398 
EUCLID 
deal with multiples and equimultiples. 11-14 are transforma 
tions of proportions corresponding to similar transformations 
{separando, alternately, &c.) in Book V. The following are 
the results, expressed with the aid of letters which here repre 
sent integral numbers exclusively. 
If a:h — c:d (a > c. b > d), then 
(a — c):{b — d) = a:b. (11) 
If a : a' = h : V = c : c f ..then each of the ratios is equal to 
(a + b -f- c + ..,) : (a -f- b' -j- c +...). (12) 
If a:b = c:d, then a : c = b : d. (13) 
If a:b = d:e and b : c = e :f, then, ex aequali, 
a:c = d :/. (14) 
If 1 : m = a : via (expressed by saying that the third 
number measures the fourth the same number of times that 
the unit measures the second), then alternately 
1 :a — m:via. (15) 
The last result is used to prove that ab = ba ; in other 
words, that the order of multiplication is indifferent (16), and 
this is followed by the propositions that b:c = ab:ac (17) 
and that a:b ■= ac:bc (18), which are again used to prove 
the important proposition (19) that, if a:b = c:d, then 
ad — be, a theorem which corresponds to VI. 16 for straight 
lines. 
Zeuthen observes that, while it was necessary to use the 
numerical definition of proportion to carry the numerical 
theory up to this point, Proposition 19 establishes the necessary 
point of contact between the two theories, since it is now 
shown that the definition of proportion in V, Def. 5, has, 
when applied to numbers, the same import as that in VII, 
Def. 20, and we can henceforth without hesitation borrow any 
of the propositions established in Book Y. 1 
Propositions 20, 21 about ‘the least numbers of those which 
have the same ratio with them ’ prove that, if m, n are such 
numbers and a, b any other numbers in the same ratio, m 
1 Zeuthen, ‘ Sur la constitution des livres arithmétiques des Éléments 
d’Euclide ’ ( Oversigt over det kgl. Danske Videnskabernes Selskabs Forhand- 
linger, 1910, pp. 412, 418).