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B What are ways that someone would study synthetic geometry?

  1. Jun 16, 2017 #1
    By the above question, I mean how would one effectively study synthetic geometry (geometry that makes no reference to explicit formulas or coordinate systems, like described in Euclid's Elements)? Do you just read through the propositions, try to reconstruct them later and perhaps more or are there some practice problems?

    Analytic geometry, for example, uses algebra to determine numerical aspects of geometrical figures. It was developed by Rene Descartes and Pierre de Fermat independently of one another. Studying and practicing analytic geometry seems, to me, a bit more straightforward at the moment.

    For example, you probably first learn about points, lines, line segments, and rays, how to label them, how to use the Cartesian coordinate system and so on. Perhaps this statement is naive but studying synthetic geometry seems completely different from analytic geometry. Anyway, if anyone has any advice, thank you.
    Last edited by a moderator: Jun 16, 2017
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  3. Jun 16, 2017 #2


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    I recall in school, doing a bunch of proofs by construction (with straight edge and a compass). Is that what you are referring to?
  4. Jun 16, 2017 #3
    Never really thought about doing this type of geometry as a "modern" thing since I would assume straightedge and compass. But there is apparently such a thing as computational synthetic geometry.
  5. Jun 17, 2017 #4
    Yes. Did you simply do a bunch of proofs or was there a particular way of studying the subject? In other words, was there practice geometry problems involving only a compass and straightedge (in the fashion of Euclid, Eudoxus, etc.)?
  6. Jun 17, 2017 #5


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    Perhaps you could find a used textbook on the subject. So this was 9th grade Geometry, and we did start out with doing the Axioms, Postulates, etc and using those to prove or disprove statements. With the compass/straight edge: we would do things like duplicate (transfer) lengths and angles, find midpoints of line segments, divide a line segment into equal lengths, bisect angles, and other stuff. Then on the tests we would be given problems to perform. It has been over 30 years, but that is what I recall from the "pure geometry" portion of the class.
  7. Jun 17, 2017 #6
    I have found a PDF of a textbook by Robin Hartshorne called Geometry: Euclid and Beyond and also some really helpful (haven't gone through them all the way but seem thorough to me) documents and a booklet on the basics of Euclidean Geometry from Fine Arts Math Centre. So it appears I am on the right track.
  8. Jun 17, 2017 #7

    Stephen Tashi

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    There is "Elementary Geometry from an Advanced Standpoint" by Edwin Moise - if you wish to study plane geometry in a rigorous style. The style of Euclid is not rigorous by modern standards, but "there's something to it". There are plane geometry textbooks from the 1960's that were used in high schools and are written in the style of Euclid .
  9. Jun 17, 2017 #8
    I see what you mean by Euclid not being rigorous, for example the definition of a point, something "which has no part" is almost useless and so is the definition of a line. There's also Hilbert's axioms which fix an unstated assumption in the very first proposition of The Elements that circles intersect (and much more). He presented his axioms in 1900 in "The Foundations of Geometry". Apparently it is "second-order". I don't know what that means yet. There's also Tarski's axioms, which are "first-order". In a geometry class I had a couple years ago, we did some two column proofs involving angles and similarity of triangles but never touched the foundations, which in my mind, doesn't make any sense. And of course, we did traditional analytic geometry. Also, thanks for the recommendation; I'll have to check it out.
  10. Jun 20, 2017 #9


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    Hey Mikaelochi.

    Co-ordinate free systems are studied with tensors and geometry. Differential geometry looks at this sort of thing and once you get enough calculus and linear algebra under your belt, then this should be a good avenue.
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