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 the 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
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 us to get 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, therefore this project will have educational value for everyone.
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 very an inquisitive child, so he asked visiting sailors about other countries. When he grew up a little, he felt cramped on the small island, which he crawled 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 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: in Greece tall buildings weren't built. 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.
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 common divisor, are classified as non-composite-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, 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 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 the general fact of the consistency of all these various methods evidence, 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 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. ,
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 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, 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, 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 corresponded to the seven organs human body and with 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 right triangle 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 we will be left with equal areas, 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. 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 in history 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.
The life story of Pythagoras is difficult to separate from the legends that present him as a perfect sage and a great initiate into all the mysteries of the Greeks and barbarians. Herodotus also called him “the greatest Hellenic sage.”
The main sources on the life and teachings of Pythagoras are the works of the Neoplatonist philosopher Iamblichus (242-306) " About Pythagorean life"; Porphyria (234-305) " Life of Pythagoras"; Diogenes Laertius (200-250) book. 8, " Pythagoras" These authors relied on the writings of earlier authors, of which it should be noted that Aristotle's student Aristoxenus (370-300 BC) was from Tarentum, where the Pythagoreans had a strong position.
Thus, the earliest known sources about the teachings of Pythagoras did not appear until 200 years after his death. Pythagoras himself did not leave any writings, and all information about him and his teachings is based on the works of his followers, who are not always impartial.
Pythagoras' parents were Mnesarchus and Parthenides from the island of Samos. Mnesarchus was a stone cutter (Diogenes Laertius); according to Porphyry, he was a rich merchant from Tyre, who received Samian citizenship for distributing grain in a lean year. The first version is preferable, since Pausanias gives the genealogy of Pythagoras in the male line from Hippasus from the Peloponnesian Phlius, who fled to Samos and became the great-grandfather of Pythagoras. Parthenides, later renamed Pyphaida by her husband, came from the noble family of Ankeus, the founder of the Greek colony on Samos.
The birth of a child was allegedly predicted by Pythia in Delphi, which is why Pythagoras got his name, which means “ the one announced by the Pythia" In particular, Pythia told Mnesarchus that Pythagoras would bring as much benefit and goodness to people as no one else had brought and would not bring in the future. Therefore, to celebrate, Mnesarchus gave his wife a new name, Pyphaidas, and named the child Pythagoras. Pyphaida accompanied her husband on his trips, 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 supplied Pythagoras letter of recommendation to Pharaoh Amasis, thanks to which he was allowed to study and initiated into the sacraments forbidden to other foreigners.
« The Pythagoreans formed a large community (there were more than three hundred of them), but it constituted only a small part of the city, which was no longer governed according to the same customs and mores. However, as long as the Crotonians owned their land, and Pythagoras was with them, it remained government structure, which existed from the founding of the city, 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 their parents, and that they provide their property for common use, and it is separated from the property of relatives. When the relatives began this hostility, the rest readily joined the conflict... After many years... the Crotonians were overcome by regret and repentance, and they decided to return to the city those Pythagoreans who were still alive.»
Many Pythagoreans died, the survivors scattered throughout Italy and Greece. The German historian F. Schlosser notes regarding the defeat of the Pythagoreans: “ The attempt to transfer caste and clerical life to Greece and, contrary to the spirit of the people, to change its political structure and morals according to the requirements of an abstract theory ended in complete failure.»
According to Porphyry, Pythagoras himself died as a result of the anti-Pythagorean rebellion in Metapontus, but other authors do not confirm this version, although they readily convey the story that the dejected philosopher starved himself to death in the sacred temple.
The teachings of Pythagoras should be divided into two components: the scientific approach to understanding the world and the religious and mystical way of life preached by Pythagoras. The merits of Pythagoras in the first part are not known for certain, since everything created by followers within the school of Pythagoreanism was later attributed to him. The second part prevails in the teachings of Pythagoras, and it is this part that remained in the minds of most ancient authors.
The merit of the Pythagoreans was the promotion of ideas about the quantitative laws of the development of the world, which contributed to the development of mathematical, physical, astronomical and geographical knowledge. Numbers are the basis of things, Pythagoras taught, to know the world means to know the numbers that control it. By studying numbers, the Pythagoreans developed numerical relationships and found them in all areas human activity. Numbers and proportions were studied in order to know and describe the human soul, and, having learned it, to manage the process of transmigration of souls with the ultimate goal of sending the soul to some higher divine state.
