When did Pythagoras live? Pythagoras - biography, information, personal life

Name: Pythagoras of Samos

Years of life: 569 BC - 495 BC

State: Ancient Greece

Field of activity: Mathematician, Philosopher

Greatest Achievement: One of the greatest mathematicians who proved many theorems. Founder of the Pythagorean school.

He was born on the island of Samos (Greece), in 569 BC. According to various sources, the death of Pythagoras is recorded between 500 BC. and 475 BC in Metaponte (Italy).

Personal life of Pythagoras

His father, Mnesarchus, was a merchant of precious stones. His mother's name was Pyphaida. Pythagoras had two or three brothers.

Some historians say that Pythagoras was married to a woman named Theano and had a daughter, Miya, as well as a son named Thelaugus, who succeeded as a teacher of mathematics and may have taught Empedocles.

Others say that Theano was one of Pythagoras' students, not his wife, and it is possible that Pythagoras never married or had children.

Pythagoras was well educated, he played the lyre throughout his life, knew poetry and read Homer. He was interested in mathematics, philosophy, astronomy and music, and was greatly influenced by Pherecydes (philosophy), (mathematics and astronomy) and Anaximander (philosophy, geometry).

Pythagoras abandoned Samos around 535 BC. and went to Egypt to study with the priests in the temples. Many of the beliefs that Pythagoras later pursued in Italy were borrowed from the Egyptian priests, such as secret signs, the pursuit of purity, and not eating legumes or wearing animal skins as clothing.

Ten years later, when Persia invaded, Pythagoras was captured and sent to Babylon (now Iraq), where he met priests who taught him sacred rites. Iamblichus (250-330 AD), a Syrian philosopher, wrote about Pythagoras: “He also achieved perfection in arithmetic, music and other mathematical sciences, which were taught by the Babylonians...”.

In 520 BC. Pythagoras, now a free man, left Babylon and returned to Samos, and after some time opened a school called "Semicircle". However, his teachings were not popular with the rulers of the island of Samos, and their desire for Pythagoras' involvement in politics failed, so Pythagoras left and settled in Crotona, a Greek colony in southern Italy, around 518 BC.

There he founded a philosophical and religious school, where his many followers lived and worked.

School of Pythagoras

The Pythagoreans lived by special rules of behavior, including the rules that stated when to say what to wear and what to eat. Pythagoras was the head of the society, and his followers, both men and women who also lived there, were known as mathematicians. They had no personal belongings and were vegetarians.

Another group of followers, who lived separately from the school, had the right to own personal property and not be vegetarians. They all worked together. Pythagoras believed:
All things are numbers. Mathematics is the basis of everything, and geometry is the highest form of mathematical study. The physical world can be understood through mathematics.
The soul resides in the brain and is immortal. It passes from one being to another, sometimes from human to animal, through a series of reincarnations called transmigrations, until the soul is pure. Pythagoras believed that mathematics and music could purify.
Numbers have personality, characteristics, strengths and weaknesses.
The world depends on the interaction of opposites, such as man and woman, light and dark, heat and cold, dryness and moisture, lightness and heaviness, speed and slowness.
Certain symbols have mystical meanings.

Pythagorean theorems

All members of society were expected to observe strict loyalty and secrecy. Due to the strict secrecy among members of the Pythagorean society and the fact that they shared ideas and intellectual discoveries within the group and were closed to society, it is difficult to be sure whether all theorems attributed to Pythagoras originally belonged to him or were the property of the entire Pythagorean community .

Some of Pythagoras' students eventually wrote their theories, teachings and discoveries, but the Pythagoreans always gave honor to Pythagoras as their Teacher:

The sum of the angles of a triangle is equal to two right angles.
Pythagorean Theorem - For a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The Babylonians realized this 1000 years before the discovery, but Pythagoras proved it.
Constructing figures geometric algebra. For example, they solved various equations using geometric means.
The discovery of irrational numbers is attributed to the Pythagoreans, but it is unlikely that this was Pythagoras' idea because it does not agree with his philosophy that all things are numbers, since number for him meant the ratio of two integers.
Five regular solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron). It is believed that Pythagoras only knew how to build the first three, but not the last two.
Pythagoras taught that the Earth was a sphere at the center of the Cosmos (Universe); that the planets, stars and universe were spherical because the sphere was the most perfect figure. He also taught that the paths of the planets were circular. Pythagoras discovered that the morning star was the same as the evening star Venus.

