Measuring Angles and the Protractor Postulate
In 1733 a geometry book by a Jesuit priest named Saccheri appeared. Although the book caused some stir at the time, soon it was almost completely forgotten. We shall learn a great deal about this book later. In 1832 there appeared a geometry text containing a short appendix written by the son of the author of the text. This appendix has been described by G. B. Halsted as “… the most extraordinary two dozen pages in the whole history of thought!201C The author of the Appendix was John Bolyai, a name that will live as long as any advanced form of human civilization exists. It is Bolyai and Lobachevsky that are recognized as the cofounders of non-Euclidean geometry. More on Bolyai and Lobachevsky later. We now jump one more century to 1932. That year saw the publication of A Set of Postulates for Plane Geometry Based on Scale and Protractor written by George David Birkhoff. Although nobody would suggest this paper is as important as Bolyai’s Appendix, the paper is significant to the reader of this book. Our approach to axiomatizing plane geometry is based on BirkhofFs axiom system as given in that paper.
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