THE CONICS, BOOK I 
141 
the tangent at any point of a central conic, in relation 
to the original diameter of reference; if Q is the point of 
contact, QV the ordinate to the diameter through P, and 
if QT, the tangent at Q, meets the diameter produced in T, 
then (1) for the parabola PV = FT, and (2) for the central 
conic TP:TP'=PV; VP'. The method of proof is to take a 
point T on the diameter produced satisfying the respective 
relations, and to prove that, if TQ be joined and produced, 
any point on TQ on either side of Q is outside the curve: the 
form of proof is by reductio ad absurdum, and in each 
case it is again proved that no other straight line can fall 
between TQ and the curve. The fundamental property 
TP: TP' = PV: VP' for the central conic is then used to 
prove that GV.GT=CP 2 and QV 2 :CV. VT = p: PP' (or 
CD 2 : CP 2 ) and the corresponding properties with reference to 
the diameter DD' conjugate to PP' and v, t, the points where 
DD' is met by the ordinate to it from Q and by the tangent 
at Q respectively (Props. I. 37-40). 
Transition to пего diameter and tangent at its extremity. 
An important section of the Book follows (I. 41-50), con 
sisting of propositions leading up to what amounts to a trans 
formation of coordinates from the original diameter and the 
tangent at its extremity to any diameter and the tangent at 
its extremity; what Apollonius proves is of course that, if 
any other diameter be taken, the ordinate-property of the 
conic with reference to that diameter is of the same form as it 
is with reference to the original diameter. It is evident that 
this is vital to the exposition. The propositions leading up to 
the result in I. 50 are not usually given in our text-books of 
geometrical conics, but are useful and interesting. 
Suppose that the tangent at any point Q meets the diameter 
of reference PV in T, and that the tangent at P meets the 
diameter through Q in E. Let P be any third point on 
the curve; let the ordinate RW to PV meet the diameter 
through Q in F, and let RU parallel to the tangent at Q meet 
PV in U. Then 
(l) in the parabola, the triangle RUW — the parallelogram 
EW, and