## Curves

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.

I count all the things that need to be counted.
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### 4 Responses to Curves

1. Claire says:

I wonder if Snezana Lawrence is the same Snezana Lawrence former director of CMRC dismissed because of forgery?
http://www.independent.co.uk/news/university-to-offer-freemasonry-degree-1085065.html

• Well, I’m not the Peter Cameron who was excommunicated by the Presbyterian Church in Australia (for supporting the ordination of women), nor the Peter Cameron who wrote “The Weekend” and other great books; nor any of various other people of that name…

• Thomas says:

Her LinkedIn profile says she was doing a history of maths PhD from 1996-2002 and the jobs list starts in 2001. Not much after graduating a BSc in Architecture and Engineering in 1989 before this.

This letter to the THE is by someone with the same name associated with the Canonbury Masonic Research Centre in 1999. This paragraph is particularly interesting:

It was founded through an initiative that was undertaken by me, a non-mason, to establish a centre where information about different masonic traditions will be gathered and where such findings will be made available to those who are interested in and would like to study the various aspects of freemasonry. … As an educational institution we have sought and gained access to the masonic libraries and archives purely to study their enormous wealth of historiographical, sociological and other material, some of which relates to the specifically western esoteric movements.

Companies House lists a Snezana Lawrence as Secretary of the now-dissolved Canonbury Masonic Research Centre from 9 November 1998 to 23 May 2000.

Then this article says Snezana Lawrence of Canonbury Masonic Research Centre “has a degree in architecture and is studying for a phd in the history of mathematics”. The coincidence of first degree topic, PhD topic and PhD timing, plus the name, would seem to suggest these are the same person.

However, Claire Ferguson, I find no mention of her name in conjunction with forgery, or a reason for her leaving the Centre. This is not the subject of the article you linked. Do you have a reference for this claim?

2. Claire Ferguson says:

Of course not. But… it’s good to know people around you. Check it out!

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