Introduction to Archimedes Archimedes (Archimedes, 287-2 BC12) is one of the greatest mathematicians in the history of mathematics. Bell (E.T.Bell, 1883- 1960), a historian of modern mathematics, said: "In any list of the three greatest mathematicians in history, the other two are usually Newton and Gauss. However, compared with their great achievements and background of the times, Archimedes should be the first to be respected compared with their far-reaching influence on the present and future generations. " Archimedes' name became a symbol of wisdom among his contemporaries, and he would solve the most difficult problems with simple methods. Plutarch, a famous ancient Greek writer and historian (BC 1 century), said: Solving such a difficult problem is so simple and clear that it has never been heard of in mathematics. If anyone tries to solve these problems by himself, he will get nothing. However, if he is familiar with Archimedes' solution, he will get such an impression immediately. He will find a solution by himself. Archimedes led us to our goal in such a simple and clear way. Archimedes was born in Syracuse, Sicily at the southern tip of the Italian peninsula. His father is an astronomer and wrote some articles about the diameter of the sun and the moon. Archimedes studied in Alexandria in his early years and kept in touch with scholars in Alexandria. Archimedes devoted himself to scientific research all his life and was often immersed in selfless thinking. Plutarch once wrote: Archimedes forgot to eat and sleep, and completely ignored his concern for his body. He is often forced to take a bath, and when he takes a bath, he applies ointment. However, at this moment, he drew geometric figures on his oiled body with his fingers. Vitruvius, an ancient Roman architect, described Archimedes' discovery of the law of floating bodies. It's amazing. Once upon a time, King Hilong of Syracuse had a crown of pure gold. After the crown was made, the king doubted whether it was made of pure gold, so he asked Archimedes, who is famous for his versatility, to identify it. Archimedes thought for a long time to find the answer. He was very depressed. He bathed in the public bathroom. When he was immersed in a bathtub full of water, the water overflowed the basin and he suddenly lost weight. So he suddenly thought of something different. Although the weight is the same, the discharged water is not equal due to different volumes. According to this principle, we can not only judge whether the crown is doped with impurities, but also know the weight of stolen gold. This successful discovery surprised Archimedes. He ran out of the bath naked and shouted, "I found it". After careful experiments, he finally discovered the basic principle of hydrostatics: "Archimedes principle"-objects become lighter in liquid. It is equal to the weight of the discharged liquid. In the last few years of Archimedes' life, he showed sincere patriotism. He gave all his strength and wisdom for the safety of his motherland. When Maceiras, the leader of the Roman army, led an army to attack Syracuse, Archimedes developed his intelligence and made new machinery to counter the advanced military facilities in Rome at that time. He made many weapons. Be prepared to repel the enemy under any circumstances. If the enemy is far away from the city, he will use a huge long-range projector to launch a large number of "heavy artillery shells" and "rockets" to defeat the enemy warships. When Archimedes found that the shell had fallen too far to hit the ship, he used a projector suitable for short distances. In this way, the Roman army was frightened and could not move forward. According to Greek literature, when Roman warships approached the city gate, Archimedes used a huge fire mirror to reflect sunlight and make them burn. There is also a saying that he threw firearms to throw burning things out and burn the enemy warships. In a word, Archimedes did his best to invent all kinds of new instruments, which dealt a heavy blow to the Roman army and made great contributions to defending the motherland. Later, because of the traitor's betrayal, the ancient city of Sila fell. One theory is that Archimedes doesn't seem to know. Still obsessed with thinking about mathematics and burying his head in drawing geometric figures. When a Roman soldier rushed at him, Archimedes said seriously, "Go away and don't touch my figure." Roman soldiers felt insulted and drew their swords and stabbed Archimedes. He is 75 years old. According to Archimedes' living will, a figure was engraved on the tombstone, and the ball in the figure was cut into a cylinder, symbolizing his particularly cherished invention. Archimedes made many contributions to mathematics. Manuscripts of many of his works have been preserved to this day. Some historians of mathematics have translated his original works into modern languages. For example, Heath's English version, Zvalina's German version, esk's French version, and E.J.Dijksterhuis's Dutch masterpiece Archimedes, his works cover a wide range. This also shows that he has profound knowledge of all previous discoveries in mathematics. Most of Archimedes' preserved works are works on geometry, but there are also some works on mechanics and calculation problems, mainly on cylinders, orthogonal parabolas and measuring circles. About helicoids, about plane cones, sand calculators, about methods (some theorems about geometry in Archimedes' letter to Eratos), about floating bodies, lemmas. In the geometry of these works, he supplemented many original studies on the quadrature method of plane curves and the determination of the volume surrounded by surfaces. In these studies, he foresaw the concept of minimum division, which played an important role in mathematics in the17th century. It itself is the predecessor of calculus, but it lacks the concept of limit. Archimedes quadrature method contains the bud of integral thought. Archimedes discovered the theorem in this way, studied the quadrature problem of curves and figures, and established the result by exhaustive method: "The following is a brief proof of Archimedes, which can reveal his research method. AQ is a parabola bow, and the vertex of parabola is a (as shown in Figure 3. 