The turning point in mathematics is Descartes' variables. With variables, motion enters mathematics. With variables, dialectics enters mathematics. With variables, differential calculus and integral calculus immediately become necessary, and they immediately become necessary. Produced and generally completed by Newton and Leibniz, but not invented by them.

——Engels

　　Starting from the European Renaissance in the early 15th century, the large-scale development of industry, agriculture, navigation and merchant trade formed a new economic era. The religious reform and doubts about the confinement of church ideas, the advanced science and technology of the East through the Arab The introduction of Greek literature and the influx of a large number of Greek documents into Europe after the fall of the Byzantine Empire presented a completely new look to the intellectual class at that time. In the 16th century, Europe was in the budding period of capitalism. The productivity had been greatly developed. The development of production practice raised new topics for the natural sciences, urgently requiring the development of basic disciplines such as mechanics and astronomy, and these disciplines They are all deeply dependent on mathematics, and thus promote the development of mathematics. The various requirements put forward by science on mathematics are finally summarized into four core issues:
　　(1) The problem of reciprocal determination of speed and distance in motion [Geometric Demonstration]

        It is known that the distance  moved by an object is represented by the formula   as a function of time, Find the speed and acceleration of the object at any moment; conversely, given the formula that the acceleration table of the object is a function of time, find the speed and distance. This type of problem arises directly when studying motion. The difficulty is that the speed and acceleration under study are changing every moment. For example, to calculate the instantaneous speed of an object at a certain moment, you cannot use the time of movement to divide the distance moved like calculating the average speed, because at a given moment, the distance the object moves and the time it takes are 0, and 0/0 is meaningless. However, according to physics, there is no doubt that every moving object must have speed at every moment of its motion. The problem of finding the moving distance given the speed formula also encounters the same difficulty. Because the speed changes every moment, you cannot multiply the time of motion by the speed at any moment to get the distance an object moves.
　　(2) The problem of finding the tangent to a curve [ Geometric Demonstration ]
　　This problem itself is purely geometric and is of great importance for scientific applications. Due to the needs of studying astronomy, optics was an important scientific research in the seventeenth century. To study the passage of light through the lens, the designer of the lens must know the angle of the light incident on the lens in order to apply the law of reflection. What is important here is the light. The angle between the normal line and the normal line of the curve, and the normal line is perpendicular to the tangent line, so it always depends on finding the normal line or tangent line; another scientific problem involving the tangent line of the curve appears in the study of motion, finding the motion The direction of motion of an object at any point on its trajectory is the tangent direction of the trajectory.
　　(3)   Problems of finding length, area, volume and center of gravity [ Geometry Demonstration ]
　　These problems include finding the length of a curve (such as the distance a planet moves in a known period), the area enclosed by a curve, the volume enclosed by a curved surface, objects The center of gravity, the gravitational force of one sizable object (such as a planet) on another object. In fact, the problem of calculating the length of an ellipse was so difficult for mathematicians that for a period of time, further work on this problem failed, and no new results were obtained until the next century. Another example is the area problem. In early ancient Greece, people used the exhaustion method to find some areas and volumes, such as finding the area enclosed by the parabola in the interval [0, 1], axis and the straight line , they adopted the exhaustion method (geometric demonstration). However, the application of the exhaustive method requires a lot of skills and lacks generality, so numerical solutions are often not available. When Archimedes' work became famous in Europe, interest in finding length, area, volume, and center of gravity was revived. The method of exhaustion was modified first gradually and then fundamentally with the creation of calculus.
　　(4) Problems of finding maximum and minimum values [ Geometric Demonstration ]
　　When a cannonball is fired from the barrel, the horizontal distance it travels, that is, the range, depends on the inclination angle of the barrel to the ground, that is, the launch angle. A " practical " problem is to find the launch angle that gives maximum range. In the early seventeenth century, Galileo concluded that the maximum range (in a vacuum) was reached when the launch angle was ; he also concluded that the maximum height reached by the cannonball after being launched from various angles was different. Studying the motion of planets also involves the problem of maximum and minimum values, such as finding the farthest and closest distance of a planet from the sun.