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.