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EUCLID 
11. If a:h = c:d, 
and c:d — e:f, 
then a:b = e:f. 
Proved by taking any equimultiples of a, c, e and any other 
equimultiples of b, d,f, and using Def. 5. 
12. If a:b = c:d = e:f = ... 
then a:b = {a + c + e+ ...):{b + d+f+...). 
Proved by means of V. 1 and Def. 5, after taking equi 
multiples of a, c, e ... and other equimultiples of b, d, f.... 
13. If a:b = c:d, 
and c:d > e:f, 
then a:b > e:f. 
Equimultiples me, me of c, e are taken and equimultiples 
nd, nf of d, f such that, while me > nd, me is not greater 
than nf (Def. 7). Then the same equimultiples ma, me of 
a, c and the same equimultiples nb, nd of b, d are taken, and 
Defs. 5 and 7 are used in succession. 
14. If a:b = c:d, then, according as a > = < c, b > = < d. 
The first case only is proved; the others are dismissed with 
‘ Similarly 
If a > c, a:b > c:b. (8) 
But a : b = c: d, whence (13) c:d > c:b, and therefore (10) 
b > d. 
15. a:b = ma: mb. 
Dividing the multiples into their units, we have m equal 
ratios a: b; the result follows by 12. 
Propositions 16-19 prove certain cases of the transformation 
of proportions in the sense of Defs. 12-16. The case of 
inverting the ratios is omitted, probably as being obvious. 
For, if a:b = c:d, the application of Def. 5 proves simul 
taneously that b: a = d: c. 
16. If a:b = c:d, 
then, alternando, a:c = b:d. 
Since a:b — ma: mb, and c:d = nc: nd, 
(15)