An area interpretation of this statement is shown in Figure 5. The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. Ancient Egyptians arrow 4, in Figure 2 , concentrated along the middle to lower reaches of the Nile River arrow 5, in Figure 2 , were a people in Northeastern Africa.
The ancient civilization of the Egyptians thrived miles to the southwest of Mesopotamia. The two nations coexisted in relative peace for over years, from circa BCE to the time of the Greeks. As to the claim that the Egyptians knew and used the Pythagorean Theorem in building the great pyramids, there is no evidence to support this claim. Egypt has over pyramids, most built as tombs for their country's Pharaohs. Egypt arrow 4, in Figure 2 and its pyramids are as immortally linked to King Tut as are Pythagoras and his famous theorem.
King Tut ruled from the age of 8 for 9 years, — BC. He was born in BC and died some believe he was murdered in BC at the age of Elisha Scott Loomis — Figure 7 , an eccentric mathematics teacher from Ohio, spent a lifetime collecting all known proofs of the Pythagorean Theorem and writing them up in The Pythagorean Proposition , a compendium of proofs.
The manuscript was prepared in and published in Loomis received literally hundreds of new proofs from after his book was released up until his death, but he could not keep up with his compendium. As for the exact number of proofs, no one is sure how many there are.
Surprisingly, geometricians often find it quite difficult to determine whether certain proofs are in fact distinct proofs. He died on 11 December , and the obituary was published as he had written it, except for the date of his death and the addresses of some of his survivors. According to his autobiography, a preteen Albert Einstein Figure 8. Many known proofs use similarity arguments, but this one is notable for its elegance, simplicity and the sense that it reveals the connection between length and area that is at the heart of the theorem.
At the age of 12, I experienced a second wonder of a totally different nature: in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a school year. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question.
This lucidity and certainty made an indescribable impression upon me. For example I remember that an uncle told me the Pythagorean Theorem before the holy geometry booklet had come into my hands.
Einstein Figure 9 used the Pythagorean Theorem in the Special Theory of Relativity in a four-dimensional form , and in a vastly expanded form in the General Theory of Relatively. The following excerpts are worthy of inclusion. Special relativity is still based directly on an empirical law, that of the constancy of the velocity of light. The fact that such a metric is called Euclidean is connected with the following.
The postulation of such a metric in a three-dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean Geometry. The defining equation of the metric is then nothing but the Pythagorean Theorem applied to the differentials of the co-ordinates. Such transformations are called Lorentz transformations. From the latest results of the theory of relativity, it is probable that our three-dimensional space is also approximately spherical , that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry.
According to the general theory of relativity , the geometrical properties of space are not independent, but they are determined by matter. I wished to show that space time is not necessarily something to which one can ascribe to a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. The above excerpts — from the genius himself — precede any other person's narrative of the Theory of Relativity and the Pythagorean Theorem.
Accordingly, I now provide a less demanding excerpt, albeit one that addresses the effects of the Special and General theories of relativity.
The system of units in which the speed of light c is the unit of velocity allows to cast all formulas in a very simple form. The Pythagorean Theorem graphically relates energy, momentum and mass. Euclid of Alexandria was a Greek mathematician Figure 10 , and is often referred to as the Father of Geometry. The date and place of Euclid's birth, and the date and circumstances of his death, are unknown, but it is thought that he lived circa BCE.
His work Elements , which includes books and propositions, is the most successful textbook in the history of mathematics. In it, the principles of what is now called Euclidean Geometry were deduced from a small set of axioms. When Euclid wrote his Elements around BCE , he gave two proofs of the Pythagorean Theorem: The first, Proposition 47 of Book I, relies entirely on the area relations and is quite sophisticated; the second, Proposition 31 of Book VI, is based on the concept of proportion and is much simpler.
He may have used Book VI Proposition 31, but, if so, his proof was deficient, because the complete theory of Proportions was only developed by Eudoxus, who lived almost two centuries after Pythagoras. Euclid's Elements furnishes the first and, later, the standard reference in geometry.
It is a mathematical and geometric treatise consisting of 13 books. It comprises a collection of definitions, postulates axioms , propositions theorems and constructions and mathematical proofs of the propositions.
