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# Pythagoras of Samos

mathematical contributions of Pythagoras

by

Tweet## Sujay Chitta

on 27 May 2010#### Transcript of Pythagoras of Samos

Pytha goras of Samos circa 570 B.C. ... Pythagoras was born on a small island in the Eastern Aegan of Greece, Samos. Though much of his first 40 years were shadowy and unreliable with few recorded facts, he made influential contributions to philosophy, religious teachings, and more notably mathematics in the late 6th century BC. The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.

—Aristotle, Metaphysics 1-5 , cc. 350 BC Pythagoras made significant advancements in three main mathematical/music areas:

Music Tuning

Tetractys

and of course...

The Pythagoran Theorem Music Theory:

Pythagorean Tuning According to legend, while Pythagoras was working with different sized anvils, he discovered that all musical notes could somehow be translated into mathematical equations, depicting sound waves... it would later be used for tuning. Method: His tuning method is based on a stack of perfect fifths, tuned to the ratio 3:2... the simplest ratio after 2:1.

Example: If you start from "D", "A" would be tuned so that the frequency ratio of A:D = 3:2

if D is tuned to 288 Hz, then the A is tuned to 432 Hz or 288*3/2 Other contributions... Tetractys: According to Pythagoras, Tetractys symbolized the four major elements — earth, air, fire, and water.The four rows added up to ten, which was unity of a higher order... decimals The Tetractys represented the equal organization of space

the first row represented zero-dimensions or a single point

the second row represented the first dimension, a line of two points

the third row represented two-dimensions, a plane defined by a triangle of three points, the minimum # of points needed to make a plane

the fourth row represented the final dimension known to man, the third dimension... shown by a tetrahedron defined by four points From this "mystical" symbol is derived two sets of numbers.

One is made by addition of numbers

The other by multiplication

These sets encompass and cover the musical, geometric and arithmetic ratios of which harmony in the universe is composed.

The perfect world results from these sets... geometrically, harmonically and arithmetically arranged. "The Tetractys [also known as the decad] is an equilateral triangle formed from the sequence of the first ten numbers aligned in four rows. It is both a mathematical idea and a metaphysical symbol that embraces within itself—in seedlike form—the principles of the natural world, the harmony of the cosmos, the ascent to the divine, and the mysteries of the divine realm. So revered was this ancient symbol that it inspired ancient philosophers to swear by the name of the one who brought this gift to humanity."

— Pythagoras. And Finally................ THE PYTHAGOREAN THEOREM Why? Though widely speculated as the creator of the Pythagorean Theorem, it was previously used by the Babylonians and Indians. He is, however, creditted to have constructed the first proof. Still........... the way in which the Babylonians handled Pythagorean numbers, it is fairly certain that they knew that the principle was applicable. They probably had some kind of proof, but it is yet to be found. It is used to find the missing, third side on any right triangle when given any two sides. "a" and "b" represent the legs of the triangle and "c" represents the hypotenuse. Sources:

"The Philosophy of Pythagoras." Thebigview.com - Pondering the Big Questions. Web. 25 May 2010. <http://www.thebigview.com/greeks/pythagoras.html>.

"Pythagoras of Samos (ca. 560-ca. 480 BC) -- from Eric Weisstein's World of Scientific Biography." ScienceWorld. Web. 25 May 2010. <http://scienceworld.wolfram.com/biography/Pythagoras.html>.

"Pythagoras (Stanford Encyclopedia of Philosophy)." Stanford Encyclopedia of Philosophy. 23 Feb. 2005. Web. 25 May 2010. <http://plato.stanford.edu/entries/pythagoras/>.

"Pythagorean Theorem and Its Many Proofs." Interactive Mathematics Miscellany and Puzzles. Web. 25 May 2010. <http://www.cut-the-knot.org/pythagoras/>.

"530 BC: Pythagoras of Samos, scientist and philosopher, moves to Croton and starts researching and...." Hutchinson Chronology of World History. 08 Sep. 2005

http://threes.com/cms/images/stories/history/pythagoras.jpg http://www.guide-to-symbols.com/_images_pub3/tetractys.png http://questgarden.com/81/69/6/090428165756/images/pythag%20theorem%20pic.gif http://curriculumaid.com/Math/Geometry/unit1lesson1.html http://curriculumaid.com/Math/Geometry/unit1lesson1.html http://curriculumaid.com/Math/Geometry/unit1lesson1.html http://curriculumaid.com/Math/Geometry/unit1lesson1.html http://www.chemistrydaily.com/chemistry/Tetraktys http://www-history.mcs.st-andrews.ac.uk/Quotations/Aristotle.html

Full transcript—Aristotle, Metaphysics 1-5 , cc. 350 BC Pythagoras made significant advancements in three main mathematical/music areas:

Music Tuning

Tetractys

and of course...

