134 SCIENCE IN GRECO-ROMAN ANTIQUITY 



new life by Archimedes, who made successful and 

 fruitful applications of it. Eudoxus had contented 

 himself with showing by what lemmas a certain figure 

 may be considered as the limit of another figure in- 

 creasing progressively ; but he did not know how to 

 evaluate the successive terms of this progression. It 

 was Archimedes who first discovered the practical 

 means of effecting this calculation. He succeeded, for 

 instance, in determining the circumference of a circle 

 by defining it as the boundary of two polygonal 

 perimeters, inscribed and circumscribed, of which the 

 number of sides is indefinitely increased. 



Fig. 14. 



By an analogous process he was able to calculate 

 curvilinear areas or areas bounded by curves. He 

 showed that any segment bounded by a straight line 

 and a parabola is equal to four -thirds of the triangle 

 having the same base and the same height as the 

 segment (Fig. 14). In this demonstration the passage 

 to the limit is not directly used. In order to avoid 

 this Archimedes proves that it would be absurd to 

 suppose the area of the parabolic segment to be greater 

 or less than four-thirds of the triangle having the same 

 base and height. 



The method of exhaustion rests on a reductio ad 

 absurdum which proves its perfect logical exactitude. 



