If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.
如果一个平行六面体被一个平行于相对平面的平面所截，则底面积之比等于体积之比。
For let the parallelepipedal solid ABCD be cut by the plane FG which is parallel to the opposite planes RA, DH; I say that, as the base AEFV is to the base EHCF, so is the solid ABFU to the solid EGCD. For let AH be produced in each direction, let any number of straight lines whatever, AK, KL, be made equal to AE, and any number whatever, HM, MN, equal to EH; and let the parallelograms LP, KV, HW, MS and the solids LQ, KR, DM, MT be completed. Then, since the straight lines LK, KA, AE are equal to one another, the parallelograms LP, KV, AF are also equal to one another, KO, KB, AG are equal to one another, and further LX, KQ, AR are equal to one another, for they are opposite.
设平行六面体ABCD被平行于相对平面RA、DH的平面FG所截，需证底AEFV比底EHCF等于体ABFU比体EGCD。
[XI. 24] For the same reason the parallelograms EC, HW, MS are also equal to one another, HG, HI, IN are equal to one another, and further DH, MY, NT are equal to one another. Therefore in the solids LQ, KR, AU three planes are equal to three planes. But the three planes are equal to the three opposite; therefore the three solids LQ, KR, AU are equal to one another.
延长AH，取线段AK、KL等于AE，HM、MN等于EH，完成平行四边形LP、KV、HW、MS及立体LQ、KR、DM、MT。
For the same reason the three solids ED, DM, MT are also equal to one another. Therefore, whatever multiple the base LF is of the base AF, the same multiple also is the solid LU of the solid AU. For the same reason, whatever multiple the base NF is of the base FH, the same multiple also is the solid NU of the solid HU.
由等长线段得平行四边形及相对面相等，故立体LQ、KR、AU三组对面相等，从而全等；同理立体ED、DM、MT全等。
And, if the base LF is equal to the base NF, the solid LU is also equal to the solid NU; if the base LF exceeds the base NF, the solid LU also exceeds the solid NU; and, if one falls short, the other falls short. Therefore, there being four magnitudes, the two bases AF, FH, and the two solids AU, UH, equimultiples have been taken of the base AF and the solid AU, namely the base LF and the solid LU, and equimultiples of the base HF and the solid HU, namely the base NF and the solid NU, and it has been proved that, if the base LF exceeds the base FN, the solid LU also exceeds the solid NU, if the bases are equal, the solids are equal, and if the base falls short, the solid falls short.
因此底LF是底AF的倍量时，体LU是体AU的同倍量；底NF是底FH的倍量时，体NU是体HU的同倍量。由等量公理，底等则体等，底大则体大，故底比等于体比。