THE SQUARING OF THE CIRCLE. 1 25 



act, appear in a high degree intelligent. Ant v /fion divided the 

 circle into four equal arcs, and by joining the points of division ob- 

 tained a square ; he then divided each arc again into two equal 

 parts and thus obtained an inscribed octagon ; thence he constructed 

 an inscribed i6-gon, and perceived that the figure so inscribed 

 more and more approached the shape of a circle. In this way, he 

 said, one should proceed, until there was inscribed in the circle a 

 polygon whose sides by reason of their smallness should coincide 

 with the circle. Now this polygon could, by methods already 

 taught by the Pythagoreans, be converted into a square of equal 

 area ; and upon the basis of this fact Antiphon regarded the squar- 

 ing of the circle as solved. 



Nothing can be said against this method except that, however 

 far the bisection of the arcs is carried, the result still remains an 

 approximate one. 



The attempt of Bryson of Heraclea was better still ; for this 

 scholar did not rest content with finding a square that was very 

 little smaller than the circle, but obtained by means of circum- 

 scribed polygons another square that was very little larger than 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. Notwith- 

 standing this error, however, to Bryson belongs the merit first, of 

 having introduced into mathematics by his emphasis of the neces- 

 sity of a square which was too large and one which was too small, 

 the conception of upper and lower "limits" in approximations; 

 and secondly, by his comparison of the regular inscribed and cir- 

 cumscribed polygons with a circle, of having indicated to Archime- 

 des the way by which an approximate value of n was to be reached. 



Not long after Antiphon and Bryson, Hippocrates of Chios 

 treated the problem, which had now become more and more fa- 

 mous, 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 recti- 

 linearly bounded figure, and which therefore could be converted by 

 straight edge and compasses into a square equal in area. First, 



