It was a period in which there was an epidemic of a genius virus in Europe for scientists, explorers, inventors of many things including mathematics. Among them was Isaac Newton (1642-1727) who co-invented calculus, discovered the Binomial Theorem, and formulated a theory of universal gravitation (Smith). Newton has been regarded for almost 300 years as the founding exemplar of modern physical science, his achievements in experimental investigation being as innovative as these in mathematical research.
Before discussing his three achievements, it is important to note that Newton had some college experience but did his significant work was at home. Newton entered Trinity College, Cambridge in 1661. His interest in mathematics began in the autumn of 1663, a date which matches the beginnings of his deep mathematical studies. When the plague forced the University to close in the summer of 1665, he returned to Lincolnshire [where he was born]. For a period of less than two years, Newton began revolutionary advances in mathematics and other areas such as optics, physics, and astronomy (O’Connor and Robertson). It is important to know where most of his discoveries were done in order to understand the reasons behind his inventions.
Newton is generally credited with the generalized binomial theorem which is valid for any exponent. When he started to read the works of Dr. Wallis, Newton was led to consider how “he could interpolate the general values of the areas in the second series…” (“The Life of Isaac Newton”). With this knowledge, he investigated the arithmetical law of the coefficients of the series and obtained a general method of interpolating other series. It later occurred to him that the same process of interpolating series was applicable to the ordinates; thus, discovering the general method of reducing radical quantities composed of several terms into infinite series. This led Newton to the invention of the Binomial Theorem which he claimed was the easiest way to solve the quadratures of curves (“The Life of Isaac Newton”). The discovery of the Binomial Theorem is essential in understanding many mathematical concepts such as probability. For instance, when I flip a coin, the outcome is either a head or tail. Binomial Theorem is also used to expand polynomials.
While he was still at home, Newton laid the foundations for his next discovery, differential and integral calculus, several years before its independent discovery by Gottfried Leibniz. He termed it the method of fluxions which was based on his crucial insight that integration of a function is only the inverse procedure to differentiating it. Newton produced simple logical methods that:
Unified many separate techniques previously developed to solve apparently
unrelated problems such as finding areas, tangents, the length of curves, and
the maxima and minima of functions [taking differentiation as the basic
operation]. Newton’s De Methodis Serierum etFluxionum was...