TEXT-BOOKS OF MIDDLEOF CENTUR V 195 



his friends who was too modest to put his name to 

 it. (See our §§ 158, 160, 163.) Simpson used his 

 friend's manner of determining the fluxion of a rect- 

 angle and of illustrating fluxions of higher order. 

 Simpson defines a fluxion as foUows : 



'' The Magnitude by which any Flowing Ouantity 

 would be uniformly increased, in a given Portion of 

 Time, with the generating Celerity at any proposed 

 Position, or Instant (was it from thence to continue 

 invariable), is the Fluxion of the said Ouantity at 

 that Position, or Instant." 



The derivation of the fluxion of xf is explained 

 after the manner adopted by John Rowe, both 

 authors being indebted for it to the author of An 

 Explanation of Fluxions in a Short Essay on the 

 Theory. The same definitions and explanations of 

 the fundamentals are given by Thomas Simpson 

 in the last part of his Select Exerciscs for Young 

 Proficients in the M athernaticks , 1752. In the 

 preface to his Fluxions of 1750, Simpson touches 

 some points of philosophic interest. He says : 



'* By taking Fluxions as ineer Velocities^ the 

 Imagination is confin'd, as it were, to a Point, and 

 without proper Care insensibly involv'd in meta- 

 physical Difificulties : But according tò our Method 

 of conceiving and explaining the Matter, less 

 Caution in the Learner is necessary, and the higher 

 Orders of Fluxions are render'd much more easy 

 and intelligible — Besides, tho' Sir Isaac Newton 

 defines Fluxions to be tìic Velocities of Moiions, 

 yet he hath Recourse to the Increments, or 



