| Between two cubic numbers there are two mean proportional numbers, and the cube has to the cube the triplicate ratio of that which the side has to the side. | |
| Let A and B be cubic numbers, and let C be the side of A, and D of B.
I say that between A and B there are two mean proportional numbers, and A has to K the ratio triplicate of that which C has to D. Multiply C by itself to make E, and by D to make F, multiply D by itself to make G, and multiply the numbers C and D by F to make H and K respectively. Now, since A is a cube, and C its side, and C multiplied by itself makes E, therefore C multiplied by itself makes E and multiplied by E makes A. For the same reason also D multiplied by itself makes G and multiplied by G makes B. | |
| And, since C multiplied by the numbers C and D makes E and F respectively, therefore C is to D as E is to F. For the same reason also C is to D as F is to G. Again, since C multiplied by the numbers E and F makes A and H respectively, therefore E is to F as A is to H. But E is to F as C is to D. Therefore C is to D as A is to H. | VII.17 |
| Again, since the numbers C and D multiplied by F make H and K respectively, therefore C is to D as H is to K. Again, since D multiplied by each of the numbers F and G makes K and B respectively, therefore F is to G as K is to B. | VII.18 |
| But F is to G as C is to D, therefore C is to D as A is to H, as H is to K, and as K is to B.
Therefore H and K are two mean proportionals between A and B. I say next that A also has to B the ratio triplicate of that which C has to D. | |
| Since A, H, K, and B are four numbers in proportion, therefore A has to B the ratio triplicate of that which A has to H. | V.Def.10 |
| But A is to H as C is to D, therefore A also has to B the ratio triplicate of that which C has to D. | |
| Therefore, Between two cubic numbers there are two mean proportional numbers, and the cube has to the cube the triplicate ratio of that which the side has to the side. | |
| Q.E.D. | |
Book VIII Introduction - Proposition VIII.11 - Proposition VIII.13.