I70 LIMITS AND FLUXIONS 



the ratio of the fluxions of x and x^ thus : Let the 

 points iìi and n move so that the distance h de- 

 scribed by n shall always equal the square of the 

 distance g described by ni in the same tinne. Then 

 (AR)2 = CS, (AR-Rr)2=:Cj, and jS = 2AR x Rr- 

 (Rr)=^. But jS is described with accelerated velocity 

 when ;;/ moves uniformly, hence ^S will be 'Mess 

 than that which would be uniformly described in 

 the same time with the Velocity at the point S, 

 and greater than that which would be described 



with the Velocity at the 

 ,g point s ; and therefore 

 must be equal to the Dis- 

 tance that would be uni- 

 formly described with the 

 Velocity at another point 

 e posited somewhere be- 

 tween S and s, in the same 

 Time that the other point m is moving over the 

 Distance rR ; therefore rR : 2ARxRr-(Rr)2 : : 

 ^:^(2AR — Rr), the Distance that* would be de- 

 scribed with the Velocity of n^ at the point e, in the 

 same Time that in is moving over the Distance g : 

 Now therefore when the points r and s coincide 

 with R and S, then will e coincide with S ; . . . and 

 consequently (2AR — Rr)^^ will then . . . become 

 2ARx^, equal to h the required Distance." The 

 criticai part of this proof is ' ' when the points r and s 

 coincide with R and S, then will e coincide with S." 

 A modification of this proof is applied to x"". 



Simpson's text marks a departure from Newton 



