01Y THE GUTTING-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, 
-= y 
h h + x — 2 V hx 
which can easily be reduced to 
the equation of the parabola referred to the two fixed tangents 
as axes. 
(/3) On the cutting-off of an area {yjcopiov dnoropi7), 
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 sections rationis. 
The general case can here again be reduced to the more 
special one in which one of the fixed points is at the inter 
section 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 
4 OC .AD = the given rectangle. 
We have then to draw ON'M through 0 such that 
B'N' .AM = OC.AD, 
or B'N' :0C = AD: AM. 
But, by parallels, B'N': OC = B'M: CM; 
therefore AM: CM = AD: B'M 
= MD: B'C, 
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