Historical studies date the birth of Pythagoras to approximately 580 BC. The happy father Mnesarchus surrounds the boy with care. He had the opportunity to give his son a good upbringing and education.

The future great mathematician and philosopher already in childhood showed great abilities for science. From his first teacher Hermodamas, Pythagoras learned the basics of music and painting. To exercise his memory, Hermodamas forced him to learn songs from the Odyssey and Iliad. The first teacher instilled in young Pythagoras a love of nature and its secrets.

Several years have passed, and on the advice of his teacher, Pythagoras decides to continue his education in Egypt. With the help of his teacher, Pythagoras manages to leave the island of Samos. But it’s still a long way from Egypt. He lives on the island of Lesbos with his relative Zoil. There Pythagoras meets the philosopher Pherecydes, a friend of Thales of Miletus. From Pherecydes, Pythagoras studied astrology, eclipse prediction, the secrets of numbers, medicine and other sciences required for that time.

Then in Miletus he listens to lectures by Thales and his younger colleague and student Anaximander, an outstanding geographer and astronomer. Pythagoras acquired a lot of important knowledge during his stay at the Milesian school.

Before Egypt, he stops for some time in Phenicia, where, according to legend, he studies with the famous Sidonian priests.

Pythagoras' studies in Egypt contributed to his becoming one of the most educated people of his time. Here Pythagoras falls into Persian captivity.

According to ancient legends, while in captivity in Babylon, Pythagoras met with Persian magicians, became familiar with eastern astrology and mysticism, and became acquainted with the teachings of the Chaldean sages. The Chaldeans introduced Pythagoras to the knowledge accumulated by the eastern peoples over many centuries: astronomy and astrology, medicine and arithmetic.

Pythagoras spent twelve years in Babylonian captivity until he was freed by the Persian king Darius Hystaspes, who had heard about the famous Greek. Pythagoras is already sixty, he decides to return to his homeland in order to introduce his people to the accumulated knowledge.

Since Pythagoras left Greece, great changes have occurred there. The best minds, fleeing the Persian yoke, moved to Southern Italy, which was then called Magna Graecia, and founded the colony cities of Syracuse, Agrigentum, and Croton there. This is where Pythagoras decided to create his own philosophical school.

Quite quickly it gains great popularity among residents. Pythagoras skillfully uses the knowledge gained from traveling around the world. Over time, the scientist stops performing in churches and on the streets. Already in his home, Pythagoras taught medicine, the principles of political activity, astronomy, mathematics, music, ethics and much more. Outstanding political and statesmen, historians, mathematicians and astronomers came from his school. He was not only a teacher, but also a researcher. His students also became researchers. Pythagoras developed the theory of music and acoustics, creating the famous “Pythagorean scale” and conducting fundamental experiments on the study of musical tones: he expressed the relationships he found in the language of mathematics. The School of Pythagoras first suggested the sphericity of the Earth. The idea that the movement of celestial bodies obeys certain mathematical relationships, the ideas of “harmony of the world” and “music of the spheres,” which later led to a revolution in astronomy, first appeared in the School of Pythagoras.

The scientist also did a lot in geometry. Proclus assessed the Greek scientist's contribution to geometry as follows: “Pythagoras transformed geometry into the form of a free science, considering its principles in a purely abstract manner and examining the theorems from an immaterial, intellectual point of view. It was he who discovered the theory of irrational quantities and the design of cosmic bodies.”

In the school of Pythagoras, geometry was for the first time formalized into an independent scientific discipline. It was Pythagoras and his students who were the first to study geometry systematically - as a theoretical doctrine about the properties of abstract geometric figures, and not as a collection of applied recipes for land surveying.

The most important scientific merit of Pythagoras is considered to be the systematic introduction of proof into mathematics, and, above all, into geometry. Strictly speaking, only from this moment does mathematics begin to exist as a science, and not as a collection of ancient Egyptian and ancient Babylonian practical recipes. With the birth of mathematics, science in general was born, for “no human research can be called true science if it has not gone through mathematical proof” (Leonardo da Vinci).

So, the merit of Pythagoras was that he, apparently, was the first to come to the following thought: in geometry, firstly, abstract ideal objects should be considered, and, secondly, the properties of these ideal objects should not be established with using measurements on a finite number of objects, but using reasoning that is valid for an infinite number of objects. This chain of reasoning, which, using the laws of logic, reduces non-obvious statements to known or obvious truths, is a mathematical proof.

The discovery of the theorem by Pythagoras is surrounded by an aura of beautiful legends. Proclus, commenting on the last sentence of book 1 of Euclid’s Elements, writes: “If you listen to those who like to repeat ancient legends, you will have to say that this theorem goes back to Pythagoras; they say that in honor of this discovery he sacrificed a bull.” However, more generous storytellers turned one bull into one hecatomb, and this is already a whole hundred. And although Cicero noted that any shedding of blood was alien to the charter of the Pythagorean order, this legend firmly merged with the Pythagorean theorem and, two thousand years later, continued to evoke ardent responses.

Mikhail Lomonosov wrote about this: “Pythagoras sacrificed a hundred oxen to Zeus for the invention of one geometric rule. But if one were to follow the rules found in modern times from witty mathematicians out of his superstitious jealousy, then it would be unlikely that so many cattle would be found in the whole world.”

A.V. Voloshinov in his book about Pythagoras notes: “And although today the Pythagorean theorem is found in various particular problems and drawings: both in the Egyptian triangle in papyrus from the time of Pharaoh Amenemhet I (about 2000 BC), and in Babylonian cuneiform tablets from the era of King Hammurabi ( XVIII century BC), and in the ancient Chinese treatise “Zhou-bi suan jin” (“Mathematical treatise on the gnomon”), the time of creation of which is not precisely known, but where it is stated that in the 12th century BC the Chinese knew the properties of the Egyptian triangle, and by the 6th century BC - both the general form of the theorem, and in the ancient Indian geometric-theological treatise of the 7th-5th centuries BC “Sulva Sutra” (“Rules of the Rope”) - despite all this, the name of Pythagoras is so firmly established fused with the Pythagorean theorem that now it is simply impossible to imagine that this phrase will fall apart. The same applies to the legend of the slaughter of bulls by Pythagoras. And there is hardly any need to dissect beautiful ancient legends with a historical and mathematical scalpel.

Today it is generally accepted that Pythagoras gave the first proof of the theorem that bears his name. Alas, no traces of this evidence have survived either. Therefore, we have no choice but to consider some classical proofs of the Pythagorean theorem, known from ancient treatises. It is also useful to do this because modern school textbooks give an algebraic proof of the theorem. At the same time, the pristine geometric aura of the theorem disappears without a trace, the thread of Ariadne that led the ancient sages to the truth is lost, and this path almost always turned out to be the shortest and always beautiful.”

The Pythagorean theorem states: “A square constructed on the hypotenuse of a right triangle is equal to the sum of the squares constructed on its legs.” The simplest proof of the theorem is obtained in the simplest case of an isosceles right triangle. This is probably where the theorem began. In fact, it is enough just to look at the mosaic of isosceles right triangles to be convinced of the validity of the theorem.

In the 2nd century BC, paper was invented in China and at the same time the creation of ancient books began. This is how “Mathematics in Nine Books” arose - the most important of the surviving mathematical and astronomical works. Book IX of Mathematics contains a drawing proving the Pythagorean theorem. The key to this proof is not difficult to find. In fact, in an ancient Chinese drawing, four equal right-angled triangles with legs and hypotenuse C are laid out so that their outer contour forms a square with side A+B, and the inner contour forms a square with side C, built on the hypotenuse. If you cut out a square with side c and put the remaining 4 shaded triangles into two rectangles, then it is clear that the resulting void, on the one hand, is equal to C squared, and on the other - A + B, i.e. C=D+B. The theorem has been proven.

Mathematicians of Ancient India noticed that to prove the Pythagorean theorem it is enough to use the inside of an ancient Chinese drawing. In the treatise “Siddhanta Shiromani” (“Crown of Knowledge”), written on palm leaves, by the greatest Indian mathematician of the 12th century, Bhaskar contains a drawing with the word “look!”, characteristic of Indian proofs. Right triangles are laid here with the hypotenuse outward and square C is transferred to the “bride’s chair” square A plus square B. Particular cases of the Pythagorean theorem are found in the ancient Indian treatise “Sulva Sutra” (VII-V centuries BC).

