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 Conics, 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 diffi- 
* cult, 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 Conics and the Gutting-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 pro 
portion 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. 16 proves 
that, according as a : 6 > or < c:d, ad > or < be. Props.