A history of Greek mathematics - Wilbourhall.org

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.
•(/?) 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
AM :CM=AD: B'M
= MD:B'C,
such that
so that
B'M .MD = AD. B'C.
Hence, as before, the problem is reduced to an application
of a rectangle in the well-known manner. The complete
n 2