Pythagoras studied odd and even numbers, triangular numbers and perfect numbers. The Pythagoreans contributed to the understanding of angles, triangles, areas, proportions, polygons, and polyhedrons.
Pythagoras also related music to mathematics. He played the seven-string lyre for a long time and discovered how harmonious the vibrating strings are when the lengths of the strings are proportional to whole numbers such as 2:1, 3:2, 4:3.

The Pythagoreans also realized that this knowledge could be applied to other musical instruments.

Death of Pythagoras

He is said to have been killed by an angry mob, the Syracusans, during . It is also said that Pythagoras' school in Croton was burned, as a result of which he went to Metapontus, where he died of starvation.

At least both stories include a scene in which Pythagoras refuses to trample the legume crop in the field in order to escape and save himself, because of which he, along with other Pythagoreans, was caught, and during an unequal battle, the students and Pythagoras himself died.

The Pythagorean Theorem is a cornerstone of mathematics and remains so interesting to mathematicians that there are over 400 different proofs of its solution, including the original proof of the 20th American President Garfield.

Influenced by:

The life story of Pythagoras is difficult to separate from the legends that present him as a perfect sage and a great initiate into all the mysteries of the Greeks and barbarians. Herodotus also called him “the greatest Hellenic sage.”

The main sources on the life and teachings of Pythagoras are the works of the Neoplatonist philosopher Iamblichus (242-306) " About Pythagorean life"; Porphyria (234-305) " Life of Pythagoras"; Diogenes Laertius (200-250) book. 8, " Pythagoras" These authors relied on the writings of earlier authors, of which it should be noted that Aristotle's student Aristoxenus (370-300 BC) was from Tarentum, where the Pythagoreans had a strong position.

Thus, the earliest known sources about the teachings of Pythagoras did not appear until 200 years after his death. Pythagoras himself did not leave any writings, and all information about him and his teachings is based on the works of his followers, who are not always impartial.

Biography

Pythagoras' parents were Mnesarchus and Parthenides from the island of Samos. Mnesarchus was a stone cutter (Diogenes Laertius); according to Porphyry, he was a rich merchant from Tyre, who received Samian citizenship for distributing grain in a lean year. The first version is preferable, since Pausanias gives the genealogy of Pythagoras in the male line from Hippasus from the Peloponnesian Phlius, who fled to Samos and became the great-grandfather of Pythagoras. Parthenides, later renamed Pyphaida by her husband, came from the noble family of Ankeus, the founder of the Greek colony on Samos.

The birth of a child was allegedly predicted by Pythia in Delphi, which is why Pythagoras got his name, which means “ the one announced by the Pythia" In particular, Pythia told Mnesarchus that Pythagoras would bring as much benefit and goodness to people as no one else had brought and would not bring in the future. Therefore, to celebrate, Mnesarchus gave his wife a new name, Pyphaidas, and named the child Pythagoras. Pyphaida accompanied her husband on his travels, and Pythagoras was born in Sidon Phoenician (according to Iamblichus) around 570 BC. e.

According to ancient authors, Pythagoras met with almost all the famous sages of that era, Greeks, Persians, Chaldeans, Egyptians, and absorbed all the knowledge accumulated by humanity. In popular literature, Pythagoras is sometimes credited with the Olympic victory in boxing, confusing Pythagoras the philosopher with his namesake (Pythagoras, son of Crates of Samos), who won his victory at the 48th Games 18 years before the famous philosopher was born.

At a young age, Pythagoras went to Egypt to gain wisdom and secret knowledge from the Egyptian priests. Diogenes and Porphyry write that the Samian tyrant Polycrates provided Pythagoras with a letter of recommendation to Pharaoh Amasis, thanks to which he was allowed to study and initiated into the sacraments forbidden to other foreigners.

