Newton and Leibniz established calculus in different ways. If Newton approached the final conclusion earlier than Leibniz, then Leibniz published his own conclusion earlier than Newton. Although Newton's application of calculus far exceeded Leibniz's work, which stimulated and determined the analysis direction of almost the whole eighteenth century, Leibniz succeeded in establishing a more convenient symbol system and calculation method. The two founders of calculus, one has the cautious and rigorous academic style of the British, and the other has the philosophical and speculative mentality of the German, and is enthusiastic and generous. Due to the wrong background of yin and yang, Newton, who was too strict, delayed publishing his findings, which made Leibniz grab the first place in publishing. Newton and Leibniz's different philosophies led to their different methods of creating calculus. Newton insisted on materialist empiricism and attached special importance to experiments and inductive reasoning. When he was studying classical mechanics and the law of universal gravitation, he encountered some unsolvable mathematical problems, which could not be solved in Euclidean geometry and algebra in16th century. Therefore, Newton began to study new methods to solve the problems of curvature, area, curve length, center of gravity and maximum and minimum flow. "Newton's research adopted the method of initial ratio and final ratio. He thinks that the number of streams is the initial ratio of the original quantity or the final ratio of the vanishing quantity. The initial ratio of new students is the ratio at birth, and the final ratio of disappearing students is the ratio at disappearance. " [4] (page 180) This explanation is too vague, not an accurate mathematical concept, but an intuitive description. The physical prototype of initial ratio and final ratio is the mathematical abstraction of initial speed and final speed, and the speed of each moment in the process of object position movement is self-evident. Based on this objective fact, Newton put forward the intuitive concepts of initial ratio and final ratio. In this way, he gave a limited view.

Leibniz founded calculus by studying the tangent problem and quadrature problem. He realized from the differential triangle that the tangent of the curve depends on the ratio of the difference between the ordinate and the abscissa; Finding the area of curved edge figure depends on the sum of ordinate or the sum of infinite thin rectangles between infinite units in abscissa. Leibniz realized that sum operation and difference operation are reversible. Leibniz gave the basic theorem of calculus with the idea of infinitesimal, and developed it into higher order differential. Leibniz's infinitesimal is hierarchical, which stems from monism in his philosophy. "Leibniz pointed out in monadic theory that different monads have their perceptions.

The degree of clarity is different, and it changes from one perception to another. Development is a sequence of things consisting of lists, from low to high. [6] (p. 9 1) It can be said that Leibniz's infinitesimal classification exactly corresponds to the different levels of monism in his objective idealism philosophy system. In the process of Leibniz's study of calculus, the principle of continuity became the cornerstone of his work, which was rooted in the infinite essence of his philosophy.

The similarities between Newton and Leibniz in the creation of calculus are as follows: a new mathematics subject was created from different angles, which made calculus have a wide range of uses and can be applied to general functions; Get rid of the past geometric form by algebraic method; The reciprocal relationship between differential and inverse differential is studied.

The differences between Newton and Leibniz in the creation of calculus are as follows: Newton inherited Bacon's empiricism and was particularly fond of induction. Newton's calculus obviously has traces of a physical model born out of mechanics, which appears as a mathematical model of mechanical motion. Basic concepts such as primary quantity, vanishing quantity, instantaneous, initial ratio and final ratio all come from mechanical motion, which is a mathematical abstraction of the instantaneous state of mechanical motion. The purpose of his establishment of calculus is to solve special problems and emphasize concrete results that can be popularized. Leibniz emphasizes the general methods and algorithms that can be applied to special problems in order to deal with all kinds of problems in a unified way. Leibniz spent a lot of time on the choice of symbols and invented a suggestive symbol system. He stretched the initial letter S of sum to indicate the integral, and dx to indicate the differential of X. This set of simple and easy-to-understand symbols has been used to this day.

Newton thought calculus was a natural extension of pure geometry, and he paid attention to the application of calculus in physics. Experience, concreteness and caution are the characteristics of his works, which makes him unable to give full play to them. On the other hand, Leibniz cares about generalized calculus and tries to create and establish a perfect calculus system. He is imaginative, popular, bold and thoughtful, so he announced the birth of a new discipline without hesitation.

Newton and Leibniz were both scientific giants of their time. Calculus can become an independent discipline and bring revolutionary influence to the whole natural science, mainly relying on the work of Newton and Leibniz. From the process of Newton and Leibniz's founding calculus, we can see that when the giant's philosophical meditation is transformed into scientific conclusions, it has a far-reaching impact on scientific development.