A history of Greek mathematics - Wilbourhall.org

404 PAPPUS OF ALEXANDRIA
If the passage is genuine, it seems to indicate, what is not
elsewhere confirmed, that the Collection originally contained,
or was intended to contain, twelve Books.
Lemmas to the different treatises.
After the description of the treatises forming the Treasury
of Analysis come the collections of lemmas given by Pappus
to assist the student of each of the books (except Euclid's
Data) down to Apollonius's Conies, with two isolated lemmas
to the Surface-Loci of Euclid. It is difficult to give any
summary or any general idea of these lemmas, because they
are very numerous, extremely various, and often quite
difficult,
requiring first-rate ability and full command of all the
resources of pure geometry. Their number is also greatly
increased by the addition of alternative proofs, often requiring
lemmas of their own, and by the separate formulation of
particular cases where by the use of algebra and conventions
with regard to sign we can make one proposition cover all the
cases. The style is admirably terse, often so condensed as to
make the argument difficult to follow without some little
filling-out ; the hand is that of a master throughout. The
only misfortune is that, the books elucidated being lost
(except
the Conies and the Cutting-off of a ratio of Apollonius), it is
difficult, often impossible, to see the connexion of the lemmas
with one another and the problems of the book to which they
relate. In the circumstances, all that I can hope to do is to
indicate the types of propositions included in the lemmas and,
by way of illustration, now and then to give a proof where it
is sufficiently out of the common.
(a)
Pappus begins with Lemmas to the Sectio rationis and
Sectio spatii of Apollonius (Props. 1-21, pp. 684-704). The
first two show how to divide a straight line in a given ratio,
and how, given the first, second and fourth terms of a proportion
between straight lines, to find the third term. The
next section (Props. 3-12 and 16) shows how to manipulate
relations between greater and less ratios by transforming
them, e.g. componendo, convertendo, &c, in the same way
as Euclid transforms equal ratios in Book V ; Prop. 1 6 proves
that, according as a : b > or < c:d, ad > or < be. Props.