JURIN V. ROBINS AND PEMBERTON 103 



hour. Jurin points out that he has introduced 

 bere ali the suppositions of Newton's first Lemma^ 

 namely that, (i) the two figures tend constantly to 

 equality, (2) during one hour, i.e. a finite time, (3) 

 and come nearer to one another than to any gìven 

 difìference, (4) before the end of the hour, i.e, 

 before the end of a finite time. Jurin continues : 



" If any man shall say, that a right-angled 

 figure, inscribed in a curvilineal one, can never be 

 equal to that curvilineal figure ; much less to 

 another right-lined figure, circumscribed about the 

 curve ; I agree with him. I am ready to own that, 

 during the hour, these figures are one inscribed, 

 and the other circumscribed ; that neither of them 

 is equal to the curvilineal figure, much less one to 

 another. But then, on the other band, it must be 

 granted me, that, at the instant the hour expires, 

 there is no longer any inscribed or circumscribed 

 figure ; but each of them coincides with the curvi- 

 lineal figure, which is the limit, the limes curvi- 

 lìneus, at which they then arrive. " 



123. Jurin thereupon proceeds to Lemma 7 of 

 Book I, Section i in Newton's Principia, which, he 

 says, requires additional consideration. It relates 

 to fig. zj, where ACB is any are and "the points A 

 and B approach one another and meet." Lemma 7, 

 in Andrew Motte's translation, reads as follows : — 



''The same things being supposed ; I say, that 

 the ultimate ratio of the are, chord, and tangent, 

 any one to any other, is the ratio of equality." 



Jurin says that bere the chord AB, the arch ACB, 



