A history of Greek mathematics - Wilbourhall.org

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 — •
3 a
77-
r' 2
.
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).
(y)
The quadratrix.
The quadratrix is
taken next (chaps. 30-2), with Sporus's
criticism questioning the construction as involving a petitio