A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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By
THE CONIGS, BOOK II 149
angle (II. 50), making a given angle with the diameter through
the point of contact (II. 51, 53) II. 52 contains a Siopio-pos for ; so that ACQT= CU'R'F'.
150 APOLLONIUS OF PERGA
the portions of the asymptotes which include between them
the last problem, proving that, if the tangent to an ellipse at
a branch of the conjugate hyperbola (II. 11), the constancy of
any point P meets the major axis in T, the angle GPT is not
the rectangle contained by the straight lines drawn from any
greater than the angle ABA', where B is one extremity of the
point of the curve in fixed directions to meet the asymptotes
minor axis.
(equivalent to the Cartesian equation with reference to the
Book III begins with a series of propositions about the
asymptotes, xy = const.) (II. 12), and the fact that the curve
equality of certain areas, propositions of the same kind as, and
and the asymptotes proceed to infinity and approach con-
easily derived from, the propositions (I. 41-50) by means of
which, as already shown, the transformation of coordinates is
the diameters through Q, P respectively in E, T, then
points on two branches the property of II. 8. II. 1 7 shows that
AOPT = A0QE (III. 1, 4)
; and, if P, Q be points on adjacent
'conjugate opposites' (two conjugate double-branch hyperbolas)
have the same asymptotes. Propositions follow about
With the same notation, if R be any other point on the conic,
branches of conjugate hyperbolas, AGPE = ACQT (III. 13.).
and if we draw BU parallel to the tangent at Q meeting the
the diameters through Q, P in F, W, then AHQF = quadrilateral
HTUR (III. 2. ;
6) this is proved at once from the fact
that ABMF= quadrilateral QTUM (see I. 49, 50, or pp. 145-6
above) by subtracting or adding the area HRMQ on each
of the asymptotes (II. 21) if a chord Qq in one branch of
side. Next take any other point B', and draw B'U', F'H'B'W
;
a hyperbola meet the asymptotes in R, r and the conjugate
in the same way as before ; it is then proved that, if BU, R'W
hyperbola in Q', q', then Q'Q.Qq' = 2 CD 2 (II. 23). Of the
meet in I and B'U', R W in J, the quadrilaterals F'IBF, IUU'R'
rest of the propositions in this part of the Book the following
are equal, and also the quadrilaterals FJB'F', JU'TJR (III. 3,
may be mentioned : if TQ, TQ' are two tangents to a conic
7, 9, 10). The proof varies according to the actual positions
and V is the middle point of QQ', TV is a diameter (II. 29,
of the points in the figures.
30, 38) ; if tQ, tQ' be tangents to opposite branches of a hyperbola,
RR' the chord through t parallel to QQ', v the middle
In Figs. 1, 2 AHFQ = quadrilateral HTUR,
point of QQ', then vR, vR' are tangents to the hyperbola
(II. 40) ; in a conic, or a circle, or in conjugate hyperbolas, if
AH'F'Q = H'TU'R'.
two chords not passing through the centre intersect, they do not
bisect each other (II. 26, 41, 42). II. 44-7 show how to find
subtraction, FHH'F'= IUU'R + (IB);
a diameter of a conic and the centre of a central conic, the
axis of a parabola and the axes of a central conic. The Book
whence, if IE be added or subtracted, F'IRF = IUU'R',
concludes with problems of drawing tangents to conies in
and again, if IJ be added to both, FJR'F' = JU'UR.
certain ways, through any point on or outside the curve
(II. 49), making with the axis an angle equal to a given acute
In Fig. 3 AR'U'W = A CF'W - A CQT,
tinually nearer to one another, so that the distance separating
them can be made smaller than any given length (II. 14). II. 15
proves that the two opposite branches of a hyperbola have the
same asymptotes and II. 16 proves for the chord connecting
conjugate hyperbolas; any tangent tcPthe conjugate hyperbola
will meet both branches of the original hyperbola
and will be bisected at the point of contact (II. 19); if Q be
any point on a hyperbola, and GE parallel to the tangent
at Q meets the conjugate hyperbola in E, the tangent at
E will be parallel to GQ and GQ, GE will be conjugate
diameters (II. 20), while the tangents at Q, E will meet on one
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
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