A history of Greek mathematics - Wilbourhall.org

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144 APOLLONIUS OF PERGA
Therefore RUW, CPE, CFW are the halves of parallelograms
related as in the lemma
therefore
ARUW = A CFW - A CPE.
The same property with reference to the diameter secondary
to CPV is proved in I. 45.
It is interesting to note the exact significance of the property
thus proved for the central conic. The proposition, which is
the foundation of Apollonius's method of transformation of
coordinates, amounts to this. If OP, CQ are fixed semidiameters
and R a variable point, the area of the quadrilateral
GFRU is constant for all positions of R on the conic.
Suppose
now that CP, CQ are taken as axes of x and y respectively.
If we draw RX parallel to CQ to meet CP and RY parallel to
CP to meet CQ, the proposition asserts that (subject to the
proper convention as to sign)
ARYF+CJCXRY+ARXU = (const).
But since RX, RY, RF, RU are in fixed directions,
ARYF varies as RY 2 or x 2 ,
C3CXRY
and ARXU as RX 2
as RX .
or if.
Hence, if x, y are the coordinates of R,
ocx 2 + fixy + yy
2 — A,
RY
or xy,
which is the Cartesian equation of the conic referred to the
centre as origin and any two diameters as axes.
The properties so obtained are next used to prove that,
if UR meets the curve again in R f and the diameter through
Q in M, then RR' is bisected at M. (I. 46-8).
Taking (1) the case of the parabola, we have,
and
By subtraction,
whence
ARUW=EJEW,
AR'UW'=CJEW'.
(RWW'R) = CJF'W,
and, since the triangles are similar,
ARFM = AR'F'M,
RM — R'M.
The same result is easily obtained for the central conic.
It follows that EQ produced in the case of the parabola,