Just back from an enjoyable evening at Gresham College, where Jan van Maanen from Utrecht gave the BSHM–Gresham lecture marking the tercentenary of the death of Leibniz next month, preceded by warm-up acts by Snezana Lawrence and my colleague Kenneth Falconer.
The theme of the event was “Curves”. Snezana Lawrence took us from Archimedes and his spiral (which he used to solve one of the problems of classical antiquity, squaring the circle) to the equiangular spiral, whose analysis was made possible by the invention of calculus, and which is much commoner in nature. Jacob Bernoulli called it spira mirabilis; one of its remarkable properties is that, like a fractal, viewing it at a different scale doesn’t change its shape. Bernoulli wanted an equiangular spiral engraved on his tombstone, but an Archimedean spiral was placed there by mistake: Snezana showed us a picture of this.
Kenneth Falconer, unsurprisingly, talked about fractal curves; his message was that in not much over a century they went from esoteric to ubiquitous. The precursors were found in the 19th and early 20th century, and were Weierstrass’s function which is continuous everywhere but differentiable nowhere; Koch’s curve, a simpler geometric construction giving the same property; Cantor’s “Devil’s staircase”, the graph of a function whose derivative is zero almost everywhere but which increases continuously from 0 to 1; and Sierpiński’s triangle, a curve which branches at every point (although it doesn’t look like a curve). It was of course Benoît Mandelbrot who brought fractals into the common currency of science: as he said,
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
The first two talks were linked by the appearance of logarithmic spirals in his final example of a fractal.
After the traditional refreshment break of macaroons and Rhenish wine, we proceeded to the talk by Jan van Maanen. He discussed Leibniz as “mathematician and diplomat”. His activities as a diplomat, on behalf of his employers the Prince Bishop of Mainz and the Duke of Hanover, are well known. What was less well known to me was his attempt to make peace between the Bernoulli brothers. Jacob and Johann had studied Leibniz’s 1684 paper in Acta Eruditorum on differential calculus, and in Leibniz’s view they were great mathematicians who had really understood what he was doing; but they fell out, and a stream of public abuse and mathematical challenges passed between them. Leibniz attempted to reconcile them by various means, including having them both elected to various learned societies on condition that they stopped fighting!
How did curves come into this? Part of their fight consisted of the challenge by Johann to find the brachystochrone, or curve of quickest descent. (And here we had a reference back to the second lecture, since the solution is a cycloid, and Kenneth had used this as an example of a curve which has a tangent at all but “a few” points.) The brothers fought bitterly over priority and accusations of malpractice concerning this challenge. (Newton, on the other hand, solved it overnight and published his solution anonymously.)
Jan showed us a page of Leibniz’s 1684 paper. (Although it was published eight years after Leibniz got access to the papers of Newton and Gregory, we also saw a page of his notes from 1675 where he had invented the notation dx and the integral sign.) The paper contains the formulae for differentiating sums, products, and quotients. (In Arnol’d’s book on Newton, which I mentioned recently, he claims that Leibniz first thought that d(xy) = (dx)(dy), so that differentiation would be a ring homomorphism, but then realised his mistake; a geometer like Newton would never have made this error. I don’t know where the evidence for this is.)
It is clear that the Bernoulli brothers showed clearly the usefulness of Leibniz’s calculus in solving practical problems. The first text-book on calculus was by l’Hôpital, a student of Johann Bernoulli.