Author： Greenberg Marvin Jay
Publisher： Mathematical Association of America
ISSN： 1930-0972
Source： American Mathematical Monthly, Vol.117, Iss.3, 2010-03, pp. : 198-219
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Abstract
This survey highlights some foundational history and some interesting recent discoveries in elementary geometry that deserve to be better known, such as the hierarchies of axiom systems, Aristotle's axiom as a "missing link," Bolyai's discovery—proved and generalized by William Jagy—of the relationship of "circle-squaring" in a hyperbolic plane to Fermat primes, the undecidability, incompleteness, and consistency of elementary Euclidean geometry, and much more. A main theme is what Hilbert called "the purity of methods of proof," exemplified in his and his early twentieth century successors' works on foundations of geometry.
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