Name: Pythagoras

Date of Birth: 570 BC e.

Age: 80 years old

Date of death: 490 BC e.

Activity: philosopher, mathematician, mystic

Family status: was married

Pythagoras: biography

The biography of Pythagoras of Samos takes readers into the world ancient greek culture. This man can safely be called a legendary personality. Pythagoras was a great mathematician, mystic, philosopher, founded a religious and philosophical movement (Pythagoreanism), and was a politician who left his works as a legacy to his descendants.

Childhood and youth

It is difficult to determine the exact date of birth of Pythagoras. Historians have established the approximate period of his birth - 580 BC. Place of birth: Greek island of Samos.

The philosopher’s mother’s name was Parthenia (Parthenis, Pythias), and his father’s name was Mnesarchus. According to legend, one day a young couple visited the city of Delphi as a honeymoon. There the newlyweds met an oracle who prophesied the lovers the imminent appearance of a son. The legend said that the child would become a difficult person, famous for his wisdom, appearance, and great deeds.

Soon the prophecy began to come true, the girl gave birth to a boy and, in accordance with ancient tradition received the name Pythias. The baby is named Pythagoras in honor of the priestess of Apollo Pythia. The father of the future mathematician tried in every possible way to fulfill the divine tradition. Happy Mnesarchus erects an altar to Apollo, and surrounds the child with care and love.

Some sources also say that two more boys were raised in the family - the older brothers of the Greek philosopher: Eunost and Tyrrhenus.

Pythagoras's father was a master in processing gold stones, and the family was wealthy. Even as a child, the boy showed curiosity in various sciences and was distinguished by unusual abilities.

The first teacher of the future philosopher was Hermodamant. He taught Pythagoras the basics of music, technologies of painting, reading, rhetoric, and grammar. To help Pythagoras develop his memory, the teacher forced him to read the Odyssey and the Iliad and memorize songs from the poems.

A few years later, an 18-year-old boy with a ready-made baggage of knowledge went to Egypt to continue his education with the wise priests, but in those years it was difficult to get there: it was closed to the Greeks. Then Pythagoras temporarily stopped on the island of Lesbos and here he studied physics, dialectics, theogony, astrology, and medicine from Pherecydes of Syros.

Pythagoras lived on the island for several years, and then went to Miletus, the city where the famous Thales lived, who was noted in history as the founder of the first philosophical school in Greece.

The Milesian school allowed Pythagoras to acquire knowledge, but, following the advice of Thales, the young man went to Egypt to continue the path of education.

Here Pythagoras meets the priests, visits Egyptian temples closed to foreigners, becomes familiar with their secrets and traditions, and soon he himself receives the rank of priest. Studying in a culturally developed city made Pythagoras the most educated person of those times.

Mysticism and homecoming

Ancient legends claim that in Babylon a talented philosopher and a man of divine beauty (confirmation of this is a mathematician’s photo taken on the basis of paintings by ancient artists and sculptures) met with Persian magicians. Pythagoras became involved in the study of mystical events, learned the wisdom and peculiarities of astronomy, arithmetic, and medicine of the eastern peoples.

The Chaldeans tied supernatural ideas to the emergence of these sciences, and this approach was reflected in the subsequent sound of Pythagoras’ knowledge in the field of mathematics and philosophy.

12 years after Pythagoras’s forced stay in Babylon, the sage is freed by the Persian king, who has already heard about the famous teachings of the Greek. Pythagoras returns to his homeland, where he begins to introduce his own people to the acquired knowledge.

The philosopher quickly gained wide popularity among residents. Even women, who were prohibited from attending public meetings, came to hear him speak. At one of these events, Pythagoras met future wife.

To a person with high level knowledge, I had to work as a teacher with people of low morality. He became for the people the personification of purity, a kind of deity. Pythagoras mastered the methods of the Egyptian priests, knew how to purify the souls of listeners, and filled their minds with knowledge.

The sage spoke mainly on the streets, in temples, but after that he began to teach everyone in his own home. This is a special training system that is complex. The probationary period for students was 3-5 years. Listeners were forbidden to speak during lessons or ask questions, which trained them to be modest and patient.

Mathematics

A skillful orator and wise teacher taught people various sciences: medicine, political activity, music, mathematics, etc. Later, future famous figures, historians, government officials, astronomers, and researchers came out of the school of Pythagoras.

Pythagoras made a significant contribution to geometry. Today, the name of the popular ancient figure is known based on the study of the famous Pythagorean theorem in schools through mathematical problems. Here's what the formula for solving some Pythagorean problems looks like: a2 + b2 = c2. In this case, a and b are the lengths of the legs, and c is the length of the hypotenuse right triangle.