« The Pythagoreans formed a large community (there were more than three hundred of them), but it constituted only a small part of the city, which was no longer governed according to the same customs and mores. However, while the Crotonians owned their land, and Pythagoras was with them, the state structure that existed from the foundation of the city was preserved, although there were dissatisfied people who were waiting for an opportunity for a coup. But when they conquered Sybaris, Pythagoras left, and the Pythagoreans who ruled the conquered land did not distribute it by lot, as the majority wanted, then hidden hatred flared up, and many citizens opposed them... The relatives of the Pythagoreans were even more irritated by what they were serving right hand only to their own, and from relatives - only to parents, and that they provide their property for common use, and it is separated from the property of relatives. When the relatives began this hostility, the rest readily joined the conflict... After many years... the Crotonians were overcome by regret and repentance, and they decided to return to the city those Pythagoreans who were still alive.»

Many Pythagoreans died, the survivors scattered throughout Italy and Greece. The German historian F. Schlosser notes regarding the defeat of the Pythagoreans: “ The attempt to transfer caste and clerical life to Greece and, contrary to the spirit of the people, to change its political structure and morals according to the requirements of an abstract theory ended in complete failure.»

According to Porphyry, Pythagoras himself died as a result of the anti-Pythagorean rebellion in Metapontus, but other authors do not confirm this version, although they readily convey the story that the dejected philosopher starved himself to death in the sacred temple.

Philosophical teaching

The teachings of Pythagoras should be divided into two components: the scientific approach to understanding the world and the religious and mystical way of life preached by Pythagoras. The merits of Pythagoras in the first part are not known for certain, since everything created by followers within the school of Pythagoreanism was later attributed to him. The second part prevails in the teachings of Pythagoras, and it is this part that remained in the minds of most ancient authors.

The merit of the Pythagoreans was the promotion of ideas about the quantitative laws of the development of the world, which contributed to the development of mathematical, physical, astronomical and geographical knowledge. Numbers are the basis of things, Pythagoras taught, to know the world means to know the numbers that control it. By studying numbers, the Pythagoreans developed numerical relationships and found them in all areas of human activity. Numbers and proportions were studied in order to know and describe the human soul, and, having learned it, to manage the process of transmigration of souls with the ultimate goal of sending the soul to some higher divine state.

Despite the popular opinion that Pythagoras was supposedly a vegetarian, Diogenes Laeres writes that Pythagoras occasionally ate fish, abstained only from arable bulls and rams, and allowed other animals for food.

His contemporary Heraclitus acted as a critic of Pythagoras: “ Pythagoras, the son of Mnesarchus, was engaged in collecting information more than any other person in the world and, having taken these works for himself, passed off knowledge and fraud as his own wisdom". According to Diogenes Laertius, in the continuation of the famous saying of Heraclitus “Much knowledge does not teach the mind,” Pythagoras is mentioned among others: “otherwise it would have taught Hesiod and Pythagoras, as well as Xenophanes and Hecataeus.”

Scientific achievements

Coin with the image of Pythagoras

In the modern world, Pythagoras is considered the great mathematician and cosmologist of antiquity, but early evidence before the 3rd century. BC e. they do not mention such merits of his. As Iamblichus writes about the Pythagoreans: “ They also had the remarkable custom of attributing everything to Pythagoras and not at all arrogating to themselves the glory of discoverers, except perhaps in a few cases.»

In the 3rd century. BC e. a compilation of the sayings of Pythagoras appeared, known as the “Sacred Word”, from which the so-called “Golden Verses” later arose (sometimes they are attributed to the 4th century BC without good reason). These verses were first quoted by Chrysippus in the 3rd century. BC e. , although, perhaps, at that time the compilation had not yet developed into a finished form. The final excerpt from “Golden Verses” translated by I. Peter:

Be firm: the divine race is present in mortals,
To them, proclaiming, sacred nature reveals everything.
If this is not alien to you, you will carry out orders,
You will heal your soul and deliver you from many disasters.
Dishes, I said, leave those that I indicated in the cleansings
And be guided by true knowledge - the best charioteer.
If you, having left your body, ascend into the free ether,
You will become an incorruptible and eternal god who does not know death.

