If three straight lines are proportional, then the parallelepipedal solid formed out of the three equals the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid.

Let *A, B,* and *C* be three straight lines in proportion, so that *A* is to *B* as *B* is to *C.*

I say that the solid formed out of *A, B,* and *C* equals the solid on *B* which is equilateral, but equiangular with the aforesaid solid.

Set out the solid angle at *E* contained by the angles *DEG, GEF,* and *FED,* and make each of the straight lines *DE, GE,* and *EF* equal to *B.* Complete the parallelepipedal solid *EK.* Make *LM* equal to *A.* Construct a solid angle at the point *L* on the straight line *LM* equal to the solid angle at *E,* namely that contained by *NLO, OLM,* and *MLN.* Make *LO* equal to *B,* and *LN* equal to *C.*

Now, since *A* is to *B* as *B* is to *C,* while *A* equals *LM,* and *B* equals each of the straight lines *LO, ED,* and *C* to *LN,* therefore *LM* is to *EF* as *DE* is to *LN.* Thus the sides about the equal angles *NLM, DEF* are reciprocally proportional, therefore the parallelogram *MN* equals the parallelogram *DF.*

And, since the angles *DEF* and *NLM* are two plane rectilinear angles, and on them the elevated straight lines *LO* and *EG* are set up which equal one an other and contain equal angles with the original straight lines respectively, therefore the perpendiculars drawn from the points *G* and *O* to the planes through *NL* and *LM* and through *DE* and *EF* equal one another, therefore the solids *LH* and *EK* are of the same height.

But parallelepipedal solids on equal bases and of the same height equal one another, therefore the solid *HL* equals the solid *EK.*

And *LH* is the solid formed out of *A, B,* and *C,* and *EK* is the solid on *B,* therefore the parallelepipedal solid formed out of *A, B,* and *C* equals the solid on *B* which is equilateral, but equiangular with the aforesaid solid.

Therefore, *if three straight lines are proportional, then the parallelepipedal solid formed out of the three equals the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid.*

Q.E.D.