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.

The 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, a^{2} + b^{2} = c^{2}.
Integers that satisfy the conditions a^{2} + b^{2}
= c^{2} are called "Pythagorean triples."

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

Ancient
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: 1^{2} + 1^{2} = ()^{2}.]

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

A
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.

We 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

(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.

Pythagoras
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)

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

*Iamblichus*

(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.