| Mth 311 Advanced Calculus |
| Supplementary Note |
| Bent Petersen |
In a relatively brief period around 1670 calculus was created by Newton (1642-1727), Leibniz (1646-1716), and Gregory (1638-1675).
Newton's work was actually a bit earlier than 1670, but he did not communicate it. During the period 1665-1666 Newton invented differential and integral calculus, formulated the notion of universal physical laws holding throughout the cosmos (the deterministic clockwork universe for more than 200 years after Newton), generalized Galilei's laws of motion, formulated the law of gravitation, explained the tides, and provided a model for the solar system so accurate that it was used much later by others to predict and locate additional planets (for example, Neptune) and to determine the speed of light (from discrepancies in the observed positions of the satellites of Jupiter). In addition he did important work on the nature of light - the only part of this amazing output published for the next 20 years. Leibniz independently discovered calculus, did important work in differential equations, and developed much of the calculus notation that we still use today.
After the invention of calculus there was a tremendous flowering of new results in mathematics and of applications in science and technology. Of particular note is Fourier's (1768-1830) work on the propagation of heat. The need to make sense of Fourier's wonderful ideas, and to answer an increasing number of objections and logical problems with the immensely successful methods of calculus led to an increasing interest in providing a rigorous foundation for calculus. Cauchy (1789-1857) tried to formulate the notion of the derivative in terms of the concept of limit. He was not completely successful because he lacked a rigorous notion of limit. The modern epsilon-delta definition of the limit was first developed in 1872 by Heine (1821-1881). It is this rigorous foundation for calculus that we have in mind when we speak of advanced calculus (or real analysis or mathematical analysis).
According to W.W. Rouse Ball [3] the first person, in print at least, to distinguish carefully between convergent and divergent series (in 1667) was James Gregory (1638-1675). This concern with precision in dealing with infinite processes can be viewed as the beginning of advanced calculus. Major development did not occur however until the second half the nineteenth century.
Morris Kline's book, [2], is an excellent technical history of mathematics. There are many other excellent histories, but if you plan to own only one, then Kline's book is the one to get.