Pythagoras

"Everything has a number"-Pythagoras

Introduction to Pythagoras

Thales has an antithesis in philosophy. This man is the ancient Greek philosopher, mathematician and astronomer who first proposed that the motion of matter should conform to mathematical laws-Pythagoras (560-480 BC).

Life resume

In 580 BC, Pythagoras was born on Samos Island near Miletus, one of the major island cities in ionian islands. At this time, the archipelago was in its heyday, and it was far ahead of the Greek city-states in economy, culture and other aspects.

Pythagoras' father was a rich businessman. At the age of nine, he was sent to Tyre by his father to study with Semite Syrian scholars, where he came into contact with eastern religions and cultures. Later, I went to Asia Minor on business with my father many times.

In 55 1 BC, Pythagoras came to Miletus, Delos and other places, visited Thales, anaximander and Philkudes, and became their students. Before that, he studied poetry and music under the Samos poet Clay Filos.

In 550 BC, 30-year-old Pythagoras was disliked by the local people for propagating rational theology, wearing oriental clothes and having long hair. Since then, Samos has always been biased against Pythagoras, thinking that he is unconventional and advocates heresy. In 535 BC, Pythagoras was forced to leave his hometown for Egypt. On the way, he stopped in the coastal city of Phoenicia, studied local myths and religions, and then lived in seclusion in a temple in Tyre.

After arriving in Egypt, King Amassis recommended him to study in the temple. During the ten years from 535 BC to 525 BC, Pythagoras studied hieroglyphics, Egyptian mythology, history and religion, and publicized Greek philosophy, which was highly respected by many Greeks and many people went to his door to study.

Pythagoras returned to his hometown of Samos at the age of 49 and began to give lectures and run schools, but it did not achieve his expected results. Around 520 BC, in order to get rid of the tyranny of the monarch at that time, he left Samos with his mother and only disciple, moved to Sicily, and later settled in Crotone. There, he recruited disciples and established a religious, political and academic group.

His speech attracted people from all walks of life, and many upper-class people came to the lecture. According to the custom at that time, women were forbidden to attend public meetings. Pythagoras broke this rule and allowed them to attend lectures. Among the enthusiastic audience was his later wife Siena, who was young and beautiful and wrote a biography for him, but unfortunately it has been lost.

Pythagoras established a secret society in Claughton, a Greek territory in southern Italy. There are men and women in this society, they all have equal status, and all property belongs to the public. The organization discipline of the association is very strict, even with a strong religious color. Every student must reach a certain academic level, join an organization and go through a series of mysterious rituals in order to achieve "spiritual purification".

They have to accept long-term training and examinations, abide by many norms and commandments, and swear never to reveal the secrets and theories of the school. They believe that relying on mathematics can sublimate the soul and integrate with God. Everything contains numbers, even everything is numbers, and God rules the universe through numbers. This is the main difference between Pythagoras school and other sects.

School members share common philosophical beliefs and political ideals. They eat simple food and receive strict training. School teaching encourages self-control, abstinence, purity and obedience. They began to gain a high reputation and considerable influence in Great Greece (now southern Italy), which also aroused the envy of rival factions.

Later, due to the impact of the pro-democracy movement, the venues for community activities in Crotone were severely damaged. Pythagoras was forced to move to Linton (now taranto in southern Italy) and died in 500 BC at the age of 80. Many disciples fled back to Greece and re-established their position in Frios. Others went to taranto to continue their mathematical philosophy research and political activities until the middle of the 4th century BC. The Pythagorean school continued to prosper for two centuries.

"Everything counts."

Pythagoras school is the first school to put the concept of number in a prominent position. They attach great importance to mathematics and try to explain everything with numbers. It is claimed that number is the origin of all things in the universe, and the purpose of studying mathematics is not to use it but to explore the mysteries of nature. They abstract the number five from five apples, five fingers and so on. This is a common thing today, but it was a great progress in philosophy and practical mathematics at that time. In practical mathematics, it makes arithmetic possible. Philosophically, this discovery makes people believe that numbers are the foundation of the material world.

