THE GOLDEN AGE OF GREECE 65 



finest and richest city in the world. Its citizens aspired to suc- 

 cess in public life, and sought training to that end from the soph- 

 ists. While science was not generally cultivated as a leading 

 subject in the educational system thus developed, 1 mathematics 

 could not fail to be esteemed as a means of discipline, and several 

 of the sophists made notable contributions to its development. 



HIPPIAS OF ELIS is the first sophist to be mentioned for impor- 

 tant mathematical work. About 420 B.C. Hippias invented a 

 curve called the quadratrix, serving for the solution of two of the 

 three celebrated problems of Greek geometry ; viz. the quadrature 

 of the circle and the trisection of an angle. By means of straight 

 line and circle constructions, the solution of the quadratic equation 

 had been accomplished, though without algebraic symbolism, 

 or any recognition of negative or imaginary results. The tri- 

 section problem, like that of duplicating the cube, was equivalent 

 to the solution of the cubic equation, and could therefore not be 

 accomplished by line and circle methods. 

 The quadratrix was generated by the inter- 

 section P of two moving straight lines, one 

 MQ always parallel to its initial position 

 OA, the other OR revolving uniformly about 

 a centre 0. By means of this curve the 

 trisection problem is reduced to that of tri- 

 secting a straight line, which is elementary. 2 

 The curve meets the perpendicular lines OA and OB at C and B 

 respectively so that OC : OB = 2 : TT, where TT is the ratio of the 

 circumference of a circle to its diameter. To this quadrature 

 solution the name of the curve is due. 



Dinostratus showed that the assumptions OC : OB > 2 : TT and 

 OC:OB<2nr both lead to contradictions, therefore OC:OB = 2:w 

 a good example of the Greek reductio ad absurdum. The 

 study of a problem not capable of solution by elementary means 



1 See Freeman, " Schools of Hellas." 



J To trisect any angle as AOR, draw MQ parallel to OA and divide OM into three 

 equal parts by lines parallel to OA, meeting the curve in D and E respectively. 

 The radii OS and OT will then trisect the angle AOQ, by the definition of the curve. 

 F 



