THE CONICS, BOOK I 
137 
pai’abola parallel to AC] in the case of the hyperbola it meets 
the other half of the double cone in P'; and in the case of the 
ellipse it meets the cone itself again in P'. We draw, in 
the cases of the hyperbola and ellipse, AF parallel to PM 
to meet BG or BG produced in F. 
Apollonius expresses the properties of the three curves by 
means of a certain straight line PL drawn at right angles 
to PM in the plane of the section. 
In the case of the parabola, PL is taken such that 
PL: PA = BG 2 :BA.AG\ 
and in the case of the hyperbola and ellipse such that 
PL : PP' = BF. FG: A F\. 
In the latter two cases we join P'L, and then draw VR 
parallel to PL to meet P'L, produced if necessary, in R. 
If UK be drawn through V parallel to BG and meeting 
AB, AG in H, K respectively, HK is the diameter of the circular 
section of the cone made by a plane parallel to the base. 
Therefore QV 2 = HV. VK. 
Then (1) for the parabola we have, by parallels and similar 
triangles, 
HV-.PV = BG-.CA,