elem.5.10
对于与同一量有比的两个量，具有较大比的那个量较大；而同一量对其有较大比的那个量较小。
取量 A、B 与同一量 C：A:C 与 B:C 两个比中较大的一比对应较大的量；C:A 与 C:B 中较大的一比对应较小的量。
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For let A have to C a greater ratio than B has to C; I say that A is greater than B. For, if not, A is either equal to B or less. Now A is not equal to B; for in that case each of the magnitudes A, B would have had the same ratio to C; [V. 7] but they have not; therefore A is not equal to B.
设A与C的比大于B与C的比。假设A不大于B，则A等于或小于B。
Nor again is A less than B; for in that case A would have had to C a less ratio than B has to C; [V. 8] but it has not; therefore A is not less than B. But it was proved not to be equal either; therefore A is greater than B.
若A等于B，则A与C的比等于B与C的比（V.7），与假设矛盾，故A不等于B。
Again, let C have to B a greater ratio than C has to A; I say that B is less than A. For, if not, it is either equal or greater. Now B is not equal to A; for in that case C would have had the same ratio to each of the magnitudes A, B; [V. 7] but it has not; therefore A is not equal to B.
若A小于B，则A与C的比小于B与C的比（V.8），也与假设矛盾，故A不小于B。因此A大于B。
Nor again is B greater than A; for in that case C would have had to B a less ratio than it has to A; [V. 8] but it has not; therefore B is not greater than A. But it was proved that it is not equal either; therefore B is less than A.
设C与B的比大于C与A的比。类似地，假设B不小于A，则B等于或大于A，均导致矛盾（V.7和V.8），故B小于A。