From: Ken.Pledger@vuw.ac.nz (Ken Pledger) Subject: Re: Bachmann axiomatic geometry Date: Tue, 13 Feb 2001 10:38:17 +1300 Newsgroups: sci.math Summary: References for transformational geometry (Bachmann, Hjelmslev) In article <969hrp$42l$1@wanadoo.fr>, "Nico" wrote: > Who can help me to find the details of Bachmann's absolute geometry > axiomatic ? > > Thanks, Nico Wonderful! I'm very pleased to see that question, as Bachmann's beautiful transformational theory of geometry is so little known in English-speaking countries. I'm sorry I don't know about its status in France. You may already be aware of the standard reference (in German): Friedrich Bachmann, "Aufbau der Geometrie aus dem Spiegelungsbegriff," Springer, 2nd edition 1973. References in English are rare, but they do exist. The best I know is out of print, but in many libraries: "Fundamentals of Mathematics," Volume II "Geometry," edited by H. Behnke, F. Bachmann, K. Fladt & H. Kunle, translated by S.H. Gould; M.I.T. Press, 1974. (Has it been translated into French?) The absolute geometry you asked about is basically in Chapter 5, but parts of other chapters are relevant too. A very brief sketch is in Guenter Ewald, "Geometry: an Introduction," Wadsworth, 1971, Chapter 1, especially pp.40-51. His notation doesn't make quite clear the approach of the Bachmann school, but his treatment is certainly in that spirit. Nico doesn't need this final warning, but other people may. In the late 1960s and 1970s there was a lot of enthusiasm about introducing "transformational geometry" into schools; but the courses and text-books produced in English were IMHO very feeble indeed. Please let nobody confuse that miserable stuff with the splendid and subtle theory of Hjelmslev, Bachmann and his successors. Ken Pledger.