Chapter 1 Euclid The story of axiomatic geometry begins with Euclid, the most famous mathematician in history. We know essentially nothing about Euclid’s life, save that he was a Greek who lived and worked in Alexandria, Egypt, around 300 BCE. His best known work is the El- ements [Euc02], a thirteen-volume treatise that organized and systematized essentially all of the knowledge of geometry and number theory that had been developed in the Western world up to that time. It is believed that most of the mathematical results of the Elements were known well before Euclid’s time. Euclid’s principal achievement was not the discovery of new mathe- matical facts, but something much more profound: he was apparently the first mathemati- cian to find a way to organize virtually all known mathematical knowledge into a single coherent, logical system, beginning with a list of definitions and a small number of as- sumptions (called postulates) and progressing logically to prove every other result from the postulates and the previously proved results. The Elements provided the Western world with a model of deductive mathematical reasoning whose essential features we still emu- late today. A brief remark is in order regarding the authorship of the Elements. Scholars of Greek mathematics are convinced that some of the text that has come down to us as the Elements was not in fact written by Euclid but instead was added by later authors. For some portions of the text, this conclusion is well founded—for example, there are passages that appear in earlier Greek manuscripts as marginal notes but that are part of the main text in later edi- tions it is reasonable to conclude that these passages were added by scholars after Euclid’s time and were later incorporated into Euclid’s text when the manuscript was recopied. For other passages, the authorship is less clear—some scholars even speculate that the defini- tions might have been among the later additions. We will probably never know exactly what Euclid’s original version of the Elements looked like. Since our purpose here is primarily to study the logical development of geometry and not its historical development, let us simply agree to use the name Euclid to refer to the writer or writers of the text that has been passed down to us as the Elements and leave it to the historians to explore the subtleties of multiple authorship. 1
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