Pythagorean theorem-Pythagorean theorem

Pythagoras himself is famous for discovering Pythagoras theorem (called Pythagoras theorem in the west). This theorem has long been known to Babylonians and China people (in ancient China, a dialogue between Shang Yang and Duke Zhou was recorded in Zhou Biao Shu Jing, a mathematical work of the Western Han Dynasty during the Warring States Period). Shang Gao said: "... so fold the moment, tick three, fix four, and cross the corner five." Quotient height means that when two right-angled sides of a right-angled triangle are 3 (short side) and 4 (long side) respectively, the radius angle (chord) is 5. In the future, people will simply describe this fact as "hooking three strands, four strings and five". This is China's famous Pythagorean theorem. ), but the earliest proof can probably be attributed to Pythagoras. He proved by deduction that the square of the hypotenuse of a right triangle is equal to the sum of the squares of two right angles, that is, Pythagorean Theorem.

number theory

Pythagoras did a lot of research on number theory, and divided natural numbers into odd numbers, even numbers, prime numbers, perfect numbers, square numbers, triangular numbers and pentagonal numbers. Pythagoras believes that number provides a conceptual model for the universe. Quantity and shape determine the form of all natural objects. Numbers have not only quantity, but also geometry. In this sense, they understand number as the form and image of natural objects, and it is the total root of all things. Because there are geometric points when there are numbers, there are surfaces when there are wires, and there are three-dimensional points when there are points. When there are solids, there are four elements, namely fire, air, water and earth, which make up everything, so numbers come before things. All phenomena and laws in nature are determined by numbers, and they all obey the "harmony of numbers", that is, obey the relationship of numbers.

Pythagoras further proved his theory by explaining the relationship between numbers and physical phenomena. He once proved that when three strings are used to produce a musical tone and its fifth and eighth tones, the ratio of the lengths of the three strings is 6:4:3. Starting from the viewpoint that the sphere is the most perfect geometric shape, he thinks that the earth is spherical and puts forward the viewpoint that the sun, the moon and the planets move in a uniform circle. He also thinks that ten is the perfect number, so there must be ten luminous bodies moving in the sky.

A theory

He also has a theory that the earth revolves around a "central fire" along a fixed point of a sphere in space, and there is an "anti-earth star" on the other side to balance it. This "central fire" is the altar of the universe and people will never see it. The distance between these ten celestial bodies and the central fire has the same proportional relationship with the interval between syllables, thus ensuring the harmony of the planets and playing the music of celestial bodies.

Integer change

Pythagoras and his school have many creations in mathematics, especially interested in the changing law of integers. For example, a number whose sum of all factors (except itself) is equal to itself is called a perfect number (such as 6,28,496, etc.). ), and the number greater than its factor is called abundance; The number less than the sum of its factors is called deficit.

Other contributions of geometry

In geometry, the Pythagorean school proved that the sum of the interior angles of a triangle is equal to two right angles. Studied the golden section; The method of regular pentagon and similar polygon was found. It is also proved that there are only five regular polyhedrons-regular tetrahedron, regular hexahedron, regular octahedron, regular dodecahedron and regular icosahedron.

Everything counts.

At the same time, he arbitrarily exaggerates that immaterial and abstract numbers are the origin of the universe, and thinks that "everything is a number", "number is the essence of everything" and "the principle of existence", and the whole universe is a harmonious system of numbers and their relationships. Pythagoras mystified numbers, believing that numbers are the mother of gods, the origin of the universe, and the principle of opposition and negation in nature.

Ethics acquired by Pythagoras

In his early academic career, Pythagoras often gave speeches in various places to explain his thoughtful views to people. In addition to the theme that "number is the source of all things", he often talks about moral and ethical issues.

He said to the dignitaries in the conference hall, "We must be fair. Injustice destroys order and harmony, which is the greatest sin. Swearing is a very serious act. Don't swear casually until the critical moment, but every official should be able to guarantee that he will not lie. "

Speaking of running a family, he thinks that love for children can't be expected in return, but fathers should try their best to win their children's heartfelt love with their words and deeds. Parents' love is sacred and children should cherish it. Children should be friends of their parents, and brothers and sisters should respect and love each other. When referring to the relationship between husband and wife, he said that mutual respect is the most important thing, and both sides should be loyal to their spouses.

