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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)