DISCOVERIES OF PYTHAGORUS

Pythagoras of Samos, ancient Greek philosopher, great initiate of the Earth, political and religious figure, mathematician, founder of Pythagoreanism. His main life concept is “Everything is a Number.” This is usually indicated in encyclopedias and his biographies.

But who Pythagoras was, who is now and who Pythagoras will be in the future remains a cosmic Mystery...

He is a most brilliant scientist, a great dedicated philosopher, a sage, the founder of the famous Pythagorean school and the spiritual teacher of a number of outstanding philosophers of world renown. Pythagoras became the founder of the teachings about Numbers, the Music of the celestial spheres and the Cosmos, and created the basis of monadology and the quantum theory of the structure of matter. He made discoveries of great importance in the field of such sciences as mathematics, music, optics, geometry, astronomy, number theory, superstring theory (Earthly monochord), psychology, pedagogy, ethics.

Pythagoras developed his philosophy on the basis of knowledge of the laws of the interrelations of the visible and invisible world, the unity of spirit and matter, on the concept of the immortality of the soul and its gradual purification through transmigration (the theory of incarnation). Many legends are associated with the name of Pythagoras, and his students were able to win fame for themselves and became outstanding people, thanks to whose works we became aware of the fundamentals of the teachings of Pythagoras, his sayings, practical and ethical advice, as well as the theoretical postulates and spiritual tales of Pythagoras.

Perhaps not every one of us can remember the Pythagorean theorem, but everyone knows the saying “Pythagorean pants are equal on all sides.” Pythagoras, among other things, was a rather cunning man. The great scientist taught all his Pythagorean students a simple tactic that was very beneficial for him: if you made discoveries, attribute them to your teacher. This may be a rather controversial judgment, but it is thanks to his students that Pythagoras is credited with a truly incredible number of discoveries:

In geometry: the famous and beloved Pythagorean theorem, as well as the construction of individual polyhedra and polygons.

In geography and astronomy: he was one of the first to express the hypothesis that the Earth is round, and also believed that we are not alone in the Universe.

In music: determined that sound depends on the length of the flute or string.

In numerology: in our time, numerology has become famous and quite popular, but it was Pythagoras who combined numbers with predictions for the future.

Pythagoras taught that both the beginning and the end of everything that exists lies in a certain abstract quantity, the so-called Monad. It represents the unknowable absolute emptiness, chaos, the ancestral home of all gods and at the same time contains the fullness of existence in the form of divine Light. The Monad, like ether, permeates all things, but is not located in any one of them. This is the sum of all numbers, which is always considered as an indivisible whole, like a unit.

The Pythagoreans depicted the Monad as a figure that consists of ten points - the so-called nodes. All these ten knots, called tetractys by the Pythagoreans, create nine equilateral triangles, which personify the fullness of universal emptiness and the Life-giving Cross.

It is also believed that Pythagoras created the foundations of planimetry, introduced the widespread and mandatory use of evidence in geometry, and created the doctrine of similarity.

Pythagoras made all these discoveries more than two and a half thousand years ago! The discoveries of Pythagoras, like his faithful disciples, live and will live in the future.

History of Pythagorean Theorem

The great discoveries of Pythagoras the mathematician found their application at different times and around the world. This applies to the greatest extent to the Pythagorean theorem.

For example, in China Special attention in this regard, one should pay attention to the mathematical book Chu-pei, which says this about the famous Pythagorean triangle, which has sides 3, 4, 5: “If you decompose a right angle into its component parts, then the line connecting the ends of all its sides will be 5, then as the base will be 3 and the height 4.” The same book shows a drawing that is similar to one of the drawings in the Hindu geometry of Bashara.

The outstanding German researcher of the history of mathematics Cantor believes that the Pythagorean equality 3? + 4? = 5? already known in Egypt around 2300 BC. BC, during the reign of King Amenemhat I (according to papyrus 6619 of the Berlin Museum). According to Kantor, the harpedonapts, or the so-called “rope pullers,” built right angles using right triangles, the sides of which were 3, 4, 5. Their construction method is quite easily reproduced. If you take a piece of rope 12 m long, tie colored strips to it - one at a three-meter distance from one end, and the other 4 meters from the other, then a right angle will be enclosed between the two sides - 3 and 4 meters. One can object to the harpedonapts that this method of construction would be superfluous if we take, for example, the wooden triangle that all carpenters use. Indeed, there are Egyptian drawings, for example, depicting a carpenter's workshop, in which such a tool is found. But nevertheless, the fact remains that the Pythagorean triangle was used in ancient Egypt.

Little more information is available about the Pythagorean theorem used by the Babylonians. In the found text, which dates back to the time of Hammurabi, which is 2000 BC. e., there is an approximate definition of the hypotenuse of a right triangle. Consequently, this confirms that calculations with the sides of right triangles were already carried out in Mesopotamia, at least in some cases. Mathematician Van der Waerden from Holland, on the one hand, using the current level of knowledge about Babylonian and Egyptian mathematics, and on the other, based on a careful study of Greek sources, came to the following conclusions: “The merit of the first Greek mathematicians: Thales, Pythagoras and the Pythagoreans – not the discovery of mathematics, but its justification and systematization. They were able to turn computational recipes based on vague ideas into an exact science.”

Among the Hindus, along with the Babylonians and Egyptians, geometry was closely associated with cult. It is quite possible that the Pythagorean theorem was known in India already in the 18th century BC. e.

The “List of Mathematicians,” which Eudemus supposedly compiled, speaks of Pythagoras as follows: “Pythagoras reportedly turned the study of this branch of knowledge (geometry) into a real science, having analyzed its foundations with highest point vision and exploring her theories in a more mental and less material way."

Tree of Pythagoras

The Pythagorean tree is a type of fractal that is based on a figure known as Pythagorean Pants.

