Analytic geometry is the product of the combination of algebra and geometry. It introduces variables into mathematics, which makes it possible to express movement and change quantitatively, thus setting up a stage for the creation of calculus. Calculus, especially the germination of integral, can be traced back to ancient times. We already know that the calculation of area and volume has always been a topic of interest to mathematicians. In the works of mathematicians in ancient Greece, China and India, there are many examples of using infinitesimal process to calculate the area, volume and curve length of special shapes. In the Nine Chapters of Arithmetic and Quotient Work written by Liu Hui in ancient times, it is mentioned that if you solve the cube obliquely, you will get two obstacles. The circuitous tunnels are Ma Yang and Turtle Tunnel. It is not easy for a horse to come second and a turtle to come first. Combine two turtles and three into one, measure it with chess, and show it. When he solved the volume of cone by infinite division, he put forward Liu Hui's principle of calculating the volume of polyhedron. Zu Chongzhi and his son summarized the work of Liu Hui, a famous mathematician in Wei and Jin Dynasties, and put forward that the potential is the same but the product is different, that is, two solids with the same height. If the horizontal cross-sectional area at any height is equal, the volumes of two solids are equal, which is the famous axiom of ancestor's declaration or the principle of Liu Zu. Zu Xuan applied this principle to solve Liu Hui's unsolved spherical volume formula. Cavalieri calculated the area of many plane figures and the volume of three-dimensional figures by using the principle of ancestor shovel, which is the basic prototype of calculating geometric volume in the current middle school three-dimensional geometry textbook. Galileo established the laws of free fall and momentum in modern times. In 1996 "Dialogue on Two New Sciences", the foundation of dynamics was laid. He realized the parabolic nature of the trajectory, and asserted that the maximum range of the projectile should be reached when the launching angle is 0, and so on. Galileo himself strongly advocated the mathematicization of natural science, and his works aroused people's great enthusiasm for accurately expressing the concepts and laws of dynamics he established. Kepler, a German astronomer and mathematician, published "The New Solid Geometry of Measuring Cylinder" in 1998, and discussed the integration method of the solid volume formed by the conic curve rotating around a straight line on its plane. The essence of his method is to use the sum of countless infinitesimal elements with the same dimension to determine the area of bending deformation and the volume of rotating body. Descartes and Fermat, the founders of analytic geometry, are pioneers in introducing coordinate method into differential calculus. Descartes' so-called circle method for finding tangent in geometry is essentially an algebraic method. In the same year, Fermat put forward an algebraic method to find the maximum and minimum values in manuscripts. In, Newton wrote A Brief Introduction to Flow Number, which was the first systematic calculus document in history. However, the brief discussion of the convection number is immature in many aspects, and Newton corrected it after studying it. Finally, the earliest public expression of Newton's calculus theory appeared in the mechanical work Mathematical Principles of Natural Philosophy, which appeared in 1998.

@ Liu Hongping

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