## Proposition 17

 If a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied. Let the number A multiplied by the two numbers B and C make D and E. I say that B is to C as D is to E. Since A multiplied by B makes D, therefore B measures D according to the units in A. But the unit F also measures the number A according to the units in it, therefore the unit F measures the number A the same number of times that B measures D. Therefore the unit F is to the number A as B is to D. VII.Def.20 For the same reason the unit F is to the number A as C is to E, therefore B is to D as C is to E. VII.Def.20 (V.11) Therefore, alternately B is to C as D is to E. VII.13 Therefore, if a number multiplied by two numbers makes certain numbers, then the numbers so produced have the same ratio as the numbers multiplied. Q.E.D.
Algebraically, b : c = ab : ac.

This proposition is used very 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