156 LIMITS AND FLUXIONS 



Hodgson also says in his Introduction that most 

 books on fluxions that bave hitherto appeared 

 proceeded on the same principles as the Difìferential 

 Calculus, so that " by calling a Differential a 

 Fluxion^ and a second Differential a second Fluxion, 

 etc. , they bave . . . confusedly jumbled the Methods 

 together," although the principles are really "very 

 difìferent. " ' ' The Differential Method teaches us 

 to consider Magnitudes as made up of an infinite 

 Number of very small constituent Parts put 

 together ; whereas the Fluxionary Method teaches 

 US to consider Magnitudes as generated by Motion 

 ... ; so that to cali a Diffei'ential a Fluxion, or a 

 Fluxion a Differential is an Abuse of Terms. " In 

 the method of fluxions, " Quantities are rejected, 

 because they really vanish " ; in the differential 

 method they are rejected '* because they are in- 

 finitely small." Hodgson adds that he always used 

 the differential method '''till I became acquainted 

 with the Fluxionary Method." He considers fluxions 

 of quantities (p. 50) '*in the first Ratio of their 

 nascent Auginents, or in the last ratio of their 

 evanescent Decrements,'" and gives an able and faithful 

 exposition of Newton's ideas as found in his 

 Quadrature of Curves. He cannot think ''there is 

 any more difificulty in conceiving or forming an 

 adequate Notion of a nascent or evanescent Quantity, 

 than there is of a Mathematica! Point " (p. xi). In 

 explaining the derivation of the fluxion of the 

 product xy^z he apparently permits (p. xv) the 

 small quantity ^ to " vanish," and thereupon divides 



