A proportion is a mathematical statement expressing equality of two ratios.
, where ratios are expressed as fractions.
A Greek mathematician Eudoxus provided a definition for the meaning of the equality between two ratios.
This definition of proportion forms the subject of Euclid's Book V, where we can read: Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.Later, the realization that ratios are numbers allowed to switch from solving proportions to equations, and from transformation of proportions to algebraic transformations.
[5] If the means of the geometric proportion are equal, and the rightmost extreme is equal to the difference between the leftmost extreme and a mean, then such a proportion is called harmonic:[6]