He talked about self-discipline. He said that self-discipline is a test of human personality. For children, teenagers, the elderly and women, self-discipline is a virtue, but for young people, it is necessary. Self-discipline makes you healthy, clear-headed and strong-willed. Pythagoras talked about the importance of education from how to cultivate self-discipline. He believes that human self-discipline can only be cultivated under the guidance of reason and knowledge, and knowledge can only be acquired through education, so the importance of education cannot be ignored.

He vividly described the characteristics of education: "You can gain knowledge from others through learning, but those who teach you will not lose knowledge.". This is the characteristic of education. There are many beautiful things in the world. Good endowments can be obtained from heredity, such as healthy body, charming face and brave personality; Some things are precious, but once they are given to others, they no longer belong to you, such as wealth, such as power. More precious than all this is knowledge. As long as you study hard, you can get it without harming others, and you may change your nature. "

Admittedly, as an idealistic worldview, Pythagoras and his school's scientific exploration could not find the right direction, and even brought great negative influence to the later development of natural philosophy and science to some extent. However, these mistakes cannot conceal Pythagoras' positive role in the formation and development of natural science. Lenin told us that Pythagoras is "a connection between the germination of scientific thinking and religious myths and other fantasies."

Pythagoras' short stories

Pythagoras was once invited to a dinner of a rich politician. The host's luxurious palace-like restaurant is covered with square and beautiful marble floor tiles. These hungry VIPs complained bitterly because the big meal was delayed. This mathematician, who is good at observing and understanding, stares at these square tiles with regular arrangement and beautiful scale, but Pythagoras not only appreciates the beauty of tiles, but also thinks about their relationship with [numbers], so he picks up a brush, squats on the floor, selects a tile and draws a square with its diagonal AB as the edge. He found that the area of a square is exactly equal to the sum of the areas of two tiles. He was curious, so he made another square with the diagonal of the rectangle made of two tiles. He found that the area of this square is equal to the area of five tiles, which is the sum of the square areas with two sides. So far, Pythagoras has made a bold assumption: the square of the hypotenuse of any right triangle is exactly equal to the sum of the squares of the other two sides. That meal, the ancient Greek mathematician, never left the ground.

History of Western Philosophy-Pythagoras

Author: Bertrand Russell, England Source: History of Western Philosophy

The influence of Pythagoras on ancient and modern times is the theme of my chapter; Pythagoras is one of the most important figures in thought since the birth of mankind, whether he is clever or not. Mathematics, in the sense of proof deductive reasoning, started from him; Moreover, mathematics is closely combined with a special form of mysticism in his thought. Since then, partly because of him, mathematics has had a profound and unfortunate influence on philosophy.

Let's start with some known facts about his life. He is a native of Satsuma Island, and he began to flourish around 523 BC. Some people say that he is the son of a well-off citizen named Naisar, while others say that he is the son of Apollo. I ask readers to choose one of these two theories. In his time, Samak was ruled by the tyrant Pluckrady, an old rascal who made a fortune and owned a huge navy.

Samoyed is Miletus' commercial competitor; Its merchants are as far away as Seuss in Tarter, Spain, which is famous for its minerals. Polucladi became a tyrant of Satsuma in about 535 BC and ruled until 5 15 BC. He is not worried about moral censure; He drove away his two brothers who were engaged in authoritarian politics with him, and most of his navy was used for plunder at sea. The fact that Miletus surrendered to Persia not long ago was very beneficial to him.

In order to prevent the Persians from expanding westward, he made an alliance with King Amasis of Egypt. However, when the Persian king concentrated on conquering Egypt compared with the Sith, Polokrati realized that he would win and changed his position. He sent a fleet of his political enemies to attack Egypt; But the sailors defected and returned to Samoyed Island to attack him. Although he defeated them, he finally fell into a plot to exploit his greed and collapsed. The Persian governor of Sardis pretended to betray the Persian king and was willing to pay a large sum of money in exchange for the assistance of Polokrati. When Polucrati went to the mainland to meet the Persian governor, he was captured and crucified.

