The generation of calculus is generally divided into three stages: the concept of limit; Infinitesimal method of quadrature; Reciprocal relationship between integral and differential. The last step was completed by Newton and Leibniz. In the first two stages, a large number of European mathematicians have been traced back to Archimedes in ancient Greece and made their own contributions. For this work, ancient China was not inferior to the West, and the idea of calculus sprouted in ancient China, even worse than ancient Greece. As early as the 7th century BC, Thales, an ancient Greek scientist and philosopher, studied the area, volume and length of the ball, which included the idea of calculus. In the 4th century BC, there were definitions of finite, infinite, infinitesimal (without internal minima) and infinite (without external maxima), as well as the concepts of limit and instantaneity. In 263 AD, Liu Hui initiated secant technique to find the area of a circle and the volume of a square cone, and the pi was about 3. 14 16. His limit thought and infinitesimal method are the profound embodiment of the ancient limit thought in the world.

In the third century BC, Archimedes of ancient Greece implied the idea of modern integral calculus when he studied and solved the problems of parabolic arch area, spherical surface and spherical cap area, area under spiral, and volume of hyperbola of rotation. As the basis of differential calculus, limit theory was clearly discussed in ancient times. For example, the book Zhuangzi written by Zhuang Zhou in China records that "one foot of space is inexhaustible." Liu Hui in the Three Kingdoms period mentioned in his "Cutting Circle" that "if you cut it carefully, you will lose less, and if you cut it again, you will not even lose your circumference and body." These are simple and typical limit concepts. Although the idea of calculus can be traced back to ancient Greece, its concepts and laws came into being and developed on the basis of Kepler's and cavalieri's quadrature ideas and methods in the second half of the 6th century. These ideas and methods can be found from Liu Hui's proof of the volume formulas of cone, frustum of a cone and cylinder to Zuheng's method of calculating the volume of a sphere in the fifth century. Meng Qian Bi Tan by Shen Kuo, a great scientist in the Northern Song Dynasty, created techniques such as "gap product", "meeting circle" and "counting chess", and initiated the research of higher-order arithmetic progression's summation.

Especially from the forties in 13 to the beginning of 14, it reached the peak of ancient mathematics in China in all major fields. There are the root graph of Jiaxian Triangle, multiplication and multiplication methods, positive and negative roots method, big derivative method, big derivative method (one-time congruence group solution), superposition method (high-order arithmetic progression summation), difference method (internal difference method of high-order difference) and celestial element method (high number). The reform of computing technology and abacus calculation are outstanding achievements in the history of world mathematics. China ancient mathematics has done excellent work in the first two stages of calculus, many of which are the key to the establishment of calculus. China had all the innate conditions before the invention of calculus in the17th century, and was close to the door of calculus. Unfortunately, after the Yuan Dynasty in China, stereotyped writing of scholars made academic retrogression, and the blind exclusion of cultural autocracy and feudal rule led to the decline of science, including mathematics, and fell behind in the most critical step in the creation of calculus.

In the seventeenth century, there were many scientific problems to be solved, and these problems became the factors that prompted calculus. To sum up, there are mainly four kinds of problems: the first kind is the problem that appears directly when learning physical education, that is, the problem of finding the instantaneous speed. The second kind of problem is to find the tangent of the curve. The third kind of problem is to find the maximum and minimum of a function. The fourth problem is to find the length of the curve, the area enclosed by the curve, the volume enclosed by the surface, the center of gravity of the object, and the gravity of an object with a considerable volume acting on another object.

Mathematics first introduces a basic concept (such as astronomy, navigation, etc.) from the study of sports. ), and in the next two hundred years, this concept occupies a central position in almost all works, which is the concept of function-or the relationship between variables. With the adoption of the concept of function, calculus came into being, which is the greatest creation in all mathematics after Euclidean geometry. Focusing on solving the above four core scientific problems, calculus problems were explored by at least a dozen largest mathematicians and dozens of smaller mathematicians in the17th century. The culmination of all their contributions is the achievement of Newton and Leibniz. The works of these two masters are mainly introduced here.

In fact, before Newton and Leibniz sprinted, they had accumulated a lot of knowledge of calculus. /kloc-many famous mathematicians, astronomers and physicists in the 0/7th century did a lot of research work to solve the above problems, such as Fermat, Descartes, Roberts and Gilad Girard Desargues. Barrow and Wallis in England; Kepler in Germany; Italian cavalieri and others put forward many fruitful theories. Contributed to the creation of calculus.

For example, Fermat, Barrow and Descartes all studied the tangent of the curve and the area surrounded by the curve in depth and got some results, but they didn't realize its importance. In the first two thirds of the seventeenth century, the work of calculus was lost in details, and they were exhausted by trivial reasoning. Only a few great scholars are aware of this problem. For example, James Gregory said, "The real division of mathematics is not into geometry and arithmetic, but into universality and particularity". This universal thing was provided by two all-encompassing thinkers Newton and Leibniz.

