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

NET2EI2 (VERGINGS OR INCLINATIONS) 191
Therefore the triangles BEK, KEG, which have the angle
BEK common, are similar, and
But
Z GBK = Z GKE = Z GEE (from above).
Z HGE = IAGB= Z BCK.
Therefore in the triangles CBK, GHE two angles are
respectively equal, so that
Z GEH — Z GKB also.
But since LGKE = I CHE (from above), K, C, E, E are
concyclic.
Hence
therefore, since
Z CEH+ Z GKE = (two right angles)
Z GEE — Z GKB,
Z GKB + Z Cif# = (two right angles),
and BKE is a straight line.
It is certain, from the nature of this lemma, that Apollonius
made his construction by drawing the circle shown in the
figure.
He would no doubt arrive at it by analysis somewhat as
follows.
Suppose the problem solved, and EK inserted as required
(= h).
Bisect EK in N, and draw NE at right angles to KE
meeting BC produced in E. Draw KM perpendicular to BC,
and produce it to meet AC in L. Then, by the property of
the rhombus, LM = MK, and, since KN = NE also, MN is
parallel to LE.
Now, since the angles at M, N are right, M, K, N, E are
concyclic.
Therefore ICEK = Z.MNK = IGEK, so that C, K, E, E
are concyclic.
Therefore Z BCD = supplement of KCE = LEEK = lEKE,
and the triangles EKE, DGB are similar.
Lastly,
IEBK=IEKE-ICEK=IEEK-ICEK=IEEC=IEKC;
therefore the triangles EBK, EKC are similar, and
BE:EK = EK:EC,
or BE.EC = EK 2 .
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.