140 SCIENCE IN GRECO-ROMAN ANTIQUITY 



Once x is found, it is easy to construct the rectangle 



/a\ 2 /a \ 2 

 ax and the difference of the squares ( - J and ( x) 



<% 



Fig. 18. 



Fig. 19. 



It can be seen that the rectangle ax, diminished by 

 the square # 2 is equal to a gnomon whose surface is 

 equal to the given square b 2 (Fig. 20). 



The problem was afterwards generalized in the 

 following manner : to determine two quantities of 

 which the sum a is known and the product is con- 



^^■^■^^^^^^^^^"■^^^■"^^ 



Fig. 20. 



sidered as equal to a square b 2 . To find the unknown 

 value x one can proceed as follows : In a semi-circle 

 of radius a inscribe the right-angled triangle of which 

 b is the perpendicular dropped from the vertex of the 



