A history of Greek mathematics - Wilbourhall.org

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
:
= BF.FC:AF 2
PV.P'V
= PIj : PP', by hypothesis,
= RV:P'V
= PV. VR.PV.P'V,
whence QV 2 = PV.VR.
Neiv names, ' parabola ', ' ellipse ',
'
hyperbola \
Accordingly, in the case of the parabola, the square of the
ordinate (QV 2 ) is equal to the rectangle applied to PL and
with width equal to the abscissa (PV) ;
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 (u7r6p/3d\\€i) by the small rectangle LR
which is similar and similarly situated to the rectangle contained
by PL, PP' ;
in the case of the ellipse the corresponding rectangle falls
short (e\\ei7r€i) 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 application
of areas, and the curves are named accordingly :
(7rapa/3o\rj)
parabola
where the rectangle is exactly applied, hyperbola
(v7r€p/3o\r)) where it exceeds, and ellipse (eAAei^i?) where it
falls
short.