Commentary on Proposition XI.39¶
Back to Proposition XI.39.
This proposition is made use of in XII. 3, 4. The phraseology is interesting because we find one of the parallelogrammic faces of one of the triangular prisms called its base, and the perpendicular on this plane from that vertex of either triangular face which is not in this plane the height.
The proof is simple because we have only to complete parallelepipeds which are double the prisms respectively and then use XI. 31. It has to be borne in mind, however, that, if the parallelepipeds are not rectangular, the proof in XI. 28 is not sufficient to establish the fact that the parallelepipeds are double of the prisms, but has to be supplemented as shown in the note on that proposition. XII. 4 does, however, require the theorem in its general form.