CALCULUS. 



the produA of faflnrs in arithmetical progitflion ; and the 

 third fliews the api)lication of the ftparatioii of tlje fcales of 

 opcrat'ons to the direft and iiiv. rfe method of diffcrcnees. 

 I'or a fiirher account of this fubjcft, we muft refer to the 

 author's elaborate trcatife. See, alfo, an anal) lis of it in the 

 Monthly Review (New Scries), vol. XKxvi. p. 524 — 5J'2. 



Calculus rlt/fWcnlialu is a method of differencing quan- 

 tities, or of liiiding an infuiitcly fmall quantity, which, 

 being taken infinite times, fiiall be equal to a given qnan- 

 •tity : or, it is the arithmetic of the iufmitely fmall differences 

 of variable quantities. 



The foundation of this calculus is an infinitely fmall quau- 

 -tity, or an infinitefimal, which is a po tion of a quantity 

 incomparable to that quantity, or that is lefs than any af- 

 fiTnable one, and therefore acconiited as nothing; t!ic error 

 accruing by omitting it being kfs than any afll jnable one. 

 Hence two quantities, only diiiuing by an inlinitcfimal, are 

 reputed equal. 



Thus, in AJlronomy, the diameter of the earth is an in- 

 fiuitefimal. In refpedl of the-di'.lance of the fixed ft ars ; and 

 the fame holds in abllraft quantities. The term infinitefi- 

 mal, therefore, is merely refpcdtive, and involves a relation 

 to another quantity ; and does not denote any real ens, or 

 being. 



Now infinittfimals are called differentials, or differential 

 quantities, when they'are confidered as the differences of 

 two q'.:antities. Sir Ifaac Newton calls them moments ; con- 

 fidering them as the momtntary increments of quantities ; 

 V. g. of a hne generated by the flux of a puint ; or of a fur- 

 face by the flux of a line. The differential calculus, there- 

 fore, and the do6lrine of fluxions, arc the fame thing, under 

 different names ; the former given by M. Leibnitz, and the 

 latter by Sir Ifaac Newton ; each of whom lay claim to the 

 <iifcoVery. 



There is, indeed, a difference in the manner of expreffing 

 the quantities refulting from the different views wherein tlie 

 two autliors confider the inlinittfimals ; the one as moments, 

 the other as differences ; LeibniiK, and mofk foreigners, ex- 

 prefs the differentials of quantities by the fame letters as va- 

 riable ones, only prefixing the letter d : thus the differen- 

 tial of-.r is called d x; and that of_y, dy: now ds is a po- 

 fitive quantity, if x continually increafe ; negative, if it de- 

 creafe. 



The Englifh, with fir Ifaac Newton, inftead of d .v, write 



?; (with a dot over it) ; for dy, y, &c. which foreigners and 

 •others object againfl, on account of that confnfion of points, 

 which they imagine arlfes, when differentials are again dif- 

 ferenced ; on account of the ambiguity of the fluxionary ex- 

 preflions, owing to the pofition of the index ; becaufe the 

 flnxionaiy expreihon cannot be fo readily extended as the 

 differential notation ; and, befides, the printers are more apt 

 to overlook a point than a letter. 



Stable quantities are always exprefT.d by the firfl letters of 

 the alphabet d^ = e, db=^o, dc=^o; wherefore d {^x-\-y~d) 

 = d X -f- dy, and d {x—y -f-a) := dx — dy. So that the 

 differencing of quantities is eafily performed, by the addi- 

 tion or fubtraftiem of their compounds. 



