Achievements:
A, the balance of plane graphics or its center of gravity
1. Objects with the same weight are placed at equal distances (each at one end of the lever and equidistant from the fulcrum) and are in a state of balance; Objects of the same weight with different distances are unbalanced and will tilt to the far end.
2. When the weights placed at a certain distance are in equilibrium, if a little weight is added to one of the weights, it will lose its balance and tilt to the added end.
Second, parabola quadrature
In this paper, the quadrature problem of curves and figures is studied, and the conclusion is established by exhaustive method: "The area of any arch (that is, parabola) surrounded by the sections of straight lines and right-angled cones is four-thirds of the area of triangles with the same base height." He also verified this conclusion again by mechanical weight method, and successfully combined mathematics with mechanics.
Third, On the Ball and the Cylinder.
(On Sphere and Cylinder) The whole paper is divided into two volumes. At the beginning of the first volume, six definitions and five hypotheses are given. For example, a cone with a spherical bottom (fan cone) and an abacus bead solid composed of two cones are defined.
Fourth, "measurement of circles"
Using circumscribed circle and inscribed 96-sided circle, the approximate value of pi is obtained, 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; An exhaustive method was used.
Archimedes proved as follows. Let A be the area of the circle, C be the circumference of the circle, and T be the area of the triangle mentioned in the proposition. If A >；; T, we can make an inscribed regular polygon p with enough edges.
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Then get p > T.
Five, "On Spiral"
On spiral author: Archimedes, ancient Greece
Then the definition of spiral (now called "Archimedes spiral") is given:
Archimedean spiral, also known as "constant velocity spiral". When the point P moves at a uniform speed along the moving ray OP, the ray rotates around the point O at an equal angular speed, and the trajectory of the point P is called "archimedean spiral". Its polar coordinate equation is: r = aθ, and the distance between the arms of the spiral is always equal to 2π a.
Proposition 13-20 studies the tangent of helix, and gives the drawing method and various properties, including the calculation method of helix area.
Extended data
Character birth
In 287 BC, Archimedes was born in a small village near Syracuse in Sicily, 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 means great thinker. Influenced by his family, Archimedes was interested in mathematics and astronomy, especially the geometry of ancient Greece.
When Archimedes was born, the splendid culture of ancient Greece had gradually declined, and the economic and cultural center gradually moved to Alexandria, Egypt. But on the other hand, the emerging Roman Republic in the Italian peninsula is also expanding its power; There is also a new country, Carthage, rising in North Africa. Archimedes grew up in this era of alternating old and new forces, and the ancient city of Silas became the arena of many forces.
Reference: Baidu Encyclopedia-Archimedes