1695, the Dutch physicist b. Nieuwentyt accused Newton of being "vague" in the description of flow number and "lacking basis" in Leibniz's higher-order differential. The most shocking attack came from Bekele, a British philosopher and priest. Bekele (G. Berkeley, 1685- 1753) was the bishop of Croin (in present-day Ireland) in 1734, and in the same year, he published a pamphlet "Psychoanalyst, an essay for a pagan mathematician". Harry Bekele thinks in his book that mathematicians at that time used induction instead of deduction, and did not provide the legitimacy proof for their own methods. He concentrated on attacking the chaotic hypothesis of infinitesimal quantity in Newton's flow number theory. For example, the head-to-tail ratio method, in order to find the power flow number, Newton assumed that there was an increment, which was obtained by removing it, and then let it "disappear" to get the flow number. Bekele pointed out that the assumptions about increment here are inconsistent and are "obvious sophistry". He asked sarcastically, "What are these disappearing increments? They are neither finite, infinitesimal nor zero. Can't we call them ghosts of vanishing numbers? "The analyst's main goal is Newton's calculus, but he also strongly criticizes Leibniz's calculus and thinks that the correct conclusion is drawn from the wrong principle through" error cancellation ".
Bekele's attack on calculus theory is mainly motivated by religion, aiming at proving that the principle of flow number is not "clearer in concept" and "clearer in reasoning" than Christianity. However, many of his criticisms are to the point, objectively exposing the logical defects of early calculus and stimulating mathematicians to establish a strict foundation of calculus. In order to answer Bekele's attack, Britain has produced many works to defend Newton's flow number theory, the most typical of which is maclaurin's flow number theory, but all these defenses are weak because they insist on geometric argument. Mathematicians in continental Europe try to overcome the difficulties of calculus foundation by algebra. 18th century, the representative figures in this field are D'Alembert, Euler and Lagrange.
In The Theory of Analytic Functions (1797), Lagrange advocates defining the derivative by Taylor series: the derivative of a function is defined as an expansion.
As the starting point of the whole calculus, calculus is simplified as "pure algebraic analysis art"
On the one hand, mathematicians in the18th century tried to explore ways to make calculus strict; On the one hand, they often make bold progress regardless of the difficulties of basic problems, greatly expanding the application scope of calculus, especially the organic combination with mechanics, which has become one of the remarkable characteristics of mathematics in the18th century. The closeness of this combination is incomparable at any time in the history of mathematics. At that time, almost all mathematicians were also mechanics to varying degrees. Euler's name is associated with the basic equations of rigid body motion and fluid mechanics; Lagrange's most famous work is Analytical Mechanics (1788), which turned mechanics into a branch of analysis. Many of Laplace's most important mathematical achievements are contained in his five-volume "Celestial Mechanics". This wide application has become the source of new ideas and is greatly beneficial to mathematics itself. /kloc-a series of new branches of mathematics have grown up in the 0/8th century.
Ordinary differential equations are developed by calculus, and Newton and Leibniz have dealt with problems related to ordinary differential equations in their works.
/kloc-the highest achievement in solving ordinary differential equations in the 8th century is that Lagrange solved the general order non-homogeneous ordinary differential equations with variable coefficients by parametric variational method in the period of 1774- 1775. The parametric variational method for simple cases can be traced back to Newton and John. Bernoulli and Euler used this method to solve the second-order equation in 1739. Lagrange studies general equations.
All of them are. It is known that the general solution of the corresponding homogeneous equation is
Here is the special solution of integral constant and homogeneous equation. Lagrange regards it as a function, and sums all the expressions of WeChat business with the original equation to get the solution of the non-homogeneous equation.
Parametric variational method originates from the three-body in celestial mechanics. Three-body provides a lasting impetus for the theory of ordinary differential equations. At the core of this problem is a set of second-order equations:
Represents the mass of three spherical objects respectively, and represents the change coordinate of the center of mass of the first object, that is, the distance from to. Because it is impossible to solve the three-body equation accurately, an important direction of its research is to seek approximate solutions, that is, the so-called "perturbation theory" parameter variational method is a powerful tool of perturbation theory. Laplace's celestial mechanics has also made great contributions to the three-body and perturbation theory.
/kloc-In the 8th century, ordinary differential equations developed by solving some specific physical problems have become a new branch of mathematics, with its own goals and methods.
Lagrange was born in Turin, Italy. /kloc-at the age of 0/9, he was appointed as a professor of mathematics at the Turin artillery school. Euler and D'Alembert strongly recommended him to the Berlin Academy of Sciences. King Frederick of Prussia wrote to invite Lagrange to say, "The biggest king in Europe wants the biggest mathematician in Europe to be his companion in the palace." . From 1766 to 1787, Lagrange worked in Berlin Academy of Sciences for a long time. After Friedrich's death, he accepted the invitation of King Louis XVI of France and settled in Paris. During the French Revolution, the revolutionary government expelled all foreign academicians, but made an exception and let Lagrange stay in charge of the weights and measures reform in France.
/kloc-in the 0/8th century, some new branches, such as differential equations and variational methods, together with calculus itself, formed a vast field called "analysis", which was juxtaposed with algebra and geometry as the three major disciplines of mathematics, and its prosperity in this century far exceeded algebra and geometry. /kloc-mathematicians in the 0/8th century not only greatly expanded the field of analysis, but also endowed it with significance relative to geometry. They tried to get rid of the shackles of geometric argumentation through pure analysis, which became another major feature of mathematics in the18th century and reached its peak in Lagrange's works. Lagrange claimed in his Analytical Mechanics: "I can't find a picture in this book. The method I described requires neither drawing nor any geometric or mechanical reasoning, but only unified and regular algebraic (analytical) operations.