148 SCIENCE IN GRECO-ROMAN ANTIQUITY 



vices as those rendered later by analytical geometry. 

 " Whilst we now express the fundamental property of 

 a curve by an algebraical equation, Apollonius repre- 

 sented it by a figure ; and owing to the fact that this 

 auxiliary figure is drawn at right angles to the axis of 

 abscissae, even when the ordinates cut this axis at 

 another angle, it always remains in some degree 

 independent of the figure for the study of which it is 

 used. 1 



Another fact, no less remarkable, was brought to 



D' 



h 



F 



D 



Fig. 29. 



light by the Greek geometers (Pappus, Hultsch edit., 

 book vii, prop. 238). Given an infinite straight line 

 DD' (Fig. 29) and a point F, it can be demonstrated 

 that the geometrical locus of the points M such that 

 the ratios of the distances MF and MH from M to the 

 point F and to the straight line be constant and equal 

 to a given number, is a conic section. Inversely, being 

 given any conic section, it is always possible to find a 

 straight line and a point F which wiU allow the ratio 

 in question to be established with regard to each point 

 1 29 Zeuthen, Histoire des mathematiques, p. 168. 



