42 LIMITS AND FLUXIONS 



" Magnitude is divisible in infinitum, and the 

 Parts after this infinite Division, being infinitely 

 little, are what Analysts cali Moments or Differ- 

 ences ; And if we consider Magnitude as Indeter- 

 minate and perpetually Increasing or Decreasing, 

 then the infinitely little Increment or Decrement is 

 call'd the Fluxion of that Magnitude or Quantity : 

 And whether they be called Moments, Differences 

 or Fluxions, they are stili suppos'd to have the 

 same Proportion to their Whole's, as a Finite 

 Number has to an Infinite ; 

 or as a finite Space has to 

 an infinite Space. Now those 

 infinitely little Parts being 

 extended, are again infinitely 

 Divisible ; and these infinitely 

 little Parts of an infinitelylittle 

 Part of a given Ouantity, are 



tlG. 3. O ^ J 1 



by Geometers call'd Injìnite- 

 siince Infinitesiììiaruiìi or Fluxions of Fluxìons. 

 Again, one of those infinitely little Parts may be 

 conceiv'd to be Divided into an infinite Number of 

 Parts which are call'd Third Fluxions, etc. " 



He endeavours to justify this doctrine by illus- 

 trations. The angle of contact FAG formed by the 

 line AE and the ordinary parabola AG, is less than 

 any rectilineal angle ; the angle P'AD, formed by 

 AE with the cubical parabola AD, is infinitely 

 less than the angle FAG, and so on. Hayes 

 defines the doctrine of Fluxions as the " Arith- 

 metick of infinitely small Increments or Decrements 



