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Authors: Zvi Artstein
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justified. The requirement for absolute proof would be an encumbrance in the evolutionary struggle. There was a reason that throughout thousands of years of the development of mathematics before Thales, mathematicians did not try to prove propositions they were convinced were correct. However, after Thales had taken the first step and Greek mathematicians in subsequent generations had followed his path, the concept of proof became a cornerstone of mathematics.
Other crucial milestones in the formulation of the new concepts in mathematics were the contributions by Pythagoras and his school. Pythagoras came from the island Samos, not far from the coast of Italy. According to the tradition, he was born, so it is believed, in 572 BCE, and it is generally thought that he was a pupil of Thales in Miletus. He went to study in Egypt, and when he returned to Samos he found a regime of tyrants and left for the town of Cortona in Italy, which was then under Greek rule, and founded the Order of Pythagoras. All sorts of mysteries have been attributed to that order, and it is difficult to separate truth from myth. It was involved in local politics in Cortona and considered itself part of the elite or upper stratum. It thus came into conflict with the democratic regime that came to power in the town, and, according to popular history, Pythagoras himself was murderedin 497 BCE. The members of the order dispersed, joining various seminaries in Greece, but they continued with their mathematical activities according to the Pythagorean tradition for about two hundred years. They customarily attributed every important theory or mathematical result to the founder of the order, so it is not clear what Pythagoras's own contributions were and what should be attributed to his followers.
One of Pythagoras's best-known contributions to mathematics is the theorem named after him: in a right-angled triangle, the sum of the squares of the sides equals the square of the hypotenuse. This is one of the most famous mathematical theorems, and until now, hundreds of different proofs of it have been published. Beyond the discovery of the general property in the theorem, Pythagoras's main contribution in this case was his search for a general property. As we saw above, the Babylonian's knew about Pythagorean triangles, that is, triangles with sides whose lengths were natural numbers that satisfied the theorem, and they made a list of such. The Chinese left written instructions on how to calculate the length of a side of a triangle if the lengths of the other two sides are known, and they gave many numerical examples and illustrations of various such triangles. Calculations left by the Egyptians show that they too knew of the relation of the lengths of the sides of a right triangle in many examples of specific triangles. It did not occur to any of them to even ask whether the property applied to all right triangles or to prove Pythagoras's theorem even for those triangles for which they had calculated the figures. They knew about the connection between the lengths of the sides, but they used it only in the context of specific calculations.
Moreover, Pythagoreans (and possibly Pythagoras himself) did not merely prove the relation between the sides of the triangle but looked for,and found, a formula according to which all Pythagorean triangles can be calculated.
The formula is (in the notation of today): for all two natural numbers u and v , such that u is bigger than v , define
A = 2 uv , B = u 2 – v 2 , C = u 2 + v 2 .
A simple calculation shows that A 2 + B 2 = C 2 , or, in other words, A , B , and C are the sides of a Pythagorean triangle. The Pythagoreans proved that all Pythagorean triangles are obtained in this manner (the claim that these constitute all the triangles appeared in Euclid's book, but without a proof).
Note the conceptual leap. The Babylonians and the Chinese compiled lists of many Pythagorean triangles; the Greeks found a proof that
Other crucial milestones in the formulation of the new concepts in mathematics were the contributions by Pythagoras and his school. Pythagoras came from the island Samos, not far from the coast of Italy. According to the tradition, he was born, so it is believed, in 572 BCE, and it is generally thought that he was a pupil of Thales in Miletus. He went to study in Egypt, and when he returned to Samos he found a regime of tyrants and left for the town of Cortona in Italy, which was then under Greek rule, and founded the Order of Pythagoras. All sorts of mysteries have been attributed to that order, and it is difficult to separate truth from myth. It was involved in local politics in Cortona and considered itself part of the elite or upper stratum. It thus came into conflict with the democratic regime that came to power in the town, and, according to popular history, Pythagoras himself was murderedin 497 BCE. The members of the order dispersed, joining various seminaries in Greece, but they continued with their mathematical activities according to the Pythagorean tradition for about two hundred years. They customarily attributed every important theory or mathematical result to the founder of the order, so it is not clear what Pythagoras's own contributions were and what should be attributed to his followers.
One of Pythagoras's best-known contributions to mathematics is the theorem named after him: in a right-angled triangle, the sum of the squares of the sides equals the square of the hypotenuse. This is one of the most famous mathematical theorems, and until now, hundreds of different proofs of it have been published. Beyond the discovery of the general property in the theorem, Pythagoras's main contribution in this case was his search for a general property. As we saw above, the Babylonian's knew about Pythagorean triangles, that is, triangles with sides whose lengths were natural numbers that satisfied the theorem, and they made a list of such. The Chinese left written instructions on how to calculate the length of a side of a triangle if the lengths of the other two sides are known, and they gave many numerical examples and illustrations of various such triangles. Calculations left by the Egyptians show that they too knew of the relation of the lengths of the sides of a right triangle in many examples of specific triangles. It did not occur to any of them to even ask whether the property applied to all right triangles or to prove Pythagoras's theorem even for those triangles for which they had calculated the figures. They knew about the connection between the lengths of the sides, but they used it only in the context of specific calculations.
Moreover, Pythagoreans (and possibly Pythagoras himself) did not merely prove the relation between the sides of the triangle but looked for,and found, a formula according to which all Pythagorean triangles can be calculated.
The formula is (in the notation of today): for all two natural numbers u and v , such that u is bigger than v , define
A = 2 uv , B = u 2 – v 2 , C = u 2 + v 2 .
A simple calculation shows that A 2 + B 2 = C 2 , or, in other words, A , B , and C are the sides of a Pythagorean triangle. The Pythagoreans proved that all Pythagorean triangles are obtained in this manner (the claim that these constitute all the triangles appeared in Euclid's book, but without a proof).
Note the conceptual leap. The Babylonians and the Chinese compiled lists of many Pythagorean triangles; the Greeks found a proof that
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