Notes

Sources and links

Iamblichus, On the Pythagorean Life
Diogenes Laertius, Pythagoras
Porphyry, Life of Pythagoras
“Golden Verses” of the Pythagoreans in the Library of Alexander Kobrinsky
Besonides, Pythagorean Word

Literature

Zhmud L.Ya. Pythagoras and the early Pythagoreans. M., 2012. - 445 p. ISBN 978-5-91244-068-7
Zhmud L. Ya. Pythagoras and his school. - M.: Nauka, 1990. - ISBN 5-02-027292-2
Zhmud L. Ya. Science, philosophy and religion in early Pythagoreanism. - St. Petersburg, 1994. - 376 p. - ISBN 5-86050-066-1
Fragments of early Greek philosophers. Part 1: From epic theocosmogonies to the emergence of atomism, Ed. A. V. Lebedev. - M.: Nauka, 1989. - p. 138-149.
Leontyev A.V. The tradition of Pythagoras among Aristoxenus and Dicaearchus // Man. Nature. Society. Actual problems. Proceedings of the 11th international conference of young scientists December 27-30, 2000 - St. Petersburg University Publishing House. 2000. - pp. 298-301.
Leontyev A.V. On the question of the image of Pythagoras in the ancient tradition of the 6th-5th centuries BC. e. // Mnemon. Research and publications on the history of the ancient world. Edited by Professor E. D. Frolov. - Issue 3. - St. Petersburg, 2004.
Panchenko D. V. The Pythagorean paradox // Indo-European linguistics and classical philology - XII: Materials of readings dedicated to the memory of prof. I. M. Tronsky June 23-25, 2008 pp. 355-363.
Sigachev A. A. Pythagoras (popular science essay) // Electronic magazine “Knowledge. Understanding. Skill ». - 2010. - No. 6 - History.

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.

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. Another law of the organization was the keeping of secrets, non-compliance with which was strictly punished - even death.

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.

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.

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.

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.

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.

Pythagoras- ancient Greek idealist philosopher, mathematician, founder of Pythagoreanism, political and religious figure. His homeland was the island of Samos (hence the nickname - Samos), where he was born around 580 BC. e. His father was a gem cutter. According to ancient sources, Pythagoras was distinguished by amazing beauty from birth; when he became an adult, he wore a long beard and a diadem of gold. His talent also showed itself at an early age.

Pythagoras's education was very good; the young man was taught by many mentors, among whom were Pherecydes of Syros and Hermodamant. The next place where Pythagoras improved his knowledge was Miletus, where he met Thales, a scientist who advised him to go to Egypt. Pythagoras had with him a letter of recommendation from the pharaoh himself, but the priests shared their secrets with him only after successfully passing difficult tests. Among the sciences that he mastered well in Egypt was mathematics. For the next 12 years he lived in Babylon, where the priests also shared their knowledge with him. According to legends, Pythagoras also visited India.

The return to their homeland took place around 530 BC. e. The status of half-court and half-slave under the tyrant Polycrates did not seem attractive to him, and he lived in caves for some time, after which he moved to Proton. Perhaps the reason for his departure lay in his philosophical views. Pythagoras was an idealist, a supporter of the slave-owning aristocracy, and in his native Ionia democratic views were very popular, their adherents had considerable influence.

In Croton, Pythagoras organized his own school, which was both a political structure and a religious monastic order with its own charter and very strict rules. In particular, all members of the Pythagorean Union were not supposed to eat meat, reveal the teachings of their mentor to others, and refused to have personal property.

The wave of democratic uprisings that swept through Greece and the colonies at that time also reached Croton. After the victory of democracy, Pythagoras and his students moved to Tarentum, and later to Metapontum. When they arrived in Metapontum, a popular uprising was raging there, and Pythagoras died in one of the night battles. Then he was a very old man, he was almost 90. Along with him, his school ceased to exist, the students were dispersed throughout the country.

