THE ELEMENTS. BOOK V 
387 
4. If a :b = c:d, then ma: nb = me: nd. 
Take any equimultiples p.vic of ma, me, and any 
equimultiples q.nb, q.nd of nb, nd. Then, by 3, these equi 
multiples are also equimultiples of a, c and b, d respectively, 
so that by Def. 5, since a : b = c:d, 
p. ma > = < q. nb according as p. me > — < q. nd, 
. whence, again by Def. 5, since p, q are any integers, 
ma: nb = me: nd. 
7, 9. If a — b, then a:c = b:c) 
and c:a =c:bj ’ 
8, 10. If a > b, then a:c > 6:c] 
and c: b > c: a] ’ 
; and conversely. 
; and conversely. 
7 is proved by means of Def. 5. Take ma, mb equi 
multiples of a, b, and ne a multiple of c. Then, since a — b, 
ma > — < nc according as mb > = < nc, 
and nc > = < ma according as nc > = < mb, 
whence the results follow. 
8 is divided into two cases according to which of the two 
magnitudes a — b, b is the less. Take m such that 
m{a~h) > g or mb > c 
in the two cases respectively. Next let nc be the first 
multiple of c which is greater than mb or m(a — b) respec 
tively, so that 
Then, (i) since m (a — b) > c, we have, by addition, ma > nc. 
(ii) since mb > c, we have similarly ma > nc. 
In either case 'tab < nc, since in case (ii) m {a — b)> mb. 
Thus in either case, by the definition (7) of greater ratio, 
and 
a:c > b:c, 
c:b > c:a.