142 SCIENCE IN GRECO-ROMAN ANTIQUITY 



Pythagoras we quickly find f — \- x j and consequently 



x (Fig. 22). The rectangle ax is then easily obtained ; 



the square # 2 is then added 

 to it externally, instead of 

 being taken away as in the 

 elliptic application (Fig. 23). 

 Without labouring the 

 point, it can be seen that 

 the Ancients have treated all 

 the forms of the equation of 

 the second degree which give 

 positive roots ; for them 

 there could be no question 



of other roots, since they had no conception of them. 1 

 The constructions which we have just mentioned 



are of no use when problems arise concerning the 



quadrature of the circle, the trisection of the angle 



Fig. 22. 



<3C 





>x 



Fig. 23. 



and the duplication of the cube, which cannot be 

 solved by means of the circle and the straight line. 

 Recourse had then to be made to intercalations. 



1 29 Zeuthen, Histoire des mathematiques, p. 39. 



