41. Pythagoras earth, air, water and fire. theorem of pythagoras. The modern statementof the theorem of Pitagoras is In a triangle rectangle, the http://www.terra.es/personal2/efr1966/ipitag.htm

PYTHAGORAS Greek philosopher considered like a great man turned in different sciences. Precursor of the geometric theorem that takes its name. - I N D E X - Introduction Life and Builds Theorem of Pythagoras Definition Pythagoras Theorem ... Application Pythagoras Theorem Introduction Pythagoras was a student of the mathematics, according to commented all the things are numbers Life and Builds Philosopher and Greek mathematician born in the 580 A.C. in Samos, Iona and died around the 500 A.C. in Metapontum, Lucania. Mystical man and aristocrat whom the Pythagorean School founded, a species of sect whose symbol was the starred pentagon, and dedicated to the study of the philosophy, mathematical and astronomy. He was original of the island of Samos, located in the Aegean Sea. At the time of this philosopher the island was governed by the Polycrates tyrant. As the free spirit of Pythagoras could not agree to this form of government, an association emigrated towards the West, founding on Crotona (to the south of Italy) that did not have the character of a philosophical school but the one of a religious community. For this reason, it can say that mathematical sciences have been born in the Greek world of a corporation of religious and moral character. They met to carry out certain ceremonies, to help themselves mutually, and to even live in community. The Pythagorean School any person could enter, until women. In that then, and during long time and in many towns, the women were not admitted in the schools.

42. Pythagoras Of Samos Pythagoras' Theorem You might like to do Activity 2 before you continue. There aremany different proofs and demonstrations for this theorem of pythagoras'. http://www.mathgym.com.au/history/pythagoras/pytheor.htm

Return to MATHGYM Back P YTHAGORAS of S AMOS A Collection of Essays and Lessons for Junior and Senior High School Contents Pythagoras' Theorem - origins and proofs Plimpton 322 - Babylonian clay tablet Academic Pythagoras' Theorem You might like to do Activity 2 before you continue. Background Firstly, we need to appreciate that Pythagoras did not discover the relationship between the length of the sides of a right-angled triangle. This relationship had been known in Babylon and Egypt for centuries (if not millennia) before. Tradition has it though, that Pythagoras did find the first rigorous geometric proof. I will digress to describe an extraordinary clay tablet containing evidence of Babylonian knowledge of the relationship, at the end of this essay. Numerical relationship I would like to start my discussion on Pythagoras' Theorem not with geometry, but with number patterns as this is the method used by Pythagoras according to Proclus (410-485 A.D.) (cited in Heath [4] "Certain methods for the discovery of triangles of this kind are handed down, one of which they refer to Plato, the other to Pythagoras. [The latter] starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it makes the square of it, subtracts unity, and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side the hypotenuse. For example, taking 3, squaring it, and subtracting unity from the 9, the method takes half of the 8, namely 4; then, adding unity to it again, it makes 5, and a right-angled triangle has been found…"

43. Untitled The theorem of pythagoras is said to have many proofs. alexl@daemon.cna.tek.com(Alexander writes The theorem of pythagoras is said to have many proofs. http://www.ics.uci.edu/~eppstein/junkyard/pytho.html

From: alexl@daemon.cna.tek.com (Alexander Lopez) Newsgroups: sci.math Subject: Pythagorean theorem - proofs on the WWW? Date: 19 Apr 1996 15:40:51 -0700 Organization: Tektronix, Inc., Redmond, OR Reply-To: alexl@daemon.cna.tek.com The theorem of Pythagoras is said to have many proofs. I've found four on the WWW (used with the Geometers SketchPad). Are there any sites with more, or sites with historical background on the proofs? Alexander Lopez Software Engineering alexl@daemon.CNA.TEK.COM From: eppstein@ics.uci.edu (David Eppstein) Date: 19 Apr 1996 22:26:48 -0700 Newsgroups: sci.math Subject: Re: Pythagorean theorem - proofs on the WWW? alexl@daemon.cna.tek.com http://www.ics.ici.edu/~eppstein/junkyard/ ), reformatted somewhat from the original HTML: Euclid's Elements ( http://www.columbia.edu/~rc142/Euclid.html ). Online, in interesting colors, without all those annoying proofs. Also see D. Joyce's Java-animated version ( http://aleph0.clarku.edu/~djoyce/java/elements/elements.html ), and a manuscript excerpt from a copy in the Bodleian library made in the year 888 ( http://www.lib.virginia.edu/science/parshall/elementsamp.html

