A history of Greek mathematics - Wilbourhall.org

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