When Euler Met l’Hospital

\t\tWilliam Dunham

Leonhard Euler (1707 – 1783) was, of course, a mathematical giant whose contributions ranged far and wide across our discipline. In this talk we look to analysis and consider his treatment of l’Hospital’s rule as found in Chapter 15 of the Institutiones calculi differentialis from 1755.

Euler began with a general discussion of indeterminate forms and then took up specific, and increasingly interesting, examples. The first few were problems that could be done either with or without l’Hospital’s rule. He then moved to a curious example requiring logarithmic differentiation. And he ended with a spectacular solution to the “Basel problem” – showing that 1 + 1/4 + 1/9 + 1/16 + … = π26 – by applying l’Hospital’s rule not once nor twice, but thrice. It was Euler at his symbol manipulating best.

Most talks at the Penn Colloquium focus on cutting-edge mathematics of today. This one addresses cutting-edge mathematics from the 18th century. As such, it should be accessible to anyone with a knowledge of calculus and an interest in the history of mathematics.