The discovery of calculus, which is a branch within mathematics, is attributed to both Isaac Newton and Gottfried Leibniz. Both Isaac Newton and Gottfried Leibniz made independent contributions towards the development of the foundations of calculus. Instrumental and significant contributions were done by both Newton and Leibniz in a bid to ensure that calculus was developed in order to assist in various concepts and applications of mathematics (Kleiner 139). However, it is important to note that the foundations of calculus developed by Newton differed widely from the foundations of calculus that were developed by Leibniz. There is no doubt that calculus has assisted in calculating areas under curves and maximization of quantities through finite processes as opposed to infinite processes used by the classical mathematicians.
The differences in foundations developed by the two revolved around the variables and the applications of the formulas involved. For instance, in respect to variables Newton believed that within calculus the variables should change depending on time whereas Leibniz believed in the fact that x and y were variables were closely infinite values that arise over sequences (Boyer 52). Since Leibniz believed that the variables were ranges of sequences composed of infinite values, he introduced the concepts of dy and dx in order to assist in establishing the differences that exist amongst successive values. Notably, even though Leibniz developed the concept of dy and dx and used it to define tangency, Leibniz did not use the concept as a defining property within calculus calculations (Kleiner 152). Newton on his contribution developed the idea of the quantities of x and y for the purposes of defining finite velocities as well as comparing tangencies. Newton identified calculus functions as geometrical.
Another important point to note is the fact that Leibniz employed the concepts of notations as well as symbols throughout his development of the foundations. On a different perspective, Newton’s personal interests led to development of notations on a daily basis. It is in this perspective that Leibniz developed notations and symbols that were easily used in the generalization concepts of calculus (Boyer 69). The generalization of calculus through the notations and symbols developed by Leibniz significantly helped in the applications of calculus on multiple variables. In addition, the notations and symbols developed by Leibniz have been very influential in understanding the concepts of integral and derivative (Kleiner 157). Consequently, the current mathematical calculus concepts used much of Leibniz’s notations and symbols as opposed to the notations and symbols developed by Newton.
Furthermore, it is important to understand the contributions of Newton and Leibniz in the development of calculus in various stages or periods of the timeline. Development of calculus was attained through anticipation, development, and rigorization periods (Boyer 32). The foundations of calculus by Newton and Leibniz started in the development stage or period after the anticipation period where mathematicians used infinite processes in establishing the area under curves as well as the maximization of quantities. During the development period, Newton and Leibniz came up with methods that could assist mathematicians to calculate areas under curves as well as maximize quantities (Kleiner 163). However, since Newton and Leibniz employed different concepts, it took the mathematicians the rigorization period to justify and apply the calculus concepts as developed by the two mathematicians.
The main reason as to why mathematicians found it difficult to justify and apply concepts of calculus developed by Newton and Leibniz is because the two used “infinitesimals” variables or quantities, which were believed to be too small. Nonetheless, there is no doubt that Newton and Leibniz significantly contributed to the development of calculus, which has made calculations of areas under curves and maximization of quantities possible through finite processes and not infinite processes used earlier.
Boyer, Carl B. The history of the calculus and its conceptual development: (The concepts of the calculus). Courier Dover Publications, 1949.
Kleiner, Israel. "History of the infinitely small and the infinitely large in calculus." Educational Studies in Mathematics 48.2-3 (2001): 137-174.