Proving his famous theorem, Pythagoras constructed a figure in which there were squares on each side of a right triangle. Over time, this figure of Pythagoras turned into a whole tree. The first to construct the Pythagorean tree during the Second World War was A. Bosman, using an ordinary drawing ruler.

One of the main properties of the Pythagorean tree is that when the area of the first square is one, then at each level the sum of the areas of the squares will also be equal to one. The classic Pythagorean tree has an angle of 45 degrees, but it is also possible to construct a generalized Pythagorean tree using other angles. Such a tree is called the wind-blown tree of Pythagoras. If you draw only the segments that somehow connect certain “centers” of the triangles, then you get a naked Pythagorean tree.

The Pythagorean tree is a fractal generated as follows:

Start with a unit square. Then, selecting one of its sides as the base (in the animation, the bottom side is the base):

Construct a right triangle on the side opposite the base with the hypotenuse coinciding with this side and the aspect ratio 3:4:5. Note that the smaller leg should be to the right relative to the base (see animation).

On each side of a right triangle, construct a square with a side coinciding with this side.

Repeat this procedure for both squares, counting the sides touching the triangle as their bases.

The figure obtained after an infinite number of iterations is a Pythagorean tree.

Pythagoras of Samos(lat. Pythagoras; 570 - 490 BC BC) - ancient Greek philosopher and mathematician, creator of the religious and philosophical school of the Pythagoreans.

The life story of Pythagoras is difficult to separate from the legends representing Pythagoras as a demigod and miracle worker, a perfect sage and a great initiate into all the mysteries of the Greeks and barbarians. Herodotus also called him “the greatest Hellenic sage” (4.95). The main sources on the life and teachings of Pythagoras are the works that have come down to us: the Neoplatonist philosopher Iamblichus (242-306) “On the Pythagorean Life”; Porphyry (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 wrote about Pythagoras 200 years after his death, and Pythagoras himself did not leave his own written works, and all information about him and his teaching is based on the works of his students, who were not always impartial.

Biography

Pythagoras' parents were Mnesarchus and Parthenides from 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. 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 supposedly predicted by the Pythia in Delphi, which is why Pythagoras got his name, which means “the one whom the Pythia announced.” Parthenis 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.

Iamblichus writes that Pythagoras, at the age of 18, left his native island 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 the legends, he gained 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.

Pythagoras settled in the Greek colony of Crotone in southern Italy, where he found many followers. They were attracted not only by the occult philosophy, which 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 the caste of the wise and knowledgeable people, and to which the people obey unconditionally in some ways, like children to their parents, and in other respects consciously, submitting to moral authority. 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 and founder. 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 Metaponto as one of the attractions.

Pythagoras had a wife named Theano, a son Telaugus and a daughter.

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 Order

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 the Pythagorean doctrine. 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.

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 it ended in complete failure political structure and morals according to the requirements of abstract theory."

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-occult 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.

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 in exactly the same way as Aristotle, and mentions Pythagoras only once as the founder of a peculiar way of life.

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 circulation 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.

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, they developed numerical relationships and found them in all areas human activity. Numbers and proportions were studied in order to know and describe the human soul, and, having learned, to manage the process of transmigration of souls from ultimate goal send the soul to some higher divine state.

Scientific achievements

IN 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 a wonderful custom of attributing everything to Pythagoras and not at all taking upon themselves the glory of discoverers, except perhaps in a few cases.”

Ancient authors of our era (Diogenes Laertius; Porphyry; Athenaeus (418f); Plutarch (collection "Moralia", 1094b)) give Pythagoras the authorship of the famous theorem: the square of the hypotenuse of a 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.

In the same time, scientific merits The Pythagorean schools of mathematics and cosmology are indisputable. Aristotle's point of view, reflected in his unpreserved treatise "On the Pythagoreans", was conveyed by Iamblichus ("On General Mathematical Science", 76.19 ff). 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, according to them own opinion, were inspired by the guiding teachings of Pythagoras for in-depth study his 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.

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.

In the 3rd century. BC e. a compilation of Pythagoras' sayings 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 taken shape in its finished form.

Municipal budgetary educational institution

average comprehensive school № 91

with in-depth study of individual subjects

Leninsky district of Nizhny Novgorod

Students' Scientific Society

Pythagoras and his discoveries.

Completed by: Alexey Vorozheikin,

7th grade student

Scientific adviser:

mathematic teacher

N. Novgorod

INTRODUCTION 4

CHAPTER 1. RESEARCH METHOD.. 4

CHAPTER 2. PYTHAGORUS. 4

2.1. Childhood. 4

2.2. Teachers. 4

2.3. School of Pythagoreans. 4

2.4. Last years.. 4

CHAPTER 3. TEACHINGS OF PYTHAGORUS.. 4

3.1. Pythagoras is a philosopher. 4

3.2. Pythagoras is a mathematician. 4

3.3. Music and Pythagoras. 4

3.4. Pythagoras about space. 4

CHAPTER 4. SYMBOLS IN THE PICTURE. 4

4.1.Tetractys of Pythagoras. 4

4.2. Pyramid. 4

4.3. Globe. 4

4.4. Lyra. 4

4.5.Drawings of Pythagoras. 4

4.6. Tools..4

4.7. Pythagorean pants.. 4

CHAPTER 5. PYTHAGOREAN THEOREM.. 4

5.1. History of the Pythagorean theorem. 4

5.2. Pythagorean theorem in a school geometry course. 4

5.3. Why pants? 4

5.4. Additional proofs of the Pythagorean theorem. 4

CONCLUSION. 4

INTRODUCTION

On the Internet I found a picture where Pythagoras was depicted surrounded by various geometric bodies, objects and some symbols of unknown origin. I became interested in finding out what they are and why they are present in the picture, so I decided to start searching for information. I set myself the following goals:

1. Find out what the symbols and objects (No.) in the found painting mean and how they are connected with Pythagoras.

2. Find out where the comic formulation of the theorem “Pythagorean pants are equal on all sides” came from and how it is related to the well-known theorem from a school geometry course.

