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

THE COLLECTION. BOOKS I, II, HI 361
plete Greek text, with apparatus, Latin translation, commentary,
appendices and indices, by Friedrich Hultsch ; this
great edition is one of the first monuments of the revived
study of the history of Greek mathematics in the last half
of the nineteenth century, and has properly formed the model
for other definitive editions of the Greek text of the other
classical Greek mathematicians, e.g. the editions of Euclid,
Archimedes, Apollonius, &c, by Heiberg and others. The
Greek index in this edition of Pappus deserves special mention
because it largely serves as a dictionary of mathematical
terms used not only in Pappus but by the Greek mathematicians
generally.
(8) Summary of contents.
At the beginning of the work, Book I and the first
13 propositions
(out of 26) of Book II are missing. The first 13
propositions of Book II evidently, like the rest of the Book,
dealt with Apollonius's method of working with very large
numbers expressed in successive powers of the myriad, 10000.
This system has already been described (vol. i, pp. 40, 54-7).
The work of Apollonius seems to have contained 26 propositions
(25 leading up to, and the 26th containing, the final
continued multiplication).
Book III consists of four sections. Section (1) is a sort of
history of the problem of finding two mean 'proportionals, in
continued proportion, between two given straight lines.
It
begins with some general remarks about the distinction
between theorems and problems. Pappus observes that,
whereas the ancients called them all alike by one name, some
regarding them all as problems and others as theorems, a clear
distinction was drawn by those who favoured more exact
terminology. According to the latter a problem is that in
which it is proposed to do or construct something, a theorem
that in which, given certain hypotheses, we investigate that
which follows from and is necessarily implied by them.
Therefore he who propounds a theorem, no matter how he has
become aware of the fact which is a necessary consequence of
the premisses, must state, as the object of inquiry, the right
result and no other. On the other hand, he who propounds
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.