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CROOKED_EUCLID

                                             September 27, 2019

     "The rigid methods employed by modem geometers have
     deposed Euclid from his pinnacle of correctness. It was
     thought, until recent times, that, as Sir Henry Savile
     remarked in 1621, there were only two blemishes in
     Euclid, the theory of parallels and the theory of pro-
     portion. It is now known that these are almost the only
     points in which Euclid is free from blemish. Countless
     errors are involved in his first eight propositions. That
     is to say, not only is it doubtful whether his axioms are
     true, which is a comparatively trivial matter, but it is
     certain that his propositions do not follow from the
     axioms which he enunciates. A vastly greater number of
     axioms, which Euclid unconsciously employs, are re-
     quired for the proof of his propositions. Even in the
     first proposition of all, where he constructs an
     equilateral triangle on a given base, he uses two circles
     which are assumed to intersect. But no explicit axiom
     assures us that they do so, and in some kinds of spaces
     they do not always intersect. It is quite doubtful
     whether our space belongs to one of these kinds or
     not. Thus Euclid fails entirely to prove his point in the
     very first proposition."

  "Mathematics and the Metaphysicians" (1901)
  collected in "Mysticism and Logic and Other Essays" (1917)
  p. 94

                                               https://users.drew.edu/~jlenz/br-ml-ch5.html
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