Archimedes
Date of birth and death: 287-2 BC12.
Introduction:
The great mathematician and mechanic of ancient Greece was born in Syracuse, Sicily, and died here. He studied with Euclid's students in Alexandria, the cultural center at that time, and later kept close contact with Alexandria's scholars, so he was a member of the Alexandria School. Later generations spoke highly of Archimedes, who was often ranked as the three greatest mathematicians in history with I Newton and C·F· Gauss. His life is not a detailed record, but about him.
Life:
Archimedes (287-2 12 BC) was born in a small village near Syracuse, Greece. He was born into a noble family and was related to King Hilong of Syracuse. His family is very rich. Archimedes's father was an astronomer and mathematician, knowledgeable and humble. Archimedes was influenced by his family. He was interested in mathematics and astronomy since he was a child, especially the geometry of ancient Greece. When he was just eleven years old, with the help of his relationship with the royal family, he was sent to study in Alexandria, Egypt. Alexandria, located at the mouth of the Nile, was one of the centers of cultural trade at that time. There are magnificent museums and libraries, which are praised as "the capital of wisdom" by the world. Archimedes studied and lived here for many years. He is in close contact with many scholars. He absorbed the excellent cultural heritage of the East and ancient Greece and made great contributions in his subsequent scientific career. In 2 12 BC, Rome invaded Syracuse, and Archimedes was killed by Roman soldiers at the age of 75. Archimedes' body was buried in Sicily, and the tombstone was engraved with a cylindrical carved ball to commemorate his outstanding contribution to geometry. Archimedes' achievements.
Archimedes is undoubtedly the greatest mathematician and scientist produced by ancient Greek civilization. His outstanding contributions in many scientific fields earned him the high respect of his contemporaries.
Archimedes worked out the areas and volumes of parabolic bows, spirals and circles, as well as the volumes of complex geometric bodies such as ellipsoids and paraboloids. In the process of deriving these formulas, he skillfully used the "exhaustive method", which is what we call the method of gradually approaching the limit today, and is recognized as the originator of calculus calculation. He also used this method to estimate the value of ∏ between sum and got the solution of cubic equation. Facing the complex digital representation in ancient Greece, Archimedes put forward an important step-by-step calculation method and used it to solve many mathematical problems. Archimedes made the most outstanding achievements in mechanics, mainly focusing on statics and hydrostatics. In the process of studying machinery, he discovered the lever principle and used this principle to design and manufacture many machines. In the process of studying floating bodies, he discovered the law of buoyancy, which is the famous Archimedes principle.
Archimedes also made outstanding achievements in astronomy. He designed some spheres and connected them with strings and sticks, imitating the movements of the sun, the moon and the stars, and making them rotate by water power. In this way, solar and lunar eclipses can be vividly displayed. Archimedes thought that the earth was spherical and revolved around the sun, which was earlier than Copernicus' Heliocentrism 1800 years. Because of the conditions at that time, he did not make a thorough and systematic study on this issue, but it was remarkable to put forward such an opinion as early as the third century BC. Archimedes wrote many works. As a mathematician, he wrote some mathematical works, such as On Sphere and Cylinder, On Cone and Sphere, Quadrature of Parabola, On Spiral, etc. As a mechanic, he wrote articles about plate balance and floating bodies.
Archimedes not only made brilliant achievements in theory, but also was an engineer with practical spirit. In his life, he designed and manufactured many harmonious machines, besides the lever system, it is worth mentioning that there are weight-lifting pulleys, irrigation machines, water pumps and military projectors. The water pump known as Archimedes' water lifting screw was invented to drain water from the cabin of a big ship. Water pumps can be transported to high places by screws and have been widely used in Egypt.
Archimedes is not only a theorist, but also a practitioner. Throughout his life, he was keen on applying his scientific discoveries to practice, thus combining the two. In Egypt, about 1500 BC, people used levers to lift heavy objects, but people don't know why. Archimedes devoted himself to studying this phenomenon and discovered the lever principle.
King Heron has always been skeptical about Archimedes' theory. He asked Archimedes to turn them into living examples to convince people. Archimedes said, "Give me a fulcrum and I can move the earth." The king said, "I'm afraid this can't be realized." You'd better help me tow the big boat to the coast. " This ship was built by King Heron for the Egyptian King. It is big and heavy because it can't move. It has been stranded on the coast for many days. Archimedes readily promised. Archimedes designed a complicated lever pulley system to be installed on the ship, and handed one end of the rope to King Henon. King Hennon gently pulled the rope, and a miracle appeared. The ship moved slowly and finally sank into the sea. The king was very surprised and admired Archimedes. He sent someone to post a notice saying, "No matter what Archimedes says, you must believe him in the future."
