138 SCIENCE IN GRECO-ROMAN ANTIQUITY 



For example, to add the rectangle B to the rectangle 

 A of which one side is b, it is necessary to find a rect- 

 angle C (with sides b and x) which, being equal to B, 



can be applied to A by 

 the common side b (Fig. 

 16). 



To solve this problem 

 it is necessary to proceed 

 in the following way : On 

 the extension of one of the 

 c sides of the rectangle B 

 (Fig. 15) take the length b, 

 then from the extremity 

 of this side thus produced, 

 draw the new diagonal to 

 the point where it cuts the other side of B likewise 

 produced. We have thus all the elements for con- 

 structing the rectangle C, which evidently fulfils the 

 requirements of the problem and can be applied to 

 the rectangle A. This construction is called naQapoXrj, 

 or the application of surfaces. When made as we 

 have just seen, it is simple, but it may be elliptic 



Fig. 16. 



or hyperbolic. When elliptic, it corresponds to the 

 following problem : on a given segment a construct a 

 rectangle ax which when diminished by an unknown 

 square # 2 is equal to a given square b 2 (Fig. 17). 



