I02 LIMITS AND FLUXIONS 



end of the hour ; . . . at the end of the hour, the 

 two quantities must become equal. " Further, " by 

 taking the consideration of a finite time, Sir Isaac 

 Newton is able to assign a point of time, at which 

 he can demonstrate the quantities to be actually 

 equal." Consider, says Jurin, the ordinate to a 

 point of a hyperbola and that ordinate continued 

 to the asymptote : they do not become equal in 

 a finite time; Newton's Lemma is, **with great 

 judgment, so worded on purpose, as necessarily to 

 exclude this and such like cases. " Thus Newton's 

 inscribed and circumscribed rectangles of Principia^ 

 Lib. I, Sec. I, Lemma 2 (fig. i in our § 9), were 

 thought by Nieuwentiit and others never to be 

 capable of coincidence with the curve (say the 

 quadrant of a circle) ; but Jurin assujues the varia- 

 tion to be of such a nature that the limit 

 is actually reached, as demanded by Newton's 

 Lemma. For, suppose a point to move on the 

 horizontal radius from the circumference to the 

 centre A in one hour ; suppose also that, when 

 this moving point is at B on that radius, there 

 be two rectangles described upon AB (one in- 

 scribed, the other circumscribed), and that upon 

 every other part of the horizontal radius that is 

 equal to AB, namely the parts BC, CD, DE, taken 

 in order, rectangles be similarly erected ''at the 

 same point of time," then as the moving point 

 nears the centre, the rectangles diminish in size 

 and increase in number, and they must together 

 become equal to the quadrant at the end of the 



