146 SCIENCE IN GRECO-ROMAN ANTIQUITY 



We can generalize the scope of this problem and say 

 that the geometrical locus of the points D such that the 

 perpendicular DH to AB is the mean proportional 

 between the two segments which it determines on this 

 straight line, is a circumference of diameter AB. 



We can also, and it is here that conic sections inter- 

 vene as geometrical loci, conceive of a more complicated 

 relation of measurements ; for instance, let us suppose 

 that the segment AB being given, the segment DH is 

 the side of a square subjected to the condition of being 

 equal to a rectangle which, when applied to another 



L -r 



A\ 1 





Fig. 27. 



given segment LM, is at the same time diminished, by 

 a rectangle similar to the rectangle of dimensions LM 

 and AB (Fig. 27). 



To find any point of the locus, on the given segment 

 AB, erect at its extremity a perpendicular AM equal to 

 the second given segment LM. Construct the rectangle 

 of the dimensions AB and AM, having MB as diagonal. 

 From any point H draw a parallel to AM ; this, at the 

 point where it cuts the diagonal MB, determines the 

 rectangle which, similar to the rectangle AB x AM, 

 must be taken away from the rectangle of the dimen- 

 sions AM and AH, applied to the segment AM (= LM). 

 Then there only remains to find the side DH of the 

 square equal to the rectangle AH x AN. 