Polucladi is an art patron who beautifies Shama with many great buildings. Analion is his court poet. However, Pythagoras did not like his government, so he left Samoyed Island. It is said-not impossible-that Pythagoras has been to Egypt, where he learned most of his wisdom; In any case, it is certain that he finally settled in Croton, southern Italy.

Greek cities in southern Italy are as rich and prosperous as Samoa and Mile. Besides, they were not threatened by Persians. The two largest cities are West Barres and Croton. Luxury goods in West Barres are still popular today; According to Diodoros, its population reached 300,000 at its peak, although this is undoubtedly an exaggeration. Croton is about as big as West Barres. Both cities make a living by importing goods from Ionia to Italy. Some of these goods are used as Italian consumer goods, and some are re-exported from the west coast to Gaul and Spain. Many Greek cities in Italy fought fiercely with each other; Pythagoras arrived in Dacroton, and Croton had just been defeated by Lockley. However, shortly after Pythagoras arrived, Croton won a complete victory in the war against West Barres, and West Barres was completely destroyed (5 10 BC). West Barres and Millie have always had close business ties. Croton is famous for its medicine; There was a man in Croton, Demetrius, who used to be a doctor of Poloclaudius and later Darius. Pythagoras and his disciples established a group in Croton, which was once very influential in this city. But in the end, the citizens opposed him, so he moved to Medapenton (also in southern Italy) and died here. He soon became a mythical figure, endowed with miracles and divine power, but he was also the founder of a mathematician school. In this way, there are two diametrically opposed legends about his deeds, and it is difficult to find out the truth.

Pythagoras is one of the most interesting and difficult figures in history. His legends are not only an almost inseparable mixture of truth and absurdity, but also provide us with the most peculiar psychology in the simplest and least controversial form of these legends. In short, he can be described as a combination of Einstein and Mrs. Aidit. He established a religion, the main teachings of which are the reincarnation of the soul and the evil nature of eating beans. His religion is embodied in a religious group, which gained control of the country and established a set of saint rule in various places. But people without reform are eager to eat beans, so sooner or later they will rebel.

pythagorean theorem

Pythagorean theorem is also called quotient height theorem, or Pythagoras theorem:

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of two right angles. If the two right angles of a right triangle are A and B and the hypotenuse is C, then A

According to textual research, human beings have known this theorem for at least 4000 years!

At the beginning of China's earliest mathematical work "Weekly Parallel Calculations", there is the relevant content of this theorem: Duke Zhou asked: "I heard that doctors are good at counting, so I want to ask the ancients to set up a calendar of weeks and days." The sky cannot rise step by step, and the earth cannot be measured. How many times can I go out? "Shang Gao replied:" The counting method comes from the circle, the circle comes from the square, the square comes from the moment, and the moment comes out 998 1. Therefore, the moment is considered as three, the stock is four and the diameter is five. The outside is square, half-carved, and circular. If you get three, four and five, then the total length of the two moments is twenty plus twenty-five, which is called the product moment. Therefore, Yu rules the world because this number is born. "From the above conversation, we can clearly see that people in ancient China discovered and applied the pythagorean theorem, an important principle of mathematics, thousands of years ago.

The earliest documents in the west proved to be given by Pythagoras. It is said that when he proved Pythagorean theorem, he was ecstatic and killed a hundred cows to celebrate. Therefore, western countries also call Pythagorean Theorem "Hundred Cows Theorem". Unfortunately, Pythagoras' proof method has long been lost, and we have no way of knowing his proof method.

In fact, in earlier human activities, people have realized some special cases of this theorem. In addition to the above two examples, it is said that the ancient Egyptians also used the law of "hooking three strands, four strings and five" to determine the right angle. However, this legend has aroused the suspicion of many mathematical historians. For example, Professor M. Klein, an American mathematical historian, once pointed out: "We don't know whether the Egyptians realized the Pythagorean theorem. We know that they have rope puller (surveyor), but they tied a knot on the rope, divided the whole length into 3, 4 and 5 sections, and then used them to form a right triangle, which has never been confirmed in any literature. " However, archaeologists discovered several pieces of ancient Babylonian clay tablets, which were completed around 2000 BC. According to expert research, one of them is engraved with the following question: "A stick with a length of 30 units stands upright on the wall. How far is its lower end from the corner when its upper end slides down by 6 units? " This is a special case of a triangle with a ratio of three sides of 3:4:5. Experts also found that there was a strange number table engraved on another clay tablet, with four columns and fifteen rows of numbers. This is a Pythagorean table: the rightmost column is the serial number from 1 to 15, while the left three columns are the values of stock, hook and string respectively, and a total of 15 Pythagorean numbers are recorded. This shows that Pythagorean theorem has actually entered the treasure house of human knowledge.

