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

THE COLLECTION. BOOK IV 377
The same proposition holds when the successive circles,
instead of being placed between the large and one of the small
semicircles, come down between the two small semicircles.
Pappus next deals with special cases (1) where the two
smaller semicircles become straight lines perpendicular to the
diameter of the other semicircle at its extremities, (2) where
we replace one of the smaller semicircles by a straight line
through D at right angles to BC, and lastly (3) where instead
of the semicircle DUC we simply have the straight line
and make the first circle touch it and the two other semicircles.
DC
Pappus's propositions of course include as particular cases
the partial propositions of the same kind included in the Book
'
of Lemmas' attributed to Archimedes (Props. 5, 6) ; cf. p. 102.
Sections (3) and (4).
Methods of squaring the circle, and of
trisecting [or dividing in any ratio) any given angle.
The last sections of Book IV (pp. 234-302) are mainly
devoted to the solutions of the problems (1) of squaring or
rectifying the circle and (2) of trisecting any given angle
or dividing it into two parts in any ratio. To this end Pappus
gives a short account of certain curves which were used for
the purpose.
(a)
The Archimedean spiral.
He begins with the spiral of Archimedes, proving some
of the fundamental properties. His method of finding the
area included (1) between the first turn and the initial line,
(2) between any radius vector on the first turn and the curve,
is worth giving because it differs from the method of Archimedes.
It is the area of the whole first turn which Pappus
works out in detail. We will take the area up to the radius
vector OB, say.
With centre
and radius OB draw the circle A'BCD.
Let BC be a certain fraction, say 1 /nth, of the arc BCDA',
and CD the same fraction, 00, OD meeting the spiral in F, E
respectively.
Let KS, SV be the same fraction of a straight
line KB, the side of a square KNLR.
Draw ST; VW parallel
to KN meeting the diagonal KL of the square in U, Q respectively,
and draw MU, PQ parallel to KR.
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) :
Similarly
(sector 00D) :
OB:BG = (arc A'DCB) :
= RK : KS
= NK : KM,
OB:OG = NK: NM.
(sector OGF) = OB 2 : OG
(arc CB)
2 = NK 2 : MN
2
= (cyl. KN, NT) : (cyl. MN, NT).
(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).