If three straight lines be proportional, the parallelepipedal solid formed out of the three is equal to the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid.
若三条线段成比例，则由它们构成的平行六面体等于以中间线段为棱且与原立体等角等边的平行六面体。
Let A, B, C be three straight lines in proportion, so that, as A is to B, so is B to C; I say that the solid formed out of A, B, C is equal to the solid on B which is equilateral, but equiangular with the aforesaid solid. Let there be set out the solid angle at E contained by the angles DEG, GEF, FED, let each of the straight lines DE, GE, EF be made equal to B, and let the parallelepipedal solid EK be completed, let LM be made equal to A, and on the straight line LM, and at the point L on it, let there be constructed a solid angle equal to the solid angle at E, namely that contained by NLO, OLM, MLN; let LO be made equal to B, and LN equal to C.
设A、B、C成比例，即A:B = B:C。构造以B为棱的立体角E，并完成平行六面体EK。
Now, since, as A is to B, so is B to C, while A is equal to LM, B to each of the straight lines LO, ED, and C to LN, therefore, as LM is to EF, so is DE to LN.
取LM等于A，在L点构造与E相等的立体角，使LO等于B，LN等于C。
Thus the sides about the equal angles NLM, DEF are reciprocally proportional; therefore the parallelogram MN is equal to the parallelogram DF. [VI. 14] And, since the angles DEF, NLM are two plane rectilineal angles, and on them the elevated straight lines LO, EG are set up which are equal to one another and contain equal angles with the original straight lines respectively, therefore the perpendiculars drawn from the points G, O to the planes through NL, LM and DE, EF are equal to one another; [XI. 35, Por.] hence the solids LH, EK are of the same height.
由比例关系得LM:EF = DE:LN，故平行四边形MN等于DF（VI.14）。
But parallelepipedal solids on equal bases and of the same height are equal to one another; [XI. 31] therefore the solid HL is equal to the solid EK.
因LO与EG等长且与对应边成等角，故从G、O到各自平面的垂线相等，从而立体LH与EK等高；等底等高平行六面体相等（XI.31），故HL等于EK。