# Determining whether proofs of the Pythagorean theorem are essentially the same, or essentially different

There are great annotated lists of proofs of the Pythagorean theorem, to name just two:

There's an obvious difference between geometric and algebraic proofs (which use some algebraic notation usually including the binomial formula). There are other differences, especially the length and complexity of a proof (compare Pythagoras' and Euclid's proof here).

But even a non-expert gets the feeling that some of the proofs are "essential the same", among the geometric ones, among the algebraic ones, but even between geometric and algebraic ones.

Other pairs of proofs seem to be essentially different, at least for the non-expert (maybe not for the expert).

What I'm looking for is a diagram or table in which for each pair of proofs of the Pythagorean theorem it is indicated that (and maybe in which respect) they are essentially the same or different (optimally to which degree).

I'm aware that the question what makes two proofs essentially the same is a vague one (and subject of discussion, e.g. here and here). But assuming that the notion is not totally pointless, and that the answer at least sometimes is clear, I dare to ask this (i.e. for a reference which sums it up).

• A good place to start may be Elisha Loomis' 300-page compendium, "The Pythagorean Proposition", sub-titled "Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of 'Proofs'". (A PDF of a microfiche transfer is here.) – Blue Aug 29 '18 at 10:19
• Marvelous! That's the answer (at least a very promising one). – Hans-Peter Stricker Aug 29 '18 at 10:21

(Promoting my comment to an answer.)

A good place to start may be Elisha Loomis' 300-page compendium, "The Pythagorean Proposition" (1940), sub-titled "Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of 'Proofs'".

I believe the book is now out-of-print. A PDF of a microfiche transfer is available from the U.S. Department of Education here.