Abstract
We used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski’s formal geometry. We used proofs as close as possible to those given by Euclid, but filling Euclid’s gaps and correcting errors. Euclid Book I has 48 propositions; we proved 235 theorems. The extras were partly “Book Zero”, preliminaries of a very fundamental nature, partly propositions that Euclid omitted but were used implicitly, partly advanced theorems that we found necessary to fill Euclid’s gaps, and partly just variants of Euclid’s propositions. We wrote these proofs in a simple fragment of first-order logic corresponding to Euclid’s logic, debugged them using a custom software tool, and then checked them in the well-known and trusted proof checkers HOL Light and Coq.
Highlights
Euclid was the “gold standard” of rigor for millenia
Euclid’s “parallel postulate”, or “Euclid 5”, is a postulate rather than an axiom, because it asserts that two lines meet, i.e., there exists a point on both lines
In the interest of following Euclid fairly closely, we take both as axioms: circle–circle is used in I.1 and I.22, while line– circle is used in I.12, and both those proofs are far simpler than the proofs of line–circle and circle–circle from each other
Summary
Euclid was the “gold standard” of rigor for millenia. The Elements of Euclid set the standard of proof used by Isaac Newton in his Principia and even Abraham Lincoln claimed to. The first gap occurs in the first proposition, I.1, in which Euclid proves the existence of an equilateral triangle with a given side, by constructing the third vertex as the intersection point of two circles. Many of these have problems like those of I.9 and I.7; that is, we could fix these problems only by proving some other propositions first, and the propositions of the first half of Book I had to be proved in a different order, namely 1,3,15,5,4,10,12,7,6,8,9,11, and in some cases the proofs are much more difficult than Euclid thought After proving those early propositions, we could follow Euclid’s order better, and things went well until Prop. They are available as ancillary files to the version of this paper posted on ArXiv
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