126 THE SQUARING OF THE CIRCLE. 



Hippocrates found that the crescent-shaped plane figure produced 

 by drawing two perpendicular radii in a circle and describing upon 

 the line joining their extremities a semicircle, is exactly equal in 

 area to the triangle that is formed by this line of junction and the 

 two radii ; and upon the basis of this fact the endeavors of this un- 

 tiring scholar were directed towards converting a circle into a cres- 

 cent. Naturally he was unable to attain this object, but by his ef- 

 forts he discovered many new geometrical truths ; among others 

 being the generalised form of the theorem mentioned, which bears 

 to the present day the name of lunulae Hippocratis, the lunes of 

 Hippocrates. Thus, in the case of Hippocrates, it appears in the 

 plainest light, how precisely the insolvable problems of science are 

 qualified to advance science ; in that they incite investigators to 

 devote themselves with persistence to its study and thus to fathom 

 its utmost depths. 



Following Hippocrates in the historical line of the great Gre- 

 cian geometricians comes the systematist Euclid, whose rigid form- 

 ulation of geometrical principles has remained the standard presen- 

 tation down to the present century. The Elements of Euclid, 

 however, contain nothing relating to the quadrature of the circle 

 or to circle-computation. Comparisons of surfaces which relate to 

 the circle are indeed found in the work, but nowhere a computa- 

 tion of the circumference of a circle or of the area of a circle. This 

 palpable gap in Euclid's system was filled by Archimedes, the 

 greatest mathematician of antiquity. 



Archimedes was born in .Syracuse in the year 287 B. C., and 

 devoted his life, which was spent in that city, to the mathematical 

 and the physical sciences, which he enriched with invaluable con- 

 tributions. He lived in Syracuse till the taking of the town by 

 Marcellus, in the year 2126. C., when he fell by the hand of a Ro- 

 man soldier whom he had forbidden to destroy the figures he had 

 drawn in the sand. To the greatest performances of Archimedes 

 the successful computation of the number n unquestionably be- 

 longs. Like Bryson he started with regular inscribed and circum- 

 scribed polygons. He showed how it was possible, beginning with 

 the perimeter of an inscribed hexagon, which is equal to six radii, 



