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.

java applet or image

I say that A is to B as D is to E.

VII.16

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.17

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.

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.

Guide

 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.