A history of Greek mathematics - Wilbourhall.org

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
1
Pappus, vii, p. 644, 25-8.