Euclid's proof is given in Proposition 1 of the book Elements. Here, for the proof, corresponding squares are constructed on the hypotenuse and legs of a right triangle.

“The 10th-century Baghdad mathematician and astronomer an-Nairizy (Latinized name - Annaricius), writes Voloshinov, - in an Arabic commentary to Euclid’s Elements, gave the following proof of the Pythagorean theorem. The square on the hypotenuse is divided into five parts by Annaricius, from which the squares on the sides are made. Of course, the equality of all corresponding parts requires proof, but we leave it to the reader as obvious. It is curious that Annaricius’s proof is the simplest among the huge number of proofs of the Pythagorean theorem using the partitioning method: it involves only 5 parts (or 7 triangles). This is the smallest number of splits possible."

Pythagoras was luckier than other ancient scientists. Dozens of legends and myths, true and fictitious, real and fictitious, have been preserved about him. Much in mathematics is associated with his name, and first of all, of course, the theorem that bears his name. Currently, everyone agrees that this theorem was not discovered by Pythagoras. Its particular cases were known even before him in China, Babylonia, and Egypt. However, some believe that Pythagoras was the first to give a complete proof of this theorem, while others deny him this merit.

But, perhaps, you cannot find any other theorem that deserves so many different comparisons. In France and some regions of Germany in the Middle Ages, for some reason the Pythagorean theorem was called the “bridge of donkeys.” Among the mathematicians of the Arab East, this theorem was called the “bride’s theorem.” The fact is that in some copies of Euclid’s Elements this theorem was called the “nymph’s theorem” for the similarity of the drawing with a bee, a butterfly, which in Greek was called a nymph. But the Greeks used this word to call some other goddesses, as well as young people in general, women and brides. When translating from Greek to Arabic, the translator, without paying attention to the drawing, translated the word “nymph” as “bride” and not “butterfly”. This is how the affectionate name for the famous theorem appeared - “the bride’s theorem.”

They say - this is, of course, just a legend - that when Pythagoras proved his famous theorem, he thanked the gods by sacrificing a hundred bulls to them. This story of sacrifice, reported by Diogenes and Plutarch, is most likely fictitious, for, as is known, Pythagoras was a vegetarian and an implacable opponent of the slaughter and shedding of animal blood.

For us, Pythagoras is a mathematician. In ancient times it was different. Herodotus calls him an “outstanding sophist,” that is, a teacher of wisdom; he also points out that the followers of Pythagoras did not bury their dead in woolen clothes. It's more like religion than mathematics.

For his contemporaries, Pythagoras was primarily a religious prophet, the embodiment of the highest divine wisdom. There were many tales about Pythagoras, such as those that he had a golden thigh, that people saw him at the same time in different places. In some texts he appears as a demigod, as he imagined himself to be - the son of Hermes. Pythagoras believed that there were three types of beings - gods, mere mortals and... "similar to Pythagoras." In literature, the Pythagoreans were most often portrayed as superstitious and very picky vegetarians, but not at all mathematicians. So who was Pythagoras really: a mathematician, philosopher, prophet, saint or charlatan?

So many legends have been created around the personality of Pythagoras that it is difficult to judge what in them is at least partly true and what is fiction.

We do not even know the exact dates of his birth and death: according to some sources, Pythagoras was born around 580 and died in 500 BC. Born on the island of Samos, located off the very coast of Asia Minor, from travelers and ship captains he learned about the near and far wonderful countries of Egypt and Babylonia, the wisdom of whose priests amazed and beckoned the young Pythagoras. Very young, he left his homeland, first sailing to the shores of Egypt, where for 22 years he carefully looked at those around him and listened to the priests. In Egypt, they say, Pythagoras was captured by Cambyses, the Persian conqueror, and he was taken to Babylon. The grandiose panorama of the city, spreading its palaces and high defensive walls along both banks of the Euphrates, delighted and amazed Pythagoras. He quickly masters the complex Babylonian traditions and studies number theory from the Chaldean magicians and priests. And perhaps this is where the numerical mysticism of attributing divine power to numbers, which Pythagoras presented as philosophy, came from. After returning to Samos, he created his own school (it would be better to call it a sect, community), which pursued not only scientific, but also religious, ethical and political goals. The activities of the union were surrounded by mystery, and all scientific discoveries made by the Pythagoreans were attributed to Pythagoras himself.

Pythagoras created his school as an organization with a strictly limited number of students from the aristocracy, and it was not easy to get into it. The applicant had to pass a series of tests; According to some historians, one of these tests was a vow of five years of silence, and all this time those admitted to the school could listen to the teacher’s voice only from behind the curtain, and could only be seen when their “souls were purified by music and the secret harmony of numbers.” Another law of the organization was the keeping of secrets, non-compliance with which was strictly punished - even death. This law had a negative impact because it prevented the teaching from becoming an integral part of the culture.

The Pythagoreans woke up at dawn, sang songs, accompanying themselves on the lyre, then did gymnastics, studied music theory, philosophy, mathematics, astronomy and other sciences. Often classes were held outdoors, in the form of conversations. Among the first students of the school there were several women, including Theano, the wife of Pythagoras.

However, the aristocratic ideology sharply contradicted the ideology of ancient democracy, which prevailed in Samos at that time. The school displeased the inhabitants of the island, and Pythagoras had to leave his homeland. He moved to southern Italy - a colony of Greece - and here, in Croton, he again founded the Pythagorean union, which lasted for about two centuries.

From the very beginning, two different directions were formed in Pythagoras - “asumatics” and “mathematics”. The first direction dealt with ethical and political issues, education and training, the second - mainly with research in the field of geometry. Pythagorean philosophy contained principles, scientific achievements, views on human education, and socio-political ideas. Pythagoras defined number as a principle, giving a scientific object a universal meaning (a technique later used by other philosophies). This admiration for number is explained by the observations that were carried out in the Pythagorean Union on the phenomena of surrounding life, but it was accompanied by mystical fabrications, the beginnings of which were borrowed along with the beginnings of mathematical knowledge from the countries of the Middle East.

While studying harmony, the Pythagoreans came to the conclusion that qualitative differences in sounds are caused by purely quantitative differences in the lengths of strings or flutes. Thus, a harmonic chord when sounding three strings is obtained when the lengths of these strings are compared with the ratio of the numbers 3, 4 and 6. The same ratio was noticed by the Pythagoreans in many other cases. For example, the ratio of the number of faces, vertices and edges of a cube is equal to the ratio of numbers 6:8:12.

The Pythagoreans found the first proof in history that the diagonal of a square and its side are incommensurable. They proved it, they were amazed and... scared. It turns out that there are neither integers nor rational numbers whose square would be equal to, for example, 2. This means that there are some other numbers?! This was so contrary to their teaching, which was based only on rational numbers, that they decided (they swore by their magic number 36!) to keep their discovery secret. According to legend, Pythagoras' disciple Hippasus of Mesapont, who revealed this secret, was “punished” by the gods and died in a shipwreck.

The solution to such a difficult problem as the construction of regular polygons and polyhedra naturally made a strong impression on the people who solved it, and therefore these polyhedra in the school of Pythagoras were given mystical significance - they were considered “cosmic figures”, and each of them was given the name of one of the elements included, according to the Greeks, in the basis of existence: the tetrahedron was called fire, the octahedron - air, the icosahedron - water, the hexahedron - earth and the dodecahedron - the Universe. Of all geometric bodies, the ball was considered the most beautiful. Pythagoras believed that the Earth has a spherical shape and some kind of fire, but not the Sun, is the center of the Universe, around which the Earth rotates in a circle, and the Sun, Moon and planets have their own movement, different from the daily movement of the fixed stars.

