Among all Newton's scientific contributions, mathematical achievements occupy a prominent position. The first creative achievement of his mathematical career was the discovery of the binomial theorem. According to Newton's own recollection, he discovered this theorem in the winter of 1664 and 1665 when he was studying Dr. Wallis's "Infinite Arithmetic" and trying to modify his series for finding the area of ??a circle.
The creation of calculus is Newton's most outstanding mathematical achievement. In order to solve the problem of motion, Newton created this mathematical theory that is directly related to physical concepts. Newton called it "fluid mathematics". Some of the specific problems it handles, such as tangent problems, quadrature problems, instantaneous velocity problems, and maximum and minimum value problems of functions, have been studied before Newton. But Newton surpassed his predecessors. He stood at a higher perspective, synthesized the scattered efforts in the past, and unified the various techniques for solving infinitesimal problems since ancient Greece into two common types of algorithms-differential and integral. The reciprocal relationship between these two types of operations was established, thereby completing the most critical step in the invention of calculus, providing the most effective tool for the development of modern science, and opening up a new era in mathematics.
In 1707, Newton's algebra lectures were compiled and published as "Universal Arithmetic". He mainly discusses the basics of algebra and its application (by solving equations) to solving various types of problems. The book states the basic concepts and basic operations of algebra, uses a large number of examples to illustrate how to turn various problems into algebraic equations, and conducts in-depth discussions on the roots and properties of equations, leading to fruitful results in equation theory, such as, He derived the relationship between the roots of an equation and its discriminant, and pointed out that the coefficients of the equation can be used to determine the sum of the powers of the roots of the equation, which is the "Newton Power Sum Formula".
Newton contributed to both analytic geometry and synthetic geometry. He introduced the center of curvature in "Analytical Geometry" published in 1736, gave the concept of a close line circle (or curved circle), and proposed a curvature formula and a method for calculating the curvature of curves. He summarized many of his research results into a monograph "Enumeration of Cubic Curves", published in 1704. In addition, his mathematical work also involves many fields such as numerical analysis, probability theory and elementary number theory.
Newton is the natural founder of classical mechanical theory. He systematically summarized the work of Galileo, Kepler, Huygens and others, and obtained the famous law of universal gravitation and Newton's three laws of motion.
Newton's discovery of the law of universal gravitation was his most brilliant achievement in natural science. It was during the holidays that Newton often came to his mother's house and sat in the garden for a while. Once, as happened many times in the past, an apple fell from the tree. The accidental falling of an apple was a turning point in the history of human thought. It opened up the mind of the man sitting in the garden and caused him to ponder: What is the reason why all objects are affected by the force that almost always points towards the center of the earth? What about attraction? Newton thought. Finally, he discovered gravity, which was of epoch-making significance to mankind. He believed that the sun attracts planets, planets attract planets, and the forces that attract all objects on the ground are forces with the same nature. He also used calculus to prove that the force of the sun on the planets in Kepler's law is an attraction, proving that any If the radius of a particle moving in a curve points to a point of rest or uniform linear motion, and it sweeps around this point an area proportional to time, then the particle must be acted upon by a centripetal force directed to that point. If the square of the period of the circle is The centripetal force is inversely proportional to the square of the radius. Newton also proved through a large number of experiments that there is an attraction between any two objects, and concluded the law of universal gravitation:
F=G (m1m2 / r 2) (m1 and m2 are the forces between the two objects) mass, r is the distance between the two objects). During the same period, scientists such as Wren, Halley and Hooke were all exploring the mysteries of celestial motion. Among them, Hooke was the most prominent. He had long been aware of the inverse square law of gravity, but he lacked the mathematical ability like Newton and could not obtain the inverse square law of gravity. Give a quantitative expression.
Newton’s three laws of motion form the theoretical basis of classical mechanics.
These laws are summarized on the basis of a large number of experiments and are the basic theoretical basis for solving mechanical motion problems.
In 1687, Newton published his masterpiece "Mathematical Principles of Natural Philosophy", a classic work on mechanics. In this book, Newton started from the basic concepts of mechanics (mass, momentum, inertia, force) and basic laws (the three laws of motion), and used the sharp mathematical tool he invented, calculus, to establish the complete system of classical mechanics. The rigorous system unified the mechanics of celestial bodies and the mechanics of objects on the ground, achieving the first major synthesis in the history of physics.
In terms of optics, Newton also achieved great results. He used a prism to experiment with the colored light that white light was decomposed into, and was the first to discover the composition of white light. He conducted a precise analysis of the refractive index of various colors of light and explained the nature of dispersion phenomena. He pointed out that the difference in color of objects is caused by the different refractive index and reflectivity of light of different colors, thus revealing the mystery of color. Newton also proposed the "particle theory" of light, believing that light is formed from particles and takes the fastest linear motion path. His "particle theory" and later Huygens' "wave theory" constitute the two basic theories about light. In addition, he also made various optical instruments such as Newton's color disk and reflecting telescope.
Newton's research fields were very broad, and he made important achievements in almost every scientific field he was involved in. He studied thermometry, the observation of fixed temperatures at which water boils or solidifies, the laws of cooling of hot bodies, and other subjects which appear inferior only when compared with his own major achievements.