138 
APOLLONIUS OF PERGA 
Therefore QV 2 : PV. PA = HV .VK :PV. PA 
= BC 2 : BA . AC 
= PL: PA, by hypothesis, 
= PL.PV.PV.PA, 
whence QV 2 = PL . PV. 
(2) In the case of the hyperbola and ellipse, 
HV:PV = BF.FA, 
VK : P'V — FC-.AF. 
Therefore QV 2 : PV. P'V = HV .VK: PV .P'V 
= BF.FC-.AP 2 
— PLj : PP', by hypothesis, 
= BV:P'V 
= PV. VB : PV .P'V, 
whence QV 2 = PV. VB. 
Few names, ‘parabola’, ‘ellipse’, ‘hyperbola’. 
Accord ingly, in the case of the parabola, the square of the 
ordinate {QV 2 ) is equal to the rectangle applied to PX and 
with width equal to the abscissa (P V); 
in the case of the hyperbola the rectangle applied to PL 
which is equal to QV 2 and has its width equal to the abscissa 
PV overlaps or exceeds (vTrepPdXXei) by the small rectangle LB 
which is similar and similarly situated to the rectangle con 
tained by PL, PP'; 
in the case of the ellipse the corresponding rectangle falls 
short (eXXfirm) by a rectangle similar and similarly situated 
to the rectangle contained by PL, PP'. 
Here then we have the properties of the three curves 
expressed in the precise language of the Pythagorean applica 
tion of areas, and the curves are named accordingly: parabola 
(napa/SoXy) where the rectangle is exactly applied, hyperbola 
(vjrepPoXri) where it exceeds, and ellipse (fXXet.y\ns) where it 
falls short.