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How Many Proofs of the Pythagorean Theorem?

Date: 03/27/2003 at 19:11:14
From: Coco
Subject: The Pythagorean Theorem

I was wondering if you know the exact number of proofs of the 
Pythagorean Theorem in existence.


Date: 03/27/2003 at 23:20:16
From: Doctor Peterson
Subject: Re: The Pythagorean Theorem

Hi, Coco.

Even if I had every single proof anyone had ever written, I couldn't 
count them, because I couldn't decide how different two proofs have 
to be in order to be counted.

But someone wrote a book in which he showed 367 proofs that were 
distinct enough to bother writing about separately. This page lists 
41 of them:

  Pythagorean Theorem and its Many Proofs - Bogomolny
  http://www.cut-the-knot.org/pythagoras/index.shtml 

and has a footnote:

  W.Dunham [Mathematical Universe] cites a book The Pythagorean
  Proposition by an early 20th century professor Elisha Scott
  Loomis. The book is a collection of 367 proofs of the
  Pythagorean Theorem and has been republished by NCTM in 1968.

Again, this page

  Pythagoras' Theorem
  http://www.sunsite.ubc.ca/DigitalMathArchive/Euclid/java/html/
pythagoras.html

describes the book just a little differently:

  Elisha Loomis, The Pythagorean Proposition, National Council of
  Teachers of Mathematics, 1968. This eccentric book was first
  compiled in 1907, first published in 1928 (at a price of $2.00!),
  and reissued in this edition. It contains 365 more or less
  distinct proofs of Pythagoras' Theorem. The total effect is
  perhaps a bit overwhelming, and the quality of the figures is
  very poor, but nonetheless there are a few gems distributed
  throughout. 

The following page

  The Pythagorean Theorem - Jim Loy
  http://www.jimloy.com/geometry/pythag.htm 

says

  The book The Pythagorean Proposition, By Elisha Scott Loomis,
  is a fairly amazing book. It contains 256 proofs of the
  Pythagorean Theorem. It shows that you can devise an infinite
  number of algebraic proofs, like the first proof above. It
  shows that you can devise an infinite number of geometric
  proofs, like Euclid's proof. And it shows that there can be
  no proof using trigonometry, analytic geometry, or calculus.
  The book is out of print, by the way.

I'm not sure which number is right, but it appears that you can't 
count the number of distinct proofs, in any case.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Triangles and Other Polygons
Middle School Triangles and Other Polygons

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