JURIN V. ROBINS AND PEM BERTO N 121 



Let the area of the rectangle with EA as 

 base and CD as height = sum of the circumscribed 

 parallelograms (not drawn in the figure) standing on 

 CA and upon as many other parts of EA as can be 

 taken equal to CA and adjoining to it ; also let 

 GDD be the curve traced by the movable point D. 

 Then as the curvilineal, the inscribed, and the 

 circumscribed figures are respectively equal to EA x 

 AF, EA X C<^, EA x CD, these figures must be pro- 

 portional to AF, C<:/, and CD. These three lines 

 will ' ' be equal to one another at the end of the 

 finite time." Now since C<:/ and CD approach each 

 other, during a finite time, within less than any 

 given distance before the end of that time, these 

 three lines will, by that Leniina, be equal to one 

 another at the end of the finite time. The limit is 

 reached (p. 1 14). 



133. As a further illustration, Jurin takes a 

 rectilinear figure, the right triangle ABE, where 

 EA = AB = ^, AF=J^, EC=;r, the point C mov- 

 ing from E to A as before. Upon AC as a base, 

 imagine an inscribed rectangle (height CH), and 

 a circumscribed rectangle (height CK). As in 

 the previous figure, imagine other inscribed and 

 circumscribed rectangles, standing upon as many 

 other parts of EA as can be taken equal to CA, 

 and adjoining to it in order. When CA is an 

 aliquot part of AE, then a x Cd is the sum of the 

 inscribed rectangles and a x CD is the sum of the 

 circumscribed rectangles, where Cd—x / 2, and 

 C'D = a-x I 2. Let K^/=CD. The ordinate Kd, 



