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

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

1. ### avitalGuest

Hi.

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 http://www.cut-the-knot.org/pythagoras/index.shtml) 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).

Thanks,
Avital.

avital, Nov 30, 2006

2. ### Ken PledgerGuest

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