Plane numbers have to one another the ratio compounded of the ratios of their sides.
两个平面数之比等于它们边长之比复合而成的比。
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Let A, B be plane numbers, and let the numbers C, D be the sides of A, and E, F of B; I say that A has to B the ratio compounded of the ratios of the sides. For, the ratios being given which C has to E and D to F, let the least numbers G, H, K that are continuously in the ratios C : E, D : F be taken, so that, as C is to E, so is G to H, and, as D is to F, so is H to K.
设A、B为平面数，C、D为A的边，E、F为B的边。取连续成比例C:E和D:F的最小三个数G、H、K，使得C:E=G:H且D:F=H:K。
[VIII. 4] And let D by multiplying E make L. Now, since D by multiplying C has made A, and by multiplying E has made L, therefore, as C is to E, so is A to L.
令D乘以E得L。由于D乘以C得A，乘以E得L，故C:E=A:L。又C:E=G:H，所以G:H=A:L。
[VII. 17] But, as C is to E, so is G to H; therefore also, as G is to H, so is A to L. Again, since E by multiplying D has made L, and further by multiplying F has made B, therefore, as D is to F, so is L to B.
由于E乘以D得L，乘以F得B，故D:F=L:B。又D:F=H:K，所以H:K=L:B。
[VII. 17] But, as D is to F, so is H to K; therefore also, as H is to K, so is L to B. But it was also proved that, as G is to H, so is A to L; therefore, ex aequali, as G is to K, so is A to B.
已证G:H=A:L，由等比定理得G:K=A:B。因此A与B的比等于由C:E和D:F复合而成的比。