Since Pythagoras considered his teaching a secret and practiced only oral transmission to his students, no collected works remained after him. Some information did become clear, but it is incredibly difficult to distinguish between truth and fiction. A number of historians doubt that the famous Pythagorean theorem was proven by him, arguing that it was known to other ancient peoples.

The name of Pythagoras has always been surrounded by a large number of legends, even during his lifetime. It was believed that he could control spirits, knew how to prophesy, knew the language of animals, communicated with them, birds, under the influence of his speeches, could change their flight vector. Legends also attributed to Pythagoras the ability to heal people, including with the help of an excellent knowledge of medicinal plants. His influence on those around him was difficult to overestimate. They tell the following episode from the biography of Pythagoras: when one day he became angry with a student, he committed suicide out of grief. Since then, the philosopher has made it a rule never to take out his irritation on people again.

In addition to proving the Pythagorean theorem, this mathematician is credited with a detailed study of integers, proportions and their properties. The Pythagoreans owe significant credit for giving geometry the character of a science. Pythagoras was one of the first who was convinced that the Earth is a ball and the center of the Universe, that the planets, the Moon, the Sun move in a special way, not like stars. To a certain extent, the ideas of the Pythagoreans about the movement of the Earth became the forerunner of the heliocentric teachings of N. Copernicus.

Biography from Wikipedia

The life story of Pythagoras is difficult to separate from the legends that present him as a perfect sage and great scientist, initiated into all the mysteries of the Greeks and barbarians. Herodotus also called him “the greatest Hellenic sage.” The main sources on the life and teachings of Pythagoras are the works of the Neoplatonist philosopher Iamblichus (242-306) “ About Pythagorean life"; Porphyria (234-305) " Life of Pythagoras"; Diogenes Laertius (200-250) book. 8, " Pythagoras" These authors relied on the writings of earlier authors, of which it should be noted that Aristotle's student Aristoxenus (370-300 BC) was from Tarentum, where the Pythagorean position was strong. Thus, the earliest known sources about the teachings of Pythagoras did not appear until 200 years after his death. Pythagoras himself did not leave any writings, and all information about him and his teachings is based on the works of his followers, who are not always impartial.

Pythagoras' parents were Mnesarchus and Parthenides from the island of Samos. Mnesarchus was a stone cutter (D. L.); according to Porphyry, he was a rich merchant from Tyre, who received Samian citizenship for distributing grain in a lean year. The first version is preferable, since Pausanias gives the genealogy of Pythagoras in the male line from Hippasus from the Peloponnesian Phlius, who fled to Samos and became the great-grandfather of Pythagoras. Parthenida, later renamed Pyphaida by her husband, came from the noble family of Ankeus, the founder of the Greek colony on Samos.

The birth of a child was allegedly predicted by Pythia in Delphi, which is why Pythagoras got his name, which means “ the one announced by the Pythia" In particular, Pythia told Mnesarchus that Pythagoras would bring as much benefit and goodness to people as no one else had brought or would bring in the future. Therefore, to celebrate, Mnesarchus gave his wife a new name, Pyphaidas, and his child, Pythagoras. Pyphaida accompanied her husband on his travels, and Pythagoras was born in Sidon Phoenician (according to Iamblichus) around 570 BC. e. From an early age he discovered extraordinary talent (also according to Iamblichus).

At a young age, Pythagoras went to Egypt to gain wisdom and secret knowledge from the Egyptian priests. Diogenes and Porphyry write that the Samian tyrant Polycrates provided Pythagoras with a letter of recommendation to Pharaoh Amasis, thanks to which he was allowed to study and was initiated not only into the Egyptian achievements of medicine and mathematics, but also into the sacraments forbidden to other foreigners.

Iamblichus writes that Pythagoras left his native island at the age of 18 and, having traveled around the sages in different parts of the world, reached Egypt, where he stayed for 22 years, until he was taken to Babylon as a captive by the Persian king Cambyses, who conquered Egypt in 525 BC. . e. Pythagoras stayed in Babylon for another 12 years, communicating with magicians, until he was finally able to return to Samos at the age of 56, where his compatriots recognized him as a wise man.