Pythagorasism assumes the existence of ten “principles” that give rise to the cosmos: finitude and infinity, unity and plurality, immobility and movement, light and darkness, good and evil, etc. The first of them are positive, the second are negative. Cosmos (a concept introduced by the Pythagoreans) is harmony, tetractys, perfection, order, measure. The Universe, created by number and opposing principles (finitude - infinity), behaves logically, in proportion to necessity and measure.

A special place in the doctrine of Pythagoras was occupied by the doctrine of the soul and proper human behavior. Pythagoras identified three components of the human soul: judgment (nous), reason (phrenes) and passions (thymos). The soul is the unity of these three components, functional harmony, a complex triad. The soul is eternal with reason, and its remaining parts (judgment and passions) are common to people and animals. Pythagoras was a consistent adherent of the doctrine of metempsychosis; he believed that after the death of a person, his soul moves into other creatures, plants, etc., until it again passes to the person, and this, in turn, depends on his earthly deeds. The Pythagoreans saw souls everywhere; it seemed to them that even the air around them was full of souls that sent people dreams, illnesses or health.

In the “rules” of education, based on the idea of the immortality of the soul, the following were mandatory: worship of the gods, honoring parents, fostering friendship, courage, and respect for elders. The Golden Verses and Symbols are attributed to Pythagoras. Here are some sayings from them:

Do only what will not upset you later and will not force you to repent.

Never do something you don't know. But learn everything there is to know, and then you will lead a quiet life.

Don't neglect the health of your body. Give it the food and drink it needs on time and the exercise it needs.

Learn to live simply and without luxury.

Don’t close your eyes when you want to sleep without analyzing all your actions of the past day.

Do not pass by the scales (that is, do not violate justice).

Don't sit on the cushion (that is, don't rest on your laurels).

Don't gnaw at your heart (that is, don't indulge in melancholy).

Do not correct the fire with the sword (that is, do not irritate those who are already angry).

Do not accept swallows (that is, talkers and frivolous people) under your roof.

Thus, Pythagorasism is a kind of mixture of scientific and magical, rational and mystical.

However, the ideology that underlay the activities of the union steadily led it to death. The union consisted mainly of representatives of the aristocracy, in whose hands the government of the city of Croton was concentrated, and this had a great influence on politics. Meanwhile, democratic governance was being introduced in Athens and most of the Greek colonies, attracting an increasing number of supporters. Democratic currents also became dominant in Crotone. Pythagoras and his supporters were forced to flee from there. But this no longer saved him. While in the city of Metaponte, he, an eighty-year-old man, died in a skirmish with his opponents. His rich experience in fist fighting and the title of the first Olympic champion in this sport, won by Pythagoras in his youth, did not help him.

The fate of Pythagoras himself and his union had a sad end, but Pythagoras with its metaphysics, scientific knowledge, and views on education continued to influence the further development of science and philosophy. Undoubtedly, the school of Pythagoras played a big role in improving scientific methods for solving mathematical problems: the principle of the need for rigorous proofs was introduced into the mathematics of the firmament, which gave it the significance of a special science.

A crater on the visible side of the Moon is named after Pythagoras.

Biography of Pythagoras

Pythagoras of Samos (c. 580 - c. 500 BC) Ancient Greek mathematician and idealist philosopher. Born on the island of Samos. Received a good education. According to legend, Pythagoras, in order to familiarize himself with the wisdom of Eastern scientists, went to Egypt and seemed to live there for 22 years. Having mastered all the Egyptian sciences well, including mathematics, he moved to Babylon, where he lived for 12 years and became acquainted with the scientific knowledge of the Babylonian priests. Traditions attribute Pythagoras to visiting India. This is very likely, since Ionia and India then had trade relations. Returning to his homeland (c. 530 BC), Pythagoras tried to organize his own philosophical school. However, for unknown reasons, he soon left Samos and settled in Crotone (a Greek colony in northern Italy). Here Pythagoras managed to organize his school, which operated for almost thirty years. The school of Pythagoras, or, as it is also called, the Pythagorean Union, was at the same time a philosophical school, a political party, and a religious fraternity. The statute of the Pythagorean League was very harsh. Everyone who joined it renounced personal property in favor of the union, pledged not to shed blood, not to eat meat, and to protect the secret of the teachings of their teacher. School members were prohibited from teaching others for compensation. In his philosophical views, Pythagoras was an idealist, a defender of the interests of the slave-owning aristocracy. Perhaps this was the reason for his departure from Samos, since supporters of democratic views had a very large influence in Ionia. In social matters, by “order” the Pythagoreans understood the dominance of aristocrats. They condemned ancient Greek democracy. Pythagorean philosophy was a primitive attempt to justify the rule of the slave-owning aristocracy. At the end of the 5th century. BC e. A wave of democratic movement swept through Greece and its colonies. Democracy has won in Crotone. Pythagoras, together with his students, leaves Croton and leaves for Tarentum, and then to Metapontum. The arrival of the Pythagoreans in Metapontum coincided with the outbreak of a popular uprising there. In one of the night skirmishes, almost ninety-year-old Pythagoras died. His school ceased to exist. The disciples of Pythagoras, fleeing persecution, settled throughout Greece and its colonies. Earning their livelihood, they organized schools in which they taught mainly arithmetic and geometry. Information about their achievements is contained in the works of later scientists - Plato, Aristotle, etc.

The discovery of the fact that there is no common measure between the side and the diagonal of a square was the greatest achievement of the Pythagoreans. This fact caused the first crisis in the history of mathematics. The Pythagorean doctrine of the integer basis of everything that exists could no longer be accepted as true. Therefore, the Pythagoreans tried to keep their discovery secret and created a legend about the death of Hippasus of Mesopotamia, who dared to divulge the discovery. Pythagoras is credited with a number of other important discoveries at that time, namely: the theorem on the sum of the internal angles of a triangle; the problem of dividing a plane into regular polygons (triangles, squares and hexagons). There is information that Pythagoras built “cosmic” figures, that is, five regular polyhedra. But it is more likely that he knew only three simple regular polyhedra: cube, tetrahedron, octahedron. The Pythagorean school did a lot to give geometry the character of a science. The main feature of the Pythagorean method was the combination of geometry with arithmetic.

Pythagoras dealt a lot with proportions and progressions and, probably, with the similarity of figures, since he is credited with solving the problem: “Given two figures, construct a third, equal in size to one of the given ones and similar to the second.” Pythagoras and his students introduced the concept of polygonal, friendly, perfect numbers and studied their properties. Pythagoras was not interested in arithmetic as a practice of calculation, and he proudly declared that he “put arithmetic above the interests of the merchant.” Pythagoras was one of the first to believe that the Earth is spherical and is the center of the Universe, that the Sun, Moon and planets have their own movement, different from the daily movement of the fixed stars. Nicolaus Copernicus perceived the teaching of the Pythagoreans about the movement of the Earth as the prehistory of his heliocentric teaching. No wonder the church declared the Copernican system a “false Pythagorean doctrine.”

Thoughts and aphorisms

In the field of life, like a sower, walk with an even and constant step.
The true fatherland is where there are good morals.
Do not be a member of a learned society: the wisest, when they form a society, become commoners.
Consider numbers, weight and measure sacred, as children of graceful equality.
Measure your desires, weigh your thoughts, count your words.
Do not be surprised at anything: the gods were surprised.
If they ask: what is more ancient than the gods? - answer: fear and hope.

The truth about Pythagoras

The most that the population now knows about this respected ancient Greek fits into one phrase: “Pythagorean pants are equal on all sides.” The authors of this tease are clearly separated by centuries from Pythagoras, otherwise they would not have dared to tease. Because Pythagoras is not at all the square of the hypotenuse, equal to the sum of the squares of the legs. This is a famous philosopher.

Pythagoras lived in the sixth century BC, had a beautiful appearance, wore a long beard, and a golden diadem on his head. Pythagoras is not a name, but a nickname that the philosopher received because he always spoke correctly and convincingly, like a Greek oracle. (Pythagoras - “persuasive by speech.”) With his speeches he acquired 2,000 students, who, together with their families, formed a school-state, where the laws and rules of Pythagoras were in effect.