Despite the popular opinion that Pythagoras was supposedly a vegetarian, Diogenes Laeres writes that Pythagoras occasionally ate fish, abstained only from arable bulls and rams, and allowed other animals for food.
His contemporary Heraclitus acted as a critic of Pythagoras: “ Pythagoras, the son of Mnesarchus, was engaged in collecting information more than any other person in the world and, having taken these works for himself, passed off knowledge and fraud as his own wisdom". According to Diogenes Laertius, continued famous saying Heraclitus “Much knowledge does not teach the mind” is mentioned among others by Pythagoras: “otherwise it would have taught Hesiod and Pythagoras, as well as Xenophanes and Hecataeus.”
Coin with the image of Pythagoras
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 the remarkable custom of attributing everything to Pythagoras and not at all arrogating to themselves the glory of discoverers, except perhaps in a few cases.»
In the 3rd century. BC e. a compilation of the sayings of Pythagoras appeared, known as the “Sacred Word”, from which the so-called “Golden Verses” later arose (sometimes they are attributed to the 4th century BC without good reason). These verses were first quoted by Chrysippus in the 3rd century. BC e. , although, perhaps, at that time the compilation had not yet developed into a finished form. The final excerpt from “Golden Verses” translated by I. Peter:
Be firm: the divine race is present in mortals,
To them, proclaiming, sacred nature reveals everything.
If this is not alien to you, you will carry out orders,
You will heal your soul and deliver you from many disasters.
Dishes, I said, leave those that I indicated in the cleansings
And be guided by true knowledge - the best charioteer.
If you, having left your body, ascend into the free ether,
You will become an incorruptible and eternal god who does not know death.
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Messages about Pythagoras, ancient Greek philosopher and mathematician, the creator of the Pythagorean school is described in this article.
Pythagoras was born around 570 BC in Sidon, Phoenician, into the family of a rich merchant from Tire. Thanks to financial condition his parents, the young man met many sages of that era and absorbed their knowledge like a sponge.
At the age of 18, Pythagoras left hometown and left for Egypt. There he stayed for 22 years, learning the knowledge of the local priests. When Persian king conquered Egypt, the scientist was taken to Babylon, where he lived for another 12 years. He returned to his native land at the age of 56, and his compatriots recognized him as a sage.
Pythagoras the donkey of Southern Italy, the Greek colony - Crotone. Here he found many followers and founded his school. His students practically deified their founder and teacher. But the omnipotence of the Pythagoreans led to the outbreak of rebellions and Pythagoras moved to another Greek colony - Metapontus. This is where he died.
He was married to a woman, Theano, from whom a son, Telaugus, and a daughter, whose name is unknown, were born.
The philosophical teaching of Pythagoras consists of two parts - a scientific approach to understanding the world and an occult way of life, preached by him. He reflected on the liberation of the soul through physical and moral purification through secret teachings. The philosopher founded the mystical doctrine of the cycle of migration of the soul. The eternal soul, according to the scientist, moves from heaven into the body of an animal or a person. And it moves from body to body until the soul earns the right to return back to heaven.
Pythagoras formulated a number of instructions from his school - about behavior, circulation human lives, sacrifices, food and burials.
The Pythagoreans put forward the idea of quantitative patterns in the development of the world. And this, in turn, contributed to the development of physical, mathematical, geographical and astronomical knowledge. Pythagoras taught that the basis of the world and things is number. He developed numerical relationships that found applications in all human activities.
What contribution of Pythagoras to science, philosophy and mathematics you will learn from this article.
His contribution to geometry cannot be underestimated, making truly great discoveries. Pythagoras created his own school and, together with his students, he worked hard to give geometry a scientific character. In addition to the fact that he created the famous Pythagorean theorem (it is very important for modern science and is used at every step in solving important geometric problems) the scientist made many discoveries. Among them:
In addition to mathematical achievements, Pythagoras made significant contributions to other sciences. In astronomy and geography, he was among the first scientists who expressed the hypothesis that our planet is round. He believed that we are not the only creatures inhabiting the universe.
Pythagoras' discoveries in the field of music are also significant. He determined that the sound directly depends on the length of the string or flute. Even popular numerology today owes its existence to Pythagoras - he was the first to combine predictions for the future with numbers.
Pythagoras' contribution to philosophy was that he first introduced the term "philosophy" into scientific use. He founded his school in Italy in 532 BC. At the same time, it was both a religious monastic order and a political structure. The school had its own charter and fairly strict rules. It is interesting that all students of the school had to give up meat food and personal property, and not tell others about the teachings of their mentor.
