THE SQTTARINO OF TFTE C'lRfLE. 107 



very little sjiiallor tli:ui the circle, but obtained by means of circniiiscnibed 

 polygons another square that was very little larger tliaa the circle. Only 

 Bryson committed the error of believing that the area of the circle was 

 tlie arithmetical meau between an inscribed and a circumscribed polygon 

 of an equal number of sides. Notwithstanding this error, however, to 

 Bryson belongs the merit, tirst, of having introduced into mathematics 

 by his emphasis of the necessity of a square which was too large and 

 one which was too small, the conception of maximum and minimum 

 "limits" in approximations; and secondly, by his comparison with a 

 circle of the inscribed and circumscribed regular polygons, the merit of 

 having indicated to Archimedes the way by which an approximate 

 value for - was to be reached. 



Hippocrates of Chios. — Not long after Antiphon and Bryson, Hippo- 

 crates of Chios treated the problem, which had now become more and 

 more famous, from a new point of view, Hii)pocrates was not satisfied 

 with approximate equalities, and searched for curvilinearly bounded 

 plane figures which should be mathematically equal to a rectilinearly 

 bounded figure, and therefore could be converted by ruler and compasses 

 into a square equal in area. First, Hip])ocrates found that the crescent- 

 shaped plane figure produced by drawing two perj)endicular 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 the untiring scholar were directed towards converting a circle into a 

 crescent. Naturally he was unable to attain this object, but by his efforts 

 to this end he discovered many a new geometrical truth; among others 

 the generalized form of the theorem mentioned, which bears to the pres- 

 ent day the name of Lunnlcv Hippocratis, the lunes of Hii)pocrates. 

 Thus it appears, in the case of Hippocrates, in the plainest light, how 

 the very insolvable problems of science are qualified to advance science; 

 in that they incite investigators to devote themselves with i)ersistence 

 to its study and thus to fathom its depths. 



EuvlUVs avoidance of the problem. — FoUowiu'^ Hippocrates in the his- 

 torical line of the great Grecian geometricians comes the systematist 

 Euclid, whose rigid formulation of geometrical principles has remained 

 the standard presentation down to the i)resent century. The Elements 

 of Euclid, however, contain nothing relating to the quadrature of the 

 circle or to circle-computation. Comi)aiisous of surfaces which relate 

 to the circle are indeed found in the book, but nowhere a computation 

 of the circumference of a circle or of the area of the circle. This pal- 

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

 mathematician of antiquity. 



Archimedes's calculations. — Achimedes was born in Syracuse in the 

 year 287 b. c, and devoted his life, there spent, to the mathematical and 

 the physical sciences which he enriched with invaluable (jontributions. 

 IJe lived in Syracuse till the taking of the town by Marcellus, in the year 



