FOCUS-DIRECTRIX PROPERTY 
121 
Similarly it can be shown that 
= XK'-.A'N. 
By multiplication, XK. XK': AN. A'N = (1 ~ e 2 ): e 2 ; 
and it follows from above, ex aequali, that 
PX 2 :AX.A'X = (1 ~e 2 ):l, 
which is the property of a central conic. 
When e < 1, A and A' lie on the same side of X, while 
X lies on A A', and the conic is an ellipse; when e > 1, A and 
A' lie on opposite sides of X, while X lies on A'A produced, 
and the conic is a hyperbola. 
The case where e = 1 and the curve is a parabola is easy 
and need not be reproduced here. 
The treatise would doubtless contain other loci of types 
similar to that which, as Pappus says, was used for the 
trisection of an angle: I refer to the proposition already 
quoted (vol. i, p. 243) that, if A, B are the base angles of 
a triangle with vertex P, and ¿B = 2¿A, the locus of P 
is a hyperbola with eccentricity 2. 
Propositions included in Euclid’s Conics. 
That Euclid’s Conics covered much of the same ground as 
the first three Books of Apollonius is clear from the language 
of Apollonius himself. Confirmation is forthcoming in the 
quotations by Archimedes of propositions (1) ‘proved in 
the elements of conics’, or (2) assumed without remark as 
already known. The former class include the fundamental 
ordinate properties of the conics in the following forms: 
(1) for the ellipse, 
PX 2 : AX. A'X = P'N' 2 : AX'. A'X' = BC 2 : AC 2 ; 
(2) for the hyperbola, 
PX 2 : AX. A'N = P'N' 2 : AN'. A'X'; 
(3) for the parabola, PX 2 = p a .AY; 
the principal tangent properties of the parabola ; 
tfie property that, if there are two tangents drawn from one 
point to any conic section whatever, and two intersecting