He was the first to give a name to his line of work. The word “philosopher”, like the word “cosmos”, came to us from Pythagoras. There is a lot of cosmic in his philosophy. He argued that to understand God, man and nature, one must study algebra with geometry, music and astronomy. By the way, it is the Pythagorean system of knowledge that is called “mathematics” in Greek. As for the notorious triangle with its hypotenuse and legs, this, according to the great Greek, is more than a geometric figure. This is the “key” to all encrypted phenomena of our life. Everything in nature, said Pythagoras, is divided into three parts. Therefore, before solving any problem, it must be represented in the form of a triangular diagram. "See the triangle - and the problem is two-thirds solved."

Pythagoras did not leave behind a collection of works; he kept his teachings secret and passed them on to his students orally. As a result, the secret died with them. Some information still leaked through the centuries, but now it is difficult to say how much of it is true and how much is false. Even with the Pythagorean theorem, not everything is certain. Some historians doubt the authorship of Pythagoras, arguing that it was widely used in the household by a variety of ancient peoples.

What can we say about individual facts of the biography of the great mathematician! They said, for example, that he could force birds to change their flight direction. He talked with the bear, and she stopped attacking people, he talked with the bull, and under the influence of the conversation, he stopped touching the beans and settled at the temple. One day, while wading a river, Pythagoras offered a prayer to the spirit of the river, and a voice was heard from the water: “Greetings, Pythagoras!” They also said that he commanded the spirits: he sent them into the water and, looking at the ripples, made predictions.

His influence on people was so great that the praise from the lips of Pythagoras overwhelmed his students with delight. One day he happened to get angry with a student, and he committed suicide. The shocked philosopher never spoke irritably to anyone again.

He allegedly managed to heal people by singing to them verses from Homer's Iliad and Odyssey. He knew the medicinal properties of a huge number of plants.

In subsequent centuries, the figure of Pythagoras was surrounded by many legends: he was considered the reincarnated god Apollo, it was believed that he had a golden thigh, and he was able to bifurcate and easily teach in two different places at the same time. The fathers of the early Christian church gave Pythagoras a place of honor between Moses and Plato. Although it is not very clear why: Pythagoras became famous for his teaching about cosmic harmony and the transmigration of souls, which does not really fit into Christian dogmas. In addition, the learned man did not shy away from witchcraft, even in the 16th century. There were frequent references to the authority of Pythagoras in matters not only of science, but also of magic. Just as in Russia all janitors are philosophers, so in Ancient Greece all philosophers were mathematicians. Pythagoras was no exception in this regard.

Pythagoras and the Pythagoreans

But Pythagoras was not only a scientist. “Part-time” he was an active preacher of his own teachings. Moreover, he was a very successful preacher: on the Greek island of Crotone, in southern Italy, where Pythagoras, expelled from Samos, preached, he was popular. His followers, captivated by the ideas of their teacher, quickly realized a religious order. Moreover, the order is so numerous and powerful that it actually managed to come to power in Croton. In ancient times, Pythagoras was most famous and popular as a preacher. And he preached his own teaching, based on the concept of reincarnation (transmigration of souls), that is, the ability of the soul to survive the death of the mortal body, which means that the soul is immortal. Since in a new incarnation the soul can move repeatedly, including into the bodies of animals, Pythagoras and his followers were categorically against killing animals, eating their meat, and even categorically urged fellow citizens not to deal with those who slaughter animals or butcher their carcasses . Pythagoras said that eating meat obscures mental abilities. In general, he did not completely deny himself this, but when he retired to the temple of God for meditation and prayer, he took with him food and drink prepared in advance. His food was poppy and sesame seeds, sea onion skins, narcissus flowers, mallow leaves, barley and peas, wild honey...

Such a seemingly meager diet did not prevent the philosopher from living a long life. Scientists believe that he calculated, preached and philosophized for about a hundred years. But he himself constantly stated that he had lived many lives...

He was the first person to call himself a philosopher. Before him, smart people proudly and somewhat arrogantly called themselves sages, which meant a person who knows. Pythagoras called himself a philosopher - one who tries to find, find out.

According to Pythagoras, bloodshed was equated, no less than, with original sin, for which, as is known, the immortal soul is expelled into the mortal world, where it is destined to wander, flitting from one body to another. The soul does not like such endless reincarnations; it strives for freedom, into the heavenly spheres, but out of ignorance it invariably repeats the sinful act.

According to Pythagoras, purification can free the soul from endless reincarnations. The simplest purification consists of abstaining from excesses, from drunkenness or from eating beans. The rules of behavior must also be strictly observed: respect for elders, obedience to the law. In relationships, the Pythagoreans placed friendship at the forefront; all the property of friends should be common. For the chosen few, as they say today, the most advanced, the highest form of purification became available - philosophy, a word that, as we have already mentioned, and Cicero argued before us, was first used by Pythagoras, who called himself not a sage, but a lover of wisdom. Mathematics is one of the components of the religion of the Pythagoreans, who taught that God put number at the basis of the world order.

The Pythagoreans tried to apply the mathematical discoveries of Pythagoras to speculative physical constructions, which led to interesting results. They believed that any planet, revolving around the Earth, passing through pure upper air, or "ether", emits a tone of a certain pitch. The pitch of the sound changes depending on the speed of the planet's movement, and the speed of this movement depends on the distance to the Earth. Merging, heavenly sounds form what we call the “harmony of the spheres”, or “music of the spheres”; literature is strewn with references to the music of the spheres, like an imperial crown with diamonds. The early Pythagoreans were convinced that the Earth was flat and at the center of the cosmos. Later they “got wiser” and began to believe that the Earth has a spherical shape and, together with other planets, including the Sun, revolves around the center of space, the so-called “hearth”.

Pythagoras’ ill-wishers, concerned about the growing popularity of his teachings, nevertheless managed to expel him to Metapontum, where he died, as they now say, of a broken heart, grieving over the futility of his efforts to educate and the futility of serving humanity, so it seemed to him. The Order ruled in Crotone for almost another century until it was defeated.

It is unfair to think that the Pythagoreans left behind only delusions. They made a lot of discoveries in mathematics and geometry. Euclid used many of their discoveries in his Elements. Pythagorean ideas penetrated into Athens, they were accepted by Socrates, and later grew into a powerful ideological movement led by the great Plato and his student Aristotle.

But let's return to mathematics. The Pythagoreans were passionate about constructing regular geometric figures using compasses and rulers. Fascinated by this “construction”, they built the figures up to a regular pentagon and were puzzled by how, using the same compass and ruler, they could construct the next regular figure - a heptagon? It must be said right away that they failed.

But they not only puzzled themselves, but also puzzled all reasonable humanity, which, with a compass and ruler in hand, with wrinkled foreheads, rushed to build regular heptagons.

Not so! This Pythagorean problem remained unsolvable for more than two thousand years! It was solved only in 1796 by the 19-year-old (!) German youth Carl Friedrich Gauss (1777 - 1855), later nicknamed the king of mathematicians.

The young genius “built” the heptagon by accident, while doing completely different calculations. Gauss outlined the theory of equations for dividing a circle Xn - 1 = 0, which in many ways was the prototype of the brilliant theory of another nineteen-year-old genius - Galois. In addition to general methods for solving these equations, Gauss established a connection between the equations and the construction of regular polygons. He found all those values of n for which a regular n-gon can be constructed using a compass and ruler.

More than two thousand years have passed since the problem arose... That's how much patience and time it sometimes takes to solve it!

History of the theorem

Caricatures

History of the theorem

Let's start the historical review with ancient China. Here the mathematical book Chu-pei attracts special attention. This work talks about the Pythagorean triangle with sides 3, 4 and 5: “If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5, when the base is 3 and the height is 4.” In the same book, a drawing is proposed that coincides with one of the drawings of the Hindu geometry of Bashara.

