Mathematicians of the Greco-Roman worldThis map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bc) to Hypatia of Alexandria (c. ad 400). Their names—located on the map under their cities of birth—can be clicked to access their biographies.
Mathematicians of the Greco-Roman worldThis map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 BC) to Hypatia of Alexandria (c. AD 400). Their names—located on the map under their cities of birth—can be clicked to access their biographies.
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Attempts to deal with incommensurables eventually led to the creation of an innovative concept of proportion by Eudoxus of Cnidus (c. 400–350 BC), which Euclid preserved in his Elements (c. 300 BC). The theory of proportions remained an important component of mathematics well into the 17th century, by allowing the comparison of ratios of pairs of magnitudes of the same kind. Greek proportions, however, were very different from modern equalities, and no concept of equation could be based on it. For instance, a proportion could establish that the ratio between two line segments, say A and B, is the same as the ratio between two areas, say R and S. The Greeks would state this in strictly verbal fashion, since symbolic expressions, such as the much later A:B::R:S (read, A is to B as R is to S), did not appear in Greek texts. The theory of proportions enabled significant mathematical results, yet it could not lead to the kind of results derived with modern equations. Thus, from A:B::R:S the Greeks could deduce that (in modern terms) A + B:A − B::R + S:R − S, but they could not deduce in the same way that A:R::B:S. In fact, it did not even make sense to the Greeks to speak of a ratio between a line and an area since only like, or homogeneous, magnitudes were comparable. Their fundamental demand for homogeneity was strictly preserved in all Western mathematics until the 17th century.
When some of the Greek geometric constructions, such as those that appear in Euclid’s Elements, are suitably translated into modern algebraic language, they establish algebraic identities, solve quadratic equations, and produce related results. However, not only were symbols of this kind never used in classical Greek works but such a translation would be completely alien to their spirit. Indeed, the Greeks not only lacked an abstract language for performing general symbolic manipulations but they even lacked the concept of an equation to support such an algebraic interpretation of their geometric constructions.
For the classical Greeks, especially as shown in Books VII–XI of the Elements, a number was a collection of units, and hence they were limited to the counting numbers. Negative numbers were obviously out of this picture, and zero could not even start to be considered. In fact, even the status of 1 was ambiguous in certain texts, since it did not really constitute a collection as stipulated by Euclid. Such a numerical limitation, coupled with the strong geometric orientation of Greek mathematics, slowed the development and full acceptance of more elaborate and flexible ideas of number in the West.
