DIFFERENTIAL CALCV. 



DIFFERENTIAL COK1-TI' 



Ul 



The mcihiJ of UmiU of D'Alembert, which is now more in 

 qurtitly u*cd than anjr other, WM considered by th author bimael: 

 a* an explanation of Newton ' prime aud ultimate ratio*. It i usual 

 to attribute the cxpomitioa of this method to D'Alembert (and, oon 

 idered a* MI actual application of limit*. correctly), though evoral 

 previously to him, bad made special applications uf the principle 

 The article* in the Encyclopedic must be considered a the j.iica 

 jutijeotint. The fallowing article, DirrEBUTTLU. COEH u ' 

 explain thu mthoJ, which contains the point in which the principle* 

 of all the prrcr.lu>.r in.it.-, ami which must more or leu be in all. 

 See alao the article LIMIT. 



The two remaining method* (thow of Lauden and Lagrange) are 

 attempts to MUbliah the acieooe upon purely algebraical principles. 

 Previously to entering upon them, we must remark that none of the 

 preceding theorist* attempted to make his system furnish any ndili 

 tional security to the methods of the algebra already in use. Such a 

 it was, correct or incorrect, clear or obscure, no one gave a nu'im-ni' 

 consideration to the fact that algebra already contained difficulties of 

 precisely the same character as those which were matter of dispute 

 in the differential calculus. Taking it for granted that algebra in 

 very part stood already upon as firm a basis as the differential 

 calculus cuuld ever on any supposition be expected to do, it won a 

 matter of some interest to make the latter a pure extension of the 

 former. 



The residual analysis of Landen is a technically algebraical exhibi- 

 tion of the theory of prime and ultimate ratios. The tract in which it 

 was promulgated, ' the Residual Analysis, a new branch of Uie Alge- 

 braic Art,' &c., appeared in 1764. When it is considered that this 

 new branch of the algebraic art was only old fluxions iu a different 

 dress, the title may excite surprise, if we remember how well Landen 

 deserved his reputation. But it must be remembered that all the 

 discussion of which this article is meant to elucidate the history, arose 

 from a tendency to consider two methods as mathematically different, 

 which were not the same in the method of enunciating their first 

 principles. A something between Landen and D'Alemburt, as to 

 principle, published in 1748, was called the ' Doctrine of Ultimators, 

 containing a new Acquisition, &c. , or a Discovery of the true and 

 genuine Foundation of what has hitherto mistakenly prevailed under 

 the improper names of Fluxions and the Differential Calculus.' The 

 difference between Landen and Newton will appear in the article 

 FRACTIONS, VANISHING, and in the instances which we Khali presently 

 give. It is the LIMIT of D'Alembert supposed to be attaitud, instead 

 of being a tertainut which can be attained as nearly as we please. A 

 little difference of algebraical suppositions makes a fallacious difference 

 of form ; and though the residual analysis draws less upon the dis- 

 putable part of algebra than the method of Lagrange, the sole 

 reason of this is that the former does not go BO far into the subject 

 a* the latter. 



The method of Lagrange, first given in the public lectures at the 

 Ecole Normale. and afterwards published separately under the title 

 of Thiuric da function*, is a deduction of the whole science from 

 Taylor's Theorem, which being absolutely granted, undoubtedly all 

 the rest may be made to follow. If ^ (JT + A) can be always expanded 

 in a form of which the two first terms are ipx+<t>'s.h, and if 

 >' (z + A) be related iu the same manner to $>'./ + <f>"s. I, and <f>"U' + A) 

 to e>"* + <f'"x. A, and so on, it can be made to follow that 



h- h* 



4> (x + A) = +x + +'x. K + <t>"*.-2 + *"'<. 3 



upon principles as sound as those of algebra iu the hands of Maclauriu, 

 or Euler, or Clairaut, as elementary writers. It is our opinion that 

 Lagrangu has not been correctly understood, nor fairly dealt with, by 

 those who have compared his theory of functions with the other 

 methods. Undoubtedly any one who should maintain the unqualified 

 admisribility of Lagrange's work must assert both the major and 

 minor of the following syllogism. 



Algebraical expansion (Maria da uile is Lagrange's phrase) as 

 generally received in 1790, was founded on sound principles : the 

 tkiorit da fvMctioHt is a logical and incontestable result of such algebra ; 

 therefore, Ac. 



All the attacks upon Lagrange have denied the major of this i-vl- 

 logism, whereas it appears to us that he never intended to assert 

 more than the minor. Perceiving that the mathematical world \..-,.- 

 in the habit of calling in the aid of limits or infinitesimals, to In )]' 

 a certain alyebra in deducing certain conclusions, he showed them 

 how that very algebra, good or bad, was competent to the deduction of 

 the same conclusion* without either limits or infinitesimals ; and he 

 was correct. Notwithstanding anything in the work in question, 

 Lagrange might have admitted all that we ahull find it necessary to 

 say against hi* system, absolutely considered, in the article FUNCTIONS, 

 THEORY or. 



