A history of Greek mathematics - Wilbourhall.org

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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
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