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43 The idea that geometry is timeless led people to think that there can be no change in mathematics. There can be discovery, they thought, but not invention. It also encouraged the idea that change and moving things are foreign to pure mathematics. Euclid seemed to have purified geometry, although Archimedes later mixed it with practical things. Euclid’s books on geometry, the Elements, began to circulate roughly around 250 BCE. The earliest extant fragments and copies of these books specify no author. The earliest historical trace of a Euclid who worked on geometry is a brief critical mention in a work by the geometer Apollonius, around 185 BCE.1 Euclid was later mentioned, rarely and briefly, by Cicero and a few others.2 I don’t know where Euclid was born or where he lived, but many writers say that he lived in Alexandria, Egypt. This popular guess evolved in the following way. At around 185 BCE, Apollonius claimed to have once spent some time in Alexandria. He also briefly mentioned Euclid, in a separate passage, making no reference to where or when he lived. Five centuries later, around 320 CE, Pappus of Alexandria claimed that Apollonius “spent a long time with the pupils of Euclid at Alexandria.”3 But note, Pappus did not say that Euclid himself lived in Alexandria. Next, around 460 CE, Proclus claimed that the Egyptian king Ptolemy I “once asked Euclid if there was not a shorter road to geometry than through the Elements, and Euclid replied that there is no Royal Road to geometry.”4 It’s a great story, but there is no good reason 4 THE DEATH OF ARCHIMEDES 44 T H E D E AT H O F A RC H I M E D E S to believe that Proclus really echoed events from more than seven centuries prior, when Ptolemy reigned in Alexandria. Nevertheless, knowing nearly nothing more about the mysterious Euclid, most writers and historians echo an old guess: that Euclid lived in Alexandria. And they often invent additional details: when he was born, that he was a student of Plato, that he was summoned to work for King Ptolemy, that he worked at the Library of Alexandria, and so on. At least there is fair evidence that Euclid was really a geometer, in contrast to Pythagoras. In any case, Proclus was a pagan theologian and a mystic admirer of Plato. Accordingly, Proclus claimed that “Euclid belonged to the persuasion of Plato and was at home in this philosophy.”5 The conjecture is often illustrated by the following similarity. Plato had speculated that the five regular solids were the building blocks of the universe. And Euclid’s Elements ends with the construction of these same five solids. But really, this does not mean that Euclid admired Plato, because over the centuries many people, before and after Plato, built or discussed these geometric figures. Still, did ancient geometers echo Plato’s beliefs? The Elements elegantly conveys a wealth of geometrical knowledge, systematically organized as propositions proven on the basis of basic principles. Interestingly, the Elements proceeds without much reference to physical matters. The propositions and proofs lack expressions that explicitly refer to motion. Geometric figures are constructed but formulated mostly as expressions that do not require any movement of figures in space or relative to one another. One exception is the fourth “common notion.” It states: “Things that coincide with one another are equal to one another.” This rule could be used to show that figures can be moved so that they overlap, and if their parts match exactly then they are equal. Historian Thomas Heath commented that Euclid’s words left “no room for doubt that he regarded one figure as actually moved and placed upon the other.”6 Such displacements of figures were not specified explicitly in many of the demonstrations, but the fourth common notion was used in proposition 4 of the Elements, book 1, to move a triangle to superimpose it onto another. Afterward, that proposition is repeatedly used, explicitly or implicitly, and therefore Heath argued that the procedure of moving figures was fundamental. Still, Heath inferred that Euclid probably preferred not to move figures, because apparently Euclid chose not to move figures in some propositions where that would have sufficed. Therefore, Heath conjectured that the fourth common notion was not originally in Euclid’s Elements but had been added [18.119.104.238] Project MUSE (2024-04-26...

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