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

ON THE CUTTING-OFF OF A RATIO 179
Further, if we put for A the ratio between the lengths of the
two fixed tangents, then if h, k be those lengths,
k
h
y
h + x-2\/hx
which can easily be reduced to
©*+©-'
the equation of the parabola referred to the two fixed tangents
as axes.
180 APOLLONIUS OF PERGA
treatment of this problem in all its particular cases with their
Swpio-fxoi could present no difficulty to Apollonius.
If the two straight lines are parallel, the solution of the
problem gives a means of drawing any number of tangents
to an ellipse when two parallel tangents, their points of contact,
and the length of the parallel semi-diameter are given
(see Conies, III. 42). In the case of the hyperbola (III. 43)
the intercepts made by any tangent on the asymptotes contain
a constant rectangle. Accordingly the drawing of tangents
depends upon the particular case of our problem in which both
fixed points are the intersection of the two fixed lines.
•(/?) On the cutting-off of an area (\coptov ccttoto/jltj),
two Books.
This work, also in two Books, dealt with a similar problem,
with the difference that the intercepts on the given straight
lines measured from the given points are required, not to
have a given ratio, but to contain a given rectangle. Halley
included an attempted restoration of
this work in his edition
of the De sectione rationis.
The general case can here again be reduced to the more
special one in which one of the fixed points is at the intersection
of the two given straight lines. Using the same
figure as before, but with D taking the position shown by (D)
in the figure, we take that point such that
or
OC .
AD — the given rectangle.
We have then to draw ON'M through
B'N' .AM=OC.AD,
B'N':OC=AD:AM.
But, by parallels, B''N' : OC = B'M: CM;
therefore
so that
AM :CM=AD: B'M
= MD:B'C,
B'M .MD = AD. B'C.
such that
Hence, as before, the problem is reduced to an application
of a rectangle in the well-known manner. The complete
n 2
(y) On determinate section (SLoopicr/ievr] rofirj), two Books.
The general problem here is, Given four points A, B, G, D on
a straight line, to determine another point P on the same
straight line such that the ratio AP . CP : BP . DP has a
given value. It is clear from Pappus's account x of the contents
of this work, and from his extensive
collection of lemmas to
the different propositions in it, that the question was very
exhaustively discussed. To determine P by means of the
equation
AP.GP=\.BP.DP,
where A, B, C, D, A are given, is in itself an easy matter since
the problem can at once be put into the form of a quadratic
equation, and the Greeks would have no difficulty in reducing
it to the usual application of areas. If, however (as we may
fairly suppose), it was intended for application in further
investigations, the complete discussion of it would naturally
include not only the finding of a solution, but also the determination
of the limits of possibility and the number of possible
solutions for different positions of the point-pairs A, C and
B, D, for the cases in which the points in either pair coincide,
or in which one of the points is infinitely distant, and so on.
This agrees with what we find
in Pappus, who makes it clear
that, though we do not meet with any express mention of
series of point-pairs determined by the equation for different
values of A, yet the treatise contained what amounts to a com-
1
Pappus, vii, pp. 642-4.