If two straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial, be added together, the whole straight line is irrational : and let it be called major.
若两条线段平方不可通，且它们平方和是有理的，但所成矩形是中项线，则它们之和是无理的，称之为主线段。
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For let two straight lines AB, BC incommensurable in square, and fulfilling the given conditions [X. 33 ], be added together; I say that AC is irrational.
设AB、BC平方不可通，且满足给定条件，相加得AC。
For, since the rectangle AB, BC is medial, twice the rectangle AB, BC is also medial.
因矩形AB、BC是中项线，其二倍也是中项线。
[X. 6 and 23, Por.] But the sum of the squares on AB, BC is rational; therefore twice the rectangle AB, BC is incommensurable with the sum of the squares on AB, BC, so that the squares on AB, BC together with twice the rectangle AB, BC that is, the square on AC, is also incommensurable with the sum of the squares on AB, BC; [X. 16 ] therefore the square on AC is irrational, so that AC is also irrational.
但AB、BC平方和是有理的，故二倍矩形与平方和不可通，从而平方和加二倍矩形（即AC上的正方形）与平方和不可通。
因此AC上的正方形是无理的，故AC也是无理的。