By Julian Lowell Coolidge

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Similar reasoning may have been used to derive the Babylonian formula for the volume of a basket whose height is h and whose upper and lower bases have areas B and B' respectively. If the volume of the basket is assumed to be equal to that of a cylinder of the same height whose base has an area equal to the average of B and B', then we get the Babylonian formula for the volume of the basket, Unfortunately, this formula is correct only if B = B'. The Egyptians who lived one thousand years later were more ingenious and more successful in using the averaging principle.

One of Plato’s pupils. Here is the essential content of Eudoxus’s definition, expressed in modern language and notation: I. Two magnitudes a and b have a ratio if there exists a positive integer m such that ma > b, and there exists a positive integer n such that nb > a. II. If (a, b) and (c, d) are two ordered pairs of magnitudes that have a ratio, then they have the same ratio if for any choice of positive integers m and n, either 1) ma > nb, and mc > nd, or 2) ma = nb, and mc = nd, or 3) ma < nb, and mc < nd.

It is the product of a long process of growth. At first the only numbers known were the counting numbers, 0, 1, 2, 3, and so on. Then fractions were introduced, to make it possible to express, at least approximately, lengths that are not whole number multiples of the unit of length. Irrational numbers were invented to represent lengths that cannot be expressed by fractions. Negative numbers were brought in to permit the subtraction of any two numbers. * To see some features of the number system that have a bearing on geometric concepts that we discuss later, we shall retrace the steps in the growth of the number system.