TEXr-BOOKS OF MIDDLE OFCENTUR Y 201 



Israel Lyons, 1758 



175. Lyons was a mathematician and botanist. 

 His Trcatise of Fluxions, London, 1758, is dedi- 

 cated to Robert Smith, Master of Trinity College, 

 Cambridge, " being the first Essay of a young and 

 unpractised Writer " which ' ' ovved its first rude 

 Beginning to the early Encouragement " receìved 

 from the Master, as the author modestly states. 

 His treatment is geometrie. He says : "I reject 

 no Ouantities as infinitely smaller than the rest, 

 nor suppose difìferent Orders of Infinitesimals and 

 infinitely great Ouantities. But consider the Ratio 

 of the Fluxions as the same as that of the con- 

 temporaneous Increments, and take Part of the 

 Increment before and Part after the Fluent is 

 arrived at the Term, where we want the Fluxion, 

 since it is not the Increment after, or the Increment 

 before that we want, but at the very instant, which 

 can no otherwise be found but by considering Part 

 of the Increment before and Part after" (Preface). 

 Fluxions are defined as velocities. '*The moments 

 of quantities are the indefinitely small parts, by the 

 addition or subtraction of which, in equal particles 

 of time,they are continually increased or diminished. " 

 The author proves the proposition : **The indefi- 

 nitely small spaces described in equal indefinitely 

 small times are as the velocities," since, " when 

 the time is diminished ad infinitum, the difference 

 of the velocities at the beginning and ending of 

 that time will vanish." If two flowing quantities 



