Prop. 11. If
a : b :: c : d,
and
a : b :: e : f,
then
c : d :: e : f.
In words, if too ratios are equal to a third, they are equal to one another. After these propositions have been proved, we have a right to consider a ratio as a magnitude, for only now can we consider a ratio as something for which the axiom about magnitudes holds: things which are equal to a third are equal to one another.
We shall indicate this by writing in future the sign = instead of ::. The remaining propositions, which explain themselves, may then be stated as follows:
§ 53. Prop. 12. If
a : b = c : d = e : f,