A new question ha* arisen of late years, namely, whether the theory 

 of limit* be not al aolutely necemary to the rigorous development of 

 common algebraical forms, and whether this same theory of limits may 

 not be applied to the establishment of the differential calculus, inde- 

 pendently of any expansion. A tract of M. Ampere (a* we believe), 

 entitled ' Precis du Calcul Differentiel,' Ac., is the earliest writing we 

 are acquainted with in which this is attempted to be done. Mr. de 



Morgan* treatise on the subject, published by th* Society for the 

 Diffusion of Useful Knowledge, profane* to have th* same end in 

 view. The works of M. Cauchy have much in them by which this 

 object I* promoted, but expansion is avowedly introduced. 



We ahall now *t*te two propceition*, one geometrical, the other 

 algebraical, in the word* of the several lyttema. 



1. /nJbtitaimaU. Ao infinitely email arc of a circle is equal to it* 

 chord. 



Prime and Ultimate llatiot. If an arc of a circle diminish, the ulti- 

 mate ratio which it bear* to it* chord is one of equality ; or if it begin 

 to increase from nothing, the prime or nascent ratio of the arc and 

 chord is that of equality. Or the arc is vltimat,ln equ.il to it* chord. 



J-'liLriottt. If an arc increase from nothing with a uniform v< 

 the velocity with which the chord increase* is, at the first moment, 

 equal to that of the arc. 



Limits. If the arc of a circle (and therefore its chord) Himininh 

 without limit, the limit of the ratio of the arc to the chord is one of 

 equality. 



arc 



KtsiduMl Aiiaii/tit. When the arc of a circle = cnor j = Q 1, 



which is ascertained by clearing the numerator and denominator of 

 a factor which vanishes when arc = 0. 



Theory of Functions. When the arc is expanded in the following 

 series, 



Arc = A x chord + B x (chord)* + &c. 

 then A = 1. 



2. Infinitesimal. If an infinitely small increment dt be given to x, 

 then a? receives the infinitely small increment ' 



Prime and Ultimate Ratios. The ratio which any increment given 

 to x bears to the consequent increment of u. 4 is ultimately that of 

 1 to 3 a*. 



Fluxions. If .< be a line which increases with a velocity x, then a? 

 increases with the velocity 3 3?x. 



LimiU. The limit of the ratio obtained by dividing an increment 

 of .' J by the increment of x which produced it, on the supposition 

 that the latter increment diminishes without limit, is 3 a- 5 . 



Residual Analytii Since 





it follows that when y=x, -_ - = 3:r 2 . 



Theory of Ftincti<sn<. If (x+hf be expanded in a series of powers of 

 A, the coefficient of the first power of k is &e ! . 



DIFFERENTIAL COEFFICIENT. The expressions to which this 

 term is applied are of a degree of imjx>rtance in the science to which 

 they belong, as great as that of the letters of the alphabet in writing. 

 Without entering into the method of using them, which would be iu 

 effect to write a treatise on the diflerentinl calculus, we shall make 

 some remarks on the manner of defining and understanding the 

 lerm. 



When two magnitudes are so related that either being given the 

 other is also given, it follows that any change being made in the one, 

 :he consequent change in the value of the other can be found, and the 

 two changes can be compared as to magnitude. By this means a rough 

 notion can be formed as to the effect which a change of value in 

 one produces in the other. In such articles as CURVATURE, DIUECTIOK, 

 VELOCITY, FOHCE, &c., it is sufficiently shown that this rough notion, 

 obtained by making a sensible change in one magnitude and comparing 

 t with that produced in another, though sufficient for practical 

 >urposes, does not afford any exact and mathematical measure of the 

 .hing sought for. In each of the articles cited, it is found necessary 

 :o diminish the change originally supposed without limit, and it is not 

 .he actual ratio of two changes which we have to consider, but the 

 iniit to which that ratio npt roximates as the changes are diminished 

 without limit. We must distinctly refer the student to sensible 

 objects for an illustration of the cause why it is convenient, and 

 even necessary, to have recourse to the limit of a ratio. [See also 

 ^IMIT, RATIOS (PiUMK AND ULTIMATE), and various articles cited in 

 DIFFERENTIAL CALCULUS.] 



Let QJC be a function of x, called ;/, which when ,c is changed into 

 ; + A, becomes y -r I; so that 



I = <t> (x + h) fr: 



Mvide I- by A, ascertain the limit towards which this quotient apprux- 

 mates when k is diminished without limit, and that limit is what is 

 called the differential coefficient of y with respect to A: For instance, 

 ety a? + ;c; then 



ind 



= 2.E& + h" + h; 



d I- divided by /< gives 2* + * + 1, which, when h is diminished ' 

 ithout limit, has for the limit of its decrease, 2j- + 1, which is there- 

 fore the diflerential coefficient of x> + arwith respect to *. 



The term differential coefficient arise* thu*. The method of Leibnitz 

 ^INFINITESIMAL CALCULUS] proceeds as follows : Imagine x in the 

 receding expression to be increased by an infinitely email quantity (sec 