At the same time, there is also the inverse Pythagorean theorem, developed by other equally competent mathematicians, but today in science there are only 367 proofs of the Pythagorean theorem, which indicates its fundamental importance for geometry as a whole.

Another invention of the great Greek scientist was the “Pythagorean table”. Nowadays it is usually called the multiplication table, according to which students of the philosopher’s school were taught in those years.

An interesting discovery from past years was the mathematical relationship between the vibrating strings of the lyre and their length in musical performance. This approach can easily be applied to other instruments.

Numerology

The philosopher paid close attention to numbers, trying to understand their nature, the meaning of things and phenomena. He tied numerical properties to vital categories of existence: humanity, death, illness, suffering, etc.

It was the Pythagoreans who divided numbers into even and odd. Pythagoras saw something important (justice and equality) for life on the planet in the square of a number. Nine characterized constancy, number eight - death.

Even numbers were assigned female, odd - for male representation, and the symbol of marriage among the followers of the teachings of Pythagoras was the five (3 + 2).

Thanks to the knowledge of Pythagoras, people today have the opportunity to find out the level of compatibility with their future half, take a look at the curtain of the future. To do this, you can use the numerological system of the Pythagorean square. “Game” with certain numbers (date, day, month of birth) will allow you to build a graph that clearly shows the picture of a person’s fate.

The followers of Pythagoras believed that numbers could have an incredible effect on the world society. The main thing is to understand their chain meaning. There are good and bad numbers, such as thirteen or seventeen. Numerology, as a science, is not recognized as official; it is considered a system of beliefs and knowledge, but nothing more.

Philosophical teaching

The teachings of Pythagoras' philosophy should be divided into two parts:

1. Scientific approach of world knowledge.
2. Religiosity and mysticism.

Not all of Pythagoras' works have been preserved. The great master and sage wrote down practically nothing, but was mainly engaged in oral teaching of those wishing to learn the intricacies of this or that science. Information about the philosopher's knowledge was subsequently passed on by his followers - the Pythagoreans.

It is known that Pythagoras was a religious innovator, created a secret society, and preached acousmatic principles. He forbade his disciples to eat food of animal origin, and especially the heart, which is primarily a symbol of life. It was not allowed to touch the beans, according to legend, obtained from the blood of Dionysus-Zagreus. Pythagoras condemned the use of alcohol, foul language and other ignorant behavior.

The philosopher believed that a person can save and free his soul through physical and moral purification. His teachings can be compared with ancient Vedic knowledge, based on the quantitative transmigration of the soul from heaven into the body of an animal or human until it earns the right to return to God in heaven.

Pythagoras did not impose his philosophy on ordinary people who were only trying to comprehend the basics of the exact sciences. His specific teachings were intended for truly “enlightened”, chosen individuals.

Personal life

Returning from Babylonian captivity to his homeland in Greece, Pythagoras met an unusually beautiful girl named Feana, who secretly attended his meetings. The ancient philosopher was already in mature age(56-60 years old). The lovers got married and had two children: a boy and a girl (names unknown).

Some historical sources claim that Feana was the daughter of Brontin, a philosopher, friend and student of Pythagoras.

Death

The school of Pythagoras was located in the Greek colony of Croton (Southern Italy). A democratic uprising took place here, as a result of which Pythagoras was forced to leave the place. He went to Metapontum, but military clashes reached this town.

The famous philosopher had many enemies who did not share his principles of life. There are three versions of the death of Pythagoras. According to the first, the murderer was a man to whom a mathematician once refused to teach secret occult techniques. Being in feelings of hatred, the rejected one set fire to the building of the Pythagorean Academy, and the philosopher died saving his students.

The second legend says that in a burning house, the scientist’s followers created a bridge from own bodies, wanting to save his teacher. And Pythagoras died of a broken heart, having underestimated his efforts in the development of humanity.

A common version of the sage’s death is considered to be his death under random circumstances during a skirmish in Metapontus. At the time of his death, Pythagoras was 80-90 years old.

