Extended Abstract of Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry

摘要

This extended abstract describes a forthcoming book which should be of interest to those attending the Diagrams 2006 conference and to others generally interested in diagrammatic reasoning in the context of Euclidean Geometry. The book, Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry, is still in preparation, but will be published by CSLI press once it is completed. In 1879, the English mathematician Charles Dodgson, better know to the world under his pen name of Lewis Carroll, published a little book entitled Euclid and His Modern Rivals. Dodgson was concerned by the fact that quite a number of different nineteenth century authors had written their own treatments of planar geometry, most claiming to improve on Euclid, and each one slightly different in the order of its theorems, in which theorems it chose to include, in the proofs given of these theorems, in its treatment of the theory of parallel lines, and in other aspects. Dodgson's book was written "[i]n furtherance of the great cause which I have at heart—the vindication of Euclid's masterpiece...." It is written mostly in the form of a dream dialogue between a nineteenth century mathematician, Minos, and the ghost of Euclid. In it, they consider each of the modern rivals in turn, and conclude in each case that, while many of the rivals have interesting things to say, none of them are a more appropriate basis for the study of a beginning geometry student than Euclid's Elements.