THE QUADRATRIX OF HIPPIAS 
227 
Then, in their ultimate positions, the moving straight line 
and the moving radius will both coincide with AD] and at 
any previous instant during the motion the, moving line and 
the moving radius will by their intersection determine a point, 
as F or L. 
The locus of these points is the quadratrix. 
The property of the curve is that 
Z BAD: LEAD — (arc BED): (arc ED) = AB: FH. 
In other words, if 0 is the angle FAD made by any radius 
vector AF with AD, p the length of AF, and a the length 
of the side of the square, 
p sin 0 _ 0 
a \ 7T 
Now clearly, when the curve is once constructed, it enables 
us not only to trisect the angle EAD but also to divide it in 
any given ratio. 
For let FH be divided at F' in the given ratio. Draw F'L 
parallel to AD to meet the curve in L: join AL, and produce 
it to meet the circle in N. 
Then the angles FAN, NAD are in the ratio of FF' to F'II, 
as is easily proved. 
Thus the quadratrix lends itself quite readily to the division 
of any angle in a given ratio. 
The application of the quadratrix to the rectification of the 
circle is a more difficult matter, because it requires us to 
know the position of G, the point where the quadratrix 
intersects AD. This difficulty was fully appreciated in ancient 
times, as we shall see. 
Meantime, assuming that the quadratrix intersects AD 
in G, we have to prove the proposition which gives the length 
of the arc of the quadrant BED and therefore of the circum 
ference of the circle. This proposition is to the effect that 
(arc of quadrant BED): AB = AB: AG. 
This is proved by reductio ad absurdurn. 
If the former ratio is not equal to AB:AG, it must be 
equal to AB: AK, where AK is either (1) greater or (2) less 
than AG. 
(1) Let AK be greater than AG] and with A as centre 
Q 2