| If a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied. | ||
| Let the number A multiplied by the two numbers B and C make D and E.
I say that B is to C as D is to E. |
||
| Since A multiplied by B makes D, therefore B measures D according to the units in A. | ||
| But the unit F also measures the number A according to the units in it, therefore the unit F measures the number A the same number of times that B measures D. Therefore the unit F is to the number A as B is to D. | VII.Def.20 | |
| For the same reason the unit F is to the number A as C is to E, therefore B is to D as C is to E. | VII.Def.20
(V.11) | |
| Therefore, alternately B is to C as D is to E. | VII.13 | |
| Therefore, if a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied. | ||
| Q.E.D. | ||
This proposition is used very frequently in Books VII through IX starting with the next proposition.
Book VII Introduction - Proposition VII.16 - Proposition VII.18.