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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.

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Algebraically, b : c = ab : ac.

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

Book VII Introduction - Proposition VII.16 - Proposition VII.18.

© 1996
D.E.Joyce
Clark University