THE MATHEMATICAL SCIENCES 145 



which are similar (and not equal) to given surfaces. 

 For example, to construct on a given segment a a 

 rectangle ax, which, diminished by a rectangle similar 

 to a given rectangle cd, is equal to a given square b 2 , 

 we must have (Fig. 25) : 



DB DD' x , „ c 



- — = = - whence DB =— x. 



cad a 



AD is then equal to a — T x and the unknown rectangle 



a 



has for surface x ( a — —x j ; but as this must be equal to 

 b 2 , we have finally the equation of condition 



ax — 3 x 2 = b 2 . 

 a 



The theory of proportions also enables the magni- 

 tudes which correspond to a given problem to be found 



Fig. 26. 



in a more direct manner. For example, to construct a 

 square x 2 , equal to a given rectangle ab, comes to 

 finding a mean proportional between a and b, which is 

 easy. Taking as diameter the segment AB of the 

 length a + b (Fig. 26), describe a semicircle, then at the 

 extremity of a at H, raise a perpendicular HD = x. 

 The triangle ADB inscribed in a semicircle is right- 

 angled and we have x 2 = ab. 



