If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be called binomial.
若将两条仅平方可通分的有理线段相加，则整体为无理线段，称之为二项线。
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For let two rational straight lines AB, BC commensurable in square only be added together; I say that the whole AC is irrational. For, since AB is incommensurable in length with BC— for they are commensurable in square only— and, as AB is to BC, so is the rectangle AB, BC to the square on BC, therefore the rectangle AB, BC is incommensurable with the square on BC.
设AB、BC为两条仅平方可通分的有理线段，相加得AC。
[X. 11 ] But twice the rectangle AB, BC is commensurable with the rectangle AB, BC [X. 6 ], and the squares on AB, BC are commensurable with the square on BC—for AB, BC are rational straight lines commensurable in square only— [X. 15 ] therefore twice the rectangle AB, BC is incommensurable with the squares on AB, BC.
因AB与BC长度不可通分（仅平方可通分），且AB比BC等于AB、BC所成矩形比BC上的正方形，故AB、BC所成矩形与BC上的正方形不可通分。
[X. 13 ] And, componendo, twice the rectangle AB, BC together with the squares on AB, BC, that is, the square on AC [II. 4 ], is incommensurable with the sum of the squares on AB, BC.
但二倍AB、BC所成矩形与AB、BC所成矩形可通分，且AB、BC上的正方形与BC上的正方形可通分（因AB、BC为仅平方可通分的有理线段），故二倍AB、BC所成矩形与AB、BC上的正方形不可通分。
[X. 16 ] But the sum of the squares on AB, BC is rational; therefore the square on AC is irrational, so that AC is also irrational.
合比，二倍AB、BC所成矩形与AB、BC上的正方形之和（即AC上的正方形）与AB、BC上的正方形之和不可通分；而AB、BC上的正方形之和为有理，故AC上的正方形为无理，从而AC为无理。