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

Therefore
THE COLLECTION. BOOK IV 385
A 2tt-4 &* —
(surface of hemisphere) 2n \iv
(segment ABC) -,
*
"" (sector I)ABC) J
The second part of the last section of Book IV (chaps. 36-41,
pp. 270-302) is mainly concerned with the problem of trisecting
any given angle or dividing it into parts in any given
ratio. Pappus begins with another account of the distinction
between plane, solid and linear problems (cf . Book III, chaps.
20-2) according as they require for their solution (1) the
straight line and circle only, (2) conies or their equivalent,
(3) higher curves still, 'which have a more complicated and
forced (or unnatural) origin, being produced from more
irregular surfaces and involved motions. Such are the curves
which are discovered in the so-called loci on surfaces, as
well as others more complicated still and many in number
discovered by Demetrius of Alexandria in his Linear considerations
and by Philon of Tyana by means of the interlacing
of plectoids and other surfaces of all sorts, all of which
curves possess many remarkable properties peculiar to them.
Some of these curves have been thought bv the more recent
writers to be worthy of considerable discussion ;
one of them is
that which also received from Menelaus the name of the
paradoxical curve. Others of the same class are spirals,
quadratrices, cochloids and cissoids.' He adds the often-quoted
reflection on the error committed by geometers when they
'
solve a problem by means of an inappropriate class ' (of
curve or its equivalent), illustrating this by the use in
Apollonius, Book V, of a rectangular hyperbola for finding the
feet of normals to a parabola passing through one point
(where a circle would serve the purpose), and by the assumption
by Archimedes of a solid vevcris in his book On Spirals
(see above, pp. 65-8).
Trisection (or division in any ratio) of any angle.
The method of trisecting any angle based on a certain vevo-i y
is next described, with the solution of the vevo-i$ itself by
1523 ? C C
386 PAPPUS OF ALEXANDRIA
means of a hyperbola which has to be constructed from certain
data, namely the asymptotes and a certain point through
which the curve must pass (this easy construction is given in
Prop. 33, chap. 41-2). Then the problem is directly solved
(chaps. 43, 44) by means of a hyperbola in two ways practically
equivalent, the hyperbola being determined in the one
case by the ordinary Apollonian property, but in the other by
means of the focus-directrix property. Solutions follow of
the problem of dividing any angle in a given ratio by means
(1) of the quadratrix, (2) of the spiral of Archimedes (chaps.
45, 46). All these solutions have been sufficiently described
above (vol. i, pp. 235-7, 241-3, 225-7).
Some problems follow (chaps. 47-51) depending on these
results, namely those of constructing an isosceles triangle in
which either of the base angles has a given ratio to the vertical
angle (Prop. 37), inscribing in a circle a regular polygon of
any number of sides (Prop. 38), drawing a circle the circumference
of which shall be equal to a given straight line (Prop.
39), constructing on a given straight line AB a segment of
a circle such that the arc of the segment may have a given
ratio to the base (Prop. 40), and constructing an angle incommensurable
with a given angle (Prop. 41).
Section (5). Solution of the v ever is of Archimedes, * On Spirals',
Pro}). 8, by means of conies.
Book IV concludes with the solution of the vevcri? which,
according to Pappus, Archimedes unnecessarily assumed in
On Spirals, Prop. 8. Archimedes's assumption is this. Given
a circle, a chord (BC) in it less than the diameter, and a point
A on the circle the perpendicular from which to BC cuts BC
again
in a point D such that BD > DO and meets the circle
in E, it is possible to draw through A a straight line ARP
cutting BC in R and the circle in P in such a way that RP
shall be equal to DE (or, in the phraseology of yeva-ei?, to
place between the straight line BC and the circumference
of the circle a straight line equal to DE and verging
towards A).
Pappus makes the problem rather more general by not
requiring PR to be equal to DE, but making it of any given