Pythagoras of Samos is an ancient Greek mathematician, philosopher and mystic, the founder of the Pythagorean school. The years of his life are 570-490. BC e. Our article will present to your attention the biography of Pythagoras, his main achievements, as well as Interesting Facts about this great man.
It is difficult to separate the life story of this thinker from the legends that represented him as a perfect sage, as well as initiated into the mysteries of the barbarians and Greeks. Herodotus called this man "the greatest Hellenic sage." Below you will be presented with the biography of Pythagoras and his works, which should be treated with a certain degree of doubt.
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' 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 (more on this below). 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, Pyphaidas, and named his son Pythagoras. Pyphaida accompanied her husband on trips. Pythagoras was born in Sidon Phoenician around 570 BC. e.
This thinker, according to ancient authors, met with many famous sages of that time: Egyptians, Chaldeans, Persians, Greeks, absorbing the knowledge accumulated by humanity. Sometimes in popular literature Pythagoras is also credited with the Olympic victory in boxing competitions, confusing the philosopher with his namesake, the son of Crates, also from the island of Samos, who won the 48 games a little earlier, 18 years before the philosopher appeared on light.
Pythagoras at a young age went to the country of Egypt to gain secret knowledge and wisdom from the priests here. Porphyry and Diogenes write that Polycrates, the Samian tyrant, provided this philosopher with a letter of recommendation to Amasis (Pharaoh), because of which he began to be taught and initiated not only into the achievements of mathematics and medicine in Egypt, but also into the sacraments that were for other foreigners were forbidden.
At the age of 18, as Iamblichus writes, the biography of Pythagoras is supplemented by the fact that he left the island and got to Egypt, traveling around all kinds of sages from various parts of the world. He stayed in this country for 22 years, until Cambyses, the Persian king, took him among the captives to Babylon, who in 525 BC. e. conquered Egypt. Pythagoras stayed in Babylon for another 12 years, communicating here with magicians, until he was finally able to return to Samos at the age of 56, where his compatriots recognized him as the wisest of people.
This thinker, according to Porphyry, left his native island due to disagreements with the local tyrannical power exercised by Polycrates, at the age of 40. Since this information is based on the testimony of Aristoxenus, who lived in the 4th century BC. e., they were considered relatively reliable. In 535 BC. e. Polycrates came to power. Therefore, the date of birth of Pythagoras is considered to be 570 BC. e., if we assume that he left for Italy in 530 BC. e. According to Iamblichus, Pythagoras moved to this country during the 62nd Olympiad, that is, in the period from 532 to 529. BC e. This information correlates well with Porphyry, but completely contradicts the legend of Iamblichus about the captivity of Pythagoras in Babylon. Therefore, it is not known for sure whether this thinker visited Phenicia, Babylon or Egypt, where, according to legend, he gained eastern wisdom. short biography Pythagoras, provided to us by various authors, is very contradictory and does not allow us to draw a clear conclusion.
It is unlikely that the reason for the departure of this philosopher could have been disagreements with Polycrates; rather, he needed the opportunity to preach and put his teaching into practice, which was difficult to achieve in Ionia, as well as mainland Hellas. He went to Italy because he believed that there was more people capable of learning.
The short biography of Pythagoras, compiled by us, continues. This thinker settled in Southern Italy, in Crotona, a Greek colony, where he found numerous followers. They were attracted not only by the convincingly presented mystical philosophy, but also by a way of life that included strict morality and healthy asceticism.
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 Croton, but 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.
Pythagoras had a wife named Theano, as well as a daughter Mia and a son Telaugus (according to another version, the children's names were Arignota and Arimnest).
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.
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). The biography of this philosopher, as well as his achievements, is in many ways 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.
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.
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. To ancient writers from the beginning new era the written works of Pythagoras were unknown. 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).
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. Traditions 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. A curious episode from the life that the biography of Pythagoras tells us about (interesting facts about him are by no means exhausted by it) is this: one day he became 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.
You were presented with a biography of Pythagoras, summary the life and work of this great man. We have tried to describe events based on different opinions, since it is incorrect to judge this thinker based on only one source. The information available about him is very contradictory. Biography of Pythagoras for children usually does not take these contradictions into account. It represents in an extremely simplified and one-sided way the fate and legacy of this person. A short biography of Pythagoras for children is studied at school. We tried to reveal it in more detail in order to deepen readers’ understanding of this person.