The Pythagoran Theorem Music Theory:

Pythagorean Tuning According to legend, while Pythagoras was working with different sized anvils, he discovered that all musical notes could somehow be translated into mathematical equations, depicting sound waves... it would later be used for tuning. Method: His tuning method is based on a stack of perfect fifths, tuned to the ratio 3:2... the simplest ratio after 2:1.

Example: If you start from "D", "A" would be tuned so that the frequency ratio of A:D = 3:2

if D is tuned to 288 Hz, then the A is tuned to 432 Hz or 288*3/2 Other contributions... Tetractys: According to Pythagoras, Tetractys symbolized the four major elements — earth, air, fire, and water.The four rows added up to ten, which was unity of a higher order... decimals The Tetractys represented the equal organization of space

the first row represented zero-dimensions or a single point

the second row represented the first dimension, a line of two points

the third row represented two-dimensions, a plane defined by a triangle of three points, the minimum # of points needed to make a plane

the fourth row represented the final dimension known to man, the third dimension... shown by a tetrahedron defined by four points From this "mystical" symbol is derived two sets of numbers.

One is made by addition of numbers

The other by multiplication

These sets encompass and cover the musical, geometric and arithmetic ratios of which harmony in the universe is composed.

The perfect world results from these sets... geometrically, harmonically and arithmetically arranged. "The Tetractys [also known as the decad] is an equilateral triangle formed from the sequence of the first ten numbers aligned in four rows. It is both a mathematical idea and a metaphysical symbol that embraces within itself—in seedlike form—the principles of the natural world, the harmony of the cosmos, the ascent to the divine, and the mysteries of the divine realm. So revered was this ancient symbol that it inspired ancient philosophers to swear by the name of the one who brought this gift to humanity."

— Pythagoras. And Finally................ THE PYTHAGOREAN THEOREM Why? Though widely speculated as the creator of the Pythagorean Theorem, it was previously used by the Babylonians and Indians. He is, however, creditted to have constructed the first proof. Still........... the way in which the Babylonians handled Pythagorean numbers, it is fairly certain that they knew that the principle was applicable. They probably had some kind of proof, but it is yet to be found. It is used to find the missing, third side on any right triangle when given any two sides. "a" and "b" represent the legs of the triangle and "c" represents the hypotenuse. Sources:

"The Philosophy of Pythagoras." Thebigview.com - Pondering the Big Questions. Web. 25 May 2010. <http://www.thebigview.com/greeks/pythagoras.html>.

"Pythagoras of Samos (ca. 560-ca. 480 BC) -- from Eric Weisstein's World of Scientific Biography." ScienceWorld. Web. 25 May 2010. <http://scienceworld.wolfram.com/biography/Pythagoras.html>.

"Pythagoras (Stanford Encyclopedia of Philosophy)." Stanford Encyclopedia of Philosophy. 23 Feb. 2005. Web. 25 May 2010. <http://plato.stanford.edu/entries/pythagoras/>.

"Pythagorean Theorem and Its Many Proofs." Interactive Mathematics Miscellany and Puzzles. Web. 25 May 2010. <http://www.cut-the-knot.org/pythagoras/>.

"530 BC: Pythagoras of Samos, scientist and philosopher, moves to Croton and starts researching and...." Hutchinson Chronology of World History. 08 Sep. 2005

http://threes.com/cms/images/stories/history/pythagoras.jpg http://www.guide-to-symbols.com/_images_pub3/tetractys.png http://questgarden.com/81/69/6/090428165756/images/pythag%20theorem%20pic.gif http://curriculumaid.com/Math/Geometry/unit1lesson1.html http://curriculumaid.com/Math/Geometry/unit1lesson1.html http://curriculumaid.com/Math/Geometry/unit1lesson1.html http://curriculumaid.com/Math/Geometry/unit1lesson1.html http://www.chemistrydaily.com/chemistry/Tetraktys http://www-history.mcs.st-andrews.ac.uk/Quotations/Aristotle.html