A history of Greek mathematics - Wilbourhall.org

362 PAPPUS OF ALEXANDRIA
a problem may bid us do something which is in fact impossible,
and that without necessarily laying himself open
to blame or criticism. For it is part of the solver's duty
to determine the conditions under which the problem is
possible or impossible, and, ' if possible, when, how, and in
how many ways it is possible '. When, however, a man professes
to know mathematics and yet commits some elementary
blunder, he cannot escape censure. Pappus gives, as an
example, the case of an unnamed person who was thought to
'
be a great geometer' but who showed ignorance in that he
claimed to know how to solve the problem of the two mean
proportionals by 'plane' methods (i.e.
by using the straight
line and circle only). He then reproduces the argument of
the anonymous person, for the purpose of showing that it
does not solve the problem as its author claims. We have
seen (vol. i, pp. 269-70) how the method, though not actually
solving the problem, does furnish a series of successive approximations
to the real solution. Pappus adds a few simple
lemmas assumed in the exposition.
Next comes the passage 1 ,
already referred to, on the distinction
drawn by the ancients between (1) plane problems or
problems which can be solved by means of the
and circle, (2)
straight line
solid problems, or those which require for their
solution one or more conic sections, (3) linear problems, or
those which necessitate recourse to higher curves still, curves
with a more complicated and indeed a forced or unnatural
origin (Pefitao-fiivrji/) such as spirals, quadratrices, cochloids
and cissoids,
which have many surprising properties of their
own. The problem of the two mean proportionals, being
a solid problem, required for its solution either conies or some
equivalent, and, as conies could not be constructed by purely
geometrical means, various mechanical devices were invented
such as that of Eratosthenes (the mean-finder), those described
in the Mechanics of Philon and Heron, and that of Nicomedes
(who used the ' cochloidal ' curve). Pappus proceeds to give the
solutions of Eratosthenes, Nicomedes and Heron, and then adds
a fourth which he claims as his own, but which is
the same as that attributed by Eutocius to Sporus.
practically
All these
solutions have been given above (vol. i, pp. 258-64, 266-8).
1
Pappus, iii, p. 54. 7-22.