To a minor straight line only one straight line can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of the squares on them rational but twice the rectangle contained by them medial.
对于一条小线段，只能附加一条线段，使得该线段与整体线段平方不可通约，且与整体线段所成两线段平方和为有理，但两倍所成矩形为中项线。
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Let AB be the minor straight line, and let BC be an annex to AB; therefore AC, CB are straight lines incommensurable in square which make the sum of the squares on them rational, but twice the rectangle contained by them medial.
设AB为小线段，BC为附加于AB的线段，则AC、CB是平方不可通约的线段，且它们平方和为有理，但两倍所成矩形为中项线。
[X. 76] I say that no other straight line can be annexed to AB fulfilling the same conditions.
假设可以附加另一条线段BD满足相同条件，则AD、DB也是平方不可通约的线段，且满足前述条件。
For, if possible, let BD be so annexed; therefore AD, DB are also straight lines incommensurable in square which fulfil the aforesaid conditions.
由于AD、DB的平方和减去AC、CB的平方和等于两倍矩形AD、DB减去两倍矩形AC、CB，而前者之差为有理面积（因为两者平方和均为有理），因此两倍矩形AD、DB也减去两倍矩形AC、CB得到有理面积。
[X. 76] Now, since the excess of the squares on AD, DB over the squares on AC, CB is also the excess of twice the rectangle AD, DB over twice the rectangle AC, CB, while the squares on AD, DB exceed the squares on AC, CB by a rational area, for both are rational, therefore twice the rectangle AD, DB also exceeds twice the rectangle AC, CB by a rational area: which is impossible, for both are medial.
但两倍矩形AD、DB和两倍矩形AC、CB均为中项线，其差不可能为有理面积，矛盾。故只能附加一条这样的线段。