|If four numbers are proportional, then they are also proportional alternately.|
|Let the four numbers A, B, C, and D be proportional, so that A is to B as C is to D.
I say that they are also proportional alternately, so that A is to C as B is to D.
|Since A is to B as C is to D, therefore, A is the same part or parts of B as C is of D.||VII.Def.20|
|Therefore, alternately, A is the same part or parts of C as B is of D.||VII.10|
|Therefore A is to C as B is to D.||VII.Def.20|
|Therefore, if four numbers are proportional, then they are also proportional alternately.|
This proposition is used frequently in Books VII through IX starting with the next proposition.
Next proposition: VII.14