THE MATHEMATICAL SCIENCES 147 



It can be demonstrated that the locus of the points 

 satisfying the enunciation of the problem is an ellipse. 

 If AB = 2a, LM = 2p, AH = x, HD =y, we shaU 

 have, according to the equation of condition (p. 145), 



y 



2 = 2px — — x 2 or y 2 = 2px 



2a 



a 



When the rectangle to which the square DH 2 is 

 equal is to be increased, instead of diminished, by a 

 rectangle similar to the rectangle of dimensions AB 



Fig. 28. 



and LM, the geometrical locus is no longer an ellipse, 

 but a hyperbola (Fig. 28). 



Finally if the rectangle is not to be either diminished 

 or increased but simply applied to the segment LM, we 

 have the parabola. 



Apollonius was of the opinion that whatever the conic 

 section considered might be, the segment LM must 

 always be perpendicular to the extremity of the 

 segment AB even if the half chord HD be oblique 

 in respect to the diameter AB. Hence the name of 

 latus rectum (right side), which was given to it. For 

 this reason, geometrical algebra renders the same ser- 



