theory of fluxions is best exhibited in the work of Maclauriu, 

 'Treatise on Fluxions,' 2 vols. 4to., Edinburgh, 1742, which, for 



rigorous and consistent application of its own principles, has, in our 

 opinion, never been surpassed. It has the advantage of having been 



written in answer to acute objections of principle [BjiKKiiLJjy], and i 



ike work oil fluxions. 



