Proportion

Proportion, in arithmetic and geometry, a particular kind of relation between groups of numbers or quantities. According to the arithmetical definition, proportion is the equality of ratios; ratio is in turn a relation of two numbers to each other defined as the division of one number by the other. Thus, the ratio of 12 to 3, expressed by 12/3 or by 4, denotes that 12 contains 3 four times. The ratio of 8 to 2 is also 4, and therefore, according to the definition of proportion, the four numbers 12, 3, 8, and 2 are in proportion. This proportion is expressed as 12:3::8:2, read “12 is to 3 as 8 is to 2.” In every true proportion the product of the first and last terms (known as the extremes) is equal to the product of the second and third term (known as the means); the arithmetical rule called proportion directly depends upon this property. The object of this rule is to find a fourth number that is proportional to three given numbers; the number is found by multiplying together the second and third terms and dividing the product by the first. Continued proportion is a property of every three consecutive or equidistant terms in a geometric progression; for example, in the sequence 2, 4, 8, 16, 32 ..., 2:4::4:8 and 4:8::8:16.

In ancient Greece the theory of numbers was not adequate for an arithmetical account of geometrical magnitudes. Therefore, Greek astronomer and mathematician Eudoxus proposed a separate theory of geometrical proportion in the early 4th century bc. An account of this theory is found in books five and six of Elements, written by the Greek mathematician Euclid.