# Thomas Simpson and Maxima and Minima

Author(s):
Michel Helfgott

Thomas Simpson (1710-1761) was a self-taught English mathematician who started his working life as a weaver, his father’s trade. Quite early he showed a keen interest in mathematics and later in life became an accomplished writer of textbooks on algebra, geometry, the calculus, and other mathematical subjects. His life was quite remarkable, from being a weaver to becoming a fellow of the Royal Society in 1745 (Clarke 1929).  Nowadays, Simpson is best remembered for the numerical integration technique that bears his name.

Simpson’s most widely known book appeared in print in 1750 under the title The Doctrine and Application of Fluxions. The fact that it was reprinted as late as 1823 (Simpson 1823) attests to its wide popularity. By modern standards it is an unusual work in the sense that applications of the calculus appear rather early and pervade all of the book.

After a first section on the nature of fluxions, and how to calculate with them, Simpson discusses with great care a collection of twenty two examples about maxima and minima. Fifteen of these examples are of a geometrical nature, three are applications to kinematics, and only four are strictly mathematical. We will discuss in detail eight of them, none commonly found in contemporary Calculus textbooks, replacing the word "fluxion" with "derivative" whenever the former appears in Simpson’s book. As expected, a certain amount of editing has been necessary, but we have kept the core of Simpson’s approach and explanations. We share the belief that it is a fruitful endeavor to engage students in the solution of mathematical problems from the past (Swetz 1995).

It is to be noted that we will use the concept of function, an idea that took almost two hundred years to mature since Leibniz and Johan Bernoulli introduced it at the end of the seventeenth century. Euler, the greatest eighteenth century mathematician, used the symbol $f(x)$ starting in 1734 (Siu 1995). The notation or definition of function is nowhere to be found in Doctrine although one might surmise that it is implicitly employed, one way or another, in the work.

Furthermore, Simpson does not apply the second derivative test for extrema; it is not even stated in his work. Neither does he use the first derivative test except when discussing example XXII, as we will see later on. Despite this fact, no errors are to be found throughout the section on maxima and minima; the very nature of the problems, mostly applications to geometry and kinematics, helped Simpson avoid any pitfalls. For him it was enough to take the first derivative of the pertinent expression and then find the critical point. Of course, we can check, through the first or second derivative test, that things work well in all the examples of Simpson’s that we will discuss.

After these preliminary considerations, let us discuss in detail some of the examples from section II of Doctrine. We will state them almost verbatim, then we will provide a solution patterned on Simpson’s solution, and finally we will make some remarks pertinent to each problem.