|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
|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.|
This proposition is used very frequently in Books VII through IX starting with the next proposition.
Next proposition: VII.18