378 
PAPPUS OF ALEXANDRIA 
With 0 as centre and OE, OF as radii draw arcs of circles 
meeting OF, OB in H, G respectively. 
For brevity we will now denote a cylinder in which r is the 
radius of the base and h the height by (cyl. r, h) and the cone 
with the same base and height by (cone r, h). 
K S v R 
By the property of the spiral, 
OB:BG = (arc A'BOB): (arc CB) 
= RK; KB 
= NK: KM, 
whence OB : OG = NK : NM. 
Now 
(sector OBC): (sector OGF) = OB 2 :OG 2 = NK 2 : MN 2 
= (cyl. KN, NT): (cyl. MN, NT). 
Similarly 
(sector OCD): (sector OEM) = (cyl. ST, TW): (cyl. FT. TW), 
and so on. 
The sectors OBC, OCD... form the sector OA'DB, and the 
sectors OFG, OEH... form a figure inscribed to the spiral. 
In like manner the cylinders {KN, TN), (ST, TW) ... form the 
cylinder (KN, NL), while the cylinders (MN, NT), (FT, TW)... 
form a figure inscribed to the cone (KN, NL). 
Consequently 
(sector OA'DB) :(fig. inscr. in spiral) 
= (cyl. KN, NL): (fig. inscr. in cone KN, NL).