If a square measure a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square.
如果一个平方数度量另一个平方数，那么它们的边也互相度量；反之，如果边互相度量，那么平方数也互相度量。
本页以“平方数互测与边互测”整体图解辅助阅读；点、线、角、圆索引已按命题文字和证明步骤校订，可与证明和问答联动。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let A, B be square numbers, let C, D be their sides, and let A measure B; I say that C also measures D. For let C by multiplying D make E; therefore A, E, B are continuously proportional in the ratio of C to D.
设A、B为平方数，C、D分别为其边，且A度量B。则C乘以D得E，由第八卷命题11，A、E、B成连比例，比例等于C比D。
[VIII. 11] And, since A, E, B are continuously proportional, and A measures B, therefore A also measures E. [VIII. 7] And, as A is to E, so is C to D; therefore also C measures D.
由于A、E、B成连比例且A度量B，由第八卷命题7，A也度量E。又A比E等于C比D，由第七卷定义20，C度量D。
[VII. Def. 20] Again, let C measure D; I say that A also measures B. For, with the same construction, we can in a similar manner prove that A, E, B are continuously proportional in the ratio of C to D.
反之，设C度量D。同样构造，可类似证明A、E、B成连比例，比例等于C比D。
And since, as C is to D, so is A to E, and C measures D, therefore A also measures E. [VII. Def. 20] And A, E, B are continuously proportional; therefore A also measures B.
由于C比D等于A比E，且C度量D，由第七卷定义20，A度量E。又A、E、B成连比例，故A也度量B。