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EUCLID 
or added i. e. if one compounds or adds) is the turning of 
a:b into (a + h):h (14). Separation, SLaipecns (SieXovn = 
separando) turns a:h into (a—b):b (15). Conversion, dva- 
arpotyr] {dvaarpk^avTL — convertendo) turns a:h into a:a — h 
(16). Lastly, ex aequali (sc. distantia), Sl’ i'crov, and ex aequali 
in disturbed proportion (er rerapay pevp dvaXoyia) are defined 
(17, 18). If a:b = A:B, h:c = B : C... h:l = K : L, then 
the inference ex aequali is that a :l = A : L (proved in V. 22). 
If again a:b = B :C and b :c = A : B, the inference ex aequali 
in disturbed proportion is a: c = A : C (proved in V. 23). 
In reproducing the content of the Book I shall express 
magnitudes in general (which Euclid represents by straight 
lines) by the letters a, b, c ... and I shall use the letters 
m, n, p... to express integral numbers: thus rna, mb are 
equimultiples of a, b. 
The first six propositions are simple theorems in concrete 
arithmetic, and they are practically all proved by separating 
into their units the multiples used. 
11. ma + mb + me + ... = m (a + b + c + ...). 
15. ma — mb = m {a — b). 
5 is proved by means of 1. As a matter of fact, Euclid 
assumes the construction of a straight line equal to 1 /mth of 
ma—mb. This is an anticipation of VI. 9, but can be avoided; 
for we can draw a straight line equal to m (a — b); then, 
by 1, m{a—b)+mb = ma, or ma—mb = m{a — b). 
(2. ma + na+2 M + ... = {m + n+p+ ...)a. 
(6. ma — na—{m — n)a. 
Euclid actually expresses 2 and 6 by saying that ma ± na is 
the same multiple of a that mb ± nb is of b. By separation 
of m, n into units he in fact shows (in 2) that 
ma + na = (m + n) a, and mb + nb = (m + n) b. 
6 is proved by means of 2, as 5 by means of 1. 
3. If m .na, m.nb are equimultiples of na, nb, which are 
themselves equimultiples of a, b, then m. na, m. nb are also 
equimultiples of a, b. 
By separating m, n into their units Euclid practically proves 
that m. na = mn. a and m.nb = mn. b.