A history of Greek mathematics - Wilbourhall.org

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FOCUS-DIRECTRIX PROPERTY 121
Similarly it can be shown that
(1 *e):e = XK':A'N.
By multiplication, XK . XK' :AN. A'N = (1 - e 2 ) e2 : ;
and it follows from above, ex aequali, that
PN 2 :AN.A'N=(l~e 2 ):l,
which is the property of a central conic.
When e < 1, A and A' lie on the same side of 1, while
N lies on A A', and the conic is an ellipse ; when e > 1, A and
A / lie on opposite sides of X, while N 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 AB = 2 A A, the locus of P
is a hyperbola with eccentricity 2.
Propositions included in Euclid's Conies.
That Euclid's
Conies 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 conies ', or (2) assumed without remark as
already known. The former class include the fundamental
ordinate properties of the conies in the following forms
(1) for the ellipse,
PN 2 : AN. A'N = P'N' 2 : AN'. A'N' = BC 2 :AG 2 ;
(2) for the hyperbola,
PN 2 : AN. A'N = P'N' 2 : AN' .A'N';
(3) for the parabola, PN 2 = pa
. AN;
the principal tangent properties of the parabola
the property that, if
there are two tangents drawn from one
point to any conic section whatever, and two intersecting