Similarly(!), similarity can be understood as the combination of isometric transformations and linear scalings of the plane. Because of these generalizations, one can use one's geometric intuition about things that we would otherwise have no easy intuitive way to think about - from a modern viewpoint this is a major triumph of the vector approach over synthetic geometry. These concepts are slightly less immediate than Euclid's intuitive motions of the plane, but are immeasurably more productive in terms of generalizing to other more abstract settings than the plane. length-preserving) transformations of $\mathbb R^2$, which turn out to be everything that can be expressed using translation (addition of a constant vector) and orthogonal linear transformations. The high point of Hilbert is the "field of ends" in non-Euclidean geometry, wherein a hyperbolic plane gives rise to an ordered field $F$ defined purely by the axioms, and in turn the plane is isomorphic to, say, a Poincare disk model or upper half plane model in $F^$.Ĭongruence has a nice representation in terms of isometric (i.e. ![]() He laid out a system but left it to others to fill in the details, notably Bachmann and Pejas. Hilbert's book is available in English, Foundations of Geometry. Hartshorne, in particular, takes a synthetic approach throughout, has a separate index showing where each proposition of Euclid appears, and so on. The award page, by itself, gives a pretty good response to the original question about the status of Euclid in the modern world.Īs far as book length, there are the fourth edition of Marvin's book, Euclidean and Non-Euclidean Geometries, also Geometry: Euclid and Beyond by Robin Hartshorne. If you email me I can send you a pdf.ĮDIT: Alright, Marvin won an award for the article, which can be downloaded from the award announcement page GREENBERG. Marvin promotes what he calls Aristotle's axiom, which rules out planes over arbitrary non-Archimedean fields without leaving the synthetic framework. One of the great strengths of the article is that I am in it. I can recommend an article Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg, The American Mathematical Monthly, Volume 117, Number 3, March 2010, pages 198-219. Euclid's arguments are all "synthetic", and it seems hard to carry such arguments out in an analytic framework.Ģ) What problems exist with Euclid's elements? Why are the axioms unsatisfactory? Where does Euclid commit errors in his reasoning? I've read that the logical gaps in the Elements are so large one could drive a truck through them, but I cannot see such gaps myself. For example, the proof of SAS congruence would be quite messy. ![]() My guess would be that one could simply put everything in terms of coordinates in R^2, but then it seems to be hard to carry out usual similarity and congruence arguments. I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own sets of axioms.ġ) What is the modern axiomatization of plane geometry? For example, when mathematicians speak of a point, a line, or a triangle, what does this mean formally?
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