How to evaluate Lagrange s place in the history of mathematics and its place in the history of manki

After entering the Royal Artillery Academy in Turin, Lagrange began to systematically teach himself mathematics. Because of his hard work and rapid progress, he took up the school's mathematics teaching work before graduating. At the age of 18, he began to write mathematics, and at the age of 19, he was officially appointed as a geometry teacher at the school.

Lagrange was not a "mathematical prodigy", on the contrary, he was not interested in mathematics until he was 17 years old, and he was particularly fond of literature. When he was 17 years old, he happened to read an article written by the British astronomer Halley about Newton's achievements in calculus - "On the Merit of the Analytical Method", which made him have infinite admiration and admiration for Newton, so he decided to become a Newtonian mathematician.

Lagrange's most outstanding contribution to mathematics was to separate mathematical analysis from geometry and mechanics, to make the independence of mathematics clearer, and to make mathematics no longer just a tool of other disciplines.

There are four main aspects:

Equation solution. During his first ten years in Berlin, Lagrange spent a great deal of time working on algebraic equations and solutions to transcendental equations, making valuable contributions to the development of algebra. He submitted two prestigious **::

On Solving Numerical Equations" and "Studies on Algebraic Solutions of Equations". Solve the predecessors.

The various solutions to third- and fourth-degree algebraic equations are summarized into a set of standard methods, that is, the equations are solved by turning the Lagrangian point into an equation of the lower order (called an auxiliary equation or a pre-solution).

Permutation group. He tried to find a pre-solution function for a fifth-order equation in the hope that this function would be a solution to an equation of less than fifth, but was unsuccessful. However, his ideas already contained the concept of permutation groups, which later inspired Abel and Galois, and finally solved the problem of why general equations higher than the fourth order could not be solved by algebraic methods.

Therefore, it can also be said that Lagrange is the forerunner of group theory.

Number theory. In number theory, Lagrange also showed extraordinary talent. He answered many of the questions Fermat had raised.

For example, a positive integer is the sum of no more than 4 square numbers, etc., and he also proves the irrationality of pi. The results of these studies of Joseph Lagrange enriched the content of number theory.

Power series. In his Theory of Analytic Functions, as well as in one of his first essays as early as 1772, he made a unique attempt to lay a theoretical foundation for calculus, in which he attempted to reduce differential operations to algebraic operations, thus discarding the infinitesimal quantities that had puzzled him since Newton, and from which he wanted to establish all the analytical. However, because he did not take into account the problem of convergence of infinite series, he thought that he had gotten rid of the concept of limit, but in fact he only avoided the concept of limit, and failed to achieve his goal of algebraic and rigorous calculus.

However, his treatment of functions expressed by power series had an impact on the development of analytics and became the starting point for the theory of real variable functions.

Lagrange summarized the mathematical achievements of the 18th century, and at the same time opened the way for the study of mathematics in the 19th century, and can be called the most outstanding French mathematician. At the same time, his achievements on the motion of the moon (the three-body problem), the motion of the planets, the calculation of orbits, the problem of two moving centers, and fluid mechanics have also played a historical role in the mechanization and analysis of astronomy, and promoted the further development of mechanics and celestial mechanics, and have become pioneering or foundational research in these fields.

He has made valuable contributions to the solution of algebraic equations and transcendental equations, and has promoted the development of algebra.

In number theory, Lagrange also showed extraordinary talent. He answered many of the questions Fermat had raised.

In addition, he has made important contributions to the fields of variational methods and differential equations.

During his time at the Academy of Sciences in Berlin, Lagrange conducted extensive and in-depth research on algebra, number theory, differential equations, variational methods, and mechanics. One of his most valuable contributions is in the field of equation theory. On the basis of all the methods used by predecessors to solve the equation below the fourth degree, an in-depth study was carried out.

So it was concluded that it is not possible to solve the general nth order equation (n 4) with algebraic operations. Although he was unable to give a proof of this conclusion, it played a very important role in the later establishment of the group theory by Galois.