According to Porphyry, Pythagoras left Samos due to disagreement with the tyrannical power of Polycrates at the age of 40. Since this information is based on the words of Aristoxenus, a source of the 4th century BC. e., are considered relatively reliable. Polycrates came to power in 535 BC. e., hence the date of birth of Pythagoras is estimated at 570 BC. e., if we assume that he left for Italy in 530 BC. e. Iamblichus reports that Pythagoras moved to Italy in the 62nd Olympiad, that is, in 532-529. BC e. This information is in good agreement with Porphyry, but completely contradicts the legend of Iamblichus himself (or rather, one of his sources) about the Babylonian captivity of Pythagoras. It is not known for sure whether Pythagoras visited Egypt, Babylon or Phenicia, where, according to legend, he acquired eastern wisdom. Diogenes Laertius quotes Aristoxenus, who said that Pythagoras received his teaching, at least as regards instructions on the way of life, from the priestess Themistocleia of Delphi, that is, in places not so remote for the Greeks.

Disagreements with the tyrant Polycrates could hardly have been the reason for Pythagoras’s departure; rather, he needed the opportunity to preach his ideas and, moreover, to put his teaching into practice, which was difficult to do in Ionia and mainland Hellas, where many people experienced in matters of philosophy and politics lived. Iamblichus reports:

« His philosophy spread, all of Hellas began to admire him, and the best and wisest men came to him on Samos, wanting to listen to his teaching. His fellow citizens, however, forced him to participate in all embassies and public affairs. Pythagoras felt how difficult it was, obeying the laws of the fatherland, to simultaneously engage in philosophy, and saw that all the previous philosophers lived their lives in foreign lands. Having thought all this over, withdrawing from public affairs and, as some say, considering the low appreciation of his teachings by the Samians insufficient, he left for Italy, considering his fatherland a country where there were more people capable of learning.»

Pythagoras settled in the Greek colony of Crotone in southern Italy, where he found many followers. They were attracted not only by the mystical philosophy that he convincingly expounded, but also by the way of life he prescribed with elements of healthy asceticism and strict morality. Pythagoras preached the moral ennoblement of the ignorant people, which can be achieved where power belongs to a caste of wise and knowledgeable people, and to whom the people obey in some ways unconditionally, like children to their parents, and in other respects consciously, submitting to moral authority. Tradition ascribes to Pythagoras the introduction of the words philosophy and philosopher.

The disciples of Pythagoras formed a kind of religious order, or brotherhood of initiates, consisting of a caste of selected like-minded people who literally deified their teacher, the founder of the order. This order actually came to power in Crotone, but due to anti-Pythagorean sentiments at the end of the 6th century. BC e. Pythagoras had to retire to another Greek colony, Metapontus, where he died. Almost 450 years later, during the time of Cicero (1st century BC), the crypt of Pythagoras was shown in Metaponte as one of the attractions.

Pythagoras had a wife named Theano, a son Telaugus and a daughter Miya (according to another version, a son Arimnest and a daughter Arignot).

According to Iamblichus, Pythagoras led his secret society for thirty-nine years, then the approximate date of Pythagoras' death can be attributed to 491 BC. e., to the beginning of the era of the Greco-Persian wars. Diogenes, referring to Heraclides (IV century BC), says that Pythagoras died peacefully at the age of 80, or at 90 (according to other unnamed sources). This implies the date of death is 490 BC. e. (or 480 BC, which is unlikely). Eusebius of Caesarea in his chronography designated 497 BC. e. as the year of Pythagoras' death.

Defeat of the Pythagorean League

Among the followers and students of Pythagoras there were many representatives of the nobility who tried to change the laws in their cities in accordance with Pythagorean teaching. This was superimposed on the usual struggle of that era between the oligarchic and democratic parties in ancient Greek society. The discontent of the majority of the population, who did not share the ideals of the philosopher, resulted in bloody riots in Croton and Tarentum.

