THE MATHEMATICAL SCIENCES 141 



right angle (Fig. 21). Under these conditions we have 

 b 2 = x (a — x) and when the roots of the equation are 

 both positive they can immediately be found. 



It can be seen that the treatment of magnitudes by 

 geometrical representations is generally equivalent to 

 their treatment by algebra. There is, however, a 

 difference. Geometry is always fundamentally quali- 

 tative, while algebra is quantitative. 1 



Whilst the elliptic application is by defect, the 

 hyperbolic application is by excess and corresponds 

 to the following problem : on a given segment a 

 construct a rectangle ax which when increased by the 

 unknown square a; 2 is equal to a given square b 2 . 

 This problem is equal to the solution of the modern 



a 



equation ax + x 2 = b 2 or by adding and subtracting — , 



4 



a 2 a 2 



ax + x 2 + — = b 2 , 



It is necessary, then, to construct as before a 

 difference of squares. By means of the theorem of 



1 4 Boutroux, Ideal, p. 74, 