44. Pythagoras' Theorem The Library and Archives. Figure (bottom right) demonstrates the proof of thetheorem of pythagoras in Euclid's De Arte Geometrica late 13th century. http://www.exeter-cathedral.org.uk/Gallery/Library/L07.html

THE CATHEDRAL CHURCH OF SAINT PETER IN EXETER The Library and Archives Figure (bottom right) demonstrates the proof of the Theorem of Pythagoras in Euclid's De Arte Geometrica - late 13th century. Previous picture Next picture Return to Home Page

45. Pythagoras'Theorem Framework, Projet  Europe Des Découvertes forces the largest vessel will be emptied and its liquid will fill exactly theother two, thus proving the famous theorem of pythagoras.Another activity to http://www.inrp.fr/lamap/activites/projet/europe/grece/form7.htm

Accueil Activités The project Le projet Projet l'Europe des découvertes Framework for teachers Activités : Document de travail Katerina Garga garga@cti.gr Computer Technology Institute, Educational Technology Sector Athènes Grèce Publication : august 2001 Mise en ligne : august 2001 Titre / Title Pythagoras' Theorem Date c. 530 B.C. Domaine scientifique / Scientific field Geometry, trigonometry Nom du scientifique / Name of the scientist Pythagoras (c.582-c.507 B.C.) Données biographiques / Biographical data Description de la découverte ou de l'invention/ description of the discovery or invention . Pythagoras' Theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. Documents historiques / Historical documents Références bibliographiques / Bibliographical references M. Wertheim, Pythagora's Trousers, Traulos publications. P. Gorman (1978) and T. Stanley (1988); D. J. O'Meara, Pythagoras Revived: Mathematics and Philosophy in Late Antiquity (1989).

46. An Application Of Pythagoras Theorem The theorem of pythagoras says that if a triangle has sides of length a, b and cand the angle between the sides of length a and b is a right angle, then a^2 http://mathcentral.uregina.ca/QQ/database/qq.09.95/jones1.html

Mike Jones enigma@agt.net My niece and I were discussing geometry. We'd like to know what practical applications for the Pythagorean theorem may be. I don't often have a need to calculate the height of a mountain. Is there something closer to home? Thank You Hi Mike The most widely quoted "practical" application of the Pythagorean theorem is actually an application of its converse. The theorem of Pythagoras says that if a triangle has sides of length a, b and c and the angle between the sides of length a and b is a right angle, then a^2 + b^2 = c^2. The converse says that if a triangle has sides of length a, b and c and a^2 + b^2 = c^2 then the angle between the sides of length a and b is a right angle. Such a triple of numbers is called a Pythagorean triple, so 3,4,5 is a Pythagorean triple and so are 6,8,10 and 5,12,13. The application is in construction. It is very important when starting a building to have a square corner, and a Pythagorean triple provides an easy and inexpensive way to get one. Drive a stake at the desired corner point and another stake 3 meters from the corner along the line where you want one wall of the building. Then position a third stake so that its distance from the corner is 4 meters and the third side of the triangle formed by the three stakes is 5 meters. Since 3,4,5 is a Pythagorean triple the angle at the corner is a right angle. As a mathematician I think that an application of a theorem to some area of mathematics is "practical". The Pythagorean theorem is a starting place for trigonometry, which leads to methods, for example, for calculating the heights of mountains. The Pythagorean theorem is also an example of the somewhat rare situation where both the theorem and its converse are true. It is also useful in calculating distances. Suppose you leave home and drive 12 kilometers west then 5 kilometers north. How far are you from home "as the crow flies"?

