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

THE COLLECTION. BOOK VII 403
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utility. In justification of this statement and in order that
he may not appear empty-handed when leaving the subject ',
he will present his readers with the following.
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.
(Anticipation of Guldins Theorem,)
The enunciations are not very clearly worded, but there
is no doubt as to the sense.
Figures generated by a complete revolution of a plane figure
'
about an axis are in a ratio compounded (1) of the ratio
of the areas of the figures, and (2) of the ratio of the straight
lines similarly drawn to (i.e. drawn to meet at the same angles)
the axes of rotation from the respective centres of gravity.
Figures generated by incomplete revolutions are in the rcdio
compounded (1) of the rcdio of the areas of the figures and
(2) of the ratio of the arcs described by the centres of gravity
of the respective figures, the latter rcdio being itself compounded
(a) of the ratio of the straight lines similarly drawn {from
the respective centres of gravity to the axes of rotation) and
(b) of the ratio of the angles contained (i. e. described) about
the axes of revolution by the extremities of the said straight
lines (i.e. the centres of gravity).'
Here, obviously, we have the essence of the celebrated
theorem commonly attributed to P. Guldin (1577-1643),
'
quantitas rotunda in viam rotationis ducta producit Potestatem
Rotundam uno grado altiorem Potestate sive Quantitate
Rotata *}
Pappus adds that
these propositions, which are practically one, include any
c
number of theorems of all sorts about curves, surfaces, and
solids, all of which are proved at once by one demonstration,
and include propositions both old and new, and in particular
those proved in the twelfth Book of these Elements. 5
Hultsch attributes the whole passage (pp. 680. 30-682. 20)
to an interpolator, I do not know for what reason; but it
seems to me that the propositions are quite beyond what
1
Centrobaryca, Lib. ii, chap, viii, Prop. 3. Viemiae 1641.
Dd2
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
could be expected from an interpolator, indeed I know of
no Greek mathematician from Pappus's day onward except
Pappus himself who was capable of discovering such a proposition.
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.