Mystery of golden crown
King Hennon asked the goldsmith to make a pure gold crown for him. After it was done, the king suspected that the craftsman had mixed silver into the crown, but the crown was as heavy as the pure gold given to the goldsmith at the beginning. Did the craftsman play tricks? It is difficult to verify the authenticity without destroying the crown. This question not only stumped the king, but also made the ministers look at each other. Later, the king gave it to Archimedes. Archimedes tried many ways, but all failed. One day, he went to the bathhouse to take a bath. While sitting in the bathtub, he saw the water overflowing and felt his body being gently dragged up. He suddenly realized that he jumped out of the bathtub and didn't even care about his clothes. He shouted "Eureka" and "Eureka, I know, I found it" all the way. It turns out that if the crown is put into water, the amount of water released is not equal to the amount of water released by the same weight of gold, and it must be mixed with other metals. This is the famous law of buoyancy, that is, an object immersed in a liquid is subjected to upward buoyancy, and its size is equal to the weight of the liquid discharged by the object. Later this law was passed.
Patriot Archimedes
In his later years, Rome invaded Syracuse, and Archimedes instructed his compatriots to make many weapons for attack and defense. When Marcelle, the leader of the invading army, led his troops to attack the city, he designed a trebuchet and beat the enemy out of the water. The iron claw crane he made can lift and reverse enemy ships and throw them into the depths of the sea. Legend has it that he also led the people of Syracuse to make a big concave mirror, focusing the sunlight on nearby enemy ships. Let them burn. Roman soldiers were frightened by such frequent blows. When they saw a rope or wood thrown from the city, they exclaimed, "Archimedes is coming", and then they ran away. Rome was blocked for three years. Finally, in 2 12 BC, the Romans took advantage of Syracuse's slightly relaxed defense to attack and break into the city. At this time, Archimedes was studying a profound math problem. Archimedes trampled on his drawing and argued with him angrily. The cruel soldier didn't listen, but with a flick of his knife, a talented science superstar fell.
Rumors and contributions about him:
It is said that after he established the lever law of mechanics, he once issued a magnificent statement: "Give me a foothold and I can move the earth!" " King Shiloh of Syracuse asked the goldsmith to make a crown out of pure gold. Because it was suspected that there was silver mixed in it, Archimedes was asked to identify the five crowns. The water overflowed the basin, so he realized that although objects made of different materials have the same weight, the discharged water will not be equal because of different volumes. According to this truth, it can be judged whether the crown is adulterated. Archimedes jumped up with joy and ran home naked, shouting, "Eureka! Found it. " (Greek means "I found it") He summed up the basic principle of hydrostatics in his masterpiece "On Floating Bodies", that is, the weight of an object in a liquid is equal to the weight of the liquid, and later became famous for Archimedes' principle. During the Second Punic War, the Roman army besieged Syracuse, and Archimedes devoted all his intelligence to serving the motherland. It is said that he used a crane to catch enemy ships. Invented a wonderful machine that fired stones and fireballs. There are also some books that record that he used a huge fire mirror to reflect sunlight and burn enemy ships. This is probably an exaggeration. In short, he tried his best to give the enemy a heavy blow. Finally, Syracuse Gui was betrayed by spies and ran out of food. Archimedes died unfortunately at the hands of Roman soldiers. Archimedes' handed down works mainly include the following works. On the Ball and the Cylinder, which is his proud masterpiece. Including many great achievements. He deduced more than 50 propositions about the area and volume of balls and cylinders from several definitions and axioms. Based on several basic assumptions, the balance of plane graphics or its center of gravity is demonstrated by strict geometric methods, and the centers of gravity of several plane graphics are obtained. The sand counter designs a method that can represent any large number, and corrects some people who think that sand is uncountable. Even if it is countable, it cannot be expressed by arithmetic symbols. On Floating Bodies discusses the buoyancy of objects and studies the stability of rotating projectiles in fluid. Archimedes also put forward a "cattle problem", which contains eight unknowns. Finally, it comes down to a quadratic indefinite equation. The number of its solutions is amazing, with more than 200 thousand bits!