The great ancient Greek philosopher, politician, mathematician and astronomer Pythagoras is the founder of many scientific disciplines, teachings and concepts. His biography is complex, interesting and mysterious, so much so that it is not always possible to separate the facts from the life of the great scientist and sage from legends and fiction. However, it is generally accepted that important facts from the life of Pythagoras were written down by his students from various parts of the world.
According to scientists, Pythagoras was born around 570 BC. in the city of Sidon, the current territory of modern Lebanon. His father Mnesarchus is a wealthy jeweler and merchant who was able to create excellent conditions for his son to receive a good education and great knowledge.
The origin of the very name of Pythagoras is shrouded in legend. According to legend, one day in Delphi Honeymoon Pythagoras' young parents set off. It was here that the priestess (and according to some sources, the oracle) made a prediction that Mnesarchus would have a son and he would become famous for many centuries for his wisdom and deeds. The prophecy came true, and as gratitude to the priestess who worshiped Apollo of Pythia, the boy is named Pythagoras, which translated means predicted by the Pythia (priestess).
From the early childhood Pythagoras studied a lot, visited the best temples of Greece, and as a teenager became familiar with the works of the greatest sages of that time. According to researchers of antiquity, he personally met with many figures of that era. Among them, we note Pherecydes of Syros, an ancient Greek cosmologist, one of the most important teachers of Pythagoras. It was to him that the future philosopher owes his deep knowledge of mathematics, astronomy, and physics. An equally important place in the development of Pythagoras’s personality was occupied by communication with Hermodamant, who taught him a love of art, poetry and music through the example of the works of Homer.
The next stage of Pythagoras’ biography consists of his life experience, based on travel to foreign lands. Through Phenicia he travels to Egypt, with the ancient priests, their faith, and even, despite his status as a foreigner, visits Egyptian temples.
Later in Egypt, he creates his own school, where he teaches those who want exact sciences and philosophy. Pythagoras spent a considerable time in this country - about two decades. During this time, he gained many supporters and followers who proudly called themselves Pythagoreans. During this period of his life, Pythagoras introduced the concept of “philosopher” and considered himself one of them. According to the scientist, “sage” and “philosopher” are completely different in their meanings and purpose. A philosopher is someone who “tries to find out” everything, always.
Having behind him many outstanding discoveries made on Egyptian soil, Pythagoras, as a captive of the Persian king Cambyses, ends up in Babylon and spends twelve years there. Here he actively devotes himself to the study of Eastern culture and religion, compares the features of their development in the countries of the Middle East and Greece. After this, Pythagoras visits Phenicia, Syria and Hindustan, where he further increases his knowledge of the natural sciences and achieves new achievements and discoveries in each area.
In 530 BC. the philosopher finds himself in the southern Italian city of Croton. It was here that Pythagoras gained universal fame, he was quoted and extolled, and the founding of the Pythagorean school reached its apogee. It is also called in another way the philosophical brotherhood or union. Only those who are already well versed in mathematical sciences and have an understanding of astronomy can study here.
At the age of 60, Pythagoras falls in love with his student named Theano. Their marriage produces three children.
Unfortunately, in 500 BC. Mass persecution began against Pythagoras and his school. Scientists believe that the main reason was his refusal to take the son of a wealthy government official into his ranks. After numerous unrest and riots that enveloped the city of Croton, Pythagoras disappeared, but he did not leave science and philosophy until the end of his days.

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. Recent 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

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.

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 some of the students 15 years old 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

ABOUT youthful life Little is known about 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 the 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.

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.

TO 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 instruments and applying their weight in his mind, he returned to his own house, constructed a beam, which was attached to the wall, and attached to it at regular intervals four strings, all identical in every way. 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 a large number of 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 general fact 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 accordance with the appropriate modes, since 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 viewed 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. ,

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 (a sacred number for the Pythagoreans) and form a triangle (also having 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 numeric value, secondly, he 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.

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." Various textbooks provide different evidence 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. Figure 7 shows the usual Pythagorean figure - a right 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. Materials 11th international conference 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 of Samos (ancient Greek Πυθαγόρας ὁ Σάμιος, lat. Pythagoras; 570-490 BC). Ancient Greek philosopher, mathematician and mystic, creator of 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 (242-306) “On 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 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; 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 supposedly predicted by Pythia in Delphi, which is why Pythagoras received his name, which means “the one whom Pythia announced.” 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).

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.

IN 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, 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 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.

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 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. Tradition credits Pythagoras with introducing 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 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 Mnya (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.

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.

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.

Scientific achievements 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 a wonderful custom of attributing everything to Pythagoras and not at all taking credit for the 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 of bulls for it”.

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.

The development of ideas about cosmology is touched upon in the 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.

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 gain fame and became outstanding people, thanks to whose works we learned the basics of Pythagoras’ teachings, 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 nodes, called tetractys by the Pythagoreans, create nine equilateral triangles between themselves, 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 in different times and all over the world. This applies to the greatest extent to the Pythagorean theorem.

For example, in China, special attention in this regard should be paid to the mathematical book Chu-pei, which says so about the well-known Pythagorean triangle, having sides 3, 4, 5: “If a right angle is decomposed into its component parts, then the line connecting the ends of all its sides will be 5, while 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.