Euclid provided two very different proofs, stated below, of the Pythagorean Theorem. This is probably the most famous of all the proofs of the Pythagorean proposition.
In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle. In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.
Euclid I 47 is often called the Pythagorean Theorem , called so by Proclus — a Greek philosopher who became head of Plato's Academy and is important mathematically for his commentaries on the work of other mathematicians — and others centuries after Pythagoras and even centuries after Euclid. Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making them easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics twenty-three centuries later.
Although best known for its geometric results, Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers and the Euclidean algorithm for finding the greatest common divisor of two numbers.
Three biographies of Pythagoras have survived from antiquity, but these are all extremely late. The later lives of Pythagoras, written by the Neoplatonist philosophers Porphyrios and Iamblichos respectively, are even longer and even more inundated with legends. The philosopher Herakleitos of Ephesos lived c. The poets Ion of Chios lived c. Our later sources are a bit thornier, so we need to be somewhat more skeptical.
In the late fourth century BC, the Peripatetic philosophers Dikaiarchos, Aristoxenos, and Herakleides Pontikos all wrote accounts of the lives of Pythagoras, but none of these have survived. Pythagoras may have travelled to other lands as a young man, but the extent of his travels is unclear. In Kroton, Pythagoras established a commune in which initiates swore oaths of solemn fealty and were bound by secrecy.
In BC, Kroton won a massive military victory over the neighboring city of Sybaris and several Pythagoreans, including Milon of Kroton, were apparently major generals in the battle. The victory instigated a proposal for a new, democratic constitution—a proposal which the Pythagoreans opposed.
Two supporters of democracy named Kylon and Ninon rallied the people of Kroton against the Pythagoreans and, while the Pythagoreans were gathered in one of their meeting houses, Kylon, Ninon, and their mob of supporters, set fire to the house and murdered the Pythagoreans as they attempted to escape from the burning building.
Pythagoras himself may have been killed in this purge or he may have escaped to Metaponton, where he may have survived for several more years before eventually dying. This applies both to humans and to animals and, in Pythagorean cosmology, it was possible for a human to be reincarnated as an animal or an animal to be reincarnated as a human. Hicks :. Here Pythagoras intervenes to save a dog who is being beaten by his master, saying that the dog was a friend of his in a past life and that he recognized him by the sound of his voice.
This is obviously intended satirically, but it clearly shows that Pythagoras was associated with metempsychosis from an early date. Xenophanes lived the later part of his life in Elea in southern Italy, not far from Kroton, where Pythagoras did most of his teaching, and may have either met Pythagoras himself in his old age or known people who had known him.
Pythagoras was not the first person to teach the doctrine of metempsychosis ; certainly at least Pherekydes of Syros lived c. Nonetheless, we can be sure that Pythagoras taught this idea and, in later times, Pythagoras was the one most closely associated with it. Eudoxos of Knidos takes this even further, stating that Pythagoras not only forbade eating meat, but also refused to even go anywhere in the presence of hunters or butchers.
Earlier sources, however, contradict the idea that Pythagoras forbade meat, with many of them stating that he only forbade certain kinds of meat. It is therefore likely that Pythagoras did not issue a wholesale prohibition against all meat, but rather specifically warned against particular varieties. We hope you find the article on the Origins of Pythagoras theorem helpful. If you have any doubt regarding this article, kindly drop your comments below and we will get back to you at the earliest.
Support: support embibe. General: info embibe. Exams Engineering Origins of Pythagoras theorem. How was this theorem discovered? Was Pythagoras theorem used in ancient India? Pyramids and construction: Apart from India, the Chinese and the Egyptians also used this theorem in construction. About Pythagoras and the actual truth behind the Pythagoras theorem: Pythagoras was born in around BC, in an island called Samos in Greece.
Is Pythagoras theorem only used for right triangles? A French archeological expedition first excavated the tablet, which dates to between and B. C in what is now Iraq in , and it is currently housed in the Istanbul Archeological Museum. But it is only just now that researchers have discovered the significance of its ancient markings.
Related: The 11 most beautiful mathematical equations. According to Mansfield, Si. The tablet details a marshy field with various structures, including a tower, built upon it.
0コメント