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

ON DETERMINATE SECTION 181
plete Theory of Involution. Pappus says that the separate
cases were dealt with in which the given ratio was that of
either (1) the square of one abscissa measured from the
required point or (2) the rectangle contained by two such
abscissae to any one of the following:- (1) the square of one
abscissa, (2) the rectangle contained by one abscissa and
another separate line of given length independent of the
position of the required point, (3) the rectangle contained by
two abscissae. We learn also that maxima and minima were
investigated. From the lemmas, too, we may draw other
conclusions, e. g.
(1) that, in the case where A = 1, or AP .OP = BP .DP,
Apollonius used the relation BP .DP = AB . BO : AD . DO,
(2) that Apollonius probably obtained a double point E of the
involution determined by the point-pairs A, and B, D by
means of the relation
AB.BC.AD. DC = BE 2 : DE\
A possible application of the problem was the determination
of the points of intersection of
the given straight line with a
conic determined as a four-line locus, since A, B, C, D are in
fact the points of intersection of the given straight line with
the four lines to which the locus has reference.
(S) On Contacts or Ta agencies {kirafyai), two Books.
Pappus again comprehends in one enunciation the varieties
of problems dealt with in the treatise, which we may reproduce
as follows : Given three things, each of which may be
either a 'point, a straight line or a circle, to draw a circle
which shall pass through each of the given points (so far as it
is points that are given) and touch the straight lines or
circles. 1 The possibilities as regards the different data are
ten. We may have any one of the following: (1) three
points, (2) three straight lines, (3) two points and a straight
line, (4) two straight lines and a point, (5) two points and
a circle, (6) two circles and a point, (7) two straight lines and
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 Apollonius
r 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, vii, p. 644, 25-8.
1
Pappus, iv, pp. 194-6. 2 lb. vii, p.