If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable.
若两量之比等于两数之比，则此两量可公度。
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For let the two magnitudes A, B have to one another the ratio which the number D has to the number E; I say that the magnitudes A, B are commensurable. For let A be divided into as many equal parts as there are units in D, and let C be equal to one of them; and let F be made up of as many magnitudes equal to C as there are units in E. Since then there are in A as many magnitudes equal to C as there are units in D, whatever part the unit is of D, the same part is C of A also; therefore, as C is to A, so is the unit to D. [VII. Def. 20] But the unit measures the number D; therefore C also measures A.
设两量A、B之比等于数D比数E。将A分为D个等份，每份为C；再取E个与C相等的量组成F。
And since, as C is to A, so is the unit to D, therefore, inversely, as A is to C, so is the number D to the unit. [cf. V. 7, Por.] Again, since there are in F as many magnitudes equal to C as there are units in E, therefore, as C is to F, so is the unit to E. [VII. Def. 20] But it was also proved that, as A is to C, so is D to the unit; therefore, ex aequali, as A is to F, so is D to E. [v. 22] But, as D is to E, so is A to B; therefore also, as A is to B, so is it to F also.
由定义，C与A之比等于单位与D之比，故C度量A。同理，C与F之比等于单位与E之比。
[V. 11] Therefore A has the same ratio to each of the magnitudes B, F; therefore B is equal to F. [V. 9] But C measures F; therefore it measures B also. Further it measures A also; therefore C measures A, B. Therefore A is commensurable with B.
由A与C之比等于D与单位之比，及C与F之比等于单位与E之比，经等比得A与F之比等于D与E之比。
Therefore etc. PORISM. From this it is manifest that, if there be two numbers, as D, E, and a straight line, as A, it is possible to make a straight line [F] such that the given straight line is to it as the number D is to the number E. And, if a mean proportional be also taken between A, F, as B, as A is to F, so will the square on A be to the square on B, that is, as the first is to the third, so is the figure on the first to that which is similar and similarly described on the second.
已知A与B之比等于D与E之比，故A与B之比等于A与F之比，因此B等于F。C度量F，故C度量B，又度量A，所以A与B可公度。