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

THE CONICS, BOOKS V, VI 167
tively with the conies give the points at which the normals
passing through are normals.
Pappus criticizes the use of the rectangular hyperbola in
the case of the parabola as an unnecessary resort to a solid
'
locus '
; the meaning evidently is that the same points of
intersection can be got by means of a certain circle taking
the place of the rectangular hyperbola. We can, in fact, from
2
the equation (1) above combined with y = px, obtain the
circle
(x 2 + y
2)
- (x 1
+ ^p)x-iy 1 y = 0.
The Book concludes with other propositions about maxima
and minima. In particular V. 68-71 compare the lengths of
tangents TQ, TQ f , where Q is nearer to the axis than Q\
V. 72, 74 compare the lengths of two normals from a point
from which only two can be drawn and the lengths of other
straight lines from to the curve ; V. 75-7 compare the
lengths of three normals to an ellipse drawn from a point
below the major axis, in relation to the lengths of other
straight lines from to the curve.
Book VI is of much less interest. The first part (VI. 1-27)
relates to equal (i.e. congruent) or similar conies and segments
of conies ; it is naturally preceded by some definitions including
those of ' equal ' and ' similar ' as applied to conies and
segments of conies. Conies are said to be similar if, the same
number of ordinates being drawn to the axis at proportional
distances from the vertices, all the ordinates are respectively
proportional to the corresponding abscissae. The definition of
similar segments is the same with diameter substituted for
axis, and with the additional condition that the angles
between the base and diameter in each are equal. Two
parabolas are equal if
the ordinates to a diameter in each are
inclined to the respective diameters at equal angles and the
corresponding parameters are equal ; two ellipses or hyperbolas
are equal if the ordinates to a diameter in each are
equally inclined to the respective diameters and the diameters
as well as the corresponding parameters are equal (VI. 1. 2).
Hyperbolas or ellipses are similar when the 'figure' on a
diameter of one is similar (instead of equal) to the ' figure ' on
a diameter of the other, and the ordinates to the diameters in
168 APOLLONIUS OF PERGA
each make equal angles with them ; all parabolas are similar
(VI. 11, 12,13). No conic of one of the three kinds (parabolas,
hyperbolas or ellipses) can be equal or similar to a conic
of either of the other two kinds (VI. 3, 14, 15). Let QPQ',
qpq' be two segments of similar conies in which QQ', qq' are
the bases and PV, pv are the diameters bisecting them ; then,
if PT, pt be the tangents at P, p and meet the axes at T, t at
equal angles, and if P V : PT = pv : pt, the segments are similar
and similarly situated, and conversely (VI. 17, 18). If two
ordinates be drawn to the axes of two parabolas, or the major or
conjugate axes of two similar central conies, as PN, P'N' and
pn, p'n' respectively, such that the ratios AN: an and AN': an'
are each equal to the ratio of the respective
latera recta, the
segments PP f ,
pp' will be similar ; also PP' will not be similar
to any segment in the other conic cut off by two ordinates
other than pn, p'n' , and conversely (VI. 21, 22). If any cone
be cut by two parallel planes making hyperbolic or elliptic
sections, the sections will be similar but not equal (VI. 26, 27).
The remainder of the Book consists of problems of construction;
we are shown how in a given right cone to find
a parabolic, hyperbolic or elliptic section equal to a given
parabola, hyperbola or ellipse, subject in the case of the
hyperbola to a certain Siopio-fios or condition of possibility
(VI. 28-30); also how to find
a right cone similar to a given
cone and containing a given parabola, hyperbola or ellipse as
a section of it, subject again in the case of the hyperbola to
a certain Siopio-fjios (VI. 31-3). These problems recall the
somewhat similar problems in I. 51-9.
Book VII begins with three propositions giving expressions
for AP 2