47. A Short Course In Math The theorem of pythagoras. Pythagoras of Samos was a Greek philosopher who livedaround 530 BC, mostly in the Greek colony of Crotona in southern Italy. http://home.t-online.de/home/0414184774-0001/englisch/vermtech/pythagoras.htm

http://www-istp.gsfc.nasa.gov/stargcc/Sconcat9.htm The Theorem of Pythagoras Pythagoras of Samos was a Greek philosopher who lived around 530 BC, mostly in the Greek colony of Crotona in southern Italy. According to tradition he was the first to prove the assertion (theorem) which today bears his name: If a triangle has sides of length ( a,b,c ), with sides ( a,b ) enclosing an angle of 90 degrees ("right angle"), then a + b = c A right angle can be defined here as the angle formed when two straight lines cross each other in such a way that all 4 angles produced are equal. The theorem also works the other way around: if the lengths of the three sides ( a,b,c ) of a triangle satisfy the above relation, then the angle between sides a and b must be of 90 degrees. For instance, a triangle with sides a b c = 5 (inches, feet, meterswhatever) is right-angled, because a + b = 9 + 16 = 25 = c Ancient Egyptian builders may have known the (3,4,5) triangle and used it (with measured rods or strings) to construct right angles; even today builders may still nail together boards of those lengths to help align a corner. Many proofs exist and the easiest ones are probably the ones based on algebra, using the elementary identities discussed in the preceding section, namely

48. Pythagoras And The Pythagorean Theorem The Pythagorean Theorem. (theorem of pythagoras). Pythagoras lived duringthe sixth century BC, from about 580 until 500, 150 years before Plato. http://users.rcn.com/mborelli/pytext.html

The Pythagorean Theorem (Theorem of Pythagoras) Pythagoras lived during the sixth century B.C., from about 580 until 500, 150 years before Plato. He believed that the world is a perfect whole, a model of order and rationality. He was the first to call the world kosmos and believed that mathematics was its basis. As an illustration of this he formalized what the Babylonians were aware of by 2000 B.C., that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs. Written algebraically, this is: c = a + b By taking the square root of both sides of this equation we derive a more practical corollary: c = /a + b The hypotenuse itself is equal to the square root of the sum of the squares of the legs. Program 1 is a JavaScript translation of this equation. Furthermore, if we take the first equation, c = a + b subtract b from both sides to get c - b = a then take the square root of both sides, we derive a corollary which we can use to find the length of a leg of a right triangle given the length of the other leg and the length of the hypotenuse: /c - b = a.

49. CoSy/NeoPythagorism Encyclopedia / Search term pythagoras Articles selected 2 that begin with pythagoras 2 Pythagoras of Samos 3 Pythagoras, theorem of pythagoras of Samos http://www.cosy.com/views/pythag.htm

CoSy/Home CoSy/Current ?Wha? BobA-In-Y2K.org NeoPythagorism Various items about My view of Reality CoSy/Views/PsychoPhysics . One of the great souls in the APL community has been GerardLanglet Under the History of Mathematics TimeLine sdcc14.ucsd.edu/~fillmore/timeline.html sdcc14.ucsd.edu/~fillmore/blurbs/P322.html has the following set of Pythagorean Triples known by the Babylonians circa -1700 . Triples 120 119 169 3456 3367 4825 4800 4601 6649 13500 12709 18541 72 65 97 360 319 481 2700 2291 3541 960 799 1249 600 481 769 6480 4961 8161 60 45 75 2400 1679 2929 240 161 289 2700 1771 3229 90 56 106 Here`s a little classic APL demonstrating that the sum of squares of each of the rows of the first 2 columns equals the corresponding item of the third . ( Triples [ ; 1 ] +.* 2 ) = Triples [ ; 2 ] * 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ============================: TUE.SEP,990907 :============================ Brett A. Bradley wrote: > Well, the original Pythagoreans were vegetarian, in fact Pythagorean was > the term for vegetarianism before "vegetarian" was coined. It only seemed > logical that a Neopythagorean would be vegetarian. ` Learn something new everyday , especially on the Web . I did find a reference to the the abstinance from meat eating thru Yahoo at

50. SOME SELECTED PUBLICATIONS 5. Another generalization of the theorem of pythagoras. Spectrum. 9. The theoremof Pythagoras Generalizing from right triangles to right polygons. http://mzone.mweb.co.za/residents/profmd/publications.htm