« The Pythagoreans formed a large community (there were more than three hundred of them), but it constituted only a small part of the city, which was no longer governed according to the same customs and mores. However, while the Crotonians owned their land, and Pythagoras was with them, the state structure that existed from the foundation of the city was preserved, although there were dissatisfied people who were waiting for an opportunity for a coup. But when they conquered Sybaris, Pythagoras left, and the Pythagoreans who ruled the conquered land did not distribute it by lot, as the majority wanted, then hidden hatred flared up, and many citizens opposed them... The relatives of the Pythagoreans were even more irritated by what they were serving right hand only to their own, and from relatives - only to parents, and that they provide their property for common use, and it is separated from the property of relatives. When the relatives began this hostility, the rest readily joined the conflict... After many years... the Crotonians were overcome by regret and repentance, and they decided to return to the city those Pythagoreans who were still alive.»

Philosophical teaching

Pythagoras in a fresco by Raphael (1509)

Quite complete information about the ideas about the transmigration of souls developed by Pythagoras and the food prohibitions based on them is given by Empedocles’ poem “Purifications”.

In his surviving works, Aristotle never directly addresses Pythagoras directly, but only to “the so-called Pythagoreans.” In lost works (known from excerpts), Aristotle views Pythagoras as the founder of a semi-religious cult that forbade the eating of beans and had a golden thigh, but did not belong to the sequence of thinkers who preceded Aristotle.

Plato treated Pythagoras with the deepest reverence and respect. When the Pythagorean Philolaus first published 3 books outlining the main principles of Pythagoreanism, Plato, on the advice of friends, immediately bought them for a lot of money.

The activity of Pythagoras as a religious innovator of the 6th century. BC e. was to create a secret society that not only set itself political goals (because of which the Pythagoreans were defeated in Croton), but mainly the liberation of the soul through moral and physical purification with the help of secret teaching (mystical teaching about the cycle of migration of the soul). According to Pythagoras, the eternal soul moves from heaven into the mortal body of a person or animal and undergoes a series of migrations until it earns the right to return back to heaven.

The acusmata (sayings) of Pythagoras contain ritual instructions: about the cycle of human lives, behavior, sacrifices, burials, nutrition. Akusmats are formulated succinctly and understandably for any person; they also contain postulates of universal morality. A more complex philosophy, within the framework of which mathematics and other sciences developed, was intended for “initiates,” that is, selected people worthy of possessing secret knowledge. The scientific component of Pythagoras' teachings developed in the 5th century. BC e. through the efforts of his followers (Architas from Tarentum, Philolaus from Croton, Hippasus from Metapontus), but came to naught in the 4th century. BC e., while the mystical-religious component received its development and rebirth in the form of neo-Pythagoreanism during the Roman Empire.

As I. D. Rozhansky noted: “Despite the remnants of magical thinking, the basic idea of Pythagoras that all things are based on numbers or ratios of numbers turned out to be very fruitful.” As Stobaeus noted: “Apparently, Pythagoras revered the science of numbers most of all (sciences), he advanced it forward, taking it beyond its use in trade and expressing it, modeling all things with numbers” (1, “Proemius”, 6, p. . 20).

Despite the popular opinion that Pythagoras was supposedly a vegetarian, Diogenes Laertius writes that Pythagoras occasionally ate fish, abstained only from arable bulls and rams, and allowed other animals for food.

His contemporary Heraclitus acted as a critic of Pythagoras: “ Pythagoras, the son of Mnesarchus, was engaged in collecting information more than any other person in the world and, having taken these works for himself, passed off knowledge and fraud as his own wisdom“According to Diogenes Laertius, in the continuation of the famous saying of Heraclitus “Much knowledge does not teach the mind,” Pythagoras is mentioned among others: “otherwise it would have taught Hesiod and Pythagoras, as well as Xenophanes and Hecataeus.”

Scientific achievements

Ancient authors of our era give Pythagoras the authorship of the famous theorem: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. This opinion is based on the information of Apollodorus the calculator (personality not identified) and on poetic lines (the source of the poems is unknown):

“On the day when Pythagoras discovered his famous drawing,
He erected a glorious sacrifice for him with bulls.”

