A history of Greek mathematics - Wilbourhall.org

182 APOLLONIUS OF PERGA
a circle, (8) two circles and a straight line, (9) a point, a circle
and a straight line, (10) three circles. Of these varieties the
first two are treated in Eucl. IV ; Book I of Apollonius's
treatise treated of (3), (4), (5), (6), (8), (9), while (7), the case of
two straight lines and a circle, and (10), that of the three
circles, occupied the whole of Book II.
The last problem (10), where the data are three circles,
has exercised the ingenuity of many distinguished geometers,
including Vieta and Newton. Vieta (1540-1603) set the problem
to Adrianus Komanus (van Roomen, 1561-1615) who
solved it by means of a hyperbola. Vieta was not satisfied
with this, and rejoined with his A r pollonius
Gallus (1600) in
which he solved the problem by plane methods. A solution
of the same kind is given by Newton in his Arithmetica
Universalis (Prob. xlvii), while an equivalent problem is
solved by means of two hyperbolas in the Principia, Lemma
xvi. The problem is quite capable of a ' plane ' solution, and,
as a matter of fact, it is not difficult to restore the actual
solution of Apollonius (which of course used the 'plane' method
depending on the straight line and circle only), b}^ means of
the lemmas given by Pappus. Three things are necessary to
the solution. (1) A proposition, used by Pappus elsewhere 1
and easily proved, that, if two circles touch internally or
externally, any straight line through the point of contact
divides the circles into segments respectively similar. (2) The
proposition that, given three circles, their six centres of similitude
(external and internal) lie three by three on four straight
lines. This proposition, though not proved in Pappus, was
certainly known to the ancient geometers; it is even possible
that Pappus omitted to prove it because it was actually proved
by Apollonius in his treatise. (3) An auxiliary problem solved
by Pappus and enunciated by him as follows. 2 Given a circle
ABC, and given three points D, E, F in a straight line, to
inflect (the broken line) DAE (to the circle) so as to make BG
in a straight line with CF; in other words, to inscribe in the
circle a triangle the sides of which, when produced, pass
respectively through three given points lying in a straight
line. This problem is interesting as a typical example of the
ancient analysis followed by synthesis. Suppose the problem
1
Pappus, iv, pp. 194-6. 2 lb. vii, p.