SOME SELECTED PUBLICATIONS by Michael de Villiers Mathematical Articles International Journal for Mathematical Education in Science and Technology , 20(4), 585-603, August 1989. Imstusnews , 19, 15-16, November 1989. Spectrum , 28(2), 18-21, May 1990. Physics Teacher , 286-289, May 1991. Spectrum . International Journal for Mathematical Education in Science and Technology Mathematical Digest Imstusnews Spectrum International Journal for Mathematical Education in Science and Technology Australian Senior Mathematics Journal Pythagoras The Mathematical Gazette , 79(485), 374-378, July 1995. . Int. J. Math. Ed. Sci. Technol ., 26(2), 233-241, 1995. (Co-author: J. Meyer, UOFS). , 6(3), 169-171, Sept 1996. ). KZN AMESA Math Journal , Vol 3, No 1, 11-18. Mathematical Gazette , Nov. Mathematical Gazette , March 1999. Mathematics in School , March 1999, 18-21. Mathematics in College Mathematics Education Articles Mathematics Teacher , Vol.80, No.7, pp.528-532, October 1987. Pythagoras . 19, pp.27-30, April 1989. S.A. Tydskrif vir Opvoedkunde , 10(1), Feb 1990, 68-74 (co-author: E.C. Smith).

51. The Theorem Of Pythagoras MAT FILM The theorem of pythagoras. af James F. Blinn. Videoen begyndermed tre praktiske problemer som fører til det samme matematiske http://www.mat.dtu.dk/cinemat/The_Theorem_of_Pythagoras.html

The Theorem of Pythagoras af James F. Blinn. Videoen er en del af Project MATHEMATICS! Sidst opdateret 14. august 1995. Kommentarer til: www@mat.dtu.dk

52. An Interactive Proof Of Pythagoras' Theorem View an interactive proof of the Pythagorean theorem. Shows squares off each side of a triangle. An Interactive Proof of pythagoras' theorem. This Java applet was written by Jim Morey. http://sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Pythagoras/pythagora

UBC Mathematics Department http://www.math.ubc.ca/ An Interactive Proof of Pythagoras' theorem This Java applet was written by Jim Morey . It won grand prize in Sun Microsystem's Java programming contest in the Summer of 1995. http://www.math.ubc.ca/ Return to Interactive Mathematics page

53. Pythagoras' Theorem - By Seth Y-Maxwell Includes diagrams, history, and links all related to the Pythagorean theorem. http://www.geocities.com/CapeCanaveral/Launchpad/3740/

By: Seth Yoshioka-Maxwell You can view one of the images by clicking once on the picture you want. Pythagoras was a great Mathematician who was the first to create the music scale of today. He also created theorems. One of his most famous theorem was: a +b =c Attention If You have any information on different proofs e-mail me. I would love to add more proofs to my site. Thank you. +b =c NEW ! Pythagorean Theorem Proof Calculator Pythagorean Theorem Joke President Garfield's Proof of the Pythagorian Theorem Main page ... Notify-mail Page and graphics designed by Seth Yoshioka-Maxwell

54. The Pythagorean Theorem Department of Mathematics Education J. Wilson, EMT 669 The Pythagorean theorem Stephanie J. Morris The Pythagorean theorem was one of the earliest theorems known to ancient civilizations. The Pythagorean theorem is pythagoras' most famous mathematical contribution. http://jwilson.coe.uga.edu/emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay

Department of Mathematics Education J. Wilson, EMT 669 The Pythagorean Theorem by Stephanie J. Morris The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students. The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea. The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:

55. Pythagoras' Playground (by The Event Inventor) We will use the amazing properties of the Pythagorean theorem to explore the world around us. http://www.kyes-world.com/pythagor.htm

NEW! Solar System Felt Puzzles! Felt puzzles matching geologic and meteorologic features on the worlds of our solar system, using geometric shapes as clues. Exploring Fractions with Astronomy activity book! More fun than a barrel of aliens!!! Circles activity book! Wheels of the Universe, Giant Circles in the Sky!, As the Earth Turns - For planning adventurous learning for all ages. Details HERE! From this site, we will use the amazing properties of the Pythagorean Theorem to explore the world around us. 2500 years ago, Pythagoras of Samos and his students developed the first proof that, for a right triangle, a + b = c (the sum of the squares of the two legs of a triangle is equal to the square of the hypotenuse) as well as being responsible for many other important developments in mathematics, astronomy and music. Many of the ideas used and instruments we will be making were originally employed by the Babylonians, 1000 years before Pythagoras proved on paper why they worked! During the Dark Ages, the Arabs nurtured the science of astronomy and developed the primitive quadrant into a more advanced tool, called an ASTROLABE. The first brave mariners that dared to leave sight of land in their sailing ships did so with the help of these instruments, making it possible for them to determine their position on the Earth to within a few degrees of accuracy; close enough to find land again! Here are some investigations to help you discover the secrets that these explorers used, to enjoy the world around us:

