1. 1 Newton's "flow counting"

Newton (I Newton,1642-1727)1642 was born in a farmer's family in Woolsop village, England. 166 1 year, Newton entered Trinity College, Cambridge University, and studied under Barrow. Descartes' Geometry and Wallis' arithmetica infinitorum, these two works led Newton to the road of establishing calculus.

Newton began to study calculus in the autumn of 1664, and made a breakthrough during his escape from the plague in his hometown. 1666, Newton compiled the research results of the previous two years into a summary paper-On the number of flows, which is also the first systematic calculus document in history. In a brief discussion, Newton put forward the basic problems of calculus with kinematics as the background, and invented the "downstream number technique" (differential); Starting with the determination of the area change rate, the area is calculated by inverse differential method, and the "counter-current counting technology" is established. The reciprocal relationship between area calculation and tangent problem is clearly revealed as a general law, and the "basic theorem of calculus" as the basis of general calculus algorithm is discussed.

In this way, Newton organically unified all kinds of methods and special skills for solving infinitesimal problems since ancient times with the downstream notation, that is, differential and integral. It is in this sense that Newton founded calculus.

Newton was very cautious about publishing his own scientific works. 1687, Newton published his mechanical masterpiece Mathematical Principles of Natural Philosophy, which included his calculus theory, which was the earliest public expression of Newton's calculus theory, so this masterpiece became an epoch-making work in the history of mathematics. However, his calculus paper was not published until1the beginning of the 8th century under the repeated urging of friends.

1.2 Leibniz's calculus work

W Leibniz (1646- 17 16) was born in a family of professors in Leipzig, Germany, and received a good education as a teenager. From 1672 to 1676, Leibniz worked in Paris as the ambassador of Mainz. These four years have become the most precious time in Leibniz's scientific career, and many great achievements such as the establishment of calculus were completed or laid the foundation during this period.

1684, Leibniz sorted out and summarized his research achievements in calculus since 1673, and published the first paper on differential calculus in Teacher's Magazine, "A New Method for Finding Maximum, Minimum and Tangent" (hereinafter referred to as "the new method"), which contains differential symbols and the sum, difference and product of functions. 1686, Leibniz published his first paper on integral calculus, preliminarily discussed the reciprocal relationship between integral or quadrature problem and differential or tangent problem, including integral sign, and gave the cycloidal equation:

Leibniz's explanation of the basis of calculus is as vague as Newton's. His explanation is sometimes limited, sometimes less than any specified amount, but not zero.

1.3 Newton and Leibniz independently founded calculus.

Newton and Leibniz, as far as the creation of calculus is concerned, although they have differences in background, methods and forms, each has its own characteristics, but their achievements are equivalent. However, an outsider's pamphlet has caused "the most unfortunate chapter in the history of science": the debate on the priority of calculus invention. In this pamphlet, Swiss mathematician De Diu Lei thinks that Leibniz's calculus work borrowed from Newton, and then Leibniz is accused as a plagiarist by British mathematicians. This led to differences between European mathematicians who supported Leibniz and British mathematicians who supported Newton, and even sharply attacked each other. As a result, the two schools of mathematicians parted ways in the development of mathematics and stopped the exchange of ideas.

Long after Newton and Leibniz died, things finally became clear, and the investigation confirmed that they really finished the invention of calculus independently of each other. Newton preceded Leibniz in the time of invention; In terms of publication time, Leibniz preceded Newton.

The "Basic Theorem of Calculus" is also called Newton-Leibniz Theorem. Newton and Leibniz discovered this theorem independently. The basic theorem of calculus is the most important theorem in calculus. The relationship between differential and integral is established, and it is pointed out that differential and integral are reciprocal operations.

2. The founders of strict calculus: Cauchy and Wilstrass.

2. 1 Pioneer's efforts

After the establishment of calculus, calculus has become a powerful tool to study natural science because of the completeness of operation and the universality of application. However, many concepts in calculus are not precisely defined, especially the infinitesimal concept, which is the basis of calculus, is not clearly explained, sometimes it is zero and sometimes it is not zero, which leads to a logical dilemma.

