CALCULUS. 



cefs of tlic maci:iituJc, which has to B the triplicate ratio 

 of A + Nto B above the magnitude, which has to B the 

 triplicate rati.) of A to B, equal to the geometrical exprcf- 



fion 



3A^N+;AN'+N\ 



B= 



AnJ he thus finds, in general, 



that the excefs of the magnitude, which has to B fuch a 

 ratio, as has to the ratio of A + N to B the ratio of r to q, 

 (when r has to q any given ratio whatever,) above the 

 migiMtiide, which has to B fuch a ratio, as has to the ratio 

 of A to B, the fsnie ratio of r to 7 is geometrically ex- 



prefieJ by - • A^T" " N + 



r-q_ 

 -1 



A-r • N' + 



Zq 



rj-zq 

 37 



B— 



'■—5? 



A— ^ • N' + &c. 



r—i 

 B q 



Piecifely in the fame manner he makes it appear, that the 

 excefs of the magnitude, which has to B fuch a ratio, as has 

 to the ratio of A to B the ratio of r to q, above the magni- 

 tude, which has to B fuch a ratio, as has to the ratio of 

 ,A — N to B the ratio of." to q is geometrically exprcfled by 

 T-n '—-'I 



AT- • N- 



2? 



N' + 



-'/ 



r-jr 



iq 





B , 



r— ?'/ 



N^ 



H &c. 



r— £ 



But if A + N and A — N ftand to A in relations nearer 

 to that of equality than by any given or affigned magnitudes 

 of the fame kind, each of thefe general expreffions becomes 



1- a'^^-n 



• Thus he calls the antecedental of the 



■ •— 7 

 B — 

 magnitude, which has to B fuch a ratio, as has to the ratio 

 of A to B the ratio of r to q. 



n • 



Now if N, the antecedental of A, be denoted by A or A 

 (for the notation does not at all alter the cafe) the an- 



r — q 



N 



- • A- 



r — q 



-tecedental- 



B ? 



■ becomes 



B" 



— ? 



or - . ___ ' A and has to A or A the ratio of 



q '^^. 



B <i 

 - . — . B. If n = I and r=2, 3, 4, &c. it gives 



J r — q 



B— 



2 A A . jA'A, 4A~A , jA^A , &c. refpeaively. If 

 B B' B^ B* 



I a a o 



a_Ja, a 'a 



r = 1 and y = 2, 3, 4, &c. it gives rg-' | ' 



2B 3B 



A_l4-. &c, refpeaively. And if y = i and r =: ^Z ^. 



B 



V ?> 



V4. 





&c. it gives ^3 



a"" a 



B 



v/^ 



5/3 — I n 



A^ A 



V4 • A 



\/^-i <• 



B 



V3— ' 



A 



B' 



&c. refpcc- 



tively, and fo on. 



Mr. Glcnie then makes ufe of the lad of thefc two formii- 

 Ie, which is the 5th of theorem 3d, to come at the fame 

 coiiclullons. By it, when A Hands to B in a relation nearer 

 to that of equality than by any given or afligned magnitude 

 of the fame kind, the excefs of the magnitude, (which hag 

 to B fuch a ratio as has to the ratio of A to B the ratio of 



rtoy), above B is generally exprefTed by- -A-B or 



? 



by - . A or by - . A, fmce in this relation of (A to B), 



n 



A — B may be denoted by A or A ; which cxprefTion is al- 

 ways as the meafure or quantity of the ratio, that the faid 

 formula has to B, whatever be the relation of A to B. For 



in the fimple, duplicate, triplicate ratios, &c. it gives A, 2A, 



3A, &c. and in the fubduplicate, fubtriplicate ratios, &c. 



A, A^ &c. or A 



A 

 3 



&c. But if the ratio of 



A to A or - ' A to A, which denotes the relation be- 



1 

 twcen the quantities of the ratios, that the magnitude ex- 

 prefTed by this formula and A have refpeftively to B, be 

 compounded with the ratio of thefe magnitudes themfelves, 

 when A has to B any ratio whatever, or with that of 

 r r 



A.q r Aq • • 



— — — to A we get the ratio - • —-—7 A to A . A, which 

 ' — q ° g J — q 



B— B~7 



'— ? 



r A ? 

 is equal to the ratio of " • ^, ^ 



B^~ 



to B the ratio of the 



Af 

 antecedentals of and A as found above. 



'— ? 



B — 



It is certainly worthy of remark, that the antecedental of 



any 



fuch magnitude as ^ is a fourth proportional to B 



T — q 



B — 



the ftandard of comparifon, the magnitude itfelf, and the 

 antecedental of the magnitude of the ratio, which it has to 

 the llandard B. Thus a fourth proportional to B, A, and 



• « 



^ • A Js A or A the antecedental of A.; a. fourth pro. 

 A 



portional 



