[ 435 1 



the radius, as the fquare. of FG, or the rectangle 

 EFC, to the rectangle under AG (or AC) and FB, 

 But, EB being to B F as BF to BC, by conversion, 

 E B is to EFasBFtoFC, and alfo, by taking the 

 difference of the antecedents and of the confequents, 

 E'F is to twice AF as BF to FC j and twice AFB 

 is equal to E F C. 



Nov/, let the triangle B A H be formed, where 

 the angle BAH is greater than BAG. Here, the 

 perpendicular HI being drawn, the rectangle under 

 the fine of B A H and the tangent of ABH will be 

 to the fquare of the radius, as the rectangle EIC to 

 •lie rectangle under AC, IB. But IF is to FB 

 as 2 A F I to 2 A F B, or EFC; and 2 A F 1 

 is greater than A F? — A I? ; alfo AF?— A I? to- 

 gether with EFC, is equal to EIC; therefore, by 

 cqmpofition, the ratio of IB to BF is greater than 

 that of E I C to EFC; and the ratio of A C x I B 

 to ACxF'B greater than that of E I C to EFC: 

 alfo, by permutation, the ratio of ACxIB to EIC 

 greater than the ratio of ACxFB to EFC. But 

 the hrft of thefe ratios is the fame with that of the 

 fquare of the radius to the rectangle under the fine of 

 BAH and the tangent of ABH: and the latter is 

 the fame with that of the fquare of the radius to the 

 rectangle under the fine, of B A G and the tangent 

 of ABG ; therefore, 'the latter of thefe two rectangles 

 is greater than the other. 



Again, let the triangle BAK be formed, with the 

 angle BAK lefs than BAG, and the perpendicular 

 K L be drawn. Then the rectangle under the fine of 

 BAK and the tangent of ABK is to the fquare of 

 the radius, as the fquare of KL to the rectangle under 



Kkk 2 AC, 



