122 



LIMITS AND FLUXIONS 



B K 



K 



c e 



FiG. 6. 



e E 



drawn to the base BG, will be terminate by EF. 

 When CA is not an aliquot part of AE, if we 

 divide the base into as many parts as may be, 

 there will be left a portion Eé?, which, let us 

 cali r. Then Qd=x-\-rv.a — r^{2à) and ali these 

 ordinates will be bounded by 

 Y.ddY. In the same way, 

 Y^d =a—x-j-rxa-~r-^ (2«), 

 and the ordinate will be 

 bounded by EddF. When 

 x=a, r vanishes, Cd= | a and 

 Kd= la. Hence the inscribed 

 and circumscribed figures do 

 then become equal to each 

 other, and to the triangle 

 ABE ; again, the limit is reached. 



Jurin takes Robins to task for asserting that 

 ** equality can properly subsist only between figures 

 distinct from each other." To Robins's query, 

 *'Does Philalethes here suppose the truth of Sir 

 Isaac Newton's demonstrations to depend on this 

 actual equality of the variable quantity and its 

 limit?" Jurin answers, " I do . . . In the manner 

 Mr. Robins defines, and treats of prime and ultimate 

 ratios, I allow his demonstrations to be just without 

 this actual equality. But Sir Isaac Newton does 

 not define and treat of prime and ultimate ratios, 

 in the same manner with Mr. Robins ; nor are 

 Mr. Robins's demonstrations at ali like Sir Isaac 

 Newton's demonstration " (p. 128). The inability 

 of our imagination to pursue the rectangles in reach- 



