18 - Connections between Newton, Leibniz, and Calculus I

Summary

Introduction

Calculus, like most other well-established branches of mathematics, did not originally appear in the same form as it occurs in modern textbooks. Many mathematicians contributed to the development of calculus over many centuries, using widely varying notation and languages. A proper history of the subject can easily consume a book [1].

Although a thorough study of the history of calculus is completely unnecessary for an introductory calculus student, it is nevertheless of some interest for such students to see an overview of this subject's fascinating and colorful history. Today's calculus students will no doubt consider original papers somewhat cryptic at the very least, and maddeningly cumbersome and obscure in places. Still, there are passages from these writings which will appear comforting in their familiarity. This paper seeks to point out some of these passages and their connections with the modern elementary calculus curriculum. We concentrate on the two mathematicians generally considered to be the fathers of calculus, Sir Isaac Newton (1642–1727) and the German Gottfried Leibniz (1646–1716).

Historical Background

It would be impossible to say authoritatively when the first ideas of calculus appeared. Arguably, many early mathematicians used a form of integral calculus in approximating the area or volume of irregular objects using finitely many or infinitely many recognizable shapes such as rectangles. In particular, the Greek mathematician Archimedes is famous for estimating the area of circle using the so-called method of exhaustion, and effectively computing the value of π.

One can say with confidence, however, that the English mathematician Sir Isaac Newton (1642–1727) and the German Gottfried Leibniz (1646–1716) are most famous for their discoveries of calculus.