C A T. C U T, IT 5, 



.V 



y 



X)'2 +*Z)' + VS.V 



x"X (i'!«+?«ii) putting m for the magiiilude 

 of the ratio of k to I 

 Sec. &c. &c. Ac. &c. &c. 



Thus It is nianifelt that the fluxionary anJ difTcrential 

 calcvihare legitimately derivable from out aiidlhc fame fource, 

 i);'i. general arithmetical proportion, and are in fart a branch 

 ot it, differing only from each other in words and in the me- 

 thods of notation. Sir Ifaac Newton lias not determiiied 



the fluxion of x~ when r has to a any ratio whatfoever. 



? . . 

 Neither has Mr. Leibnitz determined the differential of 



r 



x^when r and q are not integers but have to each other the 

 ratioofany tvvohomo:reneousmagnitudeswhatioever. But they 

 are obtained immediately by this method of deriving thefe 



r 



calculi, and of conrfe the fluxion and differential of .r' in 

 every poflible relation, that r can Hand into q, wliether it be 

 that ot two integers, two furds, two lines, two furfaces, or 

 two folids, &c. 



Calculus, Anthmit'ical, or Numeral, is the method of 

 performing arithmetical computations by numbers. See 

 Arithmetic. 



Calculus, Algclrntcal. See Calculus Itteralis and 

 Algebra. 



Calculus of denvat'wns, denotes a general method of 

 confidcring quantities, deriving themfvlvcs one from the 

 other, particularly developed and illuftrated by M. L. F. A. 

 Arbogall, of the National Inllitute of France, and profeflbr 

 of mathematics at Strafburgh, in a treatife entitled " Du 

 Calcul des Derivations," &c. 4to. Stralburgh, 1800. " To 

 form an idea (fays the author) of thefe derivations, it is to 

 be obferved that quantities or funftions, which are deduced 

 the one from the other, by an uniform proci.fs of operations, 

 are derived quantities ; fuch are the fucceflive differentials. 

 This idea may be extended, by confidering quantities that 

 are derived one from another, not in thtmfelves, but folely 

 in the operations which colle£t and bind them together j the 

 quantities themfelves being any whatfoever, arbitrary and in- 

 dependent. Thus, on the fuppofition that, out of many dif- 

 ferent letter*, the firlt enters folely into a funftion, while the 

 two next enter into the derivative of that function ; that the 

 firll three, by the fame law, enter into the derivative of the 

 derivative, and fo on ; we (hall have the derivatives in the ex- 

 tended fenfe which I have given to them. In my theory, 

 the quantities defignated by different letters are not derived 

 one from another; and the derivatives which I confider are 

 lefs the derivatives of quantities than of operations ; as al- 

 gebra is lefs a calculus of quantities than of operations, anth- 

 metical or geometrical, to be performed on quantities. De- 

 rivation is the operation by which a derivative is deduced from 

 that which precedes it, or^rom the fund\ion. The method 



of derivations, in grneral, ronfifls in Teiting the taw t!i:tt 

 coniecls togetlier the paiccls of any quantities whatever ; 

 and in making ufe- of this law as a method of Cilculation for 

 p i{rin:» tioni derivative to dciivalivc." 



In ord T to form the algorithm of dmvations, the autlior 

 lias intToduceJ new figns. Accordingly, as Lcibnitr hat 

 appropriated the lyinhol rf(o denote the operation forobiain- 

 ing the ditTLieitial of a quantity, M. Arbogaft employs the 

 fy bol U 111 the procefj for obtaining tlie derivative of a 

 qiantity : thus, D (.r'") = m .r"— «, D (n') — a^ \ and the 

 d velopcment of a fun^ion, l"(a^-i), isthercfore thu* ck- 

 prcffed : 



„ , DF« . D F« 



which, according to the notation in the difFerenlial Calculus, it 

 d^-tx. (i'V'i, . . . 



F« + ^-.v H \v' -f , &c. 



ria; \.^.^.aa, 



To avoid writing the f.iclors, 1.2.3, &c- '" the denominators, 



D' 

 let Dc" generally repreftnt — ; then, 



1.2.3 — " 

 F (a-f x) = Yu -f- DF«..v 4- D 'F»;.x= + Dc^ F&.x' -f , &c. 

 i his is the k::own form for the developement of the' fiin6\ion 

 of a binonii.I ; and the obje-6\ of the pnmary articles of the 

 prcfent treatife is to find a form for the developement of the 

 funiftioii of a polynomial; viz. a.-\-$x-\-yx^-\-'ix^-\-, tic^ 

 The metliod of obtaining it may be thus briefly explained :< — 

 When a. is variable, and Da=/3, write D.?i« for the deri- 

 vative : when a. is variable, and D3; = i, write D^z for the 

 derivative : therefore, 



(?F (« +/5.V) = I? (Fa 4- D.Fa..v -I- D, =.F*.*' +, &c.) : 



but ?F (a + ^x) = (pVa. + D.^iFa.A; + DcMpFa.x" +, &Ci 



K K 



put Fa =a, DFa = D. a, &c.and^(a -fD. a..v -f Dc'.iT. 



.x' +, &c.) = Ifa. + D.ifj. X + D^'. If a. *=, &c. 

 Now, to determine D. ^a, D\ <pa. &c. we have 



D. ?><j = T)ipa. D. a 



D=. <pa = D.{V).(pa) = D (D(p<2. D. a) = D<F<J. D'. <j 

 + D'?i<i. (D. fl)', and fo on. 

 To find (p (a -{-$x -}-y«' +. &c ), put it = ?a -f D. (fx.x 

 -j- Dr. ^a. «' -f-, &c; dcvelope D.?ix, D'. (pa, Ac; and after 

 developement put D . a = /i', Dr'. a = y, Di^a=:5, &c. 

 Hence it appears that ^ (a -f ^x -|- yx' +, &c.) may be con- 

 verted into a feries of the form A -f Bx + C.x" + Dx^ +&C.J 

 and the firll term A will — (pa. 



The coefficient of the fecond term will = D.(pa, or D. A. 

 The coefficient of the third term wi I = D'.ifz, or D=.A. 

 The coefficient of the n + i will = D". (pa, or D". A ; 

 provided that, after the derivations have been made, we put in 

 the refults ^ for D. *, y for Dc'. a, &c. B for D.A. C for 

 D'".A, &c. 



The coefficient of x" or A" in the fcies A -f- Bx -f, &c. 

 equals He". (?«, which is = D(pa. JJc" — ». /S -{- D' Pa. De"— ' 

 /3' +, &c.--— Dc« iPa. /9«. 



The method of proof is this :•— 

 put T = -f yx + ^x' -f , !kc. 

 then ?(a-|-/Sx -f yx' +, S;c.) = (p (« -f- rx) 

 = (Pa+Dfa.TX -f-Dc'?a. ir'x' +< ^^■ 



but T, s-', v' -r" are funrtions of the polynomial, 



'^ + 0-'+y«\ &c. 

 .-. generally ir"=:/3'! -f D. /S'.x-f D>'. jS'.x=4-, &c. (2) 

 Write, therefore, in feries (1), for ir, t", t' — v, the values 

 refulting from (2) ; co Icct the terms afftfted with the fame 

 power of X ; and it will ealily appear that the coefficient of 

 X", or A", is fuch as we have flated it to be. 



In this method is comprehended, as is evident, the form 

 for the developement oi a polynomial raifed to any power 

 i which 