Of course, already at the beginning of my work I had hypotheses:

Hypothesis 1. Most likely, this joke was related to the proof of the theorem, because the proofs could be different. It could contain squares (all sides are equal) as a way to prove the theorem.

With the picture, things were a little more complicated. I couldn’t even imagine what the symbols under No. meant, although it is clear that the symbols carry some meaning; the artist must have carefully thought through the setting in which he depicted Pythagoras.

Hypothesis 2. The symbols in the picture are somehow connected with the activities of Pythagoras the mathematician, with his discoveries.

To achieve my goals, I had to solve the following tasks:

1. Familiarize yourself with the biography of Pythagoras, find out what discoveries he made.

2. Find alternative proofs of the Pythagorean theorem.

CHAPTER 1. RESEARCH METHOD

The main research method was the search, analysis and comparison of information from various sources. First, I conducted a survey at my school on the following questions: 1. Who is Pythagoras? 2. What discoveries did he make? 3. What do the objects surrounding Pythagoras in the picture mean (the picture was attached to the questionnaire). The purpose of the survey was to identify the level of awareness of students and teachers about Pythagoras. This would allow me to obtain the necessary information and find out the relevance of my project. The results of the survey were as follows:

The vast majority of students (80%) know about Pythagoras only that he was a mathematician. Only a few of the students 15 years and older answered that he was a philosopher and lived in Ancient Greece. Of Pythagoras' discoveries, students under 12 years old only know the multiplication table, but all students over 15 years old wrote that he proved the Pythagorean theorem. The vast majority of students (over 90%) do not know about the symbols in the picture. Only a few students over 17 explained the meaning of some objects.

Teachers know much better than students. All teachers know about the Pythagorean theorem, in addition, 30% wrote that Pythagoras proved the theorem on the sum of the angles of a triangle. However, in general, very little is known about Pythagoras among the students and teachers of our school, so this project will have educational value for everyone.

CHAPTER 2. PYTHAGORUS

2.1. Childhood

Little is known reliably about the youthful life of Pythagoras. He was born around 580 BC. e. on the island of Samos in the family of a stone carver who was quite famous. Pythagoras was a very inquisitive child, so he asked visiting sailors about other countries. When he grew a little, it became cramped for him small island, which he climbed up and down, and Pythagoras left Samos.

2.2. Teachers

In search of new knowledge, Pythagoras came to the island of Miletus to visit the sage Thales, who was already more than seventy years old. He studied mathematics with him, and when he had learned everything, Thales advised Pythagoras to go to Egypt, where he himself once received knowledge.

In Egypt, Pythagoras became a student of the Egyptian priests, and for a long time He studied various sciences with them, including geometry. When Pythagoras studied everything, he wanted to return to Greece. However, conservative Egyptian priests did not want to spread their knowledge beyond the temples, and tried to interfere with Pythagoras, who had to make a lot of efforts to leave Egypt.

Pythagoras left Egypt, but on the way he was captured by the Persians and did not reach Greece. As they say, out of the frying pan and into the fire. Pythagoras was brought to Babylon, whose monumental buildings greatly impressed the scientist: tall houses were not built in Greece. The Babylonians valued smart people, so Pythagoras quickly found a use for himself. He became a student of Babylonian magicians and sages, from whom he studied mathematics, astronomy, and various mystical sciences for a long time. After living for a long time in Babylon, Pythagoras returned to Greece.

2.3. Pythagorean school

Upon returning to his homeland, Pythagoras, driven by a thirst for activity, decides to create his own school. This is how the Pythagorean Union appeared, but in essence it was more of a sect, since the Pythagorean Union was a kind of religious movement. Only an aristocrat could become a member of the union. A very limited number of members were accepted into the union, and a huge number of rituals were invented for admission, for example, the initiate had to remain silent for five years and listen to the wisest Pythagoras from behind the curtain, without seeing his face, since he was unworthy to see the great and terrible Pythagoras until his spirit is properly cleansed. The main ideology of the Pythagoreans was the numerical philosophy that Pythagoras created.

Also, the Pythagoreans had their own secret symbols, they were the tetractys and the pentagram.

The snobbery and contempt of the Pythagoreans for the common people contradicted the democratic trends that prevailed at that time in Samosea, so the Greeks, offended by the neglect, defeated the Pythagorean union, and Pythagoras fled from the island.

2.4. Last years

Being already a very old man, Pythagoras settled in the city of Crotone, where he was able to revive his union of the Pythagoreans. However, the fate of Pythagoras himself and his union had a sad end. Past experience with mistakes has taught them nothing. They have not moved one step away from their past beliefs. In the Pythagorean league, everyone was aristocrats, and in their hands was the government of Croton. However, democratic trends were already gaining momentum in Crotona, where all free thought was suppressed, and ultimately all this led to a popular uprising. The anger of the crowd was directed precisely against Pythagoras and his supporters. Pythagoras decided to flee the city, but this did not help him. While in the city of Meraponte, 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, which he won in his youth, and all his magical skills did not help.

CHAPTER 3. TEACHINGS OF PYTHAGORE

3.1. Pythagoras - philosopher

Of course, Pythagoras came to us as a mathematician, but he was more of a philosopher. The basic concepts of Pythagoras' philosophy are extremely difficult to understand. However, there is a foundation on which he subsequently built all his teaching. Pythagoras was the first to suggest that everything that exists can be expressed in numbers or proportions, since numbers are not just designations of objects, but living entities. The philosophy of Pythagoras was an unimaginable fusion of mathematics, music and pagan religion. The philosophy of Pythagoras is so confusing that researchers have been trying to understand it for 2000 years. It is impossible to reveal all the elements of his teaching in one essay, so its main sections are given below.

