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). 
« 
Two 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. 
‘1 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, El 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 CD in a given ratio. Also EF: (arc CD) = a given ratio. 
Therefore the ratio EF: El is given. And since EF, El are 
given in position, FI is given in position. But FI is perpen 
dicular to BG. Therefore FI is in a plane given in position, 
and so therefore is I. 
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 II on the helix. 
Therefore I lies on the intersection of this surface with the 
plane through FI.