[ 34 ] 



others had demonftratcd with hmitations. In the analytical de- 

 monftration however which he has given of this thorem, 

 certain quantities are omitted as being indefinitely little ; and 

 therefore it is not delivered with that axf^fcc, which is required in 

 fubje£is of this nature. Fermat has given us a fynthetic de- 

 monftration of this theorem, which Dodor Horfely has infert- 

 ed in his notes on this trad of Newton, but it is fo tedious and 

 prolix, that even the analytical is preferable to it. I fhall here 

 give a fynthetical demonftration alfo of the fame general pro- 

 pofition on the principles of prime and ultimate ratios^ a method of 

 rea-foning which Newton fcems to have had fome idea of even at 

 the time of his writing this Analyfis, in the year 1669, though 

 probably he did not bring it to perfedion until eighteen years after, 

 when he firft publiflied the Principia. 



RULE I. QUAD, of SIMPLE CURVES. 



Plate III. Let the bafe AB of any curve AD have BD for its per- 



Fig. 2. , 



pendicular ordinate ; and let AB=X, BD=Y ; alfo let a be a given 

 quantity, and m, n, whole numbers. Then if Y=aX"S", itfhall 



m-}-n 



be, area ABD = — ^X~^. 

 m+n 



Let DC, AC, drawn through D and A parallel to AB and DB, 

 meet in C ; draw the ordinate d b indefinitely near to DB, 

 meeting CD in s ; and through d draw r p parallel to AB. Since 



Y = aX " , the moment of Y will be equal to the moment 



of 



