A history of Greek mathematics - Wilbourhall.org

192 APOLLONIUS OF PERGA
But, by similar triangles EKH, DCB,
EK:KH=DC:CB,
and, since the ratio DC:CB, as well as KH, is given, EK
is
given.
The construction then is as follows.
If k be the given length, take a straight line p such that
apply to BG a rectangle BE . EC
p:k = AB:BC:
equal to p 1 and exceeding by
a square ; then with E as centre and radius equal to p describe a
circle cutting AC produced in H and CD in K. HK is then
equal to k and, by Pappus's lemma, verges towards B.
Pappus adds an interesting solution of the same problem
with reference to a square instead of a rhombus ; the solution
is by one Heraclitus and depends on a lemma which Pappus
also gives. 1
We hear of yet other lost works by Apollonius.
(rj) A Comparison of the dodecahedron with the icosahedron.
This is mentioned by Hypsicles in the preface to the so-called
Book XIV of Euclid. Like the Conies, it appeared in two
editions, the second of which contained the proposition that,
if there be a dodecahedron and an icosahedron inscribed in
one and the same sphere, the surfaces of . the solids are in the
same ratio as their volumes ;
this was established by showing
that the perpendiculars from the centre of the sphere to
a pentagonal face of the dodecahedron and to a triangular
face of the icosahedron are equal.
(0) Marinus on Euclid's Data speaks of a General Treatise
(kcc66\ov Trpay/jLCLTeia) in which Apollonius used the word
assigned (TtTayfiivov) as a comprehensive term to describe the
datum in general. It would appear that this work must
have dealt with the fundamental principles of mathematics,
definitions, axioms, &c, and that to it must be referred the
various remarks on such subjects attributed to Apollonius by
Proclus, the elucidation of the notion of a line,
the definition
1
Pappus, vii, pp. 780-4.