A history of Greek mathematics - Wilbourhall.org

154 APOLLONIUS OF PERGA
The general propositions containing the harmonic properties
of the pole and polar of a conic are III. 37-40, which prove
that in any conic, if TQ, Tq be tangents, and if Qq the chord
of contact be bisected in V, then
(1) if any straight line through T meet the conic in R', R and
TB! = RI : IR' ;
Qq in I, then (Fig. 1) RT :
t
u
(2) if any straight line through Fmeet the conic in R, R' and
the parallel through T to Qq in 0, then (Fig. 2)
R0 :
OR = RV: VR.
Fig. 2.
The above figures represent theorem (1)
theorem (2) for the ellipse.
for the parabola and
4