Is there a good survey article listing all the theorems of Euclidean geometry that are equivalent to the parallel postulate?
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asked Mar 5, 2012 at 2:29
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$\begingroup$ Do you mean all the theorems in Euclid equivalent to that postulate? "Euclidean geometry" encompasses many theorems not in the surviving manuscripts (and of course infinitely many others...) so a complete list in this wider sense seems well out of reach. $\endgroup$– Noam D. ElkiesCommented Mar 5, 2012 at 2:43
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$\begingroup$ A fairly decent list is in the book "Higher geometry" by Efimov (Mir eds.) $\endgroup$– Andrés E. CaicedoCommented Mar 5, 2012 at 2:56
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3$\begingroup$ found this: en.wikipedia.org/wiki/… $\endgroup$– Will JagyCommented Mar 5, 2012 at 3:09
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1$\begingroup$ It has been a while since I last looked, but I remember that M. J. Greenberg has a very long list of theorems equivalent to the parallel postulate in his book "Euclidean and non-Euclidean Geometries: Development and History". $\endgroup$– Robert BryantCommented Mar 7, 2012 at 13:29
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1$\begingroup$ Here's a list of 26 with a reference to the source of the list: ics.uci.edu/~eppstein/junkyard/parallel-postulate.html $\endgroup$– Doug ChathamCommented Mar 9, 2012 at 11:33
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1 Answer
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Doug Chatham's answer is the best I received; see Theorem 23.7 of George E. Martin's "The Foundations of Geometry and the Non-Euclidean Plane".
answered Mar 22, 2012 at 8:07
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$\begingroup$ @James Propp: Greenberg's list is considerably longer, but now I remember that the list is not actually in the textbook itself; it's in the teacher's supplement to the textbook. One of the projects given in Greenberg's textbook is exactly to compile the longest list one can of theorems equivalent to the parallel postulate. The list in the teacher's supplement is intended to help the teacher cope with the lists that the students' projects will generate. $\endgroup$ Commented Apr 19, 2012 at 11:37