Binomial Theorem 1665, Newton, who was only 22 years old, discovered the binomial theorem, which is an essential step for the all-round development of calculus. Binomial theorem is widely used in combinatorial theory, higher power, higher arithmetic progression summation and difference methods. Newton's most outstanding achievement in mathematics was the creation of calculus. His outstanding achievement is to unify all kinds of special skills to solve infinitesimal problems since ancient Greece into two general algorithms-differential and integral, and establish the reciprocal relationship between these two operations. For example, area calculation can be regarded as the inverse process of finding tangent. Equation theory and variational method Newton also made classical contributions to algebra, and his generalized arithmetic greatly promoted equation theory. He found that the imaginary roots of real polynomials must appear in pairs, and found the upper bound law of polynomial roots. He expressed the sum formula of the roots of polynomials by using the coefficients of polynomials, and gave a generalization of Cartesian sign rule that limits the number of imaginary roots of real polynomials. Newton's first law of motion was discovered by Galileo. This law shows that if an object is at rest or moving in a straight line at a constant speed, it will remain at rest or continue to move in a straight line at a constant speed as long as there is no external force. This law, also known as the law of inertia, describes a property of force: force can make an object move from rest to motion, from motion to rest, and can also make an object change from one form of motion to another. This is the so-called Newton's first law. The most important problem in mechanics is how objects move under similar circumstances. Newton's second law solved this problem; This law is considered to be the most important basic law in classical physics. Newton's second law quantitatively describes that force can change the motion of an object. Indicates the time change rate of speed (i.e. acceleration A is directly proportional to force F, but inversely proportional to the mass of the object, i.e. a=F/m or F = Ma). The greater the force, the greater the acceleration; The greater the mass, the smaller the acceleration. Both force and acceleration have magnitude and direction. Acceleration is caused by force, and the direction is the same as force; If several forces act on an object, the resultant force will produce acceleration. The second law is the most important, and all the basic equations of power can be derived from it by calculus. In addition, Newton formulated the third law based on these two laws. Newton's third law points out that the interaction between two objects is always equal in size and opposite in direction. For two objects in direct contact, this law is easier to understand. The downward pressure of the book on the sub-table is equal to the upward support of the table on the book, that is, the action is equal to the reaction. So is gravity. The force that an airplane in flight pulls up the earth is numerically equal to the force that the earth pulls down the airplane. Newton's laws of motion are widely used in science and dynamics. Newton's law of motion Newton's law of motion is the general name of the three laws of motion in physics proposed by isaac newton, and it is regarded as the basis of classical physics. Newton's first law (law of inertia: all objects always keep moving in a straight line or at rest without any external force until an external force forces them to change this state. -it clarifies the relationship between force and motion, and puts forward the concept of inertia), "Newton's second law (the acceleration of an object is directly proportional to the resultant force F acting on the object, and inversely proportional to the mass of the object, and the direction of acceleration is the same as that of the resultant force. Formula: F=kma (when the unit of m is kg and the unit of a is m/s2, k= 1) Newton's third law (the acting force and reaction force between two objects on the same line are equal in magnitude and opposite in direction). ) "The contribution of optics Before Newton, Mozi, Bacon, Leonardo da Vinci and others all studied optical phenomena. The law of reflection is one of the optical laws that people have long known. When modern science rose, Galileo discovered a "new universe" through a telescope, which shocked the world. Dutch mathematician Hans sneer first discovered the law of refraction of light. Descartes put forward the particle theory of light ... Newton, Hooke and Huygens, who were almost contemporary with him, studied optics with great interest and enthusiasm like Galileo and Descartes. 1666, when Newton was on vacation at home, he got a prism, and he made a famous dispersion experiment with this prism. After a beam of sunlight passes through a prism, it is decomposed into several color spectral bands. Newton blocked the light of other colors with a slit baffle, and only let the light of one color pass through the second prism, resulting in only the light of the same color. In this way, he found that white light is composed of different colors of light, which is the first major contribution. Newton's telescope Newton tried to combine several different monochromatic lights into white light, and calculated the refractive index of different colors of light, which accurately explained the dispersion phenomenon. The mystery of the color of matter has been solved. It turns out that the color of matter is caused by the different reflectivity and refractive index of different colors of light on the object. In A.D. 1672, Newton published his research results in the Journal of Philosophy of the Royal Society, which was his first paper. Many people study optics in order to improve refractive telescopes. Newton discovered the composition of white light and thought that the dispersion phenomenon of refractive telescope lenses could not be eliminated (later, some people eliminated the dispersion phenomenon with lenses made of glass with different refractive indexes), so he designed and manufactured reflective telescopes. Newton was not only good at mathematical calculation, but also able to make all kinds of experimental equipment and do fine experiments by himself. In order to make a telescope, he designed a grinding and polishing machine and tested various grinding materials. 