## 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.
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.

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 Select from Book VII Book VII intro VII.Def.1-2 VII.Def.3-5 VII.Def.6-10 VII.Def.11-14 VII.Def.15-19 VII.Def.20 VII.Def.21 VII.Def.22 VII.1 VII.2 VII.3 VII.4 VII.5 VII.6 VII.7 VII.8 VII.9 VII.10 VII.11 VII.12 VII.13 VII.14 VII.15 VII.16 VII.17 VII.18 VII.19 VII.20 VII.21 VII.22 VII.23 VII.24 VII.25 VII.26 VII.27 VII.28 VII.29 VII.30 VII.31 VII.32 VII.33 VII.34 VII.35 VII.36 VII.37 VII.38 VII.39 Select book Book I Book II Book III Book IV Book V Book VI Book VII Book VIII Book IX Book X Book XI Book XII Book XIII Select topic Introduction Table of Contents Geometry applet About the text Euclid Web references A quick trip