第11卷命题 27 · 作相似平行六面体

Let AB be the given straight line and CD the given parallelepipedal solid; thus it is required to describe on the given straight line AB a parallelepipedal solid similar and similarly situated to the given parallelepipedal solid CD. For on the straight line AB and at the point A on it let the solid angle, contained by the angles BAH, HAK, KAB, be constructed equal to the solid angle at C, so that the angle BAH is equal to the angle ECF, the angle BAK equal to the angle ECG, and the angle KAH to the angle GCF; and let it be contrived that, as EC is to CG, so is BA to AK, and, as GC is to CF, so is KA to AH.

在给定线段AB上点A处作立体角，使其等于给定立体角C，即角BAH等于角ECF，角BAK等于角ECG，角KAH等于角GCF。

[VI. 12] Therefore also, ex aequali, as EC is to CF, so is BA to AH. [V. 22] Let the parallelogram HB and the solid AL be completed.

作比例：EC比CG等于BA比AK，GC比CF等于KA比AH，从而由等比得EC比CF等于BA比AH。

Now since, as EC is to CG, so is BA to AK, and the sides about the equal angles ECG, BAK are thus proportional, therefore the parallelogram GE is similar to the parallelogram KB. For the same reason the parallelogram KH is also similar to the parallelogram GF, and further FE to HB; therefore three parallelograms of the solid CD are similar to three parallelograms of the solid AL.

完成平行四边形HB和立体AL。由于对应角相等且边成比例，平行四边形GE相似于KB，KH相似于GF，FE相似于HB。

But the former three are both equal and similar to the three opposite parallelograms, and the latter three are both equal and similar to the three opposite parallelograms; therefore the whole solid CD is similar to the whole solid AL.

立体CD的三个平行四边形分别相似于立体AL的三个平行四边形，且各自的对立面全等相似，故整个立体CD相似于整个立体AL。