THE SQUARING OF THE CIRCLE. 107 



very little smaller tbaii tbe circle, but obtained by means of circumscribed 

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

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

 the arithmetical mean between an inscribed and a circumscribed polygon 

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

 Bryson belongs the merit, first, 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 r 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. Hippocrates 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, Hippocrates found that thq 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 the untiring scholar were directed towards converting a circle into a 

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

 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 Lumikc Eippocratis^ 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 persistence 

 to its study and thus to fathom its depths. 



EudiiVs avoidance of the problem. — Followin<^ 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 present century. The Elements 

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

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

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

 of tbe circumference of a circle or of tbe area of the circle. Tbis pal- 

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

 mathematician of antiquity. 



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

 year 287 B. c, and devoted his life, there spent, to tbe mathematical and 

 the physical sciences which he enriched with invaluable contributions. 

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



