Euclid's Elements
Book VII
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.

java applet or image 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.

Guide

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