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 DUG we simply have the straight line DC 
and make the first circle touch it and the two other semi 
circles. 
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. 6, 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 Archi 
medes. 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 0 and radius OB draw the circle A'BCD. 
Let BG be a certain fraction, say 1 / nth, of the arc BCD A', 
and CD the same fraction, OG, 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 respec 
tively, and draw MU, PQ parallel to KR.