A history of Greek mathematics - Wilbourhall.org

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 ;
if
then,
PT, pt be the tangents at P, p and meet the axes at T, t at
PT = pv : pt, the segments are similar
equal angles, and if P V :
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