Calculus is crucial for the foundation of mathematics and other science, especially in analysis, it creates a mathematical basement. It is invented in the 19th century, and many mathematicians from the past and present have proposed numerous calculus approaches. The most outstanding two calculus methods are Newton’s fluxional calculus and Leibniz’s differentiation and integration, which are widely used nowadays. Newton’s method is mainly used in Physics and Leibniz’s method contributes a lot to mathematical analysis. Though the study of Leibniz and Newton built the foundation of modern calculus, their views on calculus differed. Furthermore, infinitesimality has been essential to the development of calculus, and it is crucial for the research by Leibniz, while introducing Leibniz’s method, it will be mentioned and explained. This paper will introduce Leibniz’s and Newton’s methods briefly and compare their difference. Leibniz and Newton both used infinitely small quantities to explain the constantly changing rate. Leibniz introduced infinitesimals to solve the problem of derivatives and integrals, while Newton used the term “flow number” to describe the process of change. In addition to simplifying calculation, the introduction of infinitesimal paved the way for the subsequently developed more rigorous limit theory.
