CALCULUS. 



hiufticns and limits reft on tlie fame foimdatioii. The qiief- 

 tions that led to its difcovery have been difciidld lliice tlie 

 earlieft nra of g-eometry. The I2lh propolitioii of the I2th 

 book of Euclid's Elements is a prbblem of this kind, and 

 the firft that has come down to us. Archiincdcc>. by fimihir 

 methods, advanced to the foKition of more difficidt pro- 

 blems, than tliat vvliich inveftigated the relation of circles 

 to one another ; fuch as the relations between the furfaccs 

 and folid contents of the cylinder and fphere, the qnadra- 

 ture of the parabola, and the proportion of fpirals. M. La 

 Croix then proceeds in his hillory of the diff.rential calculus, 

 as connccled with the fevcral difcoveries made by Cavalleri, 

 Roberval, Dcfcartes, Fcrniat, Hn^^ens, Gregory de St. 

 Vincent, Pafcal, Wallis, Barrow, Leibnitz, and Newton. 

 Sec Fluxions. 



M. la Croix's treatife confills of two parts : the firfl 

 part gives an account of the diiierential calculus; 

 and the fubject of the fecond part is the integral calculus. 

 This part begins with the integration ol functions of one 

 variable quantity ; and the author has furnilhcd various for- 

 niuLi: for integrating rational and irrational functions, loga- 

 rithmic ai d exponential functions, circular funflions, &c. 

 and he then applies the integral calculus to the quadrature 

 and rcililication of curves, to the quadrature of curve fur- 

 faces, and to the content of the iolids comprehended by 

 them. He fubjoins an expoiltion of the methods which 

 Euler employed in his rcfearches concerning curves that arc 

 quadrable, reftifiable, &c. His next objetl is the integra- 

 tion of differential equations of two variable quantities ; and 

 in reference to this he has collefted all that has been written 

 on this intricate fubjecl. Accordingly this chapter of his 

 work contains methods for feparating the variable quantities 

 in differential equations of the firft older ; for invelligating 

 a faftor proper to render a differential equation of the firll 

 order integrable ; for integrating differential equations of 

 the firll order; in which the differential quantities pafs the 

 full: degree ; for obtaining particu'ar folutions of differential 

 equations of the firll order ; for rcfolving by appvo:;imation, 

 duTerential equations of the firil order ; for conltnitling, 

 geometrically, differential equations of the firft order ; for 

 integrating differential equations of the Iccond order by 

 means of transformations ; for invelligating a faflor proper 

 to render differential equations of the fccond order inte- 

 grable ; for rcfolving, by approximation, diiTerential equa- 

 tions of the fecond order ; and for integrating differential 

 equations of orders fuperior to the lecond. The fubjeft 

 of the next chapter is the integration of fractions containnig 

 two, or a g! eater nu-mbcr of variable quantities. The fifth 

 and laft chapter treats of the method of variations. 



The calculus of variations originated from certain pro- 

 blems concerning the maxima and minima of quantities 

 having been propofed by John Bernouilli, to the mathe- 

 maticians of Europe. Such a problem was that in which 

 it was required to find, of all curves paffing through two 

 fixed points, and fituatcd in the fame vertical plane, that 

 one down which a body would delcend fiom the higheft to 

 the lowcfl point in the lealt time poflible. 'i'he firll geome- 

 tricians, remarkir.g that nothing was obtained by putting 



the differential of the time, , — -=0, found that they 



could obtain a folution by making the time a minimum for 

 two fuccefTwe demerits of the curve ; thus, if x, x', x" 

 were three v<rtical abfcifl'as, and y, y', y" the correfpond- 

 iHg ordinatcs, the time v/ould be exprtffed by 



v^EEFI^JzEiE -^v V'— ■-'')j_+tj'"-.>'')' 



the differer-lial of which being taken, and put =e, gave a 



refulling equation 



V 



= i 



idant 



quantity ; and confequcntly proved the curve to be a cy- 

 cloid. —Euler, with far greater analytical knowledge thaiv 

 John Bernouilli, next treated thefe problems in a general 

 manner, in his tracl intitled, " Metliodtis invcnimdi lineas 

 curvas maximi minimive proprietate gaudcntes ; five folutio 

 pioblematis ifoperiinetici latiflimo fenfu accepti." M. La 

 Grange afterwaid gave greater generality to this calculus.by 

 making variable not only y, rly", ,t'y, &c. but x. 



The explanation of M. L:i Croix affords a clear idea 

 of the calculus of va: i.itions : 



" Suppofe (fays he) the variable quantities at firfl coi- 

 ncfted together by an equation, or by any other depend- 

 ence, to change by reafon of the form of the cqnatio:i, or 

 of the relation that refults fio.m the dependence cftablifhed 

 between them ceafing to be the fame; this circumftance 

 c.innot be exprefied in a more general manner, than by re- 

 garding the increments of x and y, as abloluttly indepen- 

 dent of each other ; fince, in eifefl, this liypothefis, not 

 defignating any particular relation between x and y, com- 

 prehends all. It follows thence, that the calculus of vari- 

 ations can only be employed for exprelTions, to which the 

 diflerential calculus lias already been applied ; and it difi'crs 

 from tlie la!l only by the independence which it fuppofes 

 between the variable quantities, which before were confi- 

 dered as connedtd by conilant relations. Tlie following 



example will illuftrate this, notion. 



The expreflioir — , 



Wv 



which belongs to the fubtangent of a curve, rcprefents a 

 determinate fnnftion of x, when y is confidertd as a funtlion 

 whofe compofition in terms of.-.- is known : and if this lall 

 changes, the firft changes alfo. There will be, perhaps, fome 

 difficulty in conceiving how we can fubmit to calculation the 

 variability of a function whicli is only the abflratl depend- 

 ence in which feveral quanlitles are with regard to each 

 other: but this di'.Hculty i;^, removed, by confidcring that 

 the connetllon between the quantities 5' and a- changes, if 

 the firft be made to vary independently of tiic fecond. Tir.is, 

 in the example before us, if we fuppofe .v to remain the 



fame, and y and J to change, the rehition between .v and 



y mufl ncceflarily have changed alfo, fince thefe quantities 



'h 

 are the mimediatc confequences of that relation : -7 , in the 



<lx 



form —^^ may alone be made to vary, fince it depends only 



fly 



on one value of _)■ .• but, if an expvtflion affedlcd by the fign 



t-v 

 /"be confidcred, y and-;" mnH be made to vaiy at the fame 



time ; for it follows from tlic theory for the formation cf in- 

 tegrals, that the value of a like funttion depends on the con- 



fecutive values of y which are deduced from thofe of-r . 



ax 



•' It is evident that, to take under this point of view tlie 

 differential of any expreffion whatever, it is fufficitiit to 

 make y, Jy, d'y, &c. vary without altering x : but, in 

 treating this latter quantity, as variable as the firft, we ar- 

 rive at refults more general and fymmetrical than what are 

 otherwife obtained, and which lead to very interefting re- 

 maj'ks on the nature of the differential forms. For thefe 

 rtafons, we fhall adopt in this chapter the method cf mak- 

 ing X, dy, dy' vary. That the fymbols of this new fpecies 

 ol diffircntiaticin, in which x and y are confidcred as inde. 

 pendent, may not be confounded with the fymbols of the 

 j B 2 firft. 



