Since17th century, the original geometry and algebra have been difficult to solve many new problems raised by production and natural science at that time, such as how to find the instantaneous velocity and acceleration of objects.

Although Newton had made some achievements in logarithm, analytic geometry and infinite series before, he could not solve these problems satisfactorily or universally. The greatest influences on Newton at that time were Descartes' Geometry and Wallis' arithmetica infinitorum.

Newton unified various special methods for solving infinitesimal problems since ancient Greece into two algorithms: downstream counting and countercurrent counting. The so-called flow is an independent variable that changes with time, such as x, y, s, u, etc. And the number of streams is the rate of change of streams, that is, the rate of change, writing, etc.

When he said differential rate, variable rate was differential rate. At the same time, he first published his binomial expansion theorem in 1676. Newton used it to discover other infinite series, and used it to calculate areas, integrals, solve equations and so on.

1684, Leibniz introduced and lengthened S as the symbol of calculus from the tangent study of curves, and the calculus founded by Newton was rapidly popularized in mainland countries.

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After the birth of calculus, mathematics ushered in an unprecedented period of prosperity, which had an important and far-reaching impact on mathematics in the18th century. Newton and Leibniz's calculus lacks a clear and rigorous logical basis, which is inevitable in the initial stage. The huge demand of science overcomes the concerns of logic.

They have too much to do, and they are eager to grab new achievements. Basic problems must be put aside first, as D'Alembert said, "Go ahead and you will have confidence." The development of the history of mathematics has repeatedly proved that free creation is always ahead of the formal and logical foundation.

Therefore, in the development of calculus, there is such a situation: on the one hand, calculus was applied to science and technology immediately after its creation and developed rapidly; On the other hand, the theory of calculus at that time was not rigorous, and there were more and more paradoxes and fallacies.

The development of mathematics has encountered a profound and disturbing crisis. For example, sometimes the infinitesimal is regarded as a finite quantity that is not zero, which is eliminated from both ends of the equation, and sometimes the infinitesimal is ignored as zero.

Because of these contradictions, there is a big debate in the field of mathematics For example, Becquerel, the Irish bishop and idealist philosopher at that time, laughed at infinitesimal as a dead fool. Becker criticized the definition of Newton's derivative.

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