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 Did you know . . .?

The Pythagorean theorem takes its name from the ancient Greek mathematician Pythagoras (569 B.C.?-500 B.C.?), who was perhaps the first to offer a proof of the theorem. But people had noticed the special relationship between the sides of a right triangle long before Pythagoras.

Right triangle with sides a, b, cThe Pythagorean theorem states that the sum of the squares of the lengths of the two other sides of any right triangle will equal the square of the length of the hypoteneuse, or, in mathematical terms, for the triangle shown at right, a2 + b2 = c2. Integers that satisfy the conditions a2 + b2 = c2 are called "Pythagorean triples."

(Illustration source: http://www.cs.ucla.edu/~klinger/dorene/Gif/math1pic1.gif)

The Plimpton tablet, with tables of Pythagorean triplesAncient clay tablets from Babylonia indicate that the Babylonians in the second millennium B.C., 1000 years before Pythagoras, had rules for generating Pythagorean triples, understood the relationship between the sides of a right triangle, and, in solving for the hypoteneuse of an isosceles right triangle, came up with an approximation of accurate to five decimal places. [They needed to do so because the lengths would represent some multiple of the formula: 12 + 12 = ()2.]

(Illustration source: http://www.swan.ac.uk/compsci/ResearchGroups/TheoryGroups/AlgMethFolder/DSTFolder/HistoryOfTables/Plimpton/Plimpton1TN.GIF)

Visual demonstration of Pythagorean theorem from the Chou Pei Suan ChingA Chinese astronomical and mathematical treatise called Chou Pei Suan Ching (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, ca. 500-200 B.C.), possibly predating Pythagoras, gives a statement of and geometrical demonstration of the Pythagorean theorem. (Click here for a link to an explanation of this demonstration.)

(Illustration source: http://www.unisanet.unisa.edu.au/07305/pythag.htm)

Ancient Indian mathematicians also knew the Pythagorean theorem, and the Sulbasutras (of which the earliest date from ca. 800-600 B.C.) discuss it in the context of strict requirements for the orientation, shape, and area of altars for religious purposes. It has also been suggested that the ancient Mayas used variations of Pythagorean triples in their Long Count calendar.

Right triangle with sides a, b, c, with squares built on each sideWe do not know for sure how Pythagoras himself proved the theorem that bears his name because he refused to allow his teachings to be recorded in writing. But probably, like most ancient proofs of the Pythagorean theorem, it was geometrical in nature. That is, such proofs are demonstrations that the combined areas of squares with sides of length a and b will equal the area of a square with sides of length c, where a, b, and c represent the lengths of the two sides and hypoteneuse of a right triangle.

(Illustration source: http://www.cs.ucla.edu/~klinger/dorene/Gif/math1pic2.gif)

Here is a link to an animated example of one such geometrical proof.

Another link will take you to a page where you can move tiles from one square to another to satisfy yourself that the Pythagorean theorem indeed works.

PythagorasPythagoras himself was not simply a mathematician. He was an important philosopher who believed that the world was ruled by harmony and that numerical relationships could best express this harmony. He was the first, for example, to represent musical harmonies as simple ratios.

Pythagoras and his followers were also a bit eccentric. Pythagoras's followers were sworn to absolute secrecy, and their devotion to their master bordered on the cult-like. Pythagoreans followed a strict moral and ethical code, which included vegetarianism because of their belief in the reincarnation of souls. They also refused to eat beans!

(Illustration source: http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Pythagoras.html)

In their own words:

Number is the ruler of forms and ideas, and the cause of gods and demons. --Pythagoras, as quoted by

(Quotation source: http://www-groups.dcs.st-andrews.ac.uk/~history/Quotations/Pythagoras.html)

Every man has been made by God in order to acquire knowledge and contemplate.

(Quotation source: http://www-groups.dcs.st-andrews.ac.uk/~history/Quotations/Pythagoras.html)

The Pythagoreans say that there is but one number, the mathematical, but things of sense are not separated from this, for they are composed of it; indeed, they construct the whole heaven out of numbers, but not out of unit numbers, for they assume that the unities have quantity; but how the first unity was so constituted as to have quantity, they seem at a loss to say. b 31. All, as many as regard the one as the element and first principle of things, except the Pythagoreans, assert that numbers are based on the unit; but the Pythagoreans assert, as has been remarked, that numbers have quantity.--Aristotle, Metaphysics, xii. 6; 1080 b 16.

(Quotation source: http://history.hanover.edu/texts/presoc/pythagor.htm)

Fun links:

Easy to grasp animated demonstration of the Pythagorean theorem: http://sunsite.univie.ac.at/MathAnim/pythanim.gif

Another animated proof. Just sit back and enjoy the show!: http://www.nadn.navy.mil/MathDept/mdm/pyth.html

And yet another: http://cecasun.utc.edu/~cpmawata/geom/geom7.htm

And links to even more: http://www.geocities.com/primes135/java_anim.html

One more! Which do you like best?: http://sunsite.univie.ac.at/MathAnim/pythanim.gif

A proof by President Garfield (scroll down to find): http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html

But what's it good for?: http://www.pbs.org/wgbh/nova/proof/puzzle/use.html

For more information:

A brief history: http://www.geom.umn.edu/~demo5337/Group3/hist.html

Many proofs of the Pythagorean theorem: http://www.cut-the-knot.com/pythagoras/

And again: http://www.mcn.net/~jimloy/pythag.html

History and several proofs: http://jwilson.coe.uga.edu/emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay.1/Pythagorean.html

Dr. Math's Pythagorean theorem page: http://forum.swarthmore.edu/dr.math/faq/faq.pythagorean.html

Uses of the Pythagorean theorem: http://www.geocities.com/primes135/uses.html

About Pythagoras: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html

Ancient chinese mathematical texts: http://saxakali.com/COLOR_ASP/developcm3.htm

The Pythagorean theorem in Babylonian mathematics: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_Pythagoras.html

The Indian Sulbasutras: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html

By Laura Smoller, UALR Department of History.

May 2001.