Calculus War: Bernoulli’s Remarkable Journey in Mathematics

Controversies in Mathematics History

Jacob Bernoulli (1655-1705) was part of a family that produced no less than eight supremely gifted Mathematicians, begging the question of “nature or nurture?”. Jacob advocated the Leibnizian notation of the calculus, particularly the ratio dy/dx for derivatives as opposed to the y (dot) of I. Newton (1643-1727) or the y’ of J.L. Lagrange (1736-1813). Jacob sided with Leibniz during the so-called “Calculus Wars”, i.e. the priority dispute over the Calculus. However, when one looks closely, the Calculus controversy should have started much earlier as the Greeks were using a form of the limiting process (the method of exhaustion of Eudoxus (355BC)), and this method relies on proof by contradiction or reductio ad absurdum.

Such a proof gave rise to one of the most beautiful proofs in Mathematics, namely of Euclid’s (325-265BC) proof of the infinite number of primes, which I often derive at UCL just to demonstrate the beauty of a Mathematical argument and also to gage whether I have latent Mathematicians in my Engineering classes. Remaining with the controversy, Descartes (1596-1650) was calculating tangents to curves by using a technique akin to what we would call today differentiation a long time before Newton or Leibniz.

The Calculus of Variations

Jacob, along with his brother Johann (1667-1748), created the very powerful mathematics technique known today as the Calculus of Variations, which I am again fortunate to teach during a module at UCL to my enthusiastic students. This theory was further developed by the “Giants” L. Euler (1707-1783) and Lagrange, and we have today the Euler-Lagrange equation. I always get goosebumps when two or more “Giants” of the past are meshed together in an often mind-blowing theory or technique, for example we have the Cayley(1789-1857)-Hamilton (1805-1865) theorem, the Cauchy (1789-1857)-Riemann (1826-1866) equations or the Gauss (1777-1855)-Jacobi (1804-181) method. Discussing Johann Bernoulli, there is a wonderful little anecdote regarding him and Euler, which I would like to relay.

The Power of Reading in Mathematics

Johann gave the young Euler a Mathematical textbook and suggested he read it, and if he were to run into difficulties, then he should come to visit him every Sunday afternoon. Euler never once took Johann up on his offer and read the book cover to cover without needing help, yet history advises us that Johann was Euler’s tutor. A truly marvellous proof of Euler’s brilliance can be deemed when one looks at his solution to the so-called Basel problem, which had baffled Mathematicians of the day and made him a “Mathematical Superstar” overnight.

Remaining with “Giants” of the past who were asked to read books by their mentors and then to seek advice, one must mention Riemann, who read A. Legendre’s (1752-1833) 900-page book on the theory of numbers in a short period and then he was described as “already singing like a canary” by his professors.

Alas, again, space has caught up with me!