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The Theorem of Pythagoras

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The Theorem of Pythagoras The Most Famous Theorem The Theorem of Pythagoras feature Understanding History – Who, When and Where Does mathematics have a history? In this article the author studies the tangled and multi-layered past of a famous result through the lens of modern thinking, look- ing at contributions from schools of learning across the world, and connecting the mathematics recorded in archaeological finds with that taught in the classroom. Shashidhar Jagadeeshan magine, in our modern era, a very important theorem being Iattributed to a cult figure, a new age guru, who has collected a band of followers sworn to secrecy. The worldview of this cult includes number mysticism, vegetarianism and the transmigration of souls! One of the main preoccupations of the group is mathematics: however, all new discoveries are ascribed to the guru, and these new results are not to be shared with anyone outside the group. Moreover, they celebrate the discovery of a new result by sacrificing a hundred oxen! I wonder what would be the current scientific community’s reaction to such events. This is the legacy associated with the most ‘famous’ theorem of all times, the Pythagoras Theorem. In this article, we will go into the history of the theorem, explain difficulties historians have with dating and authorship and also speculate as to what might have led to the general statement and proof of the theorem. Vol. 1, No. 1, June 2012 | At Right Angles 5 Making sense of the history Why Pythagoras? Greek scholars seem to be in Often in the history of ideas, especially when there agreement that the first person to clearly state PT has been a discovery which has had a significant in all its generality, and attempt to establish its influence on mankind, there is this struggle to find truth by the use of rigorous logic (what we now call mathematical proof), was perhaps Pythagoras out who discovered it first. This- search is very of Samos. We actually know often coloured by various biases- and obscured by the lack of au Pythagoras very little about Pythagoras, Theorem (PT) - thentic information and scholar and what we do know was ship. The Pythagoras Theorem In right-angled written by historians centu suffers from the same fate. In this triangles, the square ries after he died. article I hope to give a summary on the side Legend has it that Pythagoras of current understanding and, not subtending the was born around 572 B.C. on being an expert historian, would right angle is equal to the island of Samos on the like to state right at the beginning the sum of the squares Aegean Sea. He was perhaps that I might have left out some - on the sides containing a student of Thales, a famous major contribution. the right angle. Greek philosopher and math Before proceeding further let us, ematician who was born half a century before Pythagoras. at the cost of redundancy,Elements recall the Pythagoras Theorem as stated It is believed that Pythagoras by Euclid (I.47 of the ) travelled to Egypt, Babylon- and refer to it from now on as PT and even to India before he (P here can stand for Pythagoras or ‘Preeminent’!). returned to Croton, a Greek settlement in south east Italy. Here he seems to have gathered a group Obviously the first challenge for historians is the of followers forming what we call the Pythagorean name. sect, with beliefs and practices as described in - the introduction. It is believed that many Greek philosophers (Plato, for instance) were deeply in fluenced by Pythagoras, so much so that Bertrand Russell felt that he should be considered one of the most influential Western philosophers. We will return to the Pythagorean School after we take a detour and look at contributions outside the Greek world (this is often difficult for many TheEurocentric problem historians of dating! to swallow!) As students of history we must realise that the greatest challenge historians of antiquity face is that of giving accurate dates to events. There are many reasons for this, including the fact that many- cultures were oral, records of events were burnt, languages of some cultures have yet to be deci phered and very often, as mentioned earlier, our only knowledge about people and events are from historians referring to them many years later. So Pythogoras (Approx 572 BC to 475 BC) I have tried to use a very conservative and broad timeline. 