<P> The existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients . This is a comparatively recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients . The reasons for this are twofold . First, there was the previously mentioned reluctance to accept irrational numbers as true numbers . Second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century . </P> <P> Book V of Euclid's Elements has 18 definitions, all of which relate to ratios . In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them . The first two definitions say that a part of a quantity is another quantity that "measures" it and conversely, a multiple of a quantity is another quantity that it measures . In modern terminology, this means that a multiple of a quantity is that quantity multiplied by an integer greater than one--and a part of a quantity (meaning aliquot part) is a part that, when multiplied by an integer greater than one, gives the quantity . </P> <P> Euclid does not define the term "measure" as used here, However, one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity measures the second . Note that these definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII . </P> <P> Definition 3 describes what a ratio is in a general way . It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself . Euclid defines a ratio as between two quantities of the same type, so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area . Definition 4 makes this more rigorous . It states that a ratio of two quantities exists when there is a multiple of each that exceeds the other . In modern notation, a ratio exists between quantities p and q if there exist integers m and n so that mp> q and nq> p . This condition is known as the Archimedes property . </P>

What is the ratio of 1 to 3