Cantor(the leading German historian of mathematics) believes that equality 3 2 + 4 2 = 5 2 was already known to the Egyptians still around 2300 BC. e., during the time of the king Amenemhet I(according to papyrus 6619 of the Berlin Museum). According to Cantor, the harpedonaptes, or “rope pullers,” built right angles using right triangles with sides 3, 4 and 5. Their method of construction can be very easily reproduced. Let's take a rope 12 m long and tie a colored strip to it at a distance of 3 m. from one end and 4 meters from the other. The right angle will be enclosed between sides 3 and 4 meters long. It could be objected to the Harpedonaptians that their method of construction becomes superfluous if one uses, for example, a wooden square, which is used by all carpenters. Indeed, Egyptian drawings are known in which such a tool is found, for example, drawings depicting a carpenter's workshop.

A little more is known about the Pythagorean theorem Babylonians. In one text related to time Hammurabi, i.e. by 2000 BC. e., an approximate calculation of the hypotenuse of a right triangle is given. From this we can conclude that in Mesopotamia they were able to perform calculations with right triangles, at least in some cases. Based, on the one hand, on the current level of knowledge about Egyptian and Babylonian mathematics, and on the other, on a critical study of Greek sources, Van der Waerden (Dutch mathematician) came to the following conclusion: “The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, is not the discovery of mathematics, but its systematization and justification. In their hands, computational recipes based on vague ideas turned into an exact science.”

Geometry Hindus, like the Egyptians and Babylonians, was closely associated with the cult. It is very likely that the theorem on the square of the hypotenuse was already known in India around the 18th century BC. e.

In the first Russian translation of Euclidean Elements, made by F. I. Petrushevsky, the Pythagorean theorem is stated as follows: "In right triangles, the square of the side opposite the right angle is equal to the sum of the squares of the sides containing the right angle."

It is now known that this theorem was not discovered by Pythagoras. However, some believe that Pythagoras was the first to give its full proof, while others deny him this merit. Some attribute to Pythagoras the proof which Euclid gives in the first book of his Elements. On the other hand, Proclus claims that the proof in the Elements belongs to Euclid himself. As we see, the history of mathematics has preserved almost no reliable data about the life of Pythagoras and his mathematical activities. But the legend even tells us the immediate circumstances that accompanied the discovery of the theorem. They say that in honor of this discovery, Pythagoras sacrificed 100 bulls.

Caricatures

Students of the Middle Ages considered the proof of the Pythagorean theorem very difficult and called it Dons asinorum - donkey bridge, or elefuga - flight of the “poor”, since some “poor” students who did not have serious mathematical training fled from geometry. Weak students who memorized theorems by heart, without understanding, and were therefore nicknamed “donkeys,” were unable to overcome the Pythagorean theorem, which served as an insurmountable bridge for them. Because of the drawings accompanying the Pythagorean theorem, students also called it a “windmill,” composed poems like “Pythagorean pants are equal on all sides,” and drew cartoons.

The Pythagorean theorem is one of the main and, one might say, the most important theorem of geometry. Its significance lies in the fact that most of the theorems of geometry can be deduced from it or with its help. The Pythagorean theorem is also remarkable because in itself it is not at all obvious. For example, the properties of an isosceles triangle can be seen directly in the drawing. But no matter how much you look at a right triangle, you will never see that there is a simple relationship between its sides: c 2 =a 2 +b 2 .

Proof No. 1 (simplest)

A square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs.

The simplest proof of the theorem is obtained in the case of an isosceles right triangle. This is probably where the theorem began.

In fact, it is enough just to look at the mosaic of isosceles right triangles to be convinced of the validity of the theorem. For example, for ΔABC: square built on the hypotenuse AC, contains 4 original triangles, and the squares built on the sides - two each. The theorem is proven .

Evidence No. 2

Let T- right triangle with legs A , b and hypotenuse With (Fig. a). Let's prove that c 2 = a 2 + b 2 .

Let's build a square Q with the side a+b (Fig. b).On the sides of the square Q let's take the points A , IN , WITH , D so that the segments AB , Sun , CD , D.A. cut off from the square Q right triangles T 1 , T 2 , T 3 , T 4 with legs A And b. Quadrangle ABCD denoted by the letter R. Let's show that R- square with side With .

All triangles T 1 , T 2 , T 3 , T 4 equal to a triangle T(on two legs). Therefore, their hypotenuses are equal to the hypotenuse of the triangle T, i.e. the segment With. Let us prove that all the angles of this quadrilateral are right.

Let a And b- the magnitude of the acute angles of the triangle T. Then, as you know, a+b = 90°. Apex angle A quadrangle R along with angles equal a And b, makes a straight angle. That's why a+b =180°. And since a+b = 90°, That g=90°. In the same way it is proved that the remaining angles of the quadrilateral R straight. Therefore, the quadrilateral R- square with side With .

Square Q with the side a+b made up of a square R with the side With and four triangles equal to a triangle T. Therefore, their areas satisfy the equality S(Q)=S(P)+4S(T) .

Because S(Q)=(a+b) 2 ; S(P)=c 2 And S(T)=½a*b, then, substituting these expressions into S(Q)=S(P)+4S(T), we get the equality (a + b) 2 = c 2 + 4*½a*b. Because the (a+b) 2 =a 2 +b 2 +2*a*b, then equality (a+b) 2 =c 2 +4*½a*b can be written like this: a 2 +b 2 +2*a*b=c 2 +2*a*b .

From equality a 2 +b 2 +2*a*b=c 2 +2*a*b follows that c 2 = a 2 + b 2 .
etc.

Evidence No. 3

Let ΔABC- given right triangle with right angle WITH. Let's find the height CD from the vertex of a right angle WITH .

By definition of the cosine of an angle (Cosine of an acute angle of a right triangle called the ratio of the adjacent leg to the hypotenuse) сosА=AD/AC=AC/AB. From here AB*AD=AC 2. Likewise сosВ=BD/BC=BC/AB. From here AB*BD=BC 2. Adding the resulting equalities term by term and noting that AD+DB=AB, we get: AC 2 + BC 2 = AB (AD + DB) = AB 2 . The theorem is proven .

Evidence No. 4

Area of a right triangle: S=½*a*b or S=½(p*r)(for an arbitrary triangle);
p- semi-perimeter of a triangle; r- the radius of the circle inscribed in it.
r = ½*(a + b - c)- the radius of a circle inscribed in any triangle.
½*a*b = ½*p*r = ½(a + b + c)*½(a + b - c) ;
a*b = (a + b + c)*½(a + b - c) ;
a+b=x ;
a*b = ½(x + c)*(x - c)*a*b = ½(x 2 -c 2)
a*b = ½(a 2 + 2*a*b + b 2 - c 2)
a 2 + b 2 - c 2 = 0, Means
a 2 + b 2 = c 2

Evidence No. 5

Given:ΔABC- right triangle A.J.- height lowered to the hypotenuse BCED- square on the hypotenuse ABFH And ACKJ- squares built on legs.

Prove: The square of the hypotenuse is equal to the sum of the squares of the legs (Pythagorean Theorem).

Proof: 1. Let us prove that the rectangle BJLD equal to a square ABFH , ΔABD=ΔBFS(on two sides and the angle between them BF=AB; BC=BD; corner FBS=ABD).But! S ΔABC =½S BJLD, because at ΔABC and rectangle BJLD common ground BD and overall height LD. Likewise S ΔFBS =½S ABFH (B.F.- common ground, AB- total height). Hence, considering that SΔABD = SΔFBS, we have: S BJLD =S ABFH. Similarly, using the triangle equality ΔBCK And ΔACE, it is proved that S JCEL =S ACKG. So, S ABFH +S ACKJ =S BJLD + S BCED .

Currently, it is generally recognized that the success of the development of many areas of science and technology depends on the development of various areas of mathematics. An important condition for increasing production efficiency is the widespread introduction of mathematical methods into technology and the national economy, which involves the creation of new, effective methods of qualitative and quantitative research that allow solving problems posed by practice. Let us consider several elementary examples of such problems in which the Pythagorean theorem is used to solve them.

Construction

Window

In Gothic and Romanesque buildings, the upper parts of the windows are divided by stone ribs, which not only play the role of ornament, but also contribute to the strength of the windows. The figure shows a simple example of such a window in the Gothic style. The method of constructing it is very simple: From the figure it is easy to find the centers of six circular arcs, the radii of which are equal to the width of the window ( b) for external arcs and half the width ( b/2), for internal arcs. There remains a complete circle touching four arcs. Since it is enclosed between two concentric circles, its diameter is equal to the distance between these circles, i.e. b/2 and therefore the radius is b/4. And then the position of its center becomes clear. In the example considered, the radii were found without any difficulty. Other similar examples may require calculations; Let us show how the Pythagorean theorem is used in such problems.

