A history of Greek mathematics - Wilbourhall.org

THE CONWS, BOOK III 157
used to prove that the focal distances of P make equal angles
with the tangent at P (III. 48). In III. 49-52 follow the
other ordinary properties, that, if SY be perpendicular to
the tangent at P, the locus of Y is the circle on A A' as
diameter, that the lines from G drawn parallel to the focal
distances to meet the tangent at P are
equal to CA, and that
the sum or difference of the focal distances of any point is
equal to A A'.
The last propositions of Book III are of use with reference
to the locus with respect to three or four lines. They are as
follows.
1. If PP' be a diameter of a central conic, and if PQ, P'Q
drawn to any other point Q of the conic meet the tangents at
P', P in R' y
R respectively, then PR . P'R' = 4 CD 2 (III. 53).
2. If TQ, TQ' be two tangents to a conic, V the middle point
of QQ', P the
point of contact of the tangent parallel to QQ',
and R any other point on the conic, let Qr parallel to TQ'
meet Q'R in r, and Q'r parallel to TQ meet QR in r' ;
then
Qr .
QY
:
QQ' 2 = (PV 2 : PT
2 (TQ
) . . TQ': QV 2 ). (Ill 54, 56.)
3. If the tangents are tangents to opposite branches of a
hyperbola and meet in t, and if R, r, r' are taken as before,
while tq is half the chord through t
Qr .
QY
:
parallel to QQ', then
QQ' = 2 tQ . tQ' : tq 2 . (III. 55.)
The second of these propositions leads at once to the threeline
locus, and from this we easily obtain the Cartesian
equation to a conic with reference to two fixed tangents as
axes, where the lengths of the tangents are h, k, viz.
Book IV is
length.
(M-')'=»©*-
on the whole dull, and need not be noticed at
Props. 1-23 prove the converse of the propositions in
Book III about the harmonic properties of the pole and polar
for a large number of particular cases.
One of the propositions
(IV. 9) gives a method of drawing two tangents to
a conic from an external point T. Draw any two straight
lines through T cutting the conic in Q, Q' and in R, R' respec-