"The theory of proportions is credited to Eudoxus... and is expounded in Book V of Euclid's Elements. The purpose of the theory is to enable lengths (and other geometric quantities) to be treated as precisely as numbers, while only admitting the use of rational numbers. ...To simplify ...let us call lengths rational if they are rational multiples of a fixed length. Eudoxus' idea was to say that a length \lambda is determined by those rational lengths less than it and those greater than it. ...he says \lambda_1 = \lambda_2...if any rational length < \lambda_1 is also < \lambda_2, and vice versa [any rational length > \lambda_2 is also > \lambda_1]. Likewise \lambda_1 < \lambda_2 if there is a rational length > \lambda_1 but < \lambda_2 [between \lambda_1 and \lambda_2]. This definition uses the rationals to give an infinitely sharp notion of length while avoiding any overt use of infinity. ... The theory of proportions was so successful that it delayed the development of a theory of real numbers for 2000 years. This was ironic, because the theory of proportion can be used to define irrational numbers just as well as lengths. It was understandable though, because the common irrational lengths... arise from constructions that are intuitively clear and finite from the geometric point of view. Any arithmetic approach to the \sqrt2, whether by sequences, decimals, or continued fractions, is infinite and therefore less intuitive. Until the nineteenth century this seemed a good reason... Then the problems of geometry came to a head, and mathematicians began to fear geometric intuition as much as they had previously feared infinity."