A history of Greek mathematics - Wilbourhall.org

380 PAPPUS OF ALEXANDRIA
principii. Its use for squaring the circle is attributed to
Dinostratus and Nicomedes. The whole substance of this
subsection is given above (vol. i, pp. 226-30).
Tivo constructions for the quadratrix by means of
'
surface-loci '.
In the next chapters (chaps. 33, 34, Props. 28, 29) Pappus
gives two alternative ways of producing the quadratrix by
'
means of surface-loci ', for which he claims the merit that
they are geometrical rather than too mechanical ' ' as the
traditional method (of Hippias) was.
(1) The first method uses a cylindrical helix thus.
Let ABC be a quadrant of a circle with centre B, and
let BD be any radius. Suppose
that EF, drawn from a point E
on the radius BD perpendicular
to BG, is (for all such radii) in
a given ratio to the arc DC.
'
I say ', says Pappus, ' that the
locus of E is a certain curve.'
Suppose a right cylinder
erected from the quadrant and
a cylindrical helix GGH drawn
upon its surface. Let DH be
the generator of this cylinder through D, meeting the
helix
in H. Draw BL, EI at right angles to the plane of the
quadrant, and draw HIL parallel to BD.
Now, by the property of the helix, EI(=DH) is to the
arc GD in a given ratio. Also EF : (arc CD) = a given ratio.
Therefore the ratio EF : EI is given, And since EF, EI are
given in position, FI is given in position. But FI is perpendicular
to BG. Therefore FI is in a plane given in position,
and so therefore is /.
But i" is also on a certain surface described by the line LH
,
which moves always parallel to the plane ABC, with one
extremity L on BL and the other extremity H on the helix.
Therefore / lies on the intersection of this surface with the
plane through FI