The motif shown in the figure is often found in Romanesque architecture. If b still denotes the width of the window, then the radii of the semicircles will be equal R = b / 2 And r = b / 4. Radius p the inner circumference can be calculated from the right triangle shown in Fig. dotted line The hypotenuse of this triangle passing through the tangency point of the circles is equal to b/4+p, one leg is equal b/4, and the other b/2-p .

According to the Pythagorean theorem we have:
(b/4+p)=(b/4)+(b/4-p)
or
b/16+ b*p/2+p=b/16+b/4-b*p+p ,
where
b*p/2=b/4-b*p .
Dividing by b and bringing similar terms, we get:
(3/2)*p=b/4, p=b/6 .

Roof

It is planned to build a gable roof on the house (sectional shape). What length should the rafters be if the beams are made AC=8 m, and AB=BF.
Solution:
Triangle ADC- isosceles AB=BC=4 m , BF=4 m Assuming that FD=1.5 m, Then:
A) From a triangle DBC: DB=2.5m

B) From a triangle ABF :

Lightning rod

A lightning rod protects from lightning all objects whose distance from its base does not exceed twice its height. Determine the optimal position of the lightning rod on a gable roof, ensuring its lowest accessible height.
Solution:
According to the Pythagorean theorem, h 2 ≥ a 2 +b 2, which means h ≥ (a 2 +b 2) ½.
Answer: h ≥ (a 2 +b 2) ½

Astronomy

This figure shows the points A And B and the path of the light beam from A To B and back. The beam path is shown with a curved arrow for clarity; in fact, the light beam is straight.

What path does the ray take? Since light travels the same path back and forth, let us ask right away: what is half the path that the beam travels? If we designate the segment AB symbol l, half the time like t, and also denoting the speed of light with the letter c, then our equation will take the form

c * t = l

Obviously? This is the product of the time spent and the speed!

Now let's try to look at the same phenomenon from a different frame of reference, from a different point of view, for example, from a spaceship flying past a running beam at a speed v. Previously, we realized that with such an observation the speeds of all bodies will change, and stationary bodies will begin to move at a speed v in the opposite direction. Let's assume that the ship is moving to the left. Then the two points between which the bunny runs will begin to move to the right at the same speed. Moreover, while the bunny runs its way, the starting point A shifts and the beam returns to a new point C .

Question: how much does the point have time to move (to turn into point C) while the light beam travels? More precisely, let's ask again about half of this displacement! If we denote half the travel time of the beam by the letter t", and half the distance A.C. letter d, then we get our equation in the form:

v * t" = d

Letter v indicates the speed of the spacecraft. Again obvious, isn't it?

Another question: how far will the light beam travel?(More precisely, what is half of this path? What is the distance to the unknown object?)

If we denote half the length of the path of light by the letter s, then we get the equation:

c * t" = s

Here c is the speed of light, and t"- this is the same time that we considered in the formulas above.

Now consider the triangle ABC. This is an isosceles triangle whose height is l. Yes, yes, the same one l, which we introduced when considering the process from a fixed point of view. Since the movement is perpendicular l, then it could not affect her.

Triangle ABC composed of two halves - identical rectangular triangles, the hypotenuses of which AB And B.C. must be connected with the legs according to the Pythagorean theorem. One of the legs is d, which we just calculated, and the second leg is s, which light passes through, and which we also calculated.
We get the equation:

s 2 = l 2 + d 2

It's just the Pythagorean theorem, right?

At the end of the nineteenth century, various assumptions were made about the existence of inhabitants of Mars similar to humans; this was a consequence of the discoveries of the Italian astronomer Schiaparelli (he discovered channels on Mars that had long been considered artificial) and others. Naturally, the question of whether it is possible to explain with the help of light signals these hypothetical creatures, caused a lively discussion. The Paris Academy of Sciences even established a prize of 100,000 francs for the first person to establish contact with any inhabitant of another celestial body; this prize is still waiting for the lucky winner. As a joke, although not entirely without reason, it was decided to transmit a signal to the inhabitants of Mars in the form of the Pythagorean theorem.

It is not known how to do this; but it is obvious to everyone that the mathematical fact expressed by the Pythagorean theorem takes place everywhere and therefore inhabitants of another world similar to us must understand such a signal.

mobile connection

Currently, there is a lot of competition among operators in the mobile communications market. The more reliable the connection, the larger the coverage area, the more consumers the operator has. When building a tower (antenna), you often have to solve the following problem: what maximum height should the antenna have so that the transmission can be received within a certain radius (for example, radius R = 200 km?, if it is known that the radius of the Earth is 6380 km.)
Solution:
Let AB= x, BC=R=200 km, OC= r =6380 km.
OB = OA + AB
OB = r + x
Using the Pythagorean theorem, we get the answer.
Answer: 2.3 km.

Introduction

When many people hear the name Pythagoras, they remember his theorem. But can we really encounter this theorem only in geometry? No, of course not! The Pythagorean theorem is found in various fields of science. For example: in physics, astronomy, architecture and others. But Pythagoras and his theorem are also sung in literature.

There are many legends, myths, stories, songs, parables, fables, anecdotes, ditties about this theorem. Below are examples of each type listed here...

Pythagoras of Samos went down in history as one of the most outstanding intellectuals of mankind. There are many unusual things in him, and it seems that fate itself has prepared for him a special path in life.

Pythagoras created his own religious and philosophical school and became famous as one of the greatest mathematicians. His intelligence and intelligence were hundreds of years ahead of the time in which he lived.

Pythagoras of Samos

Brief biography of Pythagoras

Of course, a short biography of Pythagoras will not give us the opportunity to fully reveal this unique personality, but we will still highlight the main moments of his life.

Childhood and youth

The exact date of birth of Pythagoras is unknown. Historians suggest that he was born between 586-569. BC, on the Greek island of Samos (hence his nickname - “Samos”). According to one legend, Pythagoras' parents were predicted that their son would become a great sage and educator.

Pythagoras's father was called Mnesarchus, and his mother was Parthenia. The head of the family was engaged in processing precious stones, so the family was quite wealthy.

Upbringing and education

Already at an early age, Pythagoras showed interest in various sciences and arts. His first teacher was called Hermodamant. He laid the foundations of music, painting and grammar in the future scientist, and also forced him to memorize passages from Homer's Odyssey and Iliad.

When Pythagoras turned 18, he decided to go to Egypt to gain even more knowledge and gain experience. This was a serious step in his biography, but it was not destined to come true. Pythagoras was unable to enter Egypt because it was closed to the Greeks.

Stopping on the island of Lesbos, Pythagoras began to study physics, medicine, dialectics and other sciences from Pherecydes of Syros. After living on the island for several years, he wanted to visit Miletus, where the famous philosopher Thales, who formed the first philosophical school in Greece, still lived.

Very soon, Pythagoras becomes one of the most educated and famous people of his time. However, after some time, drastic changes occur in the sage’s biography, as the Persian War began.

Pythagoras falls into Babylonian captivity and lives in captivity for a long time.

Mysticism and homecoming

Due to the fact that astrology and mysticism were popular in Babylon, Pythagoras became addicted to the study of various mystical sacraments, customs and supernatural phenomena. The entire biography of Pythagoras is full of all kinds of searches and solutions that so attracted his attention.

Having been in captivity for more than 10 years, he unexpectedly receives release personally from the Persian king, who knew firsthand about the wisdom of the learned Greek.

Once free, Pythagoras immediately returned to his homeland to tell his compatriots about the acquired knowledge.

School of Pythagoras

Thanks to his extensive knowledge, constant and oratory skills, he quickly manages to gain fame and recognition among the inhabitants of Greece.

At Pythagoras’s speeches there are always many people who are amazed at the philosopher’s wisdom and see in him almost a deity.

