A history of Greek mathematics - Wilbourhall.org

190 APOLLONIUS OF PERGA
III.
Given a circle, to insert a chord of given length verging
to a given point.
In Book I of Apollonius's work there were four cases of
I (a), two cases of III, and two of II ; the second Book contained
ten cases of I (b).
Restorations were attempted by Marino Ghetaldi (Apollonius
redivivus, Venice, 1607, and Apollonius redivivus . . . Liber
secundus, Venice, 1613), Alexander Anderson (in a Suppleonentum
Apollonii redivivi, 1612), and Samuel Horsley
(Oxford, 1770);
the last is much the most complete.
In the case of the rhombus (II) the construction of Apollonius
can be restored with certainty. It depends on a lemma given
by Pappus, which is as follows : Given a rhombus AD with
diagonal BG produced to E, if F be taken on BG such that EF
is a mean proportional between BE and EG, and if a circle be
described with E as centre and EF as radius cutting CD
in K and AG produced in H, then shall B, K, H be in one
straight line. 1
Let the circle
join HE, LE, KE.
cut AG in L, join LK meeting BG in M, and
Since now CL, GK are equally inclined to the diameter of
the circle, CL = GK. Also EL = EK, and it follows that the
triangles ECK, ECL are equal in all respects, so that
or
By hypothesis,
,
Z CKE = L CLE = Z CHE.
EB:EF=EF: EC,
EB:EK = EK:EC.
1
Pappus, vii, pp. 778-80.