The main branch of Pythagorean philosophy was numerology, which was created by Pythagoras. “Everything is a number,” he said. The main concept of Pythagoras' numerical theory, in addition to number, is the monad. The monad (from Greek unit, one) is multifaceted - it is both the unity of everything and the sum of combinations of numbers considered as a whole. The monad was compared to the seeds of a tree that has grown into many branches. Branches are like numbers - they relate to the seed of the tree in the same way that numbers relate to the monad. The Universe is also considered as a Monad. Apparently, one of the symbols of the picture (symbol No. 8) is the monad, as an integral component of the Pythagorean philosophy.

So, what is the basis of the Pythagorean number system? Numbers can be even or odd; If an odd number is divided into two parts, one will be even and the other will be odd (7=4+3). When dividing an even number, both parts obtained will be either even or odd (8=4+4, 8=5+3). A special mathematical procedure divides odd numbers into three classes: composite, non-composite, non-composite-composite.

Composite numbers include those that are divisible by themselves, by one, and by some other numbers. These are 9, 15, 21, 27, 33, etc.

Non-composite numbers are those numbers that are divisible only by themselves or by one. These are 3, 5, 7, 11, 13, 17, 19, 23, etc. Divisible numbers that do not have a common divisor are classified as non-composite. It's 9.25.

Even numbers are also divided into three classes: even-odd, even-even and odd-even. There is another division of even numbers - into perfect, superperfect and imperfect. In order to determine which of these classes a number belongs to, it must be divided into parts from the first ten and into the whole itself. The result should be whole numbers, not fractions. If the sum of the parts of a number is equal to the whole, then we can say that the number is perfect.

For example, six. Half of it is a three, the third is a two. Dividing six by itself gives one. Adding these parts together we get the integer six. Therefore, six is a perfect number. Superperfect numbers are those whose sum of parts exceeds the whole. For example, the number is 18. Half of it is 9, a third is 6, one sixth is 3, one ninth is 2, one eighteenth is 1. The total is 21, i.e. more than the whole. Therefore, the number 18 is super perfect.

Imperfect numbers are those numbers whose sum of parts is less than the whole. This is, for example, the number 8.

It was the science of numbers that was the basis of Pythagorean philosophy. Perfect numbers were a symbol of virtue, representing the mean between deficiency and excess. Virtues are rare, and perfect numbers are just as rare. Imperfect numbers are an example of vices.

However, the topic of Pythagoras' philosophy would be incomplete without mentioning Pythagoras' philosophy of music. Pythagoras was admitted to the so-called Mysteries - secret meetings of priests and magicians. Apparently, the philosophy of Pythagoras was largely based on the teachings of the priests of the Mysteries. They say that Pythagoras was not a musician, but it is he who is credited with the discovery of the diatonic scale. Having received basic information about the divine theory of music from the priests of the various Mysteries, Pythagoras spent several years pondering the laws governing consonance and dissonance. How he actually found the solution is unknown to us, but there is the following explanation.

One day, while pondering the problems of harmony, Pythagoras passed by the workshop of a coppersmith, who was bending over an anvil with a piece of metal. By noticing the difference in tones between the sounds produced by various hammers and other instruments when striking metal, and by carefully assessing the harmonies and disharmonies resulting from the combination of these sounds, Pythagoras received the first clue to the concept of musical interval on the diatonic scale. He entered the workshop and, after carefully examining the tools and applying their weight in his mind, returned to own house, constructed a beam that was attached to the wall, and attached four strings to it at regular intervals, identical in everything. To the first of them he attached a weight of twelve pounds, to the second - nine, to the third - eight, and to the fourth - six pounds. These different weights corresponded to the weight of the coppersmith's hammers.

Pythagoras discovered that the first and fourth strings, when sounded together, gave a harmonic interval of an octave, because doubling the weight had the same effect as shortening the string by half. The tension on the first string was twice that of the fourth string, and the ratio is said to be 2:1, or double. By similar reasoning, he came to the conclusion that the first and third strings give the harmony of diapente, or fifth. The tension of the first string was one and a half times greater than the third string, and their ratio was 3:2, or one and a half. Continuing this research, Pythagoras discovered that the first and second strings give the harmony of the third, the tension of the first string is one third greater than the second, their ratio is 4:3. The third and fourth strings, having the same ratio as the first and second, give the same harmony.

The key to the harmonic relationship is hidden in the famous Pythagorean tetractys, or pyramid of dots or commas (figure No. 1 in the picture). Tetractys is formed from the first four numbers: 1, 2, 3, 4, which in their proportions open the intervals of octave, diapente and diatessaron. Although the theory of harmonic intervals stated above is correct, hammers striking metal in the manner described above do not produce the tones that are attributed to them. In all likelihood, Pythagoras developed his theory of harmony by working with a monochord (an invention consisting of a single string stretched between clamps and equipped with movable frets). For Pythagoras, music was derived from the divine science of mathematics, and its harmonies were cruelly controlled by mathematical proportions. The Pythagoreans argued that mathematics demonstrated the precise method by which God established and established the universe. Numbers, therefore, precede harmony, since their immutable laws govern all harmonic proportions. After the discovery of these harmonic relationships, Pythagoras gradually initiated his followers into this teaching, as into the highest secret of his Mysteries. He divided the multiple parts of creation into big number planes or spheres, to each of which he assigned tone, harmonic interval, number, name, color and form. He then proceeded to demonstrate the accuracy of his deductions, demonstrating them on various planes of mind and substance, from the most abstract logical premises to the most concrete geometric solids. From the general fact of the consistency of all these different methods of proof, he established the unconditional existence of certain natural laws. Thus, for Pythagoras, no thing was just a thing; everything, in his opinion, had a certain essence.

