Is most of Euclid I necessary for Euclid I.47 (Pythagorean Theorem)?

Discussion in 'General Math' started by avital, Nov 30, 2006.

  1. avital

    avital Guest


    I'm going to attempt to open a study group for non-mathematicians that
    will mimick the way geometry was taught when that was the entire corpus
    of mathematical knowledge. But, instead of just going through the
    Elements page-by-page, I would prefer to create the course in a way
    where I first ask a question, let the group try to give possible
    solutions, and after all the ideas were exhausted, go to Euclid and
    show Euclid's solution, which then would lead to other questions.

    I thought the best way to start would be with what is probably the most
    important theorem of pre-Euclidean geometry, the Pythagorean Theorem. I
    read the vast commentary by Heath on I.47 who tried to explain how this
    proof is a new one not based on similarity of triangles, and thus not
    based on Eudoxus' theory of proportion, but I can't seem to find an
    explanation why the proof based on 'moving triangles around' (Proof #9
    in isn't more
    simple and intuitive. I tried to figure out which assumptions are
    implicit in that proof, and could not find any. Does anyone notice any
    such assumptions I've missed?

    To make this harder, I am sure it was not a new idea to prove things in
    similar ways, since in Plato's Meno we see a very similar solution to a
    specific case of I.47. So I would think Plato would have figured out
    that this method is extendable to the general case, and would have
    created a proof of the same spirit of Meno's proof which even a
    slave-boy would understand (or recollect, as Plato explains).

    avital, Nov 30, 2006
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  2. avital

    Ken Pledger Guest

    This beautiful proof is due to C. A. Bretschneider (1870), to
    illustrate the sort of thinking which he thought might have led to the
    theorem's discovery in the first place (about 2000 B.C. or earlier).
    I've found that a reasonably intelligent child can understand it by
    cutting the four triangles out of cardboard and rearranging them within
    the large square.

    However, think about what's needed to raise it to the Euclidean
    level of rigour. First state carefully how each of the two diagrams is
    constructed. Then prove that the yellow triangles are all congruent (by
    the side-angle-side criterion, Euclid I.4). Then use the sum of the
    angles within each yellow triangle (Euclid I.32) to prove (using Euclid
    I.13) that each angle of the blue tetragon is a right angle, so the blue
    figure is indeed a square. Put all this together with other minor
    details and you'll probably get a proof just as long as Euclid's.
    It's _very_ risky to make historical judgements along the lines "I
    would think Plato would have figured out that ...." The isosceles case
    in the Meno is very much simpler than the general case.

    Ken Pledger.
    Ken Pledger, Dec 15, 2006
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