A history of Greek mathematics - Wilbourhall.org

150 APOLLONIUS OF PERGA
the last problem, proving that, if the tangent to an ellipse at
any point P meets the major axis in T, the angle GPT is not
greater than the angle ABA', where B is one extremity of the
minor axis.
Book III begins with a series of propositions about the
equality of certain areas, propositions of the same kind as, and
easily derived from, the propositions (I. 41-50) by means of
which, as already shown, the transformation of coordinates is
effected. We have first the proposition that, if the tangents
at any points P, Q of a conic meet in 0, and if they meet
the diameters through Q, P respectively in E, T, then
AOPT = A0QE (III. 1, 4)
; and, if P, Q be points on adjacent
branches of conjugate hyperbolas, AGPE = ACQT (III. 13.).
With the same notation, if R be any other point on the conic,
and if
we draw BU parallel to the tangent at Q meeting the
diameter through P^ in U and the diameter through Q in M,
and RW parallel to the tangent at P meeting QT in H and
the
diameters through Q, P in F, W, then AHQF = quadrilateral
HTUR (III. 2. 6)
that ABMF= quadrilateral QTUM (see I. 49, 50, or pp. 145-6
; this is proved at once from the fact
above) by subtracting or adding the area HRMQ on each
side. Next take any other point B', and draw B'U', F'H'B'W
in the same way as before ; it is then proved that, if BU, R'W
meet in I and B'U', R W in J, the quadrilaterals F'IBF, IUU'R'
are equal, and also the quadrilaterals FJB'F', JU'TJR (III. 3,
7, 9, 10). The proof varies according to the actual positions
of the points in the figures.
In Figs. 1, 2 AHFQ = quadrilateral HTUR,
AH'F'Q = H'TU'R'.
•
By subtraction, FHH'F'= IUU'R + (IB);
whence, if IE be added or subtracted, F'IRF = IUU'R',
and again, if IJ be added to both, FJR'F' = JU'UR.
In Fig. 3 AR'U'W = A CF'W - A CQT,
so that
ACQT= CU'R'F'.