3.2. Pythagoras - mathematician

Pythagoras is responsible, in addition to the famous theorem, for many more mathematical discoveries. Based on the numerology of Pythagoras, such a science as number theory later appeared. Pythagoras also made discoveries:

1) sum theorems internal corners triangle;

2) construction of regular polygons and division of the plane into some of them;

3) geometric methods for solving quadratic equations;

4) dividing numbers into even and odd, simple and composite; introduction of figured, perfect and friendly numbers;

5) discovery of irrational numbers.

In the Pythagorean Union, all discoveries were attributed to Pythagoras, so now no one can determine which discoveries were made by Pythagoras and which by his students. ,

3.3. Music and Pythagoras

As already mentioned, Pythagoras considered music the most important element human life. Pythagoras owns the doctrine of the therapeutic effect of music. He did not hesitate about the influence of music on the mind and body, calling it “musical medicine.” He believed “that music greatly contributes to health if used in the appropriate modes, since the human soul and the whole world as a whole have a musical-numerical basis.”

In the evenings, choral singing took place among the Pythagoreans, accompanied by stringed instruments. “When going to bed, the Pythagoreans freed their minds from the end of the day with some special melodies and in this way ensured themselves a restful sleep, and when they got up from sleep, they relieved sleepy lethargy and numbness with the help of another kind of melodies.

Pythagoras also influenced sick people with music and singing, thus treating some diseases, however, whether this is true cannot be understood now.

Pythagoras classified the melodies used for treatment according to diseases and had his own musical recipe for each disease. It is known that Pythagoras gave a clear preference to strings musical instruments and warned his students not to listen, even fleetingly, to the sounds of the flute and cymbals, since, in his opinion, they sound harsh, solemnly mannered and somewhat undignified.

3.4. Pythagoras on space

Pythagoras thought a lot about the structure of the universe; he is the creator of a special relationship between geometric bodies and the structure of the universe. Pythagoras revealed the relationship between figures and elements. The tetrahedron (pyramid) represented fire, the cube - earth, the octahedron - air, the twenty-sided icosahedron - water. And Pythagoras represented the entire world, the “all-encompassing ether,” in the form of a pentagonal dodecahedron. According to legend, only Pythagoras was the only one who heard the music of the spheres. Pythagoras considered the Universe as a huge monochord with one string, attached at the upper end to the absolute spirit, and at the lower end to absolute matter, that is, the string is stretched between heaven and earth. Counting inwards from the periphery of the heavens, Pythagoras divided the Universe, according to one version, into 9 parts, according to another - into 12. The system of the world order was like this. The first sphere was the empiria, or the sphere of the fixed stars, which was the abode of the immortals. From the second to the twelfth were the spheres in order of Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon, fire, air, water and earth.

The Pythagoreans named the various notes of the diatonic scale based on the speed and size of the planetary bodies. Each of these gigantic spheres rushed through infinite space, it was believed, and emitted a sound of a certain tone, which arose due to the continuous displacement of ethereal dust. The theory that the planets, in their rotation around the earth, produce certain sounds, differing from each other depending on the size, speed of movement of the bodies and their distance, was generally accepted among the Greeks. So Saturn, as the most distant planet, gave the lowest sound, and the Moon, the nearest planet, the highest. The Greeks also recognized the fundamental relationship between the individual spheres of the seven planets and the seven sacred vowel sounds. The first heaven pronounces the sacred vowel sound Α (Alpha), the second heaven - the sacred sound Ε (Epsilon), the third - Η (Eta), the fourth Ι (Iota), the fifth - Ο (Omicron), the sixth - Υ (Upsilon), the seventh heaven – sacred vowel Ω (Omega). When the seven heavens sing together, they produce complete harmony. ,

CHAPTER 4. SYMBOLS IN THE PICTURE

4.1.Tetractys Pythagoras

As already stated, the goal of my project is to find the meaning of the symbols depicted in the painting. So what do these mysterious symbols mean?

At the top of the picture, above the head of Pythagoras, the famous tetractys is depicted. What is it?

Tetractys is perhaps the most mysterious figure in the whole picture. Tetractys is the most important concept of Pythagorean philosophy. As mentioned above, it consists of the first four natural numbers, which add up to ten ( sacred number for the Pythagoreans) and form a triangle (also having a mystical meaning). Each of the four numbers carries a meaning (mystical, of course). One means a point, two means a line, three means a plane and four means a body. Everything enclosed in a triangle together formed the universe in all its diversity. Tetractys was sacred to the Pythagoreans; they swore by it on the most important occasions.

The entire numerically proportional theory of Pythagoras finds its relation in the tetractys. Pythagoras believed that it contained the most important harmonic intervals that constitute the harmony of the Universe.

4.2. Pyramid

The picture clearly shows the pyramid that Pythagoras holds in his hand. It is known that Pythagoras spent a lot of time studying geometric bodies and, firstly, gave each a numerical value, and secondly, gave each body a sacred meaning.

In his youth, Pythagoras lived for a long time in Egypt. Apparently, the pyramids impressed him. He examined the pyramid as a geometric body, and decided that it had important spiritual significance (as did everything in Pythagoras). He believed that at its core the pyramid is the content of the “majestic and simple combination” on which the Order of the Universe is based. The perfect square at the base is a symbol of divine balance. Triangles converging upward at one point are not only a geometric, but also a spiritual beginning, the primary source of all things.

The top of the pyramid connects the spiritual earth and cosmic energy - this is Fire, astral Light.,

4.3. globe

There is a version that Pythagoras considered the Earth to be spherical. The ball was his favorite geometric figure (apparently because it was convenient and had no corners). Pythagoras attributed perfection to the ball. Then, according to Pythagoras, the Earth should have had the shape of a ball, that is, an ideal geometric figure. It is quite possible that Pythagoras could have placed on the globe a map of the lands known at that time, the Ecumene, that is (these are the Mediterranean and Asia Minor, the Greeks did not have the scale of Genghis Khan’s thoughts).

