20 1. Euclid Notion 3 implies that the squares on the remaining legs FB and GC are equal, and thus the legs themselves are also equal. Thus we have shown AF D AG and FB D GC , so by Common Notion 2, it follows that AB D AC . CASE 2: O lies on BC (Fig. 1.14(c)). Then O must be the point where BC is bisected, because that is where ` meets BC . In this case, we argue exactly as in Case 1, except that we can skip the first step involving triangles BMO and CMO, because we already know that BO D OC (because BC is bisected at O). The rest of the proof proceeds exactly as before to yield the conclusion that AB D AC . CASE 3: O lies outside triangle ABC (Fig. 1.14(d)). Again, the proof proceeds exactly as in Case 1, except now there are two changes: first, before drawing OF and OG, we need to extend AB beyond B, extend AC beyond C (Postulate 2), and draw OF and OG perpendicular to the extended line segments (Proposition I.12). Second, in the very last step, having shown that AF D AG and FB D GC , we now use Common Notion 3 instead of Common Notion 2 to conclude that AB D AC . Modern Axiom Systems We have seen that the discovery of non-Euclidean geometry made it necessary to rethink the foundations of geometry, even Euclidean geometry. In 1899, these efforts culminated in the development by the German mathematician David Hilbert (1862–1943) of the first set of postulates for Euclidean geometry that were sufficient (according to modern standards of rigor) to prove all of the propositions in the Elements. (One version of Hilbert’s axioms is reproduced in Appendix A.) Following the tradition established by Euclid, Hilbert did not refer to numbers or measurements in his axiom system. In fact, he did not even to refer to comparisons such as “greater than” or “less than” instead, he introduced new relationships called congruent and between and added a number of axioms that specify their properties. For example, two line segments are to be thought of as congruent to each other if they have the same length (Euclid would say they are “equal”) and a point B is to be thought of as between A and C if B is an interior point of the segment AC (Euclid would say “AB is part of AC ”). But these intuitive ideas were only the motivations for the choice of terms the only facts about these terms that could legitimately be used in proofs were the facts stated explicitly in the axioms, such as Hilbert’s Axiom II.3: Of any three points on a line there exists no more than one that lies between the other two. Although Hilbert’s axioms effectively filled in all of the unstated assumptions in Eu- clid’s arguments, they had a distinct disadvantage in comparison with Euclid’s postulates: Hilbert’s list of axioms was long and complicated and seemed to have lost the elegant simplicity of Euclid’s short list of assumptions. One reason for this complexity was the ne- cessity of spelling out all the properties of betweenness and congruence that were needed to justify all of Euclid’s assertions regarding comparisons of magnitudes. In 1932, the American mathematician George D. Birkhoff published a completely different set of axioms for plane geometry using real numbers to measure sizes of line segments and angles. The theoretical foundations of the real numbers had by then been solidly established, and Birkhoff reasoned that since numerical measurements are used ubiquitously in practical applications of geometry (as embodied in the ruler and protractor), there was no longer any good reason to exclude them from the axiomatic foundations of

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