Modern historians suggest that Pythagoras did not prove the theorem, but could have conveyed this knowledge to the Greeks, known in Babylon 1000 years before Pythagoras (according to Babylonian clay tablets recording mathematical equations). Although there is doubt about the authorship of Pythagoras, there are no weighty arguments to dispute this.

Aristotle touches on the development of ideas about cosmology in his work “Metaphysics”, but the contribution of Pythagoras is not voiced in it. According to Aristotle, the Pythagoreans studied cosmological theories in the middle of the 5th century. BC e., but, apparently, not Pythagoras himself. Pythagoras is credited with the discovery that the Earth is a sphere, but the most authoritative author on this matter, Theophrastus, gives the same discovery to Parmenides. And Diogenes Laertius reports that the opinion about the sphericity of the Earth was expressed by Anaximander of Miletus, with whom Pythagoras studied in his youth.

At the same time, the scientific merits of the Pythagorean school in mathematics and cosmology are indisputable. Aristotle’s point of view, reflected in his unpreserved treatise “On the Pythagoreans,” was conveyed by Iamblichus. According to Aristotle, the true Pythagoreans were the acousmatists, followers of the religious-mystical doctrine of the transmigration of souls. Acousmaticians viewed mathematics as a teaching coming not so much from Pythagoras as from the Pythagorean Hippasus. In turn, the Pythagorean mathematicians, in their own opinion, were inspired by the guiding teachings of Pythagoras for an in-depth study of their science.

Works of Pythagoras

Pythagoras did not write treatises. It was impossible to compile a treatise from oral instructions for the common people, and secret occult teaching for the elite could not be entrusted to a book. Iamblichus comments on the absence of Pythagoras' works:

« Their persistence in keeping their teachings secret is also remarkable: for so many years before the generation of Philolaus, it seems that no one had encountered a single Pythagorean work. Philolaus was the first of the Pythagoreans to publish three sensational books, which, it is said, Dion of Syracuse bought for a hundred minas at the direction of Plato, when Philolaus fell into extreme need.»

Diogenes lists the titles of these books attributed to Pythagoras: “On Education,” “On the State,” and “On Nature.” However, none of the authors in the first 200 years after the death of Pythagoras, including Plato, Aristotle and their successors at the Academy and Lyceum, quote from the works of Pythagoras or even indicate the existence of such works. Since the beginning of the new era, the works of Pythagoras have been unknown to ancient writers, as Plutarch, Josephus and Galen reported.

In the 3rd century. BC e. a compilation of the sayings of Pythagoras appeared, known as the “Sacred Word”, from which the so-called “Golden Verses” later arose (sometimes they are attributed to the 4th century BC without good reason). These verses were first quoted by Chrysippus in the 3rd century. BC e., although, perhaps, at that time the compilation had not yet developed into a finished form. The final excerpt from “Golden Verses” translated by I. Peter:

Be firm: the divine race is present in mortals,
To them, proclaiming, sacred nature reveals everything.
If this is not alien to you, you will carry out orders,
You will heal your soul and deliver you from many disasters.
Dishes, I said, leave those that I indicated in the cleansings
And be guided by true knowledge - the best charioteer.
If you, having left your body, ascend into the free ether,
You will become an incorruptible and eternal god who does not know death.

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 great 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 has the shape of a ball 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 the concepts of Pythagoras, bloodshed was equated, neither more nor less, 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 their hands, 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

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 values 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 the 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 A.K.J.- 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 will 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 in Wikiquote
	Pythagoras on Wikimedia Commons

When did Pythagoras live? Pythagoras - biography, information, personal life

Personal life of Pythagoras

School of Pythagoras

Pythagorean theorems

Death of Pythagoras

Biography

Philosophical teaching

Scientific achievements

Notes

Sources and links

Literature

see also

Biography from Wikipedia

Defeat of the Pythagorean League

Philosophical teaching

Scientific achievements

Works of Pythagoras

History of the theorem

History of the theorem

Caricatures

Proof No. 1 (simplest)

Evidence No. 2

Evidence No. 3

Evidence No. 4

Evidence No. 5

Construction

Window

Roof

Lightning rod

Astronomy

mobile connection

Introduction