Euclidean Geometry


As I remarked in the Preface, I assume you studied plane geometry in high school. I don’t expect that you remember the details, but I do hope you retain some feeling for how the game is played.


Equilateral Triangle Theorem Proven Euclidean Geometry Parallel Straight Line Common Notion 
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  1. 2.
    Sir Thomas L. Heath, 1861–1940, was an expert on Greek mathematics who translated some of its finest works into English. His translation of the Elements is described in the Bibliography. Volume 1 of his A History of Greek Mathematics (1921; Dover reprint, 1981) contains a detailed account of the transmission of the Elements. (Henceforth, all quotations of Euclid, De Morgan, Pappus and Heath are from Sir Thomas L. Heath: The Thirteen Books of Euclid’s Elements, 1908, 1925; Dover reprint, 3 volumes, 1956. Reprinted with permission of Cambridge University Press.)Google Scholar
  2. 3.
    Einstein once said Reported in Lincoln Barnett’s The Universe and Dr. Einstein (William Sloane Associates, 1948), p. 52.Google Scholar
  3. 6.
    what the standard of mathematical reasoning implies. “It is well known,” runs a comment added to Book X of the Elements around A.D. 450, “that the man who first made public the theory of irrationals”—this would be Hippasos of Metapontion (p. 3), who discovered that √2 is not rational—“perished in a shipwreck, in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortuitously touched on and revealed this aspect of living things, was taken to the place where he began and there is forever beaten by the waves.” (Carruccio, op. cit., p. 27.) This comment’s unmistakable tone of moral outrage reflects, I think, the commentator’s appreciation of the real significance of Hippasos’s discovery, namely, that the mind can be divided against itself. What Hippasos’s discovery signaled, and the standard of mathematical reasoning acknowledges, is humanity’s loss of intellectual innocence.Google Scholar
  4. 8.
    “... is the rigor there of Maxim of the American mathematician Robert L. Moore, 1882–1974, inventor of the “Moore method” for training mathematicians. “While he was still a graduate student,” writes F. Burton Jones in the American Mathematical Monthly (April, 1977, p. 274),... Moore conceived the basic idea that led eventually to his radical method of teaching. With his quick mind and restless spirit he found the lecture method rather boring—in fact, mind-dulling. To liven up a lecture he would run a race with his professor by seeing if he could discover the proof of an announced theorem before the lecturer had finished his presentation. Quite frequently he won the race. But in any case, he felt that he was better off for having made the attempt. So if one could get students to prove the theorems for themselves, not only would they have a deeper and longer-lasting understanding, but somehow their ability and interest would be strengthened.Google Scholar
  5. 12.
    Postulate 6. In another branch of mathematics Postulate 6 (i) is known as (a special case of) the “Jordan Curve Theorem” and (ii) is a corollary. In geometry the triangular case of (iii) is called “Pasch’s Axiom” after Moritz Pasch, 1843–1930, a pioneer in the work of ferreting out Euclid’s unstated assumptions. It was Pasch who formulated the “Pattern for a Material Axiomatic System” (p. 6).Google Scholar
  6. 16.
    as little attention to it as possible. Undermining my opinion is the fact that Euclid’s other use of superposition, in Theorem 8, can be easily avoided. “Since the alternative proof of Theorem 8 is not very difficult,” writes Mueller in Philosophy of Mathematics and Deductive Structure in Euclid’s Elements (MIT, 1981), “it seems likely that if Euclid did wish to avoid superposition, his wish was not deep-seated enough to cause him to search very hard for alternatives to it.”Google Scholar
  7. 18.
    Moritz Pasch. See the “Postulate 6” note on p. 104.Google Scholar

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