THE COLLECTION. BOOK IV 
379 
We have a similar proportion connecting a figure circum 
scribed 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. 
Therefore (area of spiral cut off by OB) = § - • nr 2 . 
Qj 
Similarly the area of the spiral cut off by any other radius 
T 
vector r' = 4— • nr' 2 , 
a 
Therefore (as Pappus proves in his next proposition) the 
first area is to the second as r 3 to r' z . 
Considering the areas cut off by the radii vectores at the 
points where the revolving line has passed through angles 
of \n, n, |-tt and 2n respectively, we see that the areas are in 
the ratio of (J) 3 , (^) 3 , (|) 3 ,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). 
(/3) 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