Pythagoras did not consider himself a musician, but he taught how to play the lyre. Pythagoras recognized only stringed instruments, considering their sound to be the most noble. Playing the lyre was as natural an activity for him as, say, eating lunch.

Many ancient instruments have seven strings, and according to legend, Pythagoras was the one who added the eighth string to Terpandra's lyre. The seven strings have always been associated with the seven organs of the human body and the seven planets.

4.5.Pythagorean drawings

In Ancient Greece, the art of writing was developed, and Pythagoras certainly knew how to write. He probably wrote down his mathematical calculations. However, the Greeks did not know paper, so he wrote on parchment. Probably, over time, the Pythagoreans accumulated a whole library, which was lost during the defeat of the union.

4.6. Tools

If you look closely at the picture, you can see drawing tools on the table. Now it is difficult to say whether they were known before Pythagoras, or whether he is the inventor of the compass and square, but he used them when constructing regular polygons. There is an opinion that compasses and squares were known back in Ancient Egypt, and Pythagoras borrowed this invention.

4.7. Pythagorean pants

“Pythagorean Pants” are visible on the side of the picture. This is the proof of his famous theorem that Pythagoras apparently found. There are many opinions on the origin of this theorem, however, Pythagoras is currently considered the discoverer not of the theorem itself, but of its proof.

CHAPTER 5. PYTHAGOREAN THEOREM

5.1. History of Pythagorean Theorem

Pythagoras made many discoveries, he brought many new things to mathematics.

However, without a doubt, his most important discovery was the theorem for which he became world famous, and which currently bears his name. The history of the appearance of this theorem has not been fully studied, however, it is currently believed that Pythagoras is not the discoverer of this theorem. It is found a thousand years before Pythagoras in the Babylonian chronicles. Pythagoras studied for a long time with the Babylonian sages, and it was probably there that he first learned about this theorem. Also, the Pythagorean theorem (more precisely, its special cases) were known in India and Ancient China. However, the ancient Indian sages did not use a full-fledged proof; they completed the drawing to a square and then the proof was reduced to visual observation. Apparently, Pythagoras was the first to find a proof of this theorem, so now it bears his name. Subsequently, other proofs of this theorem were found; now, according to some sources, there are about three hundred of these proofs, according to other sources, about five hundred.

5.2. Pythagorean theorem in a school geometry course

In modern textbooks on geometry, the Pythagorean theorem is formulated as follows: “In a right triangle, the square of the hypotenuse equal to the sum squares of legs." Different textbooks give different proofs of this theorem. This proof is given in the textbook:

https://pandia.ru/text/79/553/images/image003_63.gif" width="12" height="23">.gif" width="27" height="17 src=">·AD= AC. Similar to cos B=. Hence AB · BD = BC. Adding the resulting equalities term by term and noting that AD+DB=AB, we get: AC + BC = AB(AD+DB)=ABhttps://pandia.ru/text/79/553/images/image008_4.jpg" alt=" snap0040" width="127" height="124 id=">рис1.!}

Probably, the joke appeared precisely during the proof using the example of an isosceles right triangle, where the equality of the legs is visible visually.

5.4. Additional proofs of the Pythagorean theorem

Currently, several hundred proofs of the Pythagorean theorem are known. However, only a few dozen are widely used. I will talk about the main types of proofs of the Pythagorean theorem, some of which are not widely used.

Proofs based on the use of the concept of equal size of figures.

In Fig. 2 shows two equal squares. The length of the sides of each square is a + b. Each of the squares is divided into parts consisting of squares and right triangles. It is clear that if we subtract quadruple the area of a right triangle with legs a, b from the area of the square, then equal areas will remain, i.e. c2 = a2 + b2. These proofs are the most widely used because they are the simplest.

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Evidence by the method of completion.

The essence of this method is that equal figures are added to the squares built on the legs and to the square built on the hypotenuse in such a way that equal figures are obtained.

In Fig. 7 shows the usual Pythagorean figure - rectangular triangle ABC with squares built on its sides. Attached to this figure are triangles 1 and 2, equal to the original right triangle.

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In Fig. 8 The Pythagorean figure is completed to a rectangle, the sides of which are parallel to the corresponding sides of the squares built on the sides. Let's divide this rectangle into triangles and rectangles. From the resulting rectangle, we first subtract all the polygons 1, 2, 3, 4, 5, 6, 7, 8, 9, leaving a square built on the hypotenuse. Then from the same rectangle we subtract rectangles 5, 6, 7 and the shaded rectangles, we get squares built on the legs.

Now let us prove that the figures subtracted in the first case are equal in size to the figures subtracted in the second case.

Rice. 9 illustrates the proof given by Nassir-ed-Din (1594). Here: PCL – straight line;

KLOA = ACPF = ACED = a;

LGBO = CBMP = CBNQ = b;

AKGB = AKLO + LGBO = c;

disc"> Pythagoras and the early Pythagoreans. M., 2012. - 445 p. ISBN-068-7 Pythagoras and his school. - M.: Science, 1990. - ISBN -2 Science, philosophy and religion in early Pythagoreanism. - St. Petersburg, 1994. - 376 p. - ISBN -1 Fragments of the early Greek philosophers. Part 1: From epic theocosmogonies to the emergence of atomism, Ed. . - M.: Nauka, 1989. - p. 138-149. 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. 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. - Issue 3. - St. Petersburg, 2004. The Pythagorean paradox // Indo-European linguistics and classical philology - XII: Materials of readings dedicated to the memory of prof. June 23-25, 2008. pp. 355-363. Sigachev A. A. Pythagoras (popular science essay) // Electronic journal "Knowledge. Understanding. Skill» . - 2010. - No. 6 - History.

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 carver precious stones. 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 letter of recommendation 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 big amount 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. WITH early years 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.

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.

« 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 had 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).

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.»

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.»

