CALCULUS. 



poftloTial to B, A>, and ^^ -'"^ j^ 2A. A . and a fourth 

 BAB 



A?" 

 1 



proportional to B, _ 



and 



A y A 



B ? 



Alfo I : A' 



2 A 



A 



B .A 



zA . A ; and i 



And it is cafily (hewn, from the principles he ha« laid 



V^ V 



A^ ■ ^E 



down, that the antecedental of — — ■ „ " v 



V-B V-B ^ 



D~B~ ^"ir 



V 



,VM + MV\ B /VM + MV\ 



I tT, / or — r^ X \ r, J or that the antece- 



A^' 



^B 



Av 



A' : : J^- 3A' .A. 

 A ' 

 By means of the firfl formula in theorem i of his I'liiver- 

 fal Compaiifon, which cxprcfTes the rtiagnitiide, that has to 

 B fuch a ratio, as arifts by compounding any number of 

 ratios C to D, E to F, &:c. with the ratio of A to B, he 



,fA,C ^ AC + CAor 

 determines the antecedental ot — ~- to be -:- 



dtntal of - 's _ x 

 B"-i B"' 



(±i+^). For. tf M denote 



B 



the magnitude of the ratio A : B to the ftandard of com. 



parifon B, "v M will be the magnitude of the ratio of 



A" A" A" 



— to B. But -— : B : : the antecedental of -rr i 

 lJ=-i B-i Bi--» 



a a 



the antecedental of i; M (■uM+M'u). Wherefore the antc- 



A C + C A , and that of ^;,^;.^ to be ^^.E 



. , . A-j . A" / 



cedental oi — u — x ( 



tM + Mtx A" 

 B— ^ °' Bv-i ^ 



D 



D.F 



+ 



DF 



AE . C ^ CE . A „,. ACE + AEC + CEA 



■, and 



D F D F D F 



fo on, fuppofing the confequents D, F, &c. of the ratios 



to be invariable. And it is obftrvable, that the anteceden- 



AC 

 jgl (jf is equal to a fourth proportional to B, the mag- 



A C 



iiitude ' itfelf, and the antecedental of the magnitude 



of the ratio of A to B, together v/ith a fourth proportional 



A C 

 to D, the magnitude -=- and the antecedental of the 



maffnitude of the ratio of C to D ; that the antecedental of 

 A c^ V ACE 



is eaual to a fourth propottional to B, ^^^ and 



D F. E) F 



the antecedental of the magnitude of the ratio of A to B, 



ACE 

 together with a fourth proportional to D, ' and the 



an'ecedental of the magnitude of the ratio of C to D, to- 



ACE 

 gether with a fourth proportional to F, ■ ' and the 



antecedental of the magnitude of the ratio of E to F ; and 



fo on. 



He makes ufe of the firft formula in Theorem 2, Univer- 

 fal Compavifon, which expreffes the magnitude, tliat has to 

 B fuch a ratio, as arifes by decompounding any number of 

 ratios C to D, E to F, &c. with the ratio of A to B to de- 



^ , . AD, ADF , r c 

 termine the antecedental ot _— ' and lo on tor any 



number of ratios. , 7 , 



And here it is alfo obfervable, that the antecedental of 



AD CDA-ADC or CDA-ADC . , , ,, „ 



•^■^ or "^ 14 equal to tlie 



T" c- c^ ^ 



excefs of a fourth proportional to B, __, and the an- 

 tecedental of the magnitude of the ratio of A to 

 B above a fourth proportional to D, — -, and the 



antecedental of the magnitude of the ratio of C to D ; and 

 fo on. 



/i.M-f-Mf^, . 



The antecedents and their autecedentals will therefore 

 (land thus, B being the llandard of comparifon. 



Antecedent. Antecedental. 



a-^b; 



A or A 



a 

 2AA 2AA 



■b-°^-b- 



A I 2NA , . 



or 2AH — — (puttmg 



N forA-B), or, &c. 



A a a a 



A' or A.-^.B - 2AA or 2BA + 2NA, or &c. 



A3 ^ A- 

 B'°''^-F 



A= 

 ■A'or A.g.B' 



jA'A 3AA , 3ANA J , 



or ^-^ + ^-7- or 3A+ 



B' 



B 



B' 



6NA . 3N'A 



B 



i- ^-gT > °'^ &c. 



A? 



^nd in 3 _£tJ 

 eneral 3 '^jj 



B» 



3A'A, or jBAA+jANA, or 3BA 



4-6BNA+3N'A, or &c. 



/■■ A y .A 

 j.bT" 



a 

 ;AA 



r r-lq ANA 



ryB "^ 7; ■ ~ ' B'. 



a 



I r r-ln r-%q AN'A , „ 

 ^ -.—L.^. ___ +&c. 

 1 9 '1 B' 



a n 



>A f r-q NA r 

 or — + -.^.__4-_. 

 q q q a q 



q 2q B' 



.or fee. &c. 



Antecedent. 



