CALCULUS. 



<!/. 



~; and thus the derived hxneC\OT,i f'x, f'x, f"'x, &c. dx=x-^. 



&c. 



mav be repreleiiteu by the quantities ^-r-> —7—1 , . 1 

 '^ ' dx dx ax' 



fo that the dcvclopemcnt of/(.v + /-') takes this foim, 



dx 1.2. dx 



the celebrated theorem of Taylor. 



Cai-Cvlus, dijfn-ent'io-diJfcrcntmU is a method of differ- 

 encinir diflcrt-ntial quantities. As the fign of a differential 

 is the letter c/ prefixed to the quantity, fo that dx is the dif- 

 ferential of .V, that of a differential of dx is ddx, and the 

 differential of fl'(/» is ^i-/(/A-, &c. ; fimihir to firft, fecond, and 



third, &c. fluxions, x, x, x, &c. : thus we have degrees of 

 differentials. The differential of an ordinary quantity is a 

 differential of the firft order or degree, as dx : that of the 

 lecond degree is ddx, Sic. The rules for differentials are 

 the fame with tiiofe for fluxions. See Fluxions. 



Calculus exponcntialis, is a method of differencing ex- 

 ponential quantities, or of finding and fumming up the dif- 

 ferentials or moments of exponential quantities ; or at leaft 

 bringing them to geometrical conltruftions. 



By exponential quantity, is here underftood a power, 

 whofe exponent is variable ; v. g. a'', .v'', a}", x'J, S:c. where 

 the exponent .vdoes not denote the fame in all the points of 

 a cuive, but in fome flands for 2, in others for 3, in others 

 for 5, Sic. 



To dijfarence an exponential quantity : there is nothing re- 

 quired but to reduce the exponential qualities to logarithmic 

 «>nes ; which done, the differencing is managed as in loga- 

 rithmic quantities Thus, fuppofc the differential of the 



exponential quality .f7 required, let 



Then will j/.\-=:.'a 



, vdx dz 

 lxdy + - 



2 V d V 



That is, s,7/.i:</y + x.v? — '(/.v=:./3;. BernouiUi Opera, torn, 

 i. p. 18.3. See Exponential. 



CALcuLUs_y?ux/(3;w/. See Fluxions. 



Calculus integralis, ax fummatorius, is a method of in- 

 tegrating, or fumming up moments, or differential quanti- 

 ties; i.e. from a differential quantity given, to find the 

 quantity from whofe differencing the given differential re- 

 fults. 



The integral calculus, therefore, is the iuverfe of the dif- 

 ferential one: whence the Englifli, who ufually call the 

 differential ■cat\\\oA. Jluxions, give this calculus, which af- 

 cends from the fluxions, to the flowing or variable quanti- 

 ties : or, as foreigners exprefs it, from the differences to the 

 lums ;■ by the name of the iwuerfe method of tlvxiovis. 



Hence, the integration is known to be julUy performed, 

 if the quantity found, according to the rules of the differen- 

 tial calculus, being differenced, produce that propofed to 

 be fummed. 



Suppofc/the fign of the fum, or integral quantity ; then 

 fydx will denote the fum, or integral of the differential 

 ydx. 



To ini!';s;rjie, or fum up a dijfcrential quantity. It is de- 

 monRratod, fivll, that yi/.x=.v ; fecondly, /(</.■< 4-rt'_v)=:.x- 

 ■\-y: thiidly, / (.V </)■-!-;• i/.ie)=xy.- fourthly, /(«,<'! -J 



fifthly, /{n;m)x -dx=x — : fixthly, / 



7ft rtt 



{ydx^xdy):y' = x:y. Of thefe, the fourth and fifth'' 

 cnfes are the moll frequent; wherein the tlifferent'al quan- 

 tity is integrated, by adding a variable unity to the expu-! 

 nent, and dividing the fum by the ne«: exponent multiplied 

 into the differential of the root ; v. g. the fourth cafe, by 

 m — {i + i) dx, I. e. by mdx. 



If the differential quantity to be integrated, doth not come 

 under any of thefc formulas-, it muft either be reduced to an 

 integral finite, or an infinite feries, each of whofe terms 

 may be fummed. 



It may be here obferved, that, as in the analyfis of fiu'tes, 

 any quantity may be raifed to any degree of power; but 

 vice verfa, the root cannot be extrafted out of any number 

 required : fo in the analyfis of infinites, any variable or 

 flowing quantity may be differenced ; but, vice verfa, any 

 differential cannot be integr.Tted. And as, in the analyfis 

 of finites, we aie not yet arrived at a method of extra6fi:;g 

 the roots of all equations ; fo neither has the integral calcu- 

 lus arrived at its perfection : and as in the former we are 

 obliged to have recourfe to approximation ; fo in the latter 

 we have recourfe to infinite feries, where we cannot attain 

 to a perfeft integration. 



Tiie firft traces of the integral calculus are to he found in 

 the arithmetic of infinites of Dr. Wallis. The author, by 

 fumming feries of rational ordinates, was enabled to affign 

 the quadratures of the curves to which they belonged. 

 Newton, the inventor of the fluxionary method, advanced 

 far bevond Wallis, and afligned the quadrature of curves 

 to which tiie ordinates were irrational. What th'S great 

 man performed with regard to the doctrine o~ fluents, or 

 the integral calculus, is to be found in his treatife, " De 

 Q_uadratura Curvarum." In his " Prineipia" he concealed 

 his analyfis, and adhered to the manner of the ancients. It 

 does not appear by what method be folved the 54th propo- 

 fition of the 2d book of the " Prineipia" concerning the fo- 

 lid of leaft refinance; whether he tffefted it by a particular 

 artifice, or whether he really poffeffed the calculus of varia- 

 tions, of which Leibnitz and the Bernouilhs are now cl'- 

 teemed the inventors. The mathematicians to whom the in- 

 tegral calculus is chiefly indebted for its improvement, are 

 John BernouiUi, who integrated rational fraffions ; James 

 BernouiUi, who integrated the fluxional equation, y -f- 

 P v.v=: Qj*."" ; Cotes, who publiflied in 17 14, " Theorc- 

 mata turn Logometrica turn Trigonometrica " ; Ricati, who 

 integrated the fluxional equation _y-)-a_y' x- =^ Q^-V ; 

 Maclaurin, author of a treatife in 2 volumes, 4to. ; Simpfon, 

 Fontaine, Clairaut, D'Alembert, and Euler, whofe re- 

 fearches on the integral calculus occur in the volumes of the 

 academies of Paris, I'erlln, Turin, and Peterfburgh. The 

 laft author publifhedin 1768 his " Inllitutioncs Calculi In- 

 tegralis," highly enriched with original inventions ; and in 

 the fame year Le Seur and Jacquier publifhed a work in two 

 volumes, 410. which fuperfeded a work of M. Bougain- 

 ville, publifhed in 1754, intended as a fupplement to the 

 " Analyfe des infinitement Petits of the Marquis de L° 

 Hofpital. Condorcet, La Grange, La Place, Monge, and 

 Le Gendre, have alfo made confiderable additions to tlie 

 integral calculus. M. La Croix has likewife publifhed a 

 comprehenfive and valuable treatife on the differential and 

 integral calculus ( Traite du Calcul Differentiel, &c. 2 vols. 

 4to. Paris). In the plan of this work it is propofed to com- 

 prehend and fyllematize all that has been written on the 

 differential and integral calculus. It commences with a hif. 

 tory of the fubjeft which properly originates with the tinnes 

 of Euclid and Archimedes ; becaufe the methods of ex- 

 5 hauiliQiia 



