THE ELEMENTS. BOOK V 
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we have ma : mb — no ; nd, (H) 
whence (14), according as ma > = < no, mb > = < nd ; 
therefore (Def. 5) a : c = h : d. 
17. If a:b = c:d, 
then, separando, {a — b) : b = (c — d) : d. 
Take met, mb, me, md equimultiples of all four magnitudes, 
and nb, nd other equimultiples of b, d. It follows (2) that 
(m + n) b, (m + n)d are also equimultiples of b, d. 
Therefore, since a:h = c :d, 
ma > = <{m + n)b according as me > = <(m + n) d. (Def. 5) 
Subtracting mb from both sides of the former relation and 
md from both sides of the latter, we have (5) 
m (a — b) > = < nb according as m (c — d) > = < nd. 
Therefore (Def. 5) a — b:b = c — d:d. 
(I have here abbreviated Euclid a little, without altering the 
substance.) 
18. If a : b = c : d, 
then, componendo, {a + b):b = (c + d):d. 
Proved by reductio ad absurdum. Euclid assumes that 
a + h:b = (c + d): (d±x), if that is possible. (This implies 
that to any three given magnitudes, two of which at least 
are of the same kind, there exists a fourth proportional, an 
assumption which is not strictly legitimate until the fact has 
been proved by construction.) 
Therefore, separando (17), a\b — (c + x) : (d±x), 
whence (11), (c + x) : {d ± x) = c : d, which relations are im 
possible, by 14. 
19. If a:b = c:d, 
then (a — c) : (b — d) = a:b. 
Alternately (16), 
a:c = b:d, whence (a — c):c = (b—d):d (17). 
Alternately again, (a — c):{b — d) = c:d (16) ; 
whence (11) (a — c):{b — d) = a:b.