One of the main points in the biography of Pythagoras is the fact that he created a school based on his own principles of worldview. It was called that: the school of Pythagoreans, that is, followers of Pythagoras.

He also had his own teaching method. For example, students were prohibited from talking during classes and were not allowed to ask any questions.

Thanks to this, the students could cultivate modesty, meekness and patience.

These things may seem strange to a modern person, but we should not forget that in the time of Pythagoras the very concept schooling in our understanding simply did not exist.

Mathematics

In addition to medicine, politics and art, Pythagoras was very seriously involved in mathematics. He managed to make a significant contribution to the development of geometry.

Until now, in schools all over the world, the most popular theorem is considered to be the Pythagorean theorem: a 2 + b 2 =c 2. Every schoolchild remembers that “Pythagorean pants are equal in all directions.”

In addition, there is a “Pythagorean table”, with which it was possible to multiply numbers. In essence, this is a modern multiplication table, just in a slightly different form.

Numerology of Pythagoras

There is a remarkable thing in the biography of Pythagoras: all his life he was extremely interested in numbers. With their help, he tried to understand the nature of things and phenomena, life and death, suffering, happiness and other important issues of existence.

He associated the number 9 with constancy, 8 with death, and also paid great attention to the square of numbers. In this sense, the perfect number was 10. Pythagoras called ten a symbol of the Cosmos.

The Pythagoreans were the first to divide numbers into even and odd. Even numbers, according to the mathematician, had a feminine principle, and odd numbers had a masculine principle.

In those days when science as such did not exist, people learned about life and the world order as best they could. Pythagoras, like the great son of his time, tried to find answers to these and other questions with the help of figures and numbers.

Philosophical teaching

The teachings of Pythagoras can be divided into two categories:

Scientific approach
Religiosity and mysticism

Unfortunately, not all of Pythagoras’s works have been preserved. And all because the scientist practically did not take any notes, transferring knowledge to his students orally.

In addition to the fact that Pythagoras was a scientist and philosopher, he can rightfully be called a religious innovator. In this, Leo Tolstoy was a little like him (we published it in a separate article).

Pythagoras was a vegetarian and encouraged his followers to do so. He did not allow students to eat food of animal origin, forbade them to drink alcohol, use foul language and behave indecently.

It is also interesting that Pythagoras did not teach ordinary people who sought to obtain only superficial knowledge. He accepted as disciples only those in whom he saw chosen and enlightened individuals.

Personal life

Studying the biography of Pythagoras, one may get the mistaken impression that he had no time for his personal life. However, this is not quite true.

When Pythagoras was about 60 years old, at one of his performances he met a beautiful girl named Feana.

They got married, and from this marriage they had a boy and a girl. So the outstanding Greek was a family man.

Death

Surprisingly, none of the biographers can say unambiguously how the great philosopher and mathematician died. There are three versions of his death.

According to the first, Pythagoras was killed by one of his students whom he refused to teach. In a fit of anger, the killer set fire to the scientist's Academy, where he died.

The second version says that during the fire, the scientist’s followers, wanting to save him from death, created a bridge from their own bodies.

But the most common version of the death of Pythagoras is considered to be his death during an armed conflict in the city of Metapontus.

The great scientist lived more than 80 years, dying in 490 BC. e. During his long life he managed to do a lot, and he is quite rightly considered one of the most outstanding minds in history.

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The biography of Pythagoras is very interesting. The very fact that Pythagoras is not a name, but a nickname that the philosopher received because he always spoke correctly and convincingly, like a Greek oracle. (Pythagoras - “persuasive by speech”).

Pythagoras of Samos is a great Greek scientist. His name is familiar to every schoolchild. Very little is known about the life of Pythagoras; a large number of legends are associated with his name. Pythagoras is one of the most famous scientists, but also the most mysterious personality, a human symbol, philosopher and prophet. He was the ruler of thoughts and the preacher of the religion he created. He was deified and hated... So who are you, Pythagoras?

He was born around 580-500. BC e. on the island of Samos, far from Greece . Pythagoras's father was Mnesarchus, a gem cutter. The mother’s name is considered unknown, but when studying one of the sources, I found out that the mother’s name was Parthenisa. According to many testimonies, the born boy was fabulously handsome, and soon showed his extraordinary abilities.

Among the teachers of young Pythagoras, the names of the elder Hermodamant and Pherecydes of Syros are mentioned (although there is no firm certainty that they were Pythagoras’s first teachers). Young Pythagoras spent whole days at the feet of the elder Hermodamantus, listening to the melody of the cithara and the hexameters of Homer. Pythagoras retained his passion for the music and poetry of the great Homer throughout his life. And, being a recognized sage, surrounded by a crowd of disciples, Pythagoras began the day by singing one of Homer's songs. Pherecydes was a philosopher and was considered the founder of the Italian school of philosophy. But be that as it may, the restless imagination of young Pythagoras very soon became cramped in little Samos; on clear days he saw yellow roads running across the mainland to the big world. They called him.

He goes to Miletus, where he meets another scientist - Thales. The fame of this sage thundered throughout Hellas. Lively conversations took place during the meetings. It was Thales who advised him to go to Egypt for knowledge, which Pythagoras did.

Pythagoras left his homeland very young. First he sailed to the shores of Egypt, walked it length and breadth. He looked carefully at those around him, listened to the priests. In Egypt, they say, Pythagoras was captured by Cambyses, the Persian conqueror, and he was taken to Babylon. Pythagoras knew that this was the greatest city in the world, and he quickly became accustomed to the complex Babylonian traditions. He eagerly absorbed the speeches of the Chaldean priests. He studied number theory with the Chaldean magicians.

For 22 years he studied in the temples of Memphis and received initiation of the highest degree. Here he deeply studied mathematics, “the science of numbers or universal principles,” which he later made the center of his system. From Memphis, on the orders of Cambyses, who invaded Egypt, Pythagoras, together with the Egyptian priests, ended up in Babylon, where he spent another 12 years. Here he had the opportunity to study many religions and cults, to penetrate the mysteries of the ancient magic of the heirs of Zoroaster.

Around 530, Pythagoras finally returned to Greece and soon moved to Southern Italy, to the city of Croton. In Croton he founded the Pythagorean League, which was at once a philosophical school, a political party and a religious brotherhood.

The main Pythagorean symbol of health and identification mark was the pentagram - a star-shaped pentagon formed by the diagonals of a regular pentagon. It contained all proportions: geometric, arithmetic, golden. She was the secret sign by which the Pythagoreans recognized each other. In the Middle Ages, it was believed that the pentagram protected against “evil spirits.” The five-pointed star is about 3000 years old. Today, the five-pointed star flies on the flags of almost half the countries of the world. The inner beauty of mathematical structure was also noticed by Pythagoras. The moral principles preached by Pythagoras are still worthy of imitation today. His school contributed to the formation of an intellectual elite. The Pythagoreans lived according to certain commandments, and it would do well for us to adhere to them, although they are already about two and a half thousand years old. For example:

Don't do what you don't know;

Act in such a way that you will not be upset or repent later;

Do not rake the fire with a sword.

The school displeased the inhabitants of the island, and Pythagoras had to leave his homeland. He moved to southern Italy, a colony of Greece, and here, in Crotone, he again founded a school - the Pythagorean Union, which lasted about two centuries .

Now it is difficult to say which scientific ideas belong to Pythagoras and which belong to his pupils and followers. It remains unknown whether he discovered and proved the famous theorem that bears his name, or whether he himself was the first to prove the theorem on the sum of the angles of a triangle.

Quite quickly it gains great popularity among residents. Pythagoras skillfully uses the knowledge gained from traveling around the world. Over time, the scientist stops performing in churches and on the streets. Already in his home, Pythagoras teaches medicine, the principles of political activity, astronomy, mathematics, music, ethics and much more. Outstanding political and statesmen, historians, mathematicians and astronomers came from his school. He was not only a teacher, but also a researcher. His students also became researchers. The School of Pythagoras first suggested the sphericity of the Earth. The idea that the movement of celestial bodies obeys certain mathematical relationships first appeared precisely in the School of Pythagoras. Pythagoras lived 80 years. There are many legends about his death. According to one of them, he was killed in a street fight.

