Exercises 21 geometry. Using these ideas, he was able to replace Hilbert’s long list by only four axioms (see Appendix B). Once Birkhoff’s suggestion started to sink in, high-school text writers soon came around. Beginning with a textbook coauthored by Birkhoff himself [BB41], many high- school geometry texts were published in the U.S. that adopted axiom systems based more or less on Birkhoff’s axioms. In the 1960s, the School Mathematics Study Group (SMSG), a committee sponsored by the U.S. National Science Foundation, developed an influen- tial system of axioms for high-school courses that used the real numbers in the way that Birkhoff had proposed (see Appendix C). The use of numbers for measuring lengths and angles was embodied in two axioms that the SMSG authors called the ruler postulate and the angle measurement postulate. In one way or another, the SMSG axioms form the basis for the axiomatic systems used in most high-school geometry texts today. The axioms that will be used in this book are inspired by the SMSG axioms, although they have been modi- fied in various ways: some of the redundant axioms have been eliminated, and some of the others have been rephrased to more closely capture our intuitions about plane geometry. This concludes our brief survey of the historical events leading to the development of the modern axiomatic method. For a detailed and engaging account of the history of geometry from Euclid to the twentieth century, the book [Gre08] is highly recommended. Exercises 1A. Read all of the definitions in Book I of Euclid’s Elements, and identify which ones are descriptive and which are logical. 1B. Copy Euclid’s proofs of Propositions I.6 and I.10, and identify each of the standard six parts: enunciation, setting out, specification, construction, proof, and conclusion. 1C. Choose several of the propositions in Book I of the Elements, and rewrite the statement and proof of each in more modern, idiomatic English. (You are not being asked to change the proofs or to fill in any of the gaps all you need to do is rephrase Euclid’s proofs to make them more understandable to modern readers.) When you do your rewriting, consider the following: Be sure to include diagrams, and consider adding additional diagrams if they would help the reader follow the arguments. Many terms are used by Euclid without explanation, so make sure you know what he means by them. The following terms, for example, are used frequently by Euclid but seldom in modern mathematical writing, so once you understand what they mean, you should consider replacing them by more commonly understood terms: – a finite straight line, – to produce a finite straight line, – to describe a circle, – to apply or superpose a figure onto another, – the base and sides of an arbitrary triangle, – one angle or side is much greater than another, – a straight line standing on a straight line, – an angle subtended by a side of a triangle.

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