THE MATHEMATICAL SCIENCES 139 



In modern language the problem is expressed by 

 the equation 



ax — x 2 = b 2 



a 2 

 or again, by adding and subtracting — , 



a 2 /a 2 



4 



+ x 2 - ax) = b 2 , 



©" - (i 



The problem leads therefore to the construction of a 



3 - G - •; - "• 



Fig. 17. 



difference of squares. By putting the equation in the 

 form 



(9- 



= b* + (--x 



the length ( - — x j and the length x are easily found 



by means of the theorem of Pythagoras. 



Let a be the given segment and b the side of the 

 given square. On one of the extremities of b, raise 

 a perpendicular, then from the other describe an arc 



of circle of radius - which will cut the perpendicular. 



2 



In this way we find the side - — x and the length 

 x (Fig. 18). 



