384
EUCLID
suppose, but greater than that (the reference is clearly to
IY. 16 For.), and likewise that neither is the circle three-
fourths of the triangle circumscribed about it’. Were these
fallacies perhaps exposed in the lost Pseudaria of Euclid 1
Book V is devoted to the new theory of proportion,
applicable to incommensurable as well as commensurable
magnitudes, and to magnitudes of every kind (straight lines,
areas, volumes, numbers, times, &c,), which was due to
Eudoxus, Greek mathematics can boast no finer discovery
than this theory, which first put on a sound footing so much
of geometry as depended on the use of proportions. How far
Eudoxus himself worked out his theory in detail is unknown;
the scholiast who attributes the discovery of it to him says
that ‘ it is recognized by all ’ that Book V is, as regards its
arrangement and sequence in the Elements, due to Euclid
himself. 1 The ordering of the propositions and the develop
ment of the proofs are indeed masterly and worthy of Euclid;
as Barrow said, ‘ There is nothing in the whole body of the
elements of a more subtile invention, nothing more solidly
established, and more accurately handled, than the doctrine of
proportionals’. It is a pity that, notwithstanding the pre
eminent place which Euclid has occupied in English mathe
matical teaching, Book V itself is little known in detail; if it
were, there would, I think, be less tendency to seek for
substitutes; indeed, after reading some of the substitutes,
it is with relief that one turns to the original. For this
reason, I shall make my account of Book V somewhat full,
with the object of indicating not only the whole content but
also the course of the proofs.
Of the Definitions the following are those which need
separate mention. The definition (3) of ratio as ‘a sort of
relation (?toloc cryea-Ls) in respect of size (TrrjXLKOTrjs) between
two magnitudes of the same kind’ is as vague and of as
little practical use as that of a straight line; it was probably
inserted for completeness’ sake, and in order merely to aid the
conception of a ratio. Definition 4 (‘ Magnitudes are said to
have a ratio to one another which are capable, when multi
plied, of exceeding one another ’) is important not only because
1 Euclid, ed. Heib., vol. v, p. 282.