To difference quantities that multiply each other; the 

 rule is, firfl, multiply the differential of one faftor into the 

 other fa<ftor, the fun of the two faftors is the differential 

 fought : thus, the quantities being xy, the differential will 

 be X d y-\-y dx, i. e. d [xy] = x dy-\- y dx. Secondly, if 

 there be tlitec quantities mutually multiplying each other, 

 the faftum of the two mull then be multiplied into the dif- 

 ferential ^i the third : thus, fuppofe "v x y, let vx=:t, then 

 •vxy=^ty; coufequenlly d [v xy) == t dy -{-y d t : but dt 

 z= -v d X -\-x d 1: Thefe values, therefgre, being fubftituted 



Vol. V, 



in the antecedent differential, t dy ■{■ yd I, the refult is, d 

 (■« .V v) = -0 X dy 4- vy d s -\- x y dv. Hence it is eafy to 

 apprehend how to proceed, where the quantities are more 

 than three. 



If one variable qnantity increafe, while the ollur y de- 

 creales, it is evident y d k—x dy wiil be the diP.erentidl 

 of X y. 



To dlfTerence quantities that mutually divide each other ; 

 the riilt is, firft, multiply the differential of the divifor into 

 the dividend ; and, on ;he contrary, the differential of the 

 dividend into the divifor; fubtradl the laft ])rodiK'^t from the 

 full, and divide the remainder by the fquarc of the divifor ; 

 the quotient is the dillerciitial of the quantities mutually di- 

 viding each other. .See Fluxions and I'ukctios', under 

 whicli aitlcle' will be flatcd La Grange's method of divefl- 

 ing the principles of the differential calculus of all reference 

 to infinitely fmall or evanefcent quantities. 



M. La Croix in hii "Treatife on the Differential and In- 

 tegral Calculus," confiders the ditftrential calculus in pre- 

 cilely the fame point of view in which M. La (iJrange re- 

 garded it in the lierlin Adis for 1772, and fubfe-qucntly in 

 an cxprefs and formal treitlfe on the fiibjecfl. An ingenious 

 anonymous writer has given the fubllance and fpirit of M. 

 La Croix's reafoning, without adhering clofcly to his nota- 

 tion and method. 



Let «=/'v be any funftion of .v ; then, if for x we fub- 

 ftitute x±/}, the developement of /(.v-i-/.i) will be of thi« 

 form : 



f(x+h)=.fx+fxh + 



f"xh\ 



&C. 



the coefficientsyK,/".v, &c. being derived from the primi- 

 tive fundtionyiv, and independent of /;. 



Hence f{x+/j)-fx=fx/j+d 



&c. 



■wliich quantity reprefents the difference between fx and 

 what/v becomes when .\ is increafed to .r-f /j. Let the firft 

 term of this diffence be called the differential, and be denoted 



by the expreffion d/x; hence we have dfx=zf'xh .•.fx = jLl 



h 

 Hence to havey.v, divide the difference between two fuc- 

 ceflive values of_/'v by the increment : but, fincey'x is inde- 

 pendent of h-, h mull difappear by thif. divifion, and may be 

 repiefented by any fymbol at pkafure. Hence, for the 

 fake of uniformity in the figns, let I) be rtprcfented by 



dx .•.f'x-^~. Hence to find dfx, or the differential of^x, 



write in yjc, x-\-dx, for .v, develope /"(.•>: -j-<Zv) as far as the 

 terms affefted with the lirft power 01 dx, and fubtraftyiv.— 

 Since /(x-\-dx) — /x=::f'xdx-\-, &c. 

 /\x-i-dx)—/'x=f"xdx + , &c. 

 &c. = &c. 



/'xdx=.dfx , /"xdx^dfx 



dfx_ ,dfx\ 

 thercfore/"'*= j^^ _Uw 

 dx 

 But fince Jx is invariable, 



f{x + dx) dx—f\dxz=f"xdx' 

 hence df'xdx=:f"xdx' 



but dfxdx=:ddfx=^d'fx (d' not meaning a fymbol Jfquared, 

 but denoting the fecond differential of/* to be taken): hence 



it appears; 



, fince/";c='' (^) and =zit; that'^(^) = 



— ; dx^ -j 



dx dx 



5B 



J'fii 