14). The graph can be completed by dividing it equally at q, and now Q = 3ac. Dividing q equally repeatedly in the same way can prove the right addition of formula (1) and so on. If you do this continuously on these lines, you can prove that the parabolic arch area refers to AQ here, but Archimedes did not seek the limit, so he proved his conclusion by reducing to absurdity. The key point of this proof is that if the area is not equal to the given area S, it must be greater than it and less than it at the same time, which is unreasonable. Therefore, it is inferred that the area of parabolic bow is equal to the error estimate of pi given by Archimedes in the article Measuring Circle by circumscribed and inscribed 96 polygons. In the proof, Archimedes avoided the concept of infinitesimal. Because this concept has always been doubted by the Greeks, he considered inscribed polygons and circumscribed polygons. He established this basic principle through explanation and proof: "Given two inequalities, no matter how close the ratio of large quantity to small quantity is to 1, it is possible to: (1) find two straight lines, so that the ratio of the long one to the short one becomes smaller (greater than1); (2) Make a circle or fan-shaped similar circumscribed polygon and inscribed polygon so that the ratio of the perimeter or area of the circumscribed polygon to the perimeter or area of the inscribed polygon is less than a given ratio. " Then, as Euclid did, he proved that if the number of sides is doubled continuously, some arches will be left, which add up to be smaller than any specified area. Archimedes made a little supplement to this. That is, if the number of sides of the circumscribed polygon increases enough, the difference between the area of the polygon and the area of the circle can be smaller than any given area. Archimedes also studied the spiral and wrote The Theory of Spiral. Some people think that this is, in a sense, the most wonderful part of Archimedes' total contribution to mathematics. Many scholars foresaw the calculus method in his helix tangent method. To his credit, he defined mathematical objects from the perspective of movement. If a ray rotates around its endpoint at a uniform speed and a moving point moves along the ray from the endpoint at a uniform speed, then this point will follow a spiral. This spiral was later called "Archimedes spiral". Spiral has a basic property, which relates the length of vector diameter to the angle at which the initial straight line rotates from the initial position. This basic property appears in the proposition 14. Now it is expressed by the equation R = Aθ. Archimedes then proved that the area enclosed between the first circle and the initial line, that is, a straight line in the vector diameter o is tangent to the end of the helix', and the other straight line coming out of the fixed end is perpendicular to the straight line returning to the original position after one circle, so it intersects with the tangent line. I think that's what a straight line that intersects a tangent does. It is equal to the circumference of this circle. "This is the proposition about helix in the book. 24. Archimedes designed a counting system for large numbers in the book Calculation of Sand (on the counting of sand), which can represent numbers outside the Greek counting system at that time. Before Archimedes, the Greek calculation was expanded to no more than 10000. And called 10000 countless. Archimedes took countless as a new unit, introduced countless into calculation and put forward a higher unit. It is said that Archimedes put forward a "flock problem" to Greek mathematicians. In essence, eight positive integer solutions should be obtained from seven equations. Finally, it comes down to a quadratic indefinite equation. The number of digits of the solution of this equation is quite large. Bo Li Lemma is the earliest introduction by Archimedes, which contains 15 propositions, such as: Proposition 2, if you make a circumscribed circle and an inscribed circle of a square, the area of the circumscribed circle is equal to twice the area of the inscribed circle. Proposition 3: If two chords intersect at right angles in a circle, then the sum of four line segments divided by the square of the intersection is equal to the square of the diameter. In On Floating Bodies, Archimedes first gave the law of buoyancy for objects with smaller specific gravity than fluid, objects with the same specific gravity and objects with larger volume, which is indeed a masterpiece with epoch significance. Archimedes used many unique methods in the creation of mathematics, especially his method of finding problems according to the principles of mechanics. A letter written by Archimedes to Eratosthenes (about 274 BC-274 BC 194 BC) and a copy of other works by Archimedes were found in Constantinople (now Istanbul), the largest Turkish city. This paper describes a large number of geometric problems of calculating length, area, volume and center of gravity combined with statics and fluid mechanics. The key point is that the volume consists of the area, and the area consists of parallel lines. Each line has a weight, which is proportional to its length. So the problem can be summed up as balancing the unknown geometry with the known geometry for focusing, in which the parabolic arch area is determined by the lever principle. Examples are the area of a sphere and its crown, and the volume of a hyperbola of revolution. In fact, this is a faster detour to the integral. Archimedes confidently predicted: "Once this method is established, some people, both my contemporaries and my successors, will use this method to discover other theorems, which I did not expect. "Archimedes gave a logical proof in order to establish a method to find problems in mathematics. Archimedes' prediction finally came true after nearly 2000 years. /kloc-in the 8th century, Da-Niel Bernoulli deduced the general solution of the differential equation of string vibration in the form of trigonometric series from physical knowledge./kloc-Riemann in the middle of the 9th century determined by electrical theory that there is an algebraic function which usually has a solution on every closed Riemann surface. All Archimedes' conclusions were drawn without algebraic symbols. The process of proof is quite complicated, but with amazing originality, he combined skillful calculation skills with strict proof, and closely combined abstract theory with concrete application of engineering technology, which pushed Greek mathematics to a new stage. Because Archimedes paid attention to understanding various phenomena of things in practice in scientific research, he understood the essence through phenomena, and then after strict argumentation, he raised empirical facts into systematic theories. Archimedes also made great contributions to astronomy and mechanics. Archimedes loved astronomy all his life, but unfortunately his works on astronomy were not preserved. According to Syntaxis, Aki is more accurate for astronomical observation. He used instruments to measure the angle of view of the sun. It is said that Archimedes wrote this book about spheres. Now it has been lost. In short, all Archimedes' masterpieces are famous for their precision and preciseness. As Heath, a historian of mathematics, said, "These works are all monuments of mathematical papers without exception. The gradual enlightenment of the solution, the ingenious arrangement of the proposition order, the strict exclusion of everything that is not directly related to the purpose, and the overall embellishment-the perfect impression is so deep that it can produce a feeling of almost awe in the reader's mind. "