elem.5.7
相等的量对同一个量有相同的比，并且同一个量对相等的量也有相同的比。
A、B 相等，C 为任意量；D、E 是 A、B 的等倍量，F 是 C 的任意倍数。三行竖直排列，便于核对 A:C = B:C 的对应关系；额外标记 I、M、O、P、R、S 为推理过程中保留的辅助点位。
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Let A, B be equal magnitudes and C any other, chance, magnitude; I say that each of the magnitudes A, B has the same ratio to C, and C has the same ratio to each of the magnitudes A, B. For let equimultiples D, E of A, B be taken, and of C another, chance, multiple F. Then, since D is the same multiple of A that E is of B, while A is equal to B, therefore D is equal to E.
设A、B是相等的量，C是任意其他量。取A、B的等倍数D、E，以及C的任意倍数F。
But F is another, chance, magnitude. If therefore D is in excess of F, E is also in excess of F, if equal to it, equal; and, if less, less. And D, E are equimultiples of A, B, while F is another, chance, multiple of C; therefore, as A is to C, so is B to C.
由于A等于B，且D、E分别是A、B的相同倍数，因此D等于E。
[V. Def. 5] I say next that C also has the same ratio to each of the magnitudes A, B. For, with the same construction, we can prove similarly that D is equal to E; and F is some other magnitude. If therefore F is in excess of D, it is also in excess of E, if equal, equal; and, if less, less.
若D大于F，则E也大于F；若等于，则相等；若小于，则小于。根据定义5，A比C等于B比C。
And F is a multiple of C, while D, E are other, chance, equimultiples of A, B; therefore, as C is to A, so is C to B. [V. Def. 5] Therefore etc. PORISM.
同理，若F大于D，则也大于E；若等于，则相等；若小于，则小于。因此C比A等于C比B。