98 A SHORT HISTORY OF SCIENCE 



(3) Finally, Archimedes was a great investigator and pioneer, 

 who in each of his works carries knowledge a step forward. This 

 affects materially the form of presentation. In a most recently 

 discovered manuscript, the procedure is essentially modern as 

 contrasted with the rigid formalism of the Elements. 



CIRCLE MEASUREMENT. In this Archimedes proves three 

 theorems. 



(1) Every circle is equivalent to a right triangle having the 

 sides adjacent to the right angle equal respectively to the radius 

 and circumference of the circle. 



(2) The circle has to the square on its diameter approximately 

 the ratio 11:14. 



(3) The circumference of any circle is three times as great as the 

 diameter and somewhat more, namely less than f but more than ^. 



He proves the first theorem by showing that the assumption 

 that the circle is either larger or smaller than the triangle leads to 

 a contradiction. The second he bases on the third, at which he 

 arrives by computing successively the perimeters of both inscribed 

 and circumscribed polygons of 3, 6, 12, 24, 48 and 96 sides. All 

 this is contrary to the spirit of Euclid and essentially modern in 

 its method of successive approximation. The difficulty of the 

 achievement in view of the imperfect arithmetical notation avail- 

 able can hardly be overrated. 



QUADRATURE OF THE PARABOLA. Of special interest is his 

 quadrature of the parabola. A segment is formed by drawing any 

 chord PQ of the parabola : it is known that if a line is drawn from 

 the middle point R of the chord parallel to the axis of the parabola, 

 the tangent at the point S where this line meets the curve will be 

 parallel to the chord, and the perpendicular from S to the chord 

 is greater than any other which can be drawn from a point of the 



nown of more than human sagacity ; he yet would not deign to leave behind him 

 any commentary or writing on such subjects ; but, repudiating as sordid and ignoble 

 the whole trade of engineering, and every sort of art that lends itself to mere use 

 and profit, he placed his whole affection and ambition in those purer speculations 

 where there can be no reference to the vulgar needs of life ; studies, the superiority 

 of which to all others is unquestioned, and in which the only doubt can be whether 

 the beauty and grandeur of the subjects examined, or the precision and cogency of 

 the methods and means of proof, most deserve our admiration." 



