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Eucleodian geometry

  1. May 30, 2012 #1
    please suggest what books should I do to strong my concepts in eucleodian geometry (ELEMENTARY & PRE COLLEGE LEVEL)?
    ( in case for more than 1 book please refer the order in which they should be done)
    Also tell me , whether doing only ADVANCED EUCLEODIAN GEOMETRY by Roger A. JOHNSON IS ENOUGH OR NOT.
  2. jcsd
  3. May 30, 2012 #2
    Hi Simpy,

    It might help if you tell us what your current knowledge / experience with geometry is and also the reason you want to study geometry.

    A. If you did well in your high school geometry course and want to explore the subject further, great idea! It is very satisfying to bust out a Euclid-style proof of a geometric fact when everyone else is still fumbling around with their coordinates/vectors. Also, you get the hang of thinking geometrically rather than trying to reduce everything to some kind of algebraic manipulations which is what 99% of students try to do. Here are a few books to consider:
    "Geometry Revisited", by Coxeter. This is a really great place to start.
    "Introduction to Geometry" by Coxeter. This one is a bit harder, denser and with more content than the previous book. Most people would probably benefit from reading the previous book first even though its content is more or less a subset of what is in this book. Work through this one and you'll be a geometry pro. I found the chapter on inversions w.r.t. circles particularly helpful when I was studying complex analysis.
    The book you mentioned has a good topic list, though I hadn't heard of it before.
    Kiselev's geometry is well regarded on amazon, but check out the reviews before buying. His volume 2 is about solid geometry which most students never learn, but really should.

    B. If you feel that geometry was the weak link in your high school math education and you just want to be prepared for what college throws at you then I must say that in my experience, very few college students know anything about geometry other than a few basic formulas for area/perimeter and really basic facts about lines, angles, and trigonometry. (Two points determine a line, etc.) Also, you should know what coordinate axes are (the Cartesian plane). If this paragraph describes you, then the books above are probably all overkill. That said, I do think students would understand calculus better if only they had a basic understanding of solid geometry. Multivariable calculus is just easier if you can visualize what is going on in the problems.
  4. May 30, 2012 #3


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    My recommendations are first this book:


    Then, or better in companionship with, this book:


    Now that I have looked at the book you mention by Johnson, it looks very nice.

    On page one however he recommends review of basic Euclidean geometry, and there is no better way than to read Euclid himself. Hartshorne's book is a companion to Euclid, especially the first chapter, which is a great introduction to the first say 4-5 chapters of Euclid.

    The book by Johnson contains many nice topics that are not in Euclid, but most are likely in Hartshorne. Johnson's book however starts right out assuming something much more sophisticated than anything in Euclid, namely the theory of real numbers and trigonometry.

    In fact I believe a close study of Euclid in its original purely geometric form is the best background for understanding what the real numbers are, something few elementary students know.

    Basically real numbers are ratios of ordered pairs of segments on the Euclidean line, but with two additional axioms not made in Euclid in the first few chapters.

    1) "Archimedes" axiom, that given any three points A,B,C in sequence on the line, one can find a finite number of copies of the segment AB which when laid end to end, will eventually contain C.

    2) Dedekind's axiom. Not only does the removal of any point separate the Euclidean line into two open intervals, but conversely, any separation of the Euclidean line into two open intervals occurs by removing a point.

    Moreover the basic theory of trigonometry arises from the theorem of Pythagoras, and its two generalizations in Euclid, Propositions II.12, II.13, the "Law of cosines".

    In Euclid's chapter III, one finds essentially Newton'e definition of a tangent line in Prop. III.16, and a result equivalent to the principle of similar triangles in Prop. III.35, proved in a form that is more general than just for the case of real numbers, i.e. also for non Archimedean fields.

    Of course chapter III assumes a theory of area or "content" from Euclid's chapter I, but Hartshorne explains very clearly Hilbert's approach to filling in this theory rigorously without any further assumptions, as well as clarifying Euclid's own axioms. I.e. the modern approach does similarity first and then area.

    On my webpage you will also find shorter notes for a 2 week course I gave last summer on the topic, for bright young kids. notes from epsilon camp.


    But I like the book by Johnson you mention, and it is inexpensive. I still recommend reviewing Euclid, in the Green Lion edition linked above.
    I am advocating really understanding Euclid rather than pursuing the refinements. it is the fundamentals that are used most, and are most important.

    So now you have several opinions. You should make some exploration and decide for yourself.
    Last edited by a moderator: May 6, 2017
  5. May 30, 2012 #4
    Following advice is for learning from a problem solving approach, specifically in trying to approach olympiad type problems: I think the best starting point is "Geometry Revisited" by H.S.M Coxeter. Going through all the problems in it is a must; they are the best part of any Euclidean geometry study. At a slightly more advanced problem solving level, though theory is pretty much the same, try working through the problems in "Crossing the Bridge" by "Gerry Leversha" (UKMT). Supplement these with more involved problems from olympiad type problem books like "Mathematical Olympiad Treasures" by Titu Andreescu.
  6. May 30, 2012 #5
    I certainly respect your opinion. But I would like to know if someone who is trying to learn geometry at a "PRE COLLEGE LEVEL" could benefit from euclid's elements?

