Euclid’s Elements

Book VII

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.

VII.Def.15

Since A multiplied by B makes D, therefore B measures D according to the units in A.

VII.Def.20

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
V.11

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.13

Therefore, alternately B is to C as D is to E.

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.

Guide

Symbolically, b : c = ab : ac.

This proposition is used very frequently in Books VII through IX starting with the next proposition.

Referenced by

Book VIIVII.18, VII.19, VII.22, VII.34
Book VIIIVIII.2, VIII.5, VIII.10, VIII.11, VIII.12, VIII.18, VIII.19, VIII.20
Book IXIX.1, IX.2, IX.4, IX.5