Introduction to Geometry
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Chapter 1: Triangles
[edit | edit source]In this chapter we review some of the well-known propositions of elementary geometry, stressing the role of symmetry. We refer to Euclid’s propositions by his own numbers, which have been used throughout the world for more than two thousand years. Since the time of F. Commandino (1509-1575), who translated the works of Archimedes, Apollonius, and Pappus, many other theorems in the same spirit have been discovered. Such results were studied in great detail during the nineteenth century. As the present tendency is to abandon them in favor of other branches of mathematics, we shall be content to mention a few that seem particularly interesting.
Section 1.1: Euclid
[edit | edit source]About 300 BC, Euclid of Alexandria wrote a treatise in thirteen books called the Elements. Of the author (sometimes regrettably confused with the earlier philosopher, Euclid of Megara) we know very little. Proclus (410-485 AD) said that he “put together the Elements, collecting many of Eudoxus’ theorems, perfecting many of Theaetetus’ and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy, who once asked him if there was in geometry any shorter way than that of the Elements, and he answered that there was no royal road to geometry.” Heath quotes a story by Stobaeus, to the effect that someone who had begun to read geometry with Euclid asked him “What shall I get by learning these things?” Euclid called his slave and “Give him a dime, since he must make gain out of what he learns.”
Of the thirteen books, the first six may be very briefly described as dealing respectively with triangles, rectangles, circles, polygons, proportion, and similarity.