History and Etymology
It is impossible to trace the origin of the concept of ratio, since the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society. However, it is possible to trace the origin of the word "ratio" to the Ancient Greek λόγος (logos). Early translators rendered this into Latin as ratio ("reason"; as in the word "rational"). (A rational number may be expressed as the quotient of two integers.) A more modern interpretation of Euclid's meaning is more akin to computation or reckoning. Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.
Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers. The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.
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.
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