362 
PAPPUS OF ALEXANDRIA 
a problem may bid us do something which is in fact im 
possible, 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 pro 
fesses to know mathematics and yet commits some elementary 
blunder, he cannot escape censure. Pappus gives, as an 
example, the case of an unnamed person ‘ whc/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 approxi 
mations to the real solution. Pappus adds a few simple 
lemmas assumed in the exposition. 
Next comes the passage 1 , already referred to, on the dis 
tinction drawn by the ancients between (1) plane problems or 
problems which can be solved by means of the straight line 
and circle, (2) 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 (/3e(3La(rfjievT]v) such as spirals, quadratrices, cochloids 
and cissoids, which have m^iy surprising properties of their 
own. The problem of the two mean proportionals, being 
a solid problem, required for its solution either conics or some 
equivalent, and, as conics 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 Pinion 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 practically 
the same as that attributed by Eutocius to Sporus. All these 
solutions have been given above (vol. i, pp. 258-64, 266-8). 
1 Pappus, iii, p. 54. 7-22.