Commentary on Proposition XI.25¶
Back to Proposition XI.25.
It is to be observed that, as the word parallelogrammic was used in Book I. without any definition of its meaning, so παραλληλεπίπεδος, parallelepipedal, is here used without explanation. While it means simply “with parallel planes,” i.e. “faces,” the term is appropriated to the particular solid which has six plane faces parallel two and two. The proper translation of στερεὸν παραλληλεπίπεδον is parallelepipedal solid, not solid parallelepiped, as it is usually translated. Still less is the solid a parallelopiped, as the word is not uncommonly written.
The opposite faces in each set of parallelepipedal solids in this proposition are not only equal but equal and similar. Euclid infers that the solids in each set are equal from Def. 10; but, as we have seen in the note on Deff. 9, 10, though it is true, where no solid angle in the figures is contained by more than three plane angles, that two solid figures are equal and similar which are contained by the same number of equal and similar faces, similarly arranged, the fact should have been proved. To do this, we have only to prove the proposition, given above in the note on XI. 21, that two trihedral angles are equal if the three face angles of the one are respectively equal to the three face angles in the other, and all are arranged in the same order, and then to prove equality by applying one figure to the other as is done by Simson in his proposition C.
Application will also, of course, establish what is assumed by Euclid of the solids formed by the multiples of the original solids, namely that, if LF $\gtreqqless$ NF, the solid LU $\gtreqqless$ the solid NU.