THE MATHEMATICAL SCIENCES 149 



of the curve. Further, according as this constant ratio 



MF 



^-r— is smaller than, greater than, or equal to 1, the conic 



MH 



is an ellipse, a hyperbola or a parabola. 1 



It is useless to enter into the details of the demon- 

 strations, our aim being merely to show that the geomet- 



Fig. 30. 



rical algebra of the Greeks, even in their most perfect 

 works, remained faithful to its primary inspiration. 



Let us add also that it was owing to conic sections 

 that the study and investigation of loci was generalized. 

 Among the problems considered by Pappus there are 

 a number of this kind : from a point P drop the 

 perpendiculars a, b, c, d on four straight lines. Find 

 the locus of the points P such that the rectangle ah may 

 be equal (or similar) to the rectangle cd (Fig. 30). 

 1 3 Boutroux, Analyse, I, p. 250. 



