4 NEWTON S PKINCIPIA. 



geometers, as well as their namesakes) still proceed. But 

 speculative mathematicians being aware of the general 

 properties of the lines they have to examine, and these 

 being regularly formed, which the boundary of the field 

 is not, they could calculate the relations to each other of 

 the sides of the rectangles into which they divided the 

 figure, and could thus form series of rectilinear figures 

 diminishing in size, and which series might be carried to 

 any length so as ultimately to exhaust the curvilinear area. 

 Thus ABC being a semicircle, it was easy to find the area 



of the semihexagon or three equilateral triangles ADF, 

 FDE, and DEC, and then of the triangles FBExS, 

 and again of the triangles F O B x 6, and so on ; so that 

 the radius AD being called r, there was obtained a 



series of this form, | r 2 V3 + f r 2 (2- A/3) +~ ^2 

 /2 Va 2 ^~2^\ +* &c. : And thus we have also the 



approximation to the length of the circle. 



But the extreme cumbrousness of this calculus, which 

 is still more unmanageable in other curves where the 

 radii are not, as in the case of the circle, equal, made it 

 necessary to find some other method ; and geometricians 

 accordingly examined the laws by which the areas increase 

 in each curve, so that by adding all those innumerable 

 increments together their sum might give the exact space 



* The three first terms give 3.10582 ; the seven first come very near the 

 ordinary approximation, 3.14159, for they give 3.14144. 