It is doubtful whether Archimedes solved this problem at that time. In addition, there is a very important book "Letter to Eratosthenes", which discusses the methods to solve mechanical problems. This is a scroll of parchment manuscript discovered by Danish linguist J·L· Heiberg in 1906 in Istanbul. It was originally written in Greek, but it was later erased and rewritten with religious words. Fortunately, the original handwriting was not wiped clean. After careful identification, it proved to be Archimedes' work. There are some things I have seen elsewhere, including things that used to be considered lost. Later, it was published internationally in the name of Archimedes method. This paper mainly talks about the method of finding problems according to mechanical principles. He regards an area or volume as something with weight, divides it into many very small strips or pieces, and then balances these "elements" with the known area or volume to find the center of gravity and fulcrum. The required area or volume can be calculated by lever law. He regards this method as a tentative work before strict proof. After getting the result, he will prove it by reducing to absurdity. In this way, he achieved many brilliant achievements. Archimedes' method has the idea of modern integral theory. However, he did not explain whether this "element" is finite or infinite, nor did he get rid of the dependence on geometry, let alone use the limit method. His thought has epoch-making significance and is the pioneer of modern integral calculus. There are many other inventions. No ancient scientist, like Archimedes, combined skillful calculation skills with strict proof, and closely combined abstract theory with concrete application of engineering technology.
Later, Archimedes became a great scholar, as well as a mathematician and mechanic, enjoying the reputation of "the father of mechanics". The reason is that he discovered the lever principle through a lot of experiments, and deduced many lever propositions by geometric methods, and gave strict proofs. Among them, the famous Archimedes Principle has also made brilliant achievements in mathematics. Although Archimedes has only a dozen works, most of them are geometric works. This plays a decisive role in promoting the development of mathematics. The Calculation of Sand Particles is a book that specially introduces the calculation method and theory. Archimedes used his peculiar imagination to calculate the number of sand grains in a big sphere full of the universe, established a new counting method of order of magnitude, determined a new unit, and put forward a model for representing arbitrary large numbers, which is closely related to logarithmic operation. The measurement of the circle uses circumscribed circle and inscribed circle 96. It is found that pi is 22/7 < π < 223/7 1, which is the earliest value in the history of mathematics, and the error limit is clearly pointed out. He also proved that the area of a circle is equal to the area of a regular triangle with a circumference as the base and a high radius. By exhaustive method, the ball and cylinder skillfully prove that the surface area of the ball is equal to 4 times the area of the great circle of the ball; The volume of a sphere is four times that of a cone. The base of this cone is equal to the great circle of the sphere, which is higher than the radius of the sphere. Archimedes also pointed out that if there is an inscribed sphere in an equilateral cylinder, the total area of the cylinder and its volume are the surface area and volume of the sphere respectively. In this book, he also put forward the famous "Archimedes axiom" and "parabola quadrature method", and studied the quadrature problem of curves and graphs. The exhaustive method establishes a conclusion: "The area of any arch (i.e. parabola) surrounded by a straight line and the cross section of a right-angled cone is four-thirds of the area of a triangle with the same base and height." He also verified this conclusion again with the method of mechanical weight, and successfully combined mathematics with mechanics. On Spiral is Archimedes' outstanding contribution to mathematics. He made clear the definition of spiral and the calculation method of its area. In the same book, Archimedes also derived the geometric method of summation of geometric series and arithmetic series. Plane balance is the earliest work of mechanical science, which is about determining the center of gravity of plane and three-dimensional graphics. Floating Body is the first monograph on hydrostatics. Archimedes successfully applied mathematical reasoning to analyze the balance of floating body, and expressed the law of floating body balance with mathematical formula. It is about determining the volume of cone formed by parabola and hyperbola rotation and the volume of sphere formed by ellipse rotation around its long axis and short axis. 1906, Danish mathematical historian Heiberg discovered Archimedes' letter to erato Sese and some other works of Archimedes. Through research, it is found that these letters and manuscripts contain the idea of calculus, but what he lacks is the concept of limit. However, the essence of his thought extended to the field of infinitesimal analysis, which matured in the17th century and predicted the birth of calculus. Because of his outstanding contribution, E.T. Bell of the United States commented on Archimedes in Mathematical Figures: Any published list of the three greatest mathematicians in history will definitely include Archimedes, while the other two are usually Newton and Gauss. However, based on their brilliant achievements and the background of the times,