1668, he made the first prototype of reflective telescope, which is the second largest contribution. 167 1 year, Newton presented the improved reflective telescope to the royal society, which made him famous and was elected as a member of the royal society. Reflecting telescope's invention laid the foundation of modern large-scale optical astronomical telescope. At the same time, Newton also carried out a lot of observation experiments and mathematical calculations, such as studying the abnormal refraction phenomenon of glacier stone discovered by Huygens, the color phenomenon of soap bubbles discovered by Hooke, the optical phenomenon of Newton's ring and so on. Newton also put forward the "particle theory" of light, thinking that light is formed by particles and takes the fastest straight-line motion path. His "particle theory" and Huygens' "wave theory" later formed two basic theories about light. In addition, he also made Newton color wheel and other optical instruments. Newton is a master of classical mechanical theory. He systematically summarized the work of Galileo, Kepler and Huygens, and got the famous laws of gravity and Newton's three laws of motion. Before Newton, astronomy was the most prominent subject. But why do planets have to orbit the sun according to certain rules? Astronomers cannot fully explain this problem. The discovery of gravity shows that the movements of stars in the sky and objects on the ground are governed by the same law-mechanical law. Long before Newton discovered the law of gravity, many scientists had seriously considered this problem. For example, Kepler realized that there must be a force at work that makes the planet move along an elliptical orbit. He thinks this force is similar to magnetic force, just as a magnet attracts iron. 1659, Huygens found that a centripetal force was needed to keep the object moving in a circular orbit by studying the movement of the pendulum. Hooke and others thought it was gravity, and tried to deduce the relationship between gravity and distance. 1664, Hooke found that when comets approached the sun, their orbits were curved due to the sun's gravity. 1673, huygens deduced the law of centripetal force; 1679, Hooke and Halley deduced from centripetal force law and Kepler's third law that the gravitational force for maintaining planetary motion is inversely proportional to the square of distance. Newton himself recalled that around 1666, he had considered the problem of gravity when he lived in his hometown. The most famous saying is that Newton often sits in the garden for a while during holidays. Once, as happened many times in the past, an apple fell from the tree ... The accidental landing of an apple was a turning point in the history of human thought, which opened the thinking of people sitting in the garden and caused him to think deeply: What is the reason why almost all objects are attracted by the center of the earth? Newton mused. Finally, he discovered the gravity which is of epoch-making significance to mankind. Newton's genius lies in that he solved the mathematical argument problem that Hooke and others could not solve. 1679, Hooke wrote to Newton and asked him if he could prove that the planet moves in an elliptical orbit according to the law of centripetal force and the law that gravity is inversely proportional to the square of distance. Newton didn't answer the question. 1685, when Harley visited Newton, Newton had discovered the law of universal gravitation: there is gravitation between two objects, which is inversely proportional to the square of the distance and directly proportional to the product of the masses of the two objects. At that time, accurate data such as radius of the earth and the distance between the sun and the earth were available for calculation. Newton proved to Harley that the gravity of the earth is the centripetal force that makes the moon move around the earth, and also proved that the planetary motion conforms to Kepler's three laws of motion under the action of solar gravity. At the urging of Harley, at the end of 1686, Newton wrote an epoch-making masterpiece, Mathematical Principles of Natural Philosophy. The Royal Society is short of funds to publish this book. Later, one of the greatest works in the history of science was published in 1687 with Harley's support. In this book, from the basic concepts of mechanics (mass, momentum, inertia, force) and basic laws (three laws of motion), Newton not only demonstrated the law of universal gravitation mathematically, but also established classical mechanics as a complete and rigorous system, unified celestial mechanics with ground object mechanics, and realized the first large-scale synthesis in the history of physics. Newton's three laws of material immortality say that the quality of material is immortal; The law of conservation of energy refers to the conservation of energy of matter; Law of conservation of momentum. Newton's formula assumes that X 1 represents the distance from the object to the first focus, and X2 represents the distance from the light image to the second focus. This relationship is called Newton's formula, which is simpler and more symmetrical than1/u+1/v+1/f.