History of Pythagoras' theorem. Proof of the theorem
The history of the theorem of Pythagoras has severalmillennia. The statement that the square of the hypotenuse is equal to the sum of the squares of the legs is known long before the birth of the Greek mathematician. However, the theorem of Pythagoras, the history of creation and proof of it is connected for the majority with this scientist. According to some sources, the reason for this was the first proof of the theorem, which was cited by Pythagoras. However, some researchers deny this fact.
Music and Logic
Before I tell you how history developedPythagoras' theorem, let us briefly consider the biography of a mathematician. He lived in the VI century BC. The date of birth of Pythagoras is 570 BC. e., the place - the island of Samos. Little is known about the life of the scientist. Biographical data in ancient Greek sources are intertwined with obvious fiction. On the pages of the treatises, he appears as a great sage, who has a great command of the word and the ability to persuade. By the way, that is why the Greek mathematician was nicknamed Pythagoras, that is, "persuasive speech". According to another version, the birth of the future sage was predicted by Pythia. The father in her honor called the boy Pythagoras.
The sage learned from the great minds of the time. Among the teachers of the young Pythagoras appear Germomant and Pherekyd Syrosky. The first instilled in him a love of music, the second taught philosophy. Both these sciences will remain in the center of attention of the scientist throughout his life.
According to one of the versions, being an inquisitive young man,Pythagoras left his homeland. He went to seek knowledge in Egypt, where he stayed, according to various sources, from 11 to 22 years, and then was taken prisoner and was sent to Babylon. Pythagoras was able to benefit from his position. For 12 years he studied mathematics, geometry and magic in the ancient state. Samos Pythagoras returned only in 56 years. Here at that time the tyrant Polycrat rules. Pythagoras was unable to accept such a political system and soon went to the south of Italy, where the Greek colony Croton was located.
Today it is impossible to say exactly whether Pythagoras was in Egypt or Babylon. Perhaps he left Samos later and went straight to Croton.
The history of Pythagoras' theorem is connected with the developmentcreated by the Greek school philosopher. This religious-ethical brotherhood preached adherence to a special way of life, studied arithmetic, geometry and astronomy, and studied the philosophical and mystical side of numbers.
All discoveries of the disciples of the Greek mathematicianwere attributed to him. However, the history of the origin of Pythagoras' theorem is connected by the ancient biographers only with the philosopher himself. It is assumed that he gave the Greeks the knowledge obtained in Babylon and Egypt. There is also a version that he did discover the theorem on the ratios of the legs and hypotenuses, not knowing about the achievements of other peoples.
Pythagoras' theorem: the history of discovery
In some ancient Greek sourcesdescribes the joy of Pythagoras when he managed to prove the theorem. In honor of this event, he ordered to sacrifice the gods in the form of hundreds of bulls and arranged a feast. Some scholars, however, point out the impossibility of such an act because of the peculiarities of the views of the Pythagoreans.
It is believed that in the treatise "The Beginnings" created byEuclid, the author gives a proof of the theorem, the author of which was the great Greek mathematician. However, not everyone supported this point of view. So, even the ancient Neo-Platonist philosopher Prokl pointed out that the author of the proof given in the "Beginnings" is Euclid himself.
Be that as it may, the first person who formulated the theorem was still not Pythagoras.
Ancient Egypt and Babylon
The theorem of Pythagoras, whose historyis considered in the article, according to the German mathematician Cantor, was known as early as 2300 BC. e. in Egypt. The ancient inhabitants of the Nile Valley during the reign of Pharaoh Amenemhet I knew equality 32+ 4²= 5². It is assumed that with the help of triangles with sides 3, 4 and 5, the Egyptian "rope pullers" built right angles.
We knew the theorem of Pythagoras in Babylon. On clay tablets dating from 2000 BC. and attributed to the time of the reign of King Hammurabi, an approximate calculation of the hypotenuse of a right-angled triangle was found.
India and China
The history of the theorem of Pythagoras is connected with the ancientthe civilizations of India and China. The treatise "Chou-bi suan jing" contains indications that the Egyptian triangle (its sides are correlated as 3: 4: 5) was known in China as far back as the 12th century. BC. e., and to the VI century. BC. e. The mathematicians of this state knew the general form of the theorem.
The construction of a right angle with the aid of the Egyptian triangle was also expounded in the Indian treatise "Sulva Sutra", dating from the 7th-5th centuries. BC. e.
Thus, the history of the Pythagorean theorem at the time of the birth of the Greek mathematician and philosopher had already been counted for several hundred years.
During its existence, the theorem has become onefrom the fundamental in geometry. The history of the proof of the Pythagorean theorem probably began with an equilateral rectangular triangle. On its hypotenuse and its legs are built squares. The one that "grew" on the hypotenuse will consist of four triangles equal to the first. The squares on the legs are thus composed of two such triangles. A simple graphical representation clearly demonstrates the validity of the statement formulated in the form of the famous theorem.
Another simple proof combinesgeometry with algebra. Four identical rectangular triangles with sides a, b, c are drawn so that they form two squares: outer with side (a + b) and inner side with side c. In this case, the area of the smaller square will be c2. The area of the large is calculated from the sum of the squares of the small square and all the triangles (the area of the rectangular triangle, recall, is calculated from the formula (a * b) / 2), that is,2+ 4 * ((a * b) / 2), which is equal to c2+ 2aв. The area of a large square can be calculated and otherwise - as a product of two sides, that is (a + b)2, which is equal to a2+ 2ав + в2. It turns out:
a2+ 2ав + в2= s2+ 2ав,
a2+ in2= s2.
There are many variants of the proof of thisof the theorem. Euclid, Indian scientists, and Leonardo da Vinci worked on them. Often, the ancient sages brought up the drawings, examples of which are located above, and did not accompany them with any explanations, except for the "Look!" Note. Simplicity of the geometric proof, provided some knowledge of the comments was not required.
The history of the Pythagorean theorem, summarized inarticle, debunks the myth of its origin. However, it is difficult even to imagine that the name of the great Greek mathematician and philosopher will someday cease to be associated with it.