Philosophical teaching

Pythagoras in a fresco by Raphael (1509)

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.

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”.

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 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.

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.

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

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».

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.”

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.»

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.

Pythagoras of Samos (580-500 BC) - ancient Greek thinker, mathematician and mystic. He created the religious and philosophical school of the Pythagoreans.

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, “On the Pythagorean Life”; Porphyry "Life of Pythagoras"; Diogenes Laertius, Pythagoras. These authors relied on the writings of earlier authors, of which it should be noted that Aristotle's student Aristoxenus was from Tarentum, where the Pythagoreans had a strong position.

short biography Pythagoras:

The earliest known sources about the teachings of this thinker appeared only 200 years after his death. However, it is on them that the biography of Pythagoras is based. He himself did not leave any works to his descendants, therefore all information about his teaching and personality is based only on the works of his followers, who were not always impartial.

Pythagoras was born in Sidon Phoenician around 580 (according to other sources around 570) BC. e. Pythagoras' parents are Parthenides and Mnesarchus from the island of Samos. Pythagoras' father was, according to one version, a stone cutter, according to another, a rich merchant who received citizenship of Samos for distributing bread during a famine. The first version is preferable, since Pausanias, who testified to this, gives the genealogy of this thinker. Parthenis, his mother, was later renamed Pyphaida by her husband. She came from the family of Ankeus, a noble man who founded a Greek colony on Samos.

The great biography of Pythagoras was supposedly predetermined even before his birth, which seemed to have been predicted at Delphi by the Pythia, which is why he was called that way. Pythagoras means "he who was announced by the Pythia." This fortuneteller allegedly told Mnesarchus that the future great person will bring as much good and benefit to people as no one else subsequently. To celebrate this, the child’s father even gave a new name to his wife, Pyphaida, and called his son Pythagoras “the one who was announced by Pythia.”

There is another version of the appearance of this name. Moreover, they say that this is a nickname, and he received it for his ability to speak the truth. On behalf of the priestess-soothsayer from the temple of Apollo Pythia. And its meaning is “persuasive by speech.”

The name of his first teacher is known. It was Hermodamas. This man, who instilled in the student a love of painting and music, introduced him to the Iliad and Odyssey.

When he was eighteen years old, Pythagoras left his native island. After several years spent traveling and meeting with sages from different lands, he arrived in Egypt. His plans include studying with priests and comprehending ancient wisdom. In this he is helped by a letter of recommendation from the tyrant of Samos Polycrates to Pharaoh Amasis. Now he has access to something that many foreigners cannot even dream of: not only mathematics and medicine, but also the sacraments. Pythagoras spent 22 years here. And he left the country as a prisoner of the king of Persia, Cambyses, who conquered Egypt in 525 BC. The next 12 years were spent in Babylon.

He was able to return to his native Samos only at 56, and was recognized by his compatriots as the wisest of people. He also had followers here. Many are attracted by mystical philosophy, healthy asceticism and strict morality. Pythagoras preached the moral ennoblement of the people. It could be achieved where power is in the hands of knowledgeable and wise people, to whom the people obey unconditionally in one thing and consciously in another, as a moral authority. It is Pythagoras who is traditionally credited with introducing such words as “philosopher” and “philosophy”.

The disciples of this thinker formed a religious order, a kind of brotherhood of initiates, which consisted of a caste of like-minded people who deified the teacher. This order actually came to power in Crotone. All members of the order became vegetarians, who were forbidden to eat meat or bring sacrificial animals to the gods. Eating food of animal origin is the same as engaging in cannibalism. History has even preserved funny practices in this almost religious order. For example, they did not allow swallows to build nests under the roofs of their houses, or could not touch the white rooster, or eat beans. There is another version according to which the restriction applied only to certain types of meat.

At the end of the 6th century BC. e. Due to anti-Pythagorean sentiments, the philosopher had to go to Metapontum, another Greek colony, where he died. Here, 450 years later, during the reign of Cicero (1st century BC), the crypt of this thinker was shown as a local landmark. Like the date of his birth, the exact date of death of Pythagoras is unknown, it is only assumed that he lived to be 80 years old.

Pythagoras, according to Iamblichus, led the secret society for 39 years. Based on this, the date of his death is 491 BC. e., when the period of the Greco-Persian wars began. Referring to Heraclides, Diogenes said that this philosopher died at the age of 80, or even 90, according to other unnamed sources. That is, the date of death from here is 490 BC. e. (or, less likely, 480). In his chronology, Eusebius of Caesarea indicated 497 BC as the year of death of this thinker. e. Thus, the biography of this thinker is largely questionable.

Scientific achievements and works of Pythagoras:

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.

1) In the field of mathematics:

Pythagoras is today considered the great cosmologist and mathematician of antiquity, but early evidence does not mention such merits. Iamblichus writes about the Pythagoreans that they had a custom of attributing all achievements to their teacher. This thinker is considered by ancient authors to be the creator of the famous theorem that in a right triangle the square of the hypotenuse is equal to the sum of the squares of its legs (Pythagorean theorem). Both the biography of this philosopher and his achievements are largely dubious. The opinion about the theorem, in particular, is based on the testimony of Apollodorus the calculator, whose identity has not been established, as well as on poetic lines, the authorship of which also remains a mystery. Modern historians suggest that this thinker did not prove the theorem, but could convey this knowledge to the Greeks, which was known 1000 years ago in Babylon before the time when the biography of the mathematician Pythagoras dates back to. Although there is doubt that this particular thinker was able to make this discovery, no compelling arguments can be found to challenge this point of view. In addition to proving the above theorem, this mathematician is also credited with the study of integers, their properties and proportions.

