A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE COLLECTION. BOOK IV 379
We have a similar proportion connecting a figure circumscribed
to the spiral and a figure circumscribed to the cone.
By increasing n the inscribed and circumscribed figures can
be compressed together, and by the usual method of exhaustion
we have ultimately
(sector OA'DB) :
(area of spiral) = (cyl. KN, NL) : (cone KN, NL)
= 3:1,
or (area of spiral cut off by OB) = $ (sector OA'DB).
The ratio of the sector OA'DB to the complete circle is that
of the angle which the radius vector describes in passing from
the position OA to the position OB to
four right angles, that
is, by the property of the spiral, r : a, where r = OB, a = OA.
r
Therefore (area of spiral cut off by OB) = § - • irr
a
Similarly the area of the spiral cut off by any other radius
r
vector r = 4 — •
r' 2 3 77- .
a
Therefore (as Pappus proves in his next proposition) the
first area is to the second as r 3 to r' 3 .
Considering the areas cut off by the radii vectores at the
points where the revolving line has passed through angles
of ^tt, 7r,
f
7r and 2 it respectively, we see that the areas are in
3
the ratio of (J) ,
(J) 3 3
, ,
(f 1 or
)
1, 8, 27, 64, so that the areas of
the spiral included in the four quadrants are in the ratio
of 1, 7, 19, 37 (Prop. 22).
(P)
The conchoid of Nicomedes.
The conchoid of Nicomedes is next described (chaps. 26-7),
and it is shown (chaps. 28, 29) how it can be used to find two
geometric means between two straight lines, and consequently
to find a cube having a given ratio to a given cube (see vol. i,
pp. 260-2 and pp. 238-40, where I have also mentioned
Pappus's remark that the conchoid which he describes is the
first conchoid, while there also exist a second, a third and a
fourth which are of use for other theorems).
The quadratrix is
(y)
The quadratrix.
taken next (chaps. 30-2), with Sporus's
criticism questioning the construction as involving a petitio
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