220 LIMITS AND FLUXIONS 



converges to o, in the ultimate State before it 

 vanishes, x^'^i \ but says, when,r entirely vanishes, 

 or becomes absolutely of no Value, that then 

 x^ — o^ = o\ But being supposed no Ouantity is 

 contradicted by Algebraic Computation, which is 

 general and retains o, in a Mathematica! Sense, for 

 a Ouantity in the Scale, as much as any other 

 Figure or Literal by which Ouantity is denoted and 

 compared. . . . Waltoniensis farther observes that 

 Fluxions are the Limits to which the Ratios of the 

 Increments or Decrements of Quantities converge, 

 and are assignable from the Principles of Motion 

 only {uniform^ accelerateci^ and retarded) ; and thinks 

 the Doctrine has nothing to do with infinitely small 

 Quantities, First and Last Ratios ; and that only 

 finite Quantities need be introduced — ' to avoid 

 Disputes, and the dark Mists spread over the Pro- 

 cess, different to the demonstrative Lights of the 

 Antients.' But Motion refers to the Spaces passed 

 over, by which it is comprehended, measured, and 

 compared : And tho' Mr. Simpson has pretended to 

 deduce the Ratios of Fluxions of Quantities without 

 the use of indefinitely small Quantities (see his New 

 Doctrine and Application of Fluxions) yet the Motion 

 of his Points along the Lines answers to them by 

 the indefinitely small Spaces described together, 

 and are to the same Effect as Quantities taken in- 

 definitely small ; which Sir Isaac Newton himself 

 introduced to illustrate the Quantity of relative 

 Motion by. Fluxions^ as instantaneous Velocities, 

 are also as the Increases or Quantities of Space 



