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Proposition 13

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.
Q.E.D.

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This is the numerical analogue of proposition V.16 for magnitudes. It says that if a : b = c : d, then a : c = b : 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.

© 1996
D.E.Joyce
Clark University