Archimedes, an ancient Greek mathematician, mechanic and founder of statics and hydrostatics, was born in Syracuse, Sicily in about 287 BC and died here in 2 12 BC. Alexander studied with Euclid's students in the cultural center at that time, and later kept close contact with Alexander's scholars, so he was a member of the Alexandria School. Later generations spoke highly of Archimedes, and often ranked him with I Newton and C·F· Gauss as the three greatest mathematicians in history. His life is not recorded in detail, but many stories about him are widely circulated. 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 in it, Archimedes was asked to identify it. When he entered the bathtub to take a bath, the water overflowed outside the bathtub, so he realized that although the weight of objects made of different materials was the same, the discharged water would be different because of their 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, "Found it! Found it. " (Greek means "I found it") In his famous book On Floating Bodies, he summed up the basic principle of hydrostatics, 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. Legend has it that he grabbed the enemy's boat with a crane and smashed it to pieces; Invent wonderful machines and fire big stones and fireballs. There are also some books that record that he burned enemy ships with huge fire mirrors reflecting sunlight, which is probably an exaggeration. In a word, 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 is his masterpiece, which contains many great achievements. Starting from several definitions and axioms, he deduced more than 50 propositions about the area and volume of spheres and cylinders. The balance of plane figure or its center of gravity, starting from several basic assumptions, demonstrates the mechanical principle with strict geometric methods and finds out the centers of gravity of several plane figures. The sand counter designs a method that can represent any large number, which corrects the wrong view that sand is uncountable, and even if it can be counted, it can't be represented by arithmetic symbols. On the floating body, the buoyancy of the object is discussed and the stability of the rotating projectile in the fluid is studied. Archimedes also put forward a "herd problem", which contains eight unknowns. Finally, it comes down to a quadratic indefinite equation. The number of its solutions is amazing, * * * more than 200,000 digits! It is doubtful whether Archimedes solved this problem at that time. In addition, there is a very important work, which is a letter to Eratosthenes, the content of which is to explore ways to solve mechanical problems. This is a scroll of parchment manuscript discovered by Danish linguist J.L. Heiberg in 1906. Originally written in Greek, it was later erased and rewritten in religious words. Fortunately, the original handwriting was not wiped clean. After careful identification, it was confirmed to be Archimedes' work. Some of them have seen it in other places, and some people think it has disappeared in the past. Later, it was published internationally in the name of Archimedes Law. 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, then balances these "elements" with the known area or volume, finds the center of gravity and fulcrum, and can use the lever law to calculate the required area or volume. He regards this method as a tentative work before strict proof, and will prove it by reducing to absurdity after getting the result. 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. Nevertheless, his thought is of epoch-making significance and is the pioneer of modern integral calculus. He has 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.