56. Pythagoras' Haven pythagoras theorem asserts that for a right triangle with short sides of length a and b and long side of length c a2 + b2 = c2 Of course it has a direct geometric formulation. Click on move the node to change the shape of the triangle. http://www.math.ubc.ca/~morey/java/pyth

The following window shows a geometrical proof of Pythagoras' Theorem. The three buttons, NEXT, BACK, RESTART, allow you to go through the steps of the proof. As well, if you would like to repeat the action of the diagram simply click on the image. (The text can be retyped by clicking on the text box). Good luck understanding the proof. This will hopefully turn into a place for geometric proofs of the Pythagorean Theorem the square of the hypotenuse of a right angle triangle is equal to the sum of the squares of the other two sides. Take a look at the poorly documented program and its helpers Banner and fillTriangle this hacked from the hotjava people my home page

57. Pythagorean Theorem And Its Many Proofs triangle. The pythagoras' theorem then claims that the sum of (areasof) the two small squares equals (the area of) the large one. http://www.cut-the-knot.com/pythagoras/

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Pythagorean Theorem Let's build up squares on the sides of a right triangle. The Pythagoras' Theorem then claims that the sum of (areas of) the two small squares equals (the area of) the large one. In algebraic terms, a +b =c where c is the hypotenuse while a and b are the sides of the triangle. The theorem is of fundamental importance in the Euclidean Geometry where it serves as a basis for the definition of distance between two points. It's so basic and well known that, I believe, anyone who took geometry classes in high school couldn't fail to remember it long after other math notions got solidly forgotten. I plan to present several geometric proofs of the Pythagorean Theorem. An impetus for this page was provided by a remarkable Java applet written by Jim Morey . This constitutes the first proof on this page. There is nothing like learning while doing and, as an exercise in Java programming, I'll later offer an original Java applet. But, for now, let consider several plain HTML proofs. Remark The statement of the Theorem was discovered on a Babylonian tablet circa 1900-1600 B.C. Whether Pythagoras (c.560-c.480 B.C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. Euclid's (c 300 B.C.)

58. Babylonian Mathematics An overview of mathematics within this culture. Includes a description of the numerals used and a reference to pythagoras' theorem. http://www-groups.dcs.st-and.ac.uk/~history/Indexes/Babylonians.html

59. CTK Exchange Front Page Movie Shortcuts Personal Info Awards (Corollaries from) pythagoras' theorem pythagoras' theorem playsan important role in mathematics indeed. On this page I'll try http://www.cut-the-knot.com/pythagoras/corollary.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site (Corollaries from) Pythagoras' Theorem Pythagoras' Theorem plays an important role in mathematics indeed. On this page I'll try to collect several statements some of whose proofs depend on the Pythagorean Theorem. The Arithmetic - Geometric Means Inequality For positive a and b, (a + b)/2 (ab) with equality iff a = b One may argue that the proof followed from an algebraic identity (a + b) - (a - b) In that case, the Pythagorean Theorem furnishes an intuitive geometric illustration. Just draw two touching circles with radii a/2 and b/2 as in the diagram. As in the case Isoperimetric Inequality this too allows for two equivalent extremal problems: Among all pairs of numbers with a given product find two whose sum is minimal. Among all pairs of numbers with a given sum find two whose product is maximal. In both cases, the extremal value is attained when the two numbers coincide. The latter fact has a nice geometric illustration which also suggests another proof for the Arithmetic Mean - Geometric Mean Inequality. The former is often rewritten in a different form: with equality iff x = 1 The Arithmetic Mean - Geometric Mean Inequality for sequences of numbers was first proven when the length of the sequence was a power of 2 and from here for an arbitrary integer . (1) also extends for an arbitrary number of positive numbers: Let x i /x + x /x + ... + x

60. PythagorasGreek Philosopher Contributed To Early Mathematics, Astronomy And Musi The History of pythagoras and his proof in 3D. Page includes Diagrams, History, Links, Guestbook, Test, Calculator, and a Joke all related to pythagoras and his theorem. pythagoras was a great Mathematician who was the first to create the music scale of today. He also created theorems. One of his most famous theorem http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html

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