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. •