Magnitudes commensurable with the same magnitude are commensurable with one another also.
与同一量可公度的两个量，它们彼此也是可公度的。
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For let each of the magnitudes A, B be commensurable with C; I say that A is also commensurable with B. For, since A is commensurable with C, therefore A has to C the ratio which a number has to a number. [X. 5] Let it have the ratio which D has to E.
设量A和B均与C可公度，则A与C的比等于某两数之比，设为D比E；同样，C与B的比等于某两数之比，设为F比G。
Again, since C is commensurable with B, therefore C has to B the ratio which a number has to a number. [X. 5] Let it have the ratio which F has to G.
取连续比例数H、K、L，使得D比E等于H比K，且F比G等于K比L。
And, given any number of ratios we please, namely the ratio which D has to E and that which F has to G, let the numbers H, K, L be taken continuously in the given ratios; [cf. VIII. 4] so that, as D is to E, so is H to K, and, as F is to G, so is K to L. Since, then, as A is to C, so is D to E, while, as D is to E, so is H to K, therefore also, as A is to C, so is H to K. [V. 11] Again, since, as C is to B, so is F to G, while, as F is to G, so is K to L, therefore also, as C is to B, so is K to L.
由A比C等于D比E，且D比E等于H比K，得A比C等于H比K；同理，C比B等于K比L。
[V. 11] But also, as A is to C, so is H to K; therefore, ex aequali, as A is to B, so is H to L. [V. 22] Therefore A has to B the ratio which a number has to a number; therefore A is commensurable with B.
由等比关系，A比B等于H比L，即两数之比，故A与B可公度。