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. |
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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. | ||
Q.E.D. |
This proposition is used frequently in Books VII through IX starting with the next proposition.
Book VII Introduction - Proposition VII.12 - Proposition VII.14.