vi 
226 
THE SQUARING OF THE CIRCLE 
be used for trisecting an angle. But this becomes more doubt 
ful when the passages of Proclus are considered. Pappus’s 
authority seems to be Sporus, who was only slightly older 
than Pappus himself (towards the end of the third century A.D.), 
and who was the author of a compilation called Kqpia con 
taining, among other things, mathematical extracts on the 
quadrature of the circle and the duplication of the cube. 
Proclus’s authority, on the other hand, is doubtless Geminus, 
who was much earlier (first century B. 0.) Now not only 
does the above passage of Proclus make it possible that the 
name quadratrix may have been used by Hippias himself, 
but in another place Proclus (i.e. Geminus) says that different 
mathematicians have explained the properties of particular 
kinds of curves; 
‘ thus Apollonius shows in the case of each of the conic curves 
what is its property, and similarly Nicomedes with the 
conchoids, Hippias with the quadratrices, and Perseus with 
the spiric curves.’ 1 
This suggests that Geminus had before him a regular treatise 
by Hippias on the properties of the quadratrix (which may 
have disappeared by the time of Sporus), and that Nicomedes 
did not write any such general work on that curve; and, 
if this is so, it seems not impossible that Hippias himself 
discovered that it would serve to rectify, and therefore to 
square, the circle. 
(a) The Quadratrix of Hippias. 
The method of constructing the curve is described by 
Pappus. 2 Suppose that A BCD is 
a square, and BED a quadrant of a 
circle with centre A. 
Suppose (1) that a radius of the 
circle moves uniformly about A from 
the position HR to the position AD, 
and (2) that in the same time the 
line BG moves uniformly, always 
parallel to itself and with its ex 
tremity B moving along BA, from the position BG to the 
position AD. 
1 Proclus on Eucl. I, p. 856. 6-12. 2 Pappus, iv, pp. 252 sq. •