Pythagorean theorem is a pearl in geometry, which is full of charm. For thousands of years, people have been eager to prove it, including famous mathematicians, painters, amateur mathematicians, ordinary people, distinguished dignitaries and even the president of the country Perhaps it is precisely because Pythagorean theorem is important, simple, practical and more attractive that it has been repeatedly demonstrated for hundreds of times. 1940 published a proof album of Pythagorean theorem, which collected 367 different proof methods. In fact, that's not all. Some data show that there are more than 500 ways to prove Pythagorean theorem, and only the mathematician Hua in the late Qing Dynasty provided more than 20 wonderful ways to prove it. This is unmatched by any theorem. (The detailed proof of Pythagorean theorem is not included because the proof process is complicated. ※.)

People are interested in Pythagorean theorem because it can be generalized.

Euclid gave a generalization theorem of Pythagorean theorem in Elements of Geometry: "A straight side on the hypotenuse of a right triangle has an area equal to the sum of the areas of two similar straight sides on two right angles".

From the above theorem, the following theorem can be deduced: "If a circle is made with three sides of a right-angled triangle as its diameter, the area of the circle with the hypotenuse as its diameter is equal to the sum of the areas of two circles with two right-angled sides as its diameter".

Pythagorean theorem can also be extended to space: if three sides of a right triangle are used as corresponding sides to make a similar polyhedron, then the surface area of a polyhedron on the hypotenuse is equal to the sum of the surface areas of two polyhedrons on the right side.

If three sides of a right-angled triangle are used as balls, the surface area of the ball on the hypotenuse is equal to the sum of the surface areas of two balls made on two right-angled sides.

And so on.

appendix

First of all, briefly introduce Zhou Yi Shu Jing.

Zhou Kuai Kuai Jing is one of the ten books of calculation. Written in the second century BC, it was originally named Zhou Jie, which is the oldest astronomical work in China. It mainly expounded the theory of covering the sky and the method of four seasons calendar at that time. In the early Tang Dynasty, it was stipulated as one of imperial academy's teaching materials, so it was renamed Zhou Kuai. The main achievement of Zhouyi ·suan Jing in mathematics is the introduction of Pythagorean theorem and its application in measurement. The original book did not prove Pythagorean theorem, but the proof was given by Zhao Shuang in Zhou Zhuan Pythagorean Notes.

·suan Jing of Zhouyi adopts quite complicated fractional algorithm and Kaiping method.

Second, the story of Garfield proving Pythagorean theorem

1876 One weekend evening, on the outskirts of Washington, DC, a middle-aged man was walking and enjoying the beautiful scenery at dusk. He was then Republican Congressman Garfield of Ohio. Walking, he suddenly found two children talking about something with rapt attention on a small stone bench nearby, arguing loudly and discussing in a low voice. Driven by curiosity, Garfield followed the sound and came to the two children to find out what they were doing. I saw a little boy bend down and draw a right triangle on the ground with branches. So Garfield asked them what they were doing. The little boy said without looking up, "Excuse me, sir, if the two right angles of a right triangle are 3 and 4 respectively, what is the length of the hypotenuse?" Garfield replied, "It's five." The little boy asked again, "If the two right angles are 5 and 7 respectively, what is the length of the hypotenuse of this right triangle?" Garfield replied without thinking, "The square of the hypotenuse must be equal to the square of 5 plus the square of 7." The little boy added, "Sir, can you tell the truth?" Garfield was speechless, unable to explain, and very unhappy.

So Garfield stopped walking and immediately went home to discuss the questions the little boy gave him. After repeated thinking and calculation, he finally figured it out and gave a concise proof method.