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Home , History of geometry

CHAPTER ONE

GEOMETRICAL

A. Its Formulation

The Greeks, who were the inventors of as a postula• tory-deductive discipline, did mathematics it la grecque. This is obvi• ous to us. They, however, did not know it. If somebody were able to out to them this obvious truth, they would have denied it, precisely as Marx denied being a Marxist. The ancient Greeks would have insisted rightly that they were doing mathematics tout court. The irony of history is that modern mathematicians, in their hypostasis of historians of mathematics as well, fully agree with the ancient Greeks: They too think that is nothing more than mathematics in Greek, since the truths of mathematics are atemporal and immune to cultural influences. They are univer• sal and eternal. Math is math is math. "The TRUTH: her hallmark is ETERNITY. I "But neither thirty years, nor thirty centuries, affect the clearness, or the charm, of Geometrical truths."2 The import of this position is clear: Greek mathematics is like Egyptian mathe• matics and both are, when push comes to shove, like Babylonian, Islamic, Chinese, or Mayan mathematics. There is just one univer• sal, all-embracing, cosmic mathematics, which makes Greek mathe• matics ours as well. Hence, there is nothing to prevent us from understanding Greek mathematics while using all the mathematical paraphernalia at our disposal to remove "the trivial and the acci• dental", having to do with the idiosyncratic Greek way of doing things, in order to cut through to what is, after all, pure, unadul• terated, eternally valid mathematics. It is precisely this approach that engendered Geometrical Algebra

I "Die Wahrheit, sie besteht in Ewigkeit "reads the original of A. v. Chamisso; the English translation is by Max Delbriick in the Mathematical Intelligencer 10 (1988), p. 53. 2 C. L. Dodgson, as quoted in Benno Artmann, --7he Creation if Mathematics (Springer, 1999), p. 45. 18 CHAPTER ONE and insured its success and wide adoption. What, then, is Geometrical Algebra? And how did the development of mathematics make such a reading of Greek mathematical texts possible? In other words, how did Geometrical Algebra come into being? To answer these ques• tions, we need first to say something about the nature of Greek mathematics. According to Eudemus's lost history of , as reported in 's Commentary on the First Book qf Euclid's Elements,3 geometry originated in as land measurement, while began with the Phoenicians as calculation, in each case the generating cause being practical necessity. The myth, as perpetrated by the Greeks themselves, has it that the Greeks learned geometry from the . But Greek and Egyptian geometry are worlds apart. Whatever the Greeks may have learned from the Egyptians, they transfigured their putative inheritance, lifting it from the domain of mensuration and flood repair into the realm of an ideal, precise, and rigorous reality. After an early (and historically problematic) beginning with Thales and the Pythagoreans,4 Greek mathematics grew explosively" and, by the fifth century B.C., succeeded in discovering, i.e., proving, incommensurability, an extraordinary mathematical feat, well beyond the highest achievements of pre-Hellenic mathematics. The discov• ery of incommensurability shattered the innocence of the Pythagorean sect's (school's) naive belief that all segments were commensurable, or, which is the same, that all is . It led to Eudoxus's refor• mulation of the theory of proportions on purely geometrical grounds and to the thorough separation of geometry from arithmetic. To a large extent, Greek mathematics became Greek geometry. Numerical considerations became foreign to geometrical arguments, any attempt at mixing the two falling under sinful rubric of metabasis. The change mathematics underwent after the collapse of the view that all was number meant that geometry, being an independent

3 Cf. Morrow's translation (Princeton, 1992), pp. 5lfr. • Reliable information on Thales is scarce,and that on the mathe• matician discredited after Burkert's book (Weisheit und Wissenschqfl, 1962), translated into English ten years later as Lore and Science in Ancient f}thagoreanism. Cf., however, Leonid Zhmud, Wissenschqfl, Philosophie und Religion im fiuhen f}thagoreismus (Berlin: Akademie Verlag, 1997) for a valiant attempt at a defense of the reliability of ancient sources on Pythagoras. } Reviel Netz, TIe Shaping of Deduction in Greek Mathematics (Cambridge, 1999), pp. 272~75.