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In a recent course (New York, 30 July 2002) you mentioned a one-word proof of the Pythagorean theorem: "Behold!" Where can I find the picture that accompanies the word?
[And: Thank you for a wonderful day!]
-- Philip Wadler (email)
Here's a link to a page that has the story & a picture:
-- Sean Owens (email)
Visual proof of right triangle theorem
It is a proof from a Chinese geometry book about 2,000 years ago. How the proof works is shown in my Envisioning Information, page 84.
-- Edward Tufte
I thought it was odd in your course that you described the "Behold" right triangle proof without showing it. Then I noticed it appeared in Envisioning Information, and then I found out you have a thread on it here. I recommend that, as you are describing this proof in the course, you direct the attendees to see the "Behold" proof reproduced in your book.
I agree with Phil Wadler, the course was a day well-spent.
-- Ron Wurtz (email)
This is a quibble regarding ET and diagrams supporting proofs of the Pythagorean Theorem.
ET's lovely wire sculpture "Pythagorean Theorem" is his interpretation of the diagram described by Plato in his dialogue "Meno." In this dialogue, Socrates guides an untutored slave boy to discover a certain mathematical proof (to demonstrate that mathematics is not learned by experience but is "recollected" from innate ideas). I particularly like the mounting of the wire diagram to create a shadow simulacrum that varies slightly as the wire moves, alluding to the famous Allegory of the Cave from "Republic."
My quibble is that the theorem that Socrates and and slave-boy prove is not the full-strength Pythagorean Theorem, but merely the proposition that the diagonal of any square is the base of a square of double the area. This theorem may be read off from the diagram--the diagonal is the "missing" side of the square positioned at a 45 degree angle.
It is easily seen that this proposition follows from the Pythagorean Theorem, because, if the area of the original square is one, then the area of the square on the diagonal is two, and one squared (a squared) plus one squared (b squared) equals two (c squared).
But this theorem is not equivalent to, nor does it imply, the Pythagorean Theorem, because it only applies to triangles constructed from the diagonal bisection of a square (i.e., isosceles triangles). If the diagonal is that of a non-square rectangle, then there is no way for the diagram to demonstrate the relation of the area of the square on the diagonal to the area of the rectangle or the areas of the squares based on their sides. One may also confirm this by inspection of the passage from "Meno," which never generalizes from squares and their diagonals to triangles in general.
I like the sculpture very much but the title is misleading. An alternate title might be, "Doubling the Square," or, more philosophically, "All True Knowledge is Innate."
-- Jonathan Cohen (email)
In fact the Socrates theorem can be used to prove the Pythagorean theorem. See proof #64 of http://www.cut-the-knot.org/pythagoras/index.shtml. That page is a smorgasbord of visual and partly visual proofs. Several proofs there are claimed to go better than one word -- zero words.
-- Theodore Norvell (email)
There is a large amount of mythology and misinformation about the history of the Pythagorean theorem. I doubt much of what I read on websites like cuttheknot and others -- the mathematical proofs are usually decent, but the history is often way off.
I also have heard and read the rumor that Bhaskara II (a.k.a. Bhaskaracharya) drew a picture and said/wrote "Behold!" as a proof of the Pythagorean theorem. While I find this plausible, I've spend a fair amount of time looking through his Lilavati and Bijaganita (the two named sources of this proof) and I have not found anything like this. I did find a reference to this story in a 19th century history book, but that book gave no reference. If anyone has a precise reference for this Behold proof of Bhaskara II, I'd love to know! Otherwise, I'll continue to consider it a legend.
Another rumor is that that the Babylonians knew the Pythagorean theorem around roughly 2000-1800 B.C.E. As far as I know, this rumor is based on a tablet that has a list of Pythagorean triples, such as (3,4,5), (5,12,13), etc.. Such a list does not demonstrate a general knowledge of the theorem.
As far as I know, the earliest written statement of the general Pythagorean theorem is in the Shulba Sutras of India, around 800-300 B.C.. I believe that Indian mathematicians were aware -- as was Socrates -- of the simple proof of the isosceles Pythagorean theorem that is brought up by the above posts. The Indians of the Vedic period were motivated by the construction of altars, I think. The Chinese proof seems close in date to Euclid. A version can be found in the Zhou Bi Suan Jing. In Chinese, it is called the Gougu theorem -- I think the best place to read about it is in Liu Hui's commentary on the "Nine Chapters on the Mathematical Art" -- 3rd century A.D..
While Proof #64 at cuttheknot advertises that Socrates' result -- the isosceles Pythagorean theorem -- leads to the full Pythagorean theorem, I would disagree with this analysis. The dissection given there is essentially a standard dissection that leads to the full Pythagorean theorem, with or without Socrates' result.
Anyways - that's my professional opinion, as a mathematician but admittedly not a math historian. Since I teach this material pretty often, but I'm not an expert in ancient mathematics, I would certainly appreciate any more information on primary sources for the Pythagorean theorem.
-- Marty Weissman (email)
Here is a brief program written in Excel, modeling the behavior of the Pythagorean Theorem. The little booklet was written afterwards, documenting how much math and thought went into the brief program. Lots there, when one sits down and actually does the work!
-- Michael Round (email)
Regarding Babylonian knowledge of the Pythagorean theorem: One respondent (correctly) critiqued Plimpton 322 (a list of Pythagorean triplets) as evidence that the Babylonians knew the Pythagorean theorem. In fact, Pythagorean triplets only incidentally relate to geometry; they are a number theoretic concept.
However, the Babylonians seemed to have been aware of the theorem in general. Many tablets (dating to ca. 1600 BC) include problems of finding the dimensions of a rectangle given its area and diagonal (see Katz et al, "The Mathematics of Egypt, Mesopotamia, China, India, and Islam" for the relevant texts).
What is interesting is that both the Babylonians and ancient Indians stated the form in terms of the diagonal of a rectangle, and not as its relates to right triangles. The reformulation of the theorem as a property of right triangles seem to be original with the Greeks.
-- Jeff Suzuki (email)