History of the theorem of Pythagoras. Proof of the theorem

The history of the Pythagorean theorem is several thousand years old. 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 given by Pythagoras. However, some researchers refute this fact.

Music and Logic

Before we describe how the history of Pythagoras' theorem evolved, let us briefly discuss 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. In 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.

30-year training

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 at 56 years old. Here at that time the tyrant Polycrat rules. Pythagoras could not 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.

Pythagoreans

The history of Pythagoras' theorem is connected with the development of the school created by the Greek 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 the discoveries of the disciples of the Greek mathematician were 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 the hypotenuse, not knowing about the achievements of other peoples.

Pythagoras' theorem: the history of discovery

Some ancient Greek sources describe 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 of the "Principle", created by Euclid, 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 Neoplatonic philosopher Prokl pointed out that the author of the proof given in the "Beginnings" is Euclid himself.

Whatever it was, but the first person who formulated the theorem, after all, was not Pythagoras.

Ancient Egypt and Babylon

The theorem of Pythagoras, whose history of creation is 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 3 ² + 4 ^² = 5 ^² . 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 Pythagoras' theorem is also connected with the ancient civilizations of India and China. The treatise "Chou-bi suan jin" 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 described in the Indian treatise "Sulva Sutra," dated VII-V 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.

Evidence

During its existence, the theorem has become one of the fundamental in geometry. The history of the proof of Pythagoras' 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 form two such triangles. A simple graphic representation clearly demonstrates the validity of the statement formulated in the form of the famous theorem.

Another simple proof combines geometry with algebra. Four identical rectangular triangles with sides a, b, c are drawn so that they form two squares: the outer with the side (a + b) and the inner with the side c. In this case, the area of the smaller square will be c ² . 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, with ² + 4 * ((a *) / 2), which is c ² + 2ав. 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 a ² + 2ab + to ² . It turns out:

And ² + 2ав + in ² = с ² + 2ав,

And ² + in ² = c ² .

There are many variants of the proof of this 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 did not require.

The history of the Pythagorean theorem, summarized in the article, debunks the myth of its origin. However, it is difficult to even imagine that the name of the great Greek mathematician and philosopher will someday cease to be associated with it.