THE QUADRATRIX OF HIPPIAS 
229 
1 Pappus, iv, pp. 252. 26-254. 22. 
flourished about 350 B.C., that is to say, some time before 
Euclid, it is worth while to note certain propositions which 
are assumed as known. These are, in addition to the theorem 
of Eucl. VI. 33, the following: (1) the circumferences of 
circles are as their respective radii ; (2) any arc of a circle 
is greater than the chord subtending it; (3) any arc of a 
circle less than a quadrant is less than the portion of the 
tangent at one extremity of the arc cut off by the radius 
passing through the other extremity. (2) and (3) are of 
course equivalent to the facts that, if a be the circular measure 
of an angle less than a right angle, sin a < a. < tan a.] 
Even now we have only rectified the circle. To square it 
we have to use the proposition (1) in Archimedes’s Measure 
ment of a Circle, to the effect that the area of a circle is equal 
to that of a right-angled triangle in which the perpendicular 
is equal to the radius, and the base to the circumference, 
of the circle. This proposition is proved by the method of 
exhaustion and may have been known to Dinostratus, who 
was later than Eudoxus, if not to Hippias. 
The criticisms of Sporus, 1 2 in which Pappus concurs, are 
worth quoting : 
(1) ‘The very thing for which the construction is thought 
to serve is actually assumed in the hypothesis. For how is it 
possible, with two points starting from B, to make one of 
them move along a straight line to A and the other along 
a circumference to D in an equal time, unless you first know 
the ratio of the straight line AB to the circumference BED ? 
In fact this ratio must also be that of the speeds of motion. 
For, if you employ speeds not definitely adjusted (to this 
ratio), how can you make the motions end at the same 
moment, unless this should sometime happen by pure chance 1 
Is not the thing thus shown to be absurd 1 
(2) ‘ Again, the extremity of the curve which they employ 
for squaring the circle, I mean the point in which the curve 
cuts the straight line AD, is not found at all. For if, in the 
figure, the straight lines CB, BA are made to end their motion 
together, they will then coincide with AD itself and will not 
cut one another any more. In fact they cease to intersect 
before they coincide with AD, and yet it was the intersection 
of these lines which was supposed to give the extremity of the