    I say this because I would think they would find the proofs a bit difficult to work through.

    I am curious though, have you had high school level kids work through the elements before with success?
    Last edited by a moderator: May 6, 2017
  7. May 30, 2012 #6
    I agree with Vargo 110% you need to explain what you want to do with your geometry.

    Euclidian geometry underlies so much of modern science and technology - crystallography, mineralogy, computer graphics, and much more.

    Then we can offer better advice.

    Meanwhile if you are making the transition from high school to university maths then

    Elementary Geometry by John Roe is a good introduction to the modern subject
  8. May 30, 2012 #7


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    Diffy, emphatically yes. Euclid is by far better than other books for high schoolers for precisely the reasons I gave. It does not assume sophisticated notions such as real numbers and trig, instead it offers an intuitive background for these concepts. It assumes almost nothing except an ability to think clearly an logically, and it helps develop this ability.

    In my opinion the decline of mathematics in the USA dates from around 1900 when well meaning but to me mistaken people began to dismiss Euclid from the curriculum, in favor of "easier" treatments. The so called Birkhoff approach, via real numbers and angle measure, is actually much more sophisticated and harder to appreciate than Euclid's original one. Birkhoff's approach assumes something more difficult to treat something more elementary, just backwards for good pedagogy.

    In answer to your specific question on experience with this approach, I taught through the first 4 chapters of Euclid last summer in 2 weeks to a group of 28 bright kids aged 8-10 years. they loved it. Most of them knew nothing of proof before, but after the course went forward they were able to make some real proofs on their own. (We also taught logic and proof techniques separately). Even some who struggled with making proofs seemed to appreciate seeing the proofs presented and discussed. To keep it concrete, we also constructed cardboard models of Platonic solids. Kids really enjoyed learning to construct a pentagon, and use that construction to make a dodecahedron. On the contrary, the easier watered down treatment of Euclid I learned in high school did not include learning to construct a pentagon, as if it were somehow too advanced. Actually it is quite easy.

    First construct two perpendicular diameters, then connect the center A of one of the radii to the extremity B of the other diameter. Thus AB is the hypotenuse of a right triangle whose legs are respectively a diameter and a radius of your circle.

    Then put the point of your compass down on A, the other end on the center O of the circle, and copy off that half-radius length AO onto the hypotenuse AB, meeting that hypotenuse at C.

    Then AC is the length of a half-radius. The other length CB on that hypotenuse is the length you want. Put your compass point down on the edge of the circle at B, the other end on the point C, and copy off that length BC onto the arc of the circle, getting an arc BD, whose secant length equals BC.

    Then repeat BD around the circle, and you will have a regular decagon. I.e. the segment BD is the side of a regular decagon. Connecting alternate vertices gives a regular pentagon. I myself did not know this before teaching this class. But once you understand something, as you do after reading a master, you can teach it to anyone.

    here is an animation:


    here is another animation of essentially Euclid's proof of Pythagoras.


    Read this article by Hartshorne on teaching Euclid. Although he taught an undergraduate college class, his comments are of interest in broader context.

    Attached Files:

    Last edited: May 30, 2012
  9. May 30, 2012 #8
    I may be barking up the wrong tree here but does this not suggest simpy is at high school and moving on to university/college and already has some geometry?
  10. May 31, 2012 #9
    thanks everybody, for your replies..
    actually I am preparaing for various entrance exams in different universites whose pattern is somewhat similar to olympiad level .
    and since my geometry portion is weakest of all and also that I wish to be a mathematician oneday I do not want to leave any topic weak
    And as for my previous knowledge I know nothing much than basic geometry -simiarity criterion.
    In my country this is the topic to which a school student is exposed at maximum.
  11. May 31, 2012 #10

    Thank you. I read the article and I also forwarded it to my friends who teach math at a high school level. If nothing else at least it provides a nice history of the development of modern geometry.

    It also highlights the vast difference of what Geometry was originally and what it has evolved to today.

    As a thought exercise, I still wonder, what would happen if you took High school students. Taught them geometry in its purest form without measue, but with magnitude (as the article puts it).

    How they would hold up to the "State" standards of the US. Not that those are any good indication of knowledge. It is just curious to think about.

    Imagine a State exam asking for a proof of congruent triangles. The corrector expecting a list of meaningless "theorems" instead gets a Euclidean proof. It would probably be marked as incorrect.
  12. May 31, 2012 #11
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