THE ELEMENTS. BOOK V 
385 
it shows that the magnitudes must be of the same kind, 
but because, while it includes incommensurable as well as 
commensurable magnitudes, it excludes the relation of a finite 
magnitude to a magnitude of the same kind which is either 
infinitely great or infinitely small; it is also practically equiva 
lent to the principle which underlies the method of exhaustion 
now known as the Axiom of Archimedes. Most important 
of all is the fundamental definition (5) of magnitudes which 
are in the same ratio: ‘ Magnitudes are said to be in the same 
ratio, the first to the second and the third to the fourth, when, 
if any equimultiples whatever be taken of the first and third, 
and any equimultiples whatever of the second and fourth, the 
former equimultiples alike exceed, are alike equal to, or alike 
fall short of, the latter equimultiples taken in corresponding 
order.’ Perhaps the greatest tribute to this marvellous defini 
tion is its adoption by Weierstrass as a definition of equal 
numbers. For a most attractive explanation of its exact 
significance and its absolute sufficiency the reader should turn 
to De Morgan’s articles on Ratio and .Proportion in the Penny 
Cyclopaedia. 1 The definition (7) of greater ratio is an adden 
dum to Definition 5 : ‘ When, of the equimultiples, the multiple 
of the first exceeds the multiple of the second, but the 
multiple of the third does not exceed the multiple of the 
fourth, then the first is said to have a greater ratio to 
the second than the third has to the fourth ’; this (possibly 
for brevity’s sake) states only one criterion, the other possible 
criterion being that, while the multiple of the first is equal 
to that of the second, the multiple of the third is less than 
that of the fourth. A proportion may consist of three or 
four terms (Defs. 8, 9, 10); ‘corresponding’ or ‘homologous’ 
terms are antecedents in relation to antecedents and conse 
quents in relation to consequents (11). Euclid proceeds to 
define the various transformations of ratios. Alternation 
{kva\\d£, alternando) means taking the alternate terms in 
the proportion a: h = c : d, i.e. transforming it into a:c — b:d 
(12). Inversion (drarraXu/, inversely) means turning the ratio 
a:h into h:a (13). Composition of a ratio, avvOeats \6yov 
(componendo is in Greek a-wdem, ‘to one who has compounded 
1 Vol. xix (1841). I have largely reproduced the articles in The 
Thirteen Books of Euclid's Elements, vol. ii, pp. 116-24. 
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