The Pythagorean school gave Greece a galaxy of talented philosophers, physicists and mathematicians. Their name is associated in mathematics with the systematic introduction of proofs into geometry, consideration of it as an abstract science, the creation of the doctrine of similarity, the proof of the theorem bearing the name of Pythagoras, the construction of some regular polygons and polyhedra, as well as the doctrine of even and odd, simple and composite, figured and perfect numbers, arithmetic, geometric and harmonic proportions and averages.

For us, Pythagoras is a mathematician. In ancient times it was different. For his contemporaries, Pythagoras was primarily a religious prophet, the embodiment of the highest divine wisdom. Some called him a mathematician, a philosopher, others - a charlatan. Another interesting fact is that Pythagoras was the first and four times in a row to be the Olympic champion in fist fighting.

2. History of the discovery and proof of the Pythagorean theorem.

Much in mathematics is associated with his name, and first of all, of course, the theorem that bears his name. This is the Pythagorean theorem. Currently, everyone agrees that this theorem was not discovered by Pythagoras. She was known even before him. Its special cases were known in China, Babylonia, and Egypt.

The historical overview begins with ancient China. Here the mathematical book Chu-pei attracts special attention. This work talks about the Pythagorean triangle with sides 3, 4 and 5: “If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5, when the base is 3 and the height is 4.”.

Cantor (the greatest German historian of mathematics) believes that equality

3²+4²=5² was already known to the Egyptians around 2300 BC. e. According to Kantor harpedonaptes, or "rope pullers", built right angles using right triangles with sides of 3, 4 and 5. Their method of construction can be very easily reproduced. Let's take a rope 12 meters long and tie a colored strip to it at a distance of 3 meters from one end and 4 meters from the other. A right angle will be enclosed between sides 3 and 4 meters long .

The Egyptian triangle is a right triangle with an aspect ratio of 3:4:5. A feature of such a triangle, known since antiquity, is that with such a ratio of the sides, the Pythagorean theorem gives whole squares of both the legs and the hypotenuse, that is, 9:16:25. The Egyptian triangle is the simplest (and first known) of the Heronian triangles - triangles with integer sides and areas. The name of the triangle with this aspect ratio was given by the Hellenes: in the 7th - 5th centuries BC. e. Greek philosophers and public figures actively visited Egypt. For example, Pythagoras in 535 BC. e. at the insistence of Thales, he went to Egypt to study astronomy and mathematics - and, apparently, it was the attempt to generalize the ratio of squares characteristic of the Egyptian triangle to any right triangles that led Pythagoras to the proof of the famous theorem. The Egyptian triangle with an aspect ratio of 3:4:5 was actively used by land surveyors and architects to construct right angles.

Although it could be objected to the harpedonaptes that their method of construction becomes redundant if you use, for example, a wooden square, used by all carpenters. Indeed, Egyptian drawings are known in which such a tool is found, for example, drawings depicting a carpenter's workshop.

Somewhat more is known about the Pythagorean theorem among the Babylonians. In one text dating back to 2000 BC. e., an approximate calculation of the hypotenuse of a right triangle is given. From this we can conclude that in Mesopotamia they were able to perform calculations with right triangles, at least in some cases. Based, on the one hand, on the current level of knowledge about Egyptian and Babylonian mathematics, and on the other hand, on a critical study of Greek sources, Van der Waerden (Dutch mathematician) came to the following conclusion:

“The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, is not the discovery of mathematics, but its systematization and justification. In their hands, computational recipes based on vague ideas turned into an exact science.”

However, some believe that Pythagoras was the first to give its full proof, while others deny him this merit. But, perhaps, you cannot find any other theorem that deserves so many different comparisons. In France and some areas of Germany in the Middle Ages, the Pythagorean theorem was called the “bridge of donkeys.” It turns out that weak students who memorized theorems by heart, without understanding, and were therefore nicknamed “donkeys,” were unable to overcome the Pythagorean theorem. Among the mathematicians of the Arab East, this theorem was called the “bride’s theorem.” The fact is that in some copies of Euclid’s Elements this theorem was called the “nymph’s theorem” for the similarity of the drawing with a bee, a butterfly, which in Greek was called a nymph. But the Greeks used this word to call some other goddesses, as well as young women and brides in general. When translating from Greek, the Arabic translator, without paying attention to the drawing, translated the word “nymph” as “bride” and not “butterfly”. This is how the affectionate name for the famous theorem appeared - “the bride’s theorem.”

In the Middle Ages, the Pythagorean theorem defined the limit of, if not the maximum possible, then at least good mathematical knowledge.

Today it is generally accepted that Pythagoras gave the first proof of the theorem that bears his name. Alas, no traces of this evidence have survived either. The theorem states: A square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs.

Thus, Pythagoras did not discover this property of a right triangle; he was probably the first to generalize and prove it, thereby transferring it from the field of practice to the field of science. The Pythagorean theorem was included in the Guinness Book of Records as the theorem with the most evidence. This indicates the continued interest in it on the part of the wider mathematical community. The Pythagorean theorem has been the source of many generalizations and fertile ideas. The depth of this ancient truth, apparently, is far from exhausted.

Judging by the brief biography of Pythagoras, his life was filled with amazing events, and his contemporaries considered him perhaps the most outstanding scientist of all times and peoples, initiated into all the secrets of the Universe.

Historical evidence of the origin of Pythagoras has been preserved. His father was Mnesarchus, a native of Tyre, who received citizenship of Samos, and his mother was Parthenides or Pyphaidas, who was a relative of Ankeus, the founder of the Greek colony on Samos.

Education

If you follow the official biography of Pythagoras, then at the age of 18 he went to Egypt, to the court of Pharaoh Amasis, to whom he was sent by the Samian tyrant Polycrates. Thanks to his patronage, Pythagoras was taught by the Egyptian priests and was admitted to the temple libraries. It is believed that the sage spent about 22 years in Egypt.

Babylonian captivity

Pythagoras came to Babylon as a prisoner of King Cambyses. He stayed in the country for about 12 years, studying with local magicians and priests. At the age of 56, he returned to his native Samos.

Philosophical school

Evidence indicates that after all his wanderings, Pythagoras settled in Crotona (Southern Italy). There he founded a philosophical school, more like a kind of religious order (the followers of Pythagoras believed that transmigration of the soul and reincarnation were possible; they believed that a person should earn a place in the world of the Gods by good deeds, and until this happens, the soul will continue to return to Earth, “ “inhabiting” the body of an animal or a person), where not only knowledge was promoted, but also a special way of life.

It was Pythagoras and his students, whose teacher’s authority was unquestionable, who introduced the words “philosophy” and “philosopher” into circulation. This order actually came to power in Crotone, but due to the spread of anti-Pythagorean sentiment, the philosopher was forced to leave for the city of Metapontus, where he died in approximately 491 BC.

Personal life

The name of Pythagoras' wife is known - Theano. It is also known that the philosopher had a son and daughter.

Discoveries

It was Pythagoras, as most researchers believe, who discovered the famous theorem that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.

Pythagoras's eternal opponent was Heraclitus, who believed that “much knowledge” is not a sign of a real philosophical mind. Aristotle never quoted Pythagoras in his works, but Plato considered Pythagoras the greatest philosopher of Greece, bought the works of the Pythagoreans and often quoted their opinions in his works.

Other biography options

It is interesting that the birth of Pythagoras was predicted by the Delphic Pythia (hence the name, because “Pythagoras” translated from Greek means “predicted by Pythia”). The boy's father was warned that his son would be born unusually gifted and would bring a lot of benefit to people.
Many biographers describe the life of Pythagoras differently. There are certain discrepancies in the works of Heraclides, Ephsebius of Caesarea, Diogenes, and Porphyry. According to the latter’s works, the philosopher either died as a result of the anti-Pythagorean rebellion, or starved himself to death in one of the temples, as he was not satisfied with the results of his work.
There is an opinion that Pythagoras was a vegetarian and only occasionally allowed himself to eat fish. Asceticism in everything is one of the components of the teachings of the Pythagorean philosophical school.

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