/kloc-In the second half of the 7th century, Newton, a great British scientist, and Leibniz, a German mathematician, independently studied and completed the creation of calculus in their respective countries on the basis of their predecessors' work, although this was only a very preliminary work. Their greatest achievement is to connect two seemingly unrelated problems, one is the tangent problem (the central problem of differential calculus) and the other is the quadrature problem (the central problem of integral calculus).

Newton and Leibniz established calculus from intuitive infinitesimal, so this subject was also called infinitesimal analysis in the early days, which is the origin of the name of the big branch of mathematics. Newton's research on calculus focused on kinematics, while Leibniz focused on geometry.

Newton wrote Flow Method and Infinite Series at 167 1, and it was not published until 1736. In this book, Newton pointed out that variables are produced by the continuous motion of points, lines and surfaces, and denied that variables are static sets of infinitesimal elements. He called continuous variables flow, and the derivatives of these flows were called flow numbers. Newton's central problems in flow number technology are: knowing the path of continuous motion and finding the speed at a given moment (differential method); Given the speed of motion, find the distance traveled in a given time (integral method). Leibniz of Germany is a knowledgeable scholar. 1684, he published what is considered to be the earliest calculus literature in the world. This article has a long and strange name: a new method for finding minimax and tangents, which is also applicable to fractions and irrational numbers, and the wonderful types of calculation of this new method. It is such a vague article, but it has epoch-making significance. He is famous for containing modern differential symbols and basic differential laws. 1686, Leibniz published the first document on integral calculus. He is one of the greatest semiotics scholars in history, and his symbols are far superior to Newton's, which has a great influence on the development of calculus. The general symbols of calculus we used were carefully selected by Leibniz at that time.

Since childhood, Leibniz has clearly shown signs of being an ideological star. /kloc-at the age of 0/3, he reads difficult papers by scholastic scholars as easily as other children read novels. He put forward the infinitesimal calculus algorithm, and he published his own results, three years before Sir isaac newton submitted the manuscript to Fu Zi, who claimed that he was the first person to make this discovery.

Leibniz is an old fox, pleasing the court and being sheltered by celebrities. He had a personal relationship with Spinoza, whose philosophy left a deep impression on him, although he certainly parted ways with Spinoza's thoughts.

Leibniz has extensive correspondence with philosophers, theologians and literati. In his grand plan, he tried to reach a reconciliation between Protestantism and Catholicism and establish an alliance between Christian countries, which in his time meant the European Union. He also served as the first president of the Berlin Science Association, which later became the Prussian Academy of Sciences.

He served in the court of Hanover, but when George I became king of England, Leibniz was not invited to go with him, perhaps because of his dispute with Newton. His public influence declined, 17 16 years, and he died at the age of 70, even being ignored by the society he founded. The establishment of calculus has greatly promoted the development of mathematics. In the past, many problems that elementary mathematics was helpless were often solved by calculus, which shows the extraordinary power of calculus.

As mentioned above, the establishment of a science is by no means a person's achievement. It must be completed by one person or several people through the efforts of many people and on the basis of accumulating many achievements. So is calculus.

Unfortunately, while people appreciate the magnificent function of calculus, when they put forward who is the founder of this subject, it actually caused an uproar, resulting in a long-term opposition between European continental mathematicians and British mathematicians. British mathematics was closed to the outside world for a period of time, limited by national prejudice, and too rigidly adhered to Newton's "flow counting", so the development of mathematics fell behind for a whole hundred years.

As a matter of fact, Newton and Leibniz studied independently, and completed them in roughly the same time. More specifically, Newton founded calculus about 10 years earlier than Leibniz, but Leibniz published calculus theory three years earlier than Newton. Their research has both advantages and disadvantages. At that time, due to national prejudice, the debate about the priority of invention actually lasted from 1699 to 100 years.

It should be pointed out that this is the same as the completion of any major theory in history, and the work of Newton and Leibniz is also very imperfect. They have different views on infinity and infinitesimal, which is very vague. Newton's infinitesimal, sometimes zero, sometimes not zero but a finite small amount; Leibniz's can't justify himself. These basic defects eventually led to the second mathematical crisis. Until the beginning of19th century, the scientists of French Academy of Sciences, led by Cauchy, made a serious study of the theory of calculus and established the limit theory, which was further tightened by the German mathematician Wilstrass, making the limit theory a solid foundation of calculus. Only in this way can calculus be further developed. Any emerging and promising scientific achievements attract the vast number of scientific workers. There are also some stars in the history of calculus: Swiss Jacques Bernoulli and his brothers johann bernoulli, Euler, French Lagrange and Cauchy. ...

Euclidean geometry and algebra in ancient and medieval times were constant mathematics, and calculus was the real variable mathematics, which was a great revolution in mathematics. Calculus is the main branch of higher mathematics, and it is not limited to solving the problem of variable speed in mechanics. It gallops in the garden of modern science and technology and has made countless great achievements.