Department of Mathematics, LSU Baton Rouge
Math 4005: Geometry

Ratio

Numerous definitions of the word "ratio" can be found. (Use Google to hunt for the definitions that occur on the web.)

The earliest use of the word "ratio" cited in the Oxford English Dictionary is from Barrow's 1660 translation of Euclid's Elements, Book V, Definition 3:

Ratio (or rate) is the mutual habitude or respect of two magnitudes of the same kind each to other, according to quantity.

The Greek word for ratio is "logos".

In Heath's translation of Euclid, Definition V.3 reads:

A ratio is a sort of relation in respect of size between two magnitudes of the same kind.

The Oxford English Dictionary itself includes the following under the entry for "ratio":

2. a. Math. The relation between two similar magnitudes in respect of quantity, determined by the number of times one contains the other (integrally or fractionally).

In a paper cited elsewhere, Wu writes:

I would like to single out “ratio” for a brief discussion. This term was first introduced by Euclid in his Elements and, if he had the mathematical understanding of the real numbers as we do now, he would have said outright: “the ‘ratio of A to B’ means the quotient A/B”. But he didn’t, and human beings didn’t either for the next twenty one centuries or so. ... For this reason, the long tradition of the inability to make sense of ratio continues to encroach on school textbooks even after human beings came to a complete understanding of the real numbers round 1870. It may be 130 years too late, but now is the time for us to say unequivocally in all our school textbooks: The ratio of A to B means the quotient A/B [this being a real number--JJM]. Unfortunately, most textbooks continue to take “ratio” as a known concept and fractions are defined as “ratios of two whole numbers”. Or, prospective teachers are exhorted to “recognize that ratios are not directly measureable but they contain two units and that the order of the items in the ratio pair in a proportion is critical”. Source reference.

Proportion

The notion of proportion is often used together with ratio. (Google search for definitions of "proportion".)

The Oxford English Dictionary includes the following under the entry for "proportion":

II. In technical senses. 9. Math. An equality of ratios, esp. of geometrical ratios; a relation among quantities such that the quotient of the first divided by the second is equal to that of the third divided by the fourth.

In Heath's translation of Euclid, the term "proportional" is defined in Definition V.6:

Let magnitutes which have the same ratio be called proportional.

The Greek word translated as "proportional" is "analogon". According to Heath, this is identical in meaning to the two-word phrase "ana logon", which may be translated "in proportion". Euclid's definition seems consistent with the idea expressed in the dictionary definition above. "Proportion" is clearly used in this sense in Proposition VI:2.

Euclid and Real Numbers

In modern mathematics, a proportions are sometimes written: "a is to b as c is to d" (or "a:b::c:d"). Here, a,b,c and d might be any positive numbers, and accordingly a/b and c/d would also be numbers. The proportion is true if the two numbers a/b and c/d are equal.

In Euclid's mathematics, the comparison of ratios is not such a simple process . Definition V.5 gives conditions for two ratios to be the same. This involves an indefinite number of comparisons. (Read the definition and Joyce's commentary.)

We can translate Definition V.5 into modern terminology as follows: "a is to b as c is to d" means: for any positive integers m and n, ma > nb iff mc > nd. The latter condition has the same meaning as: for any positive integers m and n, a/b > n/m iff c/d >n/m. Thus, Euclid's definition implies that we should call the ratios a:b and c:d the same if a/b and c/d exceed exactly the same rational numbers.

Euclid, however, could not have phrased his definition this way. He did not think of the process of dividing one magnitude by another as yielding a number. He did not have any conception of a/b as a number. He would have viewed it as a comparison. In this respect, his fundamental conceptual framework differed from ours. (See the quotation from Wu, above.)