2) Aristotle's discoveries in the field of cosmology:

Aristotle in his work “Metaphysics” touches on the development of cosmology, but the contribution of Pythagoras is not voiced in any way in it. The thinker we are interested in is also credited with the discovery that the Earth is round. However, Theophrastus, the most authoritative author on this issue, gives it to Parmenides. Despite controversial issues, the merits of the Pythagorean school in cosmology and mathematics are indisputable. According to Aristotle, the real ones were the acousmatists, who followed the doctrine of the transmigration of souls. They viewed mathematics as a science that came not so much from their teacher as from one of the Pythagoreans, Hippasus.

3) Works created by Pythagoras:

This thinker did not write any treatises. It was impossible to compile a work from oral instructions addressed to the common people. And the secret occult teaching, intended for the elite, could not be entrusted to the book either. Diogenes lists some of the titles of books that allegedly belonged to Pythagoras: “On Nature,” “On the State,” “On Education.” But for the first 200 years after his death, not a single author, including Aristotle, Plato, and their successors at the Lyceum and Academy, quotes from the works of Pythagoras or even indicates their existence. The written works of Pythagoras were unknown to ancient writers from the beginning of the new era. This is reported by Josephus, Plutarch, and Galen. A compilation of the sayings of this thinker appeared in the 3rd century BC. e. It's called "The Sacred Word". Later, the “Golden Poems” arose from it (which are sometimes attributed, without good reason, to the 4th century BC, when the biography of Pythagoras is considered by various authors).

4) Pythagoras mug:

Quite a clever invention. It is not possible to fill it to the brim, because the entire contents of the mug will immediately leak out. There should be liquid in it only up to a certain level. It looks like an ordinary mug, but what distinguishes it from others is the column in the center. It was called the “greed circle.” Even today in Greece it is in deserved demand. And for those who do not know how to limit their alcohol consumption, it is even recommended.

5) Oratorical talent:

No one questions it in Pythagoras. He was a great speaker. It is known for certain that after his very first public lecture, he had two thousand students. Entire families, imbued with the ideas of their teacher, were ready to begin new life. Their Pythagorean community became a kind of state within a state. All the rules and laws developed by the Teacher were in force in their Magna Graecia. Property here was collective, even scientific discoveries, which, by the way, were attributed exclusively to Pythagoras, were attributed to his personal merits even when the teacher was no longer alive.

Pythagoras - quotes, aphorisms, sayings:

*Two things make a person godlike: living for the good of society and being truthful.

*Just as old wine is unsuitable for drinking a lot, so rude treatment is unsuitable for an interview.

*Take care of your children’s tears so that they can shed them at your grave.

*It is equally dangerous to give a sword to a madman and to a dishonest person to give power.

*Do not consider yourself a great person based on the size of your shadow at sunset.

*Of two people of equal strength, the one who is right is stronger.

*No matter how short the words “yes” and “no” are, they still require the most serious consideration.

*To learn the customs of any people, try to first learn their language.

*It is more useful to throw a stone at random than an empty word.

*Live with people so that your friends do not become enemies, and your enemies become friends.

*No one should exceed the limit in food or drink.

*Blessed be the divine number that gave birth to gods and men.

*Joke, like salt, should be consumed in moderation.

*In order to live long, buy for yourself old wine and an old friend.

*Choose the best, and habit will make it pleasant and easy.

*During anger one should neither speak nor act.

*A statue is painted by its appearance, but a man by his deeds.

*Flattery is like a weapon in a painting. It brings pleasure, but no benefit.

*Don’t chase happiness: it is always within you.

30 interesting facts about Pythagoras:

1. The name of Pythagoras is famous for his theorem. And this is this man's greatest achievement.

2. The name of the “father” of democracy has long been known. This is Plato. But he based his teaching on the ideas of Pythagoras, one might say, his grandfather.

3.According to Pythagoras, everything in the world is reflected in numbers. His favorite number was 10.

4. None of the evidence from early times contains any mention of the merits of Pythagoras as the greatest cosmologist and mathematician of antiquity. And he is considered as such today.

5.Already during his lifetime he was considered a demigod, a miracle worker and an absolute sage, a kind of Einstein of the 4th century BC. There is no more mysterious great man in history.

6. One day Pythagoras got angry with one of his students, who committed suicide out of grief. From then on, the philosopher decided never to take out his irritation on people again.

7. Legends also attributed to Pythagoras the ability to heal people, using, among other things, excellent knowledge of various medicinal plants. The influence of this personality on those around him is difficult to overestimate.

8. In fact, Pythagoras is not a name, but a nickname of the great philosopher.

9. Pythagoras was distinguished by an excellent memory and developed curiosity.

10. Pythagoras was a famous cosmologist.

11. The name of Pythagoras was always surrounded by many legends even during his lifetime. For example, it was believed that he was able to control spirits, knew the language of animals, knew how to prophesy, and birds could change the direction of their flight under the influence of his speeches.

12. Pythagoras was the first to say that the soul of a person is reborn again after his death.

13.С youth Pythagoras was drawn to travel.

14. Pythagoras had his own school, which included 3 directions: political, religious and philosophical.

15. Pythagoras conducted experiments with color on the human psyche.

16. Pythagoras tried to find the harmony of numbers in nature.

17. Pythagoras considered himself a fighter for Troy in a past life.

18. The theory of music was developed by this talented sage.

19. Pythagoras died saving his own students from a fire.

20. The lever was invented by this philosopher.

21. Pythagoras was a great orator. He taught this art to thousands of people.

22. A crater on the Moon is named after Pythagoras.

23. Pythagoras has always been considered a mystic.

24. Pythagoras believed that the secret of all essence on Earth lies in numbers.

25. Pythagoras got married when he was 60 years old. And the student of this philosopher became his wife.

26. The first lecture that Pythagoras gave brought 2000 people to him.

27.When joining the school of Pythagoras, people had to give up their property.

28. Among the followers of this sage there were quite noble people.

29. The first mentions of the life and work of Pythagoras became known only after 200 years had passed since his death.

30. The school of Pythagoras fell under the disgrace of the state.