A history of Greek mathematics - Wilbourhall.org

378 PAPPUS OF ALEXANDRIA
With 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).
N T W
By the property of the spiral,
whence
Now
(sector OBO) :
OB:BG = (arc A'DCB) :
KS
= RK :
= NK :
KM,
OB:OG = NK: NM.
(sector OGF) = OB 2 : OG
(arc CB)
2 = NK 2 : MN
2
= (cyl. KN, NT) : (cyl. MN, NT).
Similarly
(sector 00D) :
(sector OEH) = (cyl. ST, TW) :
(cyl. PT, 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), (PT, 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).