LATER BOOKS AND ARTICLES 247 



actually vanished, it is absurd to talk of any ratio 

 between them. It is true ; but we speak not here 

 of any ratio then existing between the quantities, 

 but of that ratio to which they have approached as 

 their liniit ; and that ratio stili remains. Thus, let 

 the increments of two quantities be denoted by 

 ax^ + inx and bx^ + nx ; then the limit of their ratio, 

 when ;r = o, is m : n \ for in every state of these 

 quantities, ax'^-\-mx : bx^-{-nx : : ax-{-7n : òx + n : : 

 (when x=o) m : n. As the quantities therefore 

 approach to nothing, the ratio approaches to that 

 oi m \ n as it's limit. We must therefore be careful 

 to distinguish between the ratio of two evanescent 

 quantities, and the limit of their ratio ; the former 

 ratio never arriving at the latter, as the quantities 

 vanish at the instant that such a circumstance is 

 about to take place." 



By aid of the binomial theorem, Vince finds the 

 fluxion of ;ir", when the fluxion of ;ir is given ; he then 

 finds the fluxion of xy by considering {x •\- yf = x'^ -\- 

 2xy+y'^y by which the fluxion of 2xy can be found 

 in terms of the fluxions (x-^yY, x^ aindy^. 



Agnesi — Colson — Hellins^ 1801 



214. The Analytical Institutions^ is the first cal- 

 culus that was written by a woman. The authoress 



^ Analytical Institutions , in four books : Originally written in 

 Italian, by Donna Maria Catana Agnesi, Professor of the Mathe- 

 maticks and Philosophy in the University of Bologna. Translated 

 into English by the late Rev. fohn Colson, A/. A., F.P.S., and Lticasian 

 Professor of the Mathematicks in the University of Cambridge. Now 

 first printed,from th» Translators Manuscript , under the inspection of 

 the Rev.fohn Hellins, B.D., F.R.S. Vols. i and il. London, rSoi. 