Various criticisms and attacks have not made mathematicians give up calculus, but have aroused mathematicians' efforts to establish the rigor of calculus. As a result, it also set off a rigorous movement of calculus and even the whole analysis.

/kloc-in the 0/8th century, European mathematicians tried to overcome the difficulties in the foundation of calculus by algebraic methods. The main representatives in this field are D'Alembert (17 17- 1783), Euler and Lagrange. D'Alembert gave a qualitative definition of limit and used it as the basis of calculus. He thinks that differential operation "only lies in determining the limit of the ratio we have expressed by line segments by algebraic method"; Euler put forward the infinitesimal zero theory of different orders; Lagrange also admitted that calculus can be based on limit theory, but he advocated using Taylor series to define derivatives, and thus gave us what we now call Lagrange mean value theorem. Euler and Lagrange introduced the formal viewpoint into the analysis, while D'Alembert's limit viewpoint provided a reasonable core for the rigor of calculus.

After nearly a century of trying, by the beginning of19th century, the rigor of calculus had begun to show results. Firstly, the paper "Proof of Pure Analysis" published by Czech mathematician B Porzano (1781-kloc-0/848) in 18 17 contains the correct definitions of the concepts of function continuity and derivative, the existence theorem of bounded real number sets and the conditions of sequence convergence.

2.2 Cauchy's contribution to strict calculus

/kloc-the pioneer of the truly influential analytical rigor in the 9th century was the French Mathematicians Division (A-L Cauchy, 1789- 1857). From 182 1 to 1829, Cauchy published Analysis Course, Infinitesimal Calculation Course and Differential Calculation Course. They gave a clear definition of a series of basic concepts of calculus with strict analysis as the goal. On this basis, Cauchy strictly expressed and proved the basic theorem of calculus and the mean value theorem. Cauchy's work clarified the long-standing confusion on the basic problems of calculus to a certain extent, and took a key step towards comprehensive and strict analysis.

But Cauchy's theory can only be said to be "strict", and it was soon discovered that Cauchy's theory was actually flawed. For example, Cauchy defines limit as: "When the value of the same variable gradually tends to a fixed value, and finally the difference between its value and the fixed value can be arbitrarily small, this fixed value is called the limit of all other values", in which languages such as "infinite trend" and "arbitrarily small" are only intuitive qualitative descriptions of the concept of limit, lacking quantitative analysis, and this language appears many times in other concepts and conclusions.

It should be pointed out that calculus calculation is carried out in the field of real numbers, but by the middle of19th century, there was still no clear definition of real numbers, and there was still a lack of full understanding of the real number system. In calculus calculation, mathematicians rely on the assumption that any irrational number can be arbitrarily approximated by rational number. There was a common misunderstanding at that time that all continuous functions were differentiable. Based on this, it is impossible to really lay a solid foundation of calculus in Cauchy era. These were all problems faced by mathematicians at that time.

2.3 Wilstrass's strict calculus

Another outstanding contribution to the rigor of calculus is the German mathematician Wilstrass. He defined the concept of limit quantitatively, which is the "ε-δ" method in today's limit theory. Wilstrass redefined a series of important concepts in calculus with the language he created. In particular, he introduced the concept of uniform convergence, which eliminated all kinds of objections and confusion that constantly appeared in calculus in the past.

In addition, Wilstras believes that real number is the origin of all analysis, and to make the analysis strict, we must first make the real number system itself strict. Real numbers can be converted into integers according to strict reasoning. So all the concepts of analysis can be deduced from integers. This is the program of "Analytical Arithmetic" advocated by Wilstras. Based on his contribution to the rigor of analysis, Wilstrass won the title of "Father of Modern Analysis" in the history of mathematics.

1857, Wilstrass gave the first strict definition of real number in class, but he didn't publish it. 1872, Dai Dejin (R. Dedekind,1831-16) and Cantor (B. Cantor,1829-/kloc-. This marks the completion of the analytical arithmetic movement advocated by Wilstrass.

3. Conclusion

Newton and Leibniz founded calculus alone, while Cauchy and Weisstras gave birth to strict calculus.