6 At Right Angles | Vol. 1, No. 1, June 2012 Making sense of the history Timeline of the History of the Pythogoras Theorem (PT) Why Pythagoras? Greek scholars seem to be in agreement that the first person to clearly state PT Often in the history of ideas, especially when there Sulbasutras has been a discovery which has had a significant in all its generality, and attempt to establish its Pythagorean triples. truth by the use of rigorous logic (what we now Statement of PT for a square. influence on mankind, there is this struggle to find General statement of PT. call mathematical proof), was perhaps Pythagoras out who discovered it first. This- search is very Approximation of 2. often coloured by various biases of Samos. We actually know Use of PT for geometrical constructions. Chou Pei Suan Ching - Pythagoras Statement of PT. and obscured by the lack of au very little about Pythagoras,- Plimpton 322 Diagrammatic proof Theorem (PT) for (3,4,5) triangle. thentic information and scholar and what we do know was Table of Pythagorean triples. Pythagoras ship. The Pythagoras Theorem In right-angled written by historians centu of Samos. suffers from the same fate. In this triangles, the square ries after he died. article I hope to give a summary on the side Legend has it that Pythagoras of current understanding and, not subtending the was born around 572 B.C. on being an expert historian, would right angle is equal to the island of Samos on the BC AD like to state right at the beginning the sum of the squares Aegean Sea. He was perhaps that I might have left out some - on the sides containing a student of Thales, a famous 1800 1600 800 600 500 300 200 0 major contribution. the right angle. Greek philosopher and math Before proceeding further let us, ematician who was born half a century before Pythagoras. YBC 7289 at the cost of redundancy,Elements recall Tilted square It is believed that Pythagoras the Pythagoras Theorem as stated with diagonal and Pythagorean School Euclid travelled to Egypt, Babylon approximation Statement and proof of PT. Proof of PT in complete by Euclid (I.47 of the ) - of 2. and even to India before he Use of PT in geometrical constructions. generality based on an and refer to it from now on as PT Number theoretic properties of Pythagorean triples. axiomatic system. (P here can stand for Pythagoras or ‘Preeminent’!). returned to Croton, a Greek settlement in south Discovery of irrationals and proof that 2 is irrational. Proof of the converse. east Italy. Here he seems to have gathered a group Obviously the first challenge for historians is the of followers forming what we call the Pythagorean Mesopotamia India Greece China name. sect, with beliefs and practices as described in - the introduction. It is believed that many Greek philosophers (Plato, for instance) were deeply in fluenced by Pythagoras, so much so that Bertrand Russell felt that he should be considered one of the most influential Western philosophers. Let us now try and understand the history of PT with wedge shaped markings on them. Historians We will return to the Pythagorean School after we Thechronologically. Mesopotamian contribution date these to the period of Hammurabi between take a detour and look at contributions outside 1800 BC and 1600 BC. the Greek world (this is often difficult for many You may recall from your school history that one The tablet known as YBC 7289 (tablet number TheEurocentric problem historians of dating! to swallow!) of the oldest known civilizations (Mesopotamia 7289 from the Yale Babylonian Collection) shows or Babylonia) existed in the geographical region a tilted30 2square with wedged shaped markings. The markings show calculations for the approximation As students of history we must realise that the between the Tigris and Euphrates rivers. Records of . This would not have been possible greatest challenge historians of antiquity face is of this civilization date back to 3500 BC. They that of giving accurate dates to events. There are used the sexagesimal system (base 60) and used mathematics for record keeping and astronomy. many reasons for this, including the fact that many- cultures were oral, records of events were burnt, They also seemed interested in number theory languages of some cultures have yet to be deci and geometry. phered and very often, as mentioned earlier, our We know this because they left records of their only knowledge about people and events are from work on thousands of clay tablets, five hundred - historians referring to them many years later. So Pythogoras (Approx 572 BC to 475 BC) of which seem mathematical in nature. There are Fig. 1, I have tried to use a very conservative and broad two main sources that tell us about the Mesopota YBC 7289 timeline. mian contribution to the PT. These are clay tablets 6 At Right Angles | Vol. 1, No. 1, June 2012 Vol. 1, No. 1, June 2012 | At Right Angles 7 Let us look at the entries in the - tablet. The tablet contains errors in Rows 2, 9, 13 and 15 and the origi nal entries were in bases 60, but the table below is in base 10 with the errorsd corrected.
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