## Proposition 18

 If two numbers multiplied by any number make certain numbers, then the numbers so produced have the same ratio as the multipliers. Let two numbers A and B multiplied by any number C make D and E. I say that A is to B as D is to E. Since A multiplied by C makes D, therefore C multiplied by A makes D. For the same reason also C multiplied by B makes E. VII.16 Therefore the number C multiplied by the two numbers A and B makes D and E. Therefore A is to B as Dis to E. VII.17 Therefore, if two numbers multiplied by any number make certain numbers, then the numbers so produced have the same ratio as the multipliers. Q.E.D.
Whereas the last proposition stated

b : c = ab : ac,
this one says
b : c = ba : ca.

The only difference is the order of multiplication, but VII.16 says multiplication is commutative, so that order is irrelevant.

This proposition is used in the next one and occasionally in Book VIII.

Next proposition: VII.19

Previous: VII.17

 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