On the Myriad Mathematical Traditions of Ancient Greece

To exert one’s historical imagination is to plunge into delicate deliberations that involve personal judgments and tastes. Historians can and do argue like lawyers, but their arguments are often made on behalf of a picture of the past, and these historical images obviously change over time. Why should the history of mathematics be any different? When we imagine the world of ancient Greek mathematics, the works of Euclid (Heath 1926), Archimedes (Heath 1897b), and Apollonius (Heath 1897a) easily spring to mind. Throughout most of the twentieth century, our dominant image of Greek mathematical traditions has been shaped by the high standards of rigor and creative achievement that are purportedly found in extant texts presumed to have been written by these three famous authors. Thanks to the efforts of Thomas Little Heath, the English-speaking world has long enjoyed easy access to this trio’s major works and much else besides (Heath 1921). Yet despite this plentiful source material, our conventional picture of Greek mathematics has largely been sustained by a far smaller corpus of knowledge. Our image of Greek geometry, as conveyed in countless mathematical texts and most books on the history of mathematics, has tended to stress the formal structure and methodological sophistication found in a handful of canonical works or, more accurately, selected portions of the same. Even the first two books of Euclid’s Elements – which concern the congruence properties of rectilineal figures and culminate in theorem II-14 showing how to square such a figure – have often been trivialized. Many writers have distilled their content down to a few definitions, postulates, and elementary propositions intended merely to illustrate the axiomatic-deductive method in classical geometry.