Pyramids which are of the same height and have triangular bases are to one another as the bases.
同高且底面为三角形的棱锥，其体积之比等于底面面积之比。
Let there be pyramids of the same height, of which the triangles ABC, DEF are the bases and the points G, H the vertices; I say that, as the base ABC is to the base DEF, so is the pyramid ABCG to the pyramid DEFH. For, if the pyramid ABCG is not to the pyramid DEFH as the base ABC is to the base DEF, then, as the base ABC is to the base DEF, so will the pyramid ABCG be either to some solid less than the pyramid DEFH or to a greater. Let it, first, be in that ratio to a less solid W, and let the pyramid DEFH be divided into two pyramids equal to one another and similar to the whole and into two equal prisms; then the two prisms are greater than the half of the whole pyramid. [XII. 3] Again, let the pyramids arising from the division be similarly divided, and let this be done continually until there are left over from the pyramid DEFH some pyramids which are less than the excess by which the pyramid DEFH exceeds the solid W.
假设棱锥ABCG与棱锥DEFH的体积比不等于底面ABC与DEF的面积比，则设其等于某小于棱锥DEFH的立体W之比。
[X. I] Let such be left, and let them be, for the sake of argument, DQRS, STUH; therefore the remainders, the prisms in the pyramid DEFH, are greater than the solid W. Let the pyramid ABCG also be divided similarly, and a similar number of times, with the pyramid DEFH; therefore, as the base ABC is to the base DEF, so are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH. [XII. 4] But, as the base ABC is to the base DEF, so also is the pyramid ABCG to the solid W; therefore also, as the pyramid ABCG is to the solid W, so are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH; [V. II] therefore, alternately, as the pyramid ABCG is to the prisms in it, so is the solid W to the prisms in the pyramid DEFH. [V. 16] But the pyramid ABCG is greater than the prisms in it; therefore the solid W is also greater than the prisms in the pyramid DEFH.
将棱锥DEFH反复分割为相似的小棱锥和相等的棱柱，直至剩余小棱锥之和小于棱锥DEFH超出W的部分。
But it is also less: which is impossible. Therefore the prism ABCG is not to any solid less than the pyramid DEFH as the base ABC is to the base DEF. Similarly it can be proved that neither is the pyramid DEFH to any solid less than the pyramid ABCG as the base DEF is to the base ABC. I say next that neither is the pyramid ABCG to any solid greater than the pyramid DEFH as the base ABC is to the base DEF.
同样次数分割棱锥ABCG，由XII.4知两棱锥内棱柱之比等于底面之比，结合假设推出棱柱之和与W的比例关系，导致W同时大于和小于棱柱之和的矛盾。
For, if possible, let it be in that ratio to a greater solid W; therefore, inversely, as the base DEF is to the base ABC, so is the solid W to the pyramid ABCG. But, as the solid W is to the solid ABCG, so is the pyramid DEFH to some solid less than the pyramid ABCG, as was before proved; [XII. 2, Lemma] therefore also, as the base DEF is to the base ABC, so is the pyramid DEFH to some solid less than the pyramid ABCG: [V. II] which was proved absurd. Therefore the pyramid ABCG is not to any solid greater than the pyramid DEFH as the base ABC is to the base DEF. But it was proved that neither is it in that ratio to a less solid.
类似地，若假设比例大于棱锥DEFH，则通过反比和引理推出矛盾，故原命题成立。