56 SCIENCE IN GRECO-ROMAN ANTIQUITY 



moves parallel to itself from BC to OA in the same time 

 as the radius moves from OB to OA. The curve 

 can be constructed by successive divisions of the arc 

 BA and the straight line BO. This being done, it is 

 enough to divide BO into three parts, to obtain the 

 trisection sought. (Proclus, Comm. Eucl. I, p. 356, 

 11 and p. 272, 7 ; Pappus, I, p. 253). From an 

 analytical point of view the equation of the quadratrix 

 is the natural result of the following equation in which 

 r is the radius vector of the quadratrix, a the radius 

 of the circle, and 6 the angle AOR. 



„ r , Or sin 6 



We have - = 



71 a 



hence nr = 2a0 cosec 6 



The authenticity of the discovery of Hippias has 

 often been disputed ; P. Tannery, however, after 

 detailed discussion, upholds it. 1 



Another sophist, Antiphon, likens the ultimate 

 elements of the curved line to those of the straight line, 

 and he attempts to solve problems by regarding the 

 circle as the limit of a polygon with an infinite number 

 of sides. Bryson of Heraclea takes this conception 

 and completes it by considering at the same time 

 inscribed and circumscribed polygons. But these two 

 sophists appear to have postulated that there is no 

 real difference between the straight line and the curve 

 (Simplicius : Diels, Vor. II, 594) and for this reason 

 their solutions, which might have been a guiding light, 

 remain doubtful, the more so because they bring in 

 the notion of infinity. The disputations aroused by 

 this subject became so popular that Aristophanes 

 directly alludes to them (Birds, act II, scene vi). 

 " These, said the astronomer Meton, are instruments 



1 28 Tannery, Mem. scientifiques, II, p. 1. 



