On the way back to London from Italy earlier this month, I decided to stop in Basel. It was mid-way and it had long been on the list of cities I wanted to visit. Why it was on that list started to surface as I picked a hotel on Trivago. Euler Hotel – definitely. We arrived in the evening and the boys were keen to get back home. So I only had half a day the following morning.
Basel’s old town centre is quite small and its main landmark is the Münster, a Romanesque church with a long and interesting history. As we waited for its doors to open at 10, I started touring the adjacent cloister. One of the highlights of the place is that Erasmus was buried there in 1536 – a sudden death following an attack of dysentery. But while looking for the grave in the cloister, wandering among tombs and commemorative plates of the city’s notables, one of them gave me a jolt:
Jacob Bernoulli, of course. He was born and lived in Basel his whole life, and died there on 16 August 1705 – morbo chronico, mente ad extremum integra – at the age of 50 years and 7 months.
Jacob – the eldest scion of the prodigious Bernoulli family – is one of my heroes. The author of the greatest masterwork in early probability theory, Ars Conjectandi, he is also credited as the first to discover the relationship between continuous compound interest and Euler’s number e, the base of natural logarithms. There – I suddenly realised – was a big piece of my subconscious attraction to Basel. Enchanted by my discovery, I asked my second child to pose for a photo next to the tombstone – my elder son was wandering somewhere else, supremely bored and impatiently waiting for lunch and departure.
After leaving the cloister, unable to come up with anything intelligible to say about Bernoulli, I told the kids about Erasmus and Paracelsus – another illustrious Basler. At 10 we visited the church – Erasmus’s grave is inside – and shortly after I realised my time was up – the children would have killed me if I had proposed any more ‘history stuff’. So we walked back to the Euler Hotel – Leonhard Euler was born in Basel two years after Jacob Bernoulli’s death. He was the first to use the letter e for the base of natural logarithms, apparently as the first letter of ‘exponential’, rather than of ‘Euler’. He also established the notation for π and for the imaginary number i, all beautifully joined together in Euler’s identity eiπ+1=0.
On the road to London, I kept thinking with delight at my semi-serendipitous encounter with Bernoulli. Then it struck me: I had seen that tombstone before. Back home, I checked. I was right: it was in one of the best books I have ever read, Eli Maor’s e: The Story of a Number.
As I reopened the book, it all came back to me: the Spira Mirabilis.
The logarithmic spiral is the curve r=aebθ in polar coordinates (r is the radius from the origin, θ is the angle between the radius and the horizontal axis, and a and b are parametric constants). Bernoulli had a lifelong fascination with the self-similar properties of the spiral:
But since this marvellous spiral, by such a singular and wonderful peculiarity, pleases me so much that I can scarce be satisfied with thinking about it, I have thought that it might be not inelegantly used for a symbolic representation of various matters. For since it always produces a spiral similar to itself, indeed precisely the same spiral, however it may be involved or evolved, or reflected or refracted, it may be taken as an emblem of a progeny always in all things like the parent, simillima filia matri. Or, if it is not forbidden to compare a theorem of eternal truth to the mysteries of our faith, it may be taken as an emblem of the eternal generation of the Son, who as an image of the Father, emanating from him, as light of light, remains ὁμοούσιος [consubstantial] with him, howsoever overshadowed. Or, if you prefer, since our spira mirabilis remains, amid all changes, most persistently itself, and exactly the same as ever, it may be used as a symbol, either of fortitude and constancy in adversity, or, of the human body; which after all its changes, even after death, will be restored to its exact and perfect self; so that, indeed, if the fashion of imitating Archimedes were allowed in these days, I should gladly have my tombstone bear this spiral, with the motto, Though changed, I rise again exactly the same, Eadem numero mutata resurgo.
This is the full quote from a paper by Reverend Thomas Hill published in 1875 (p. 516-517), from which Maor’s book takes an extract (p. 126-127), taken in turn from another book. Hill did not quote the source, but the original Latin quote can be found here (p. 185-186, available here), with the indication that it comes from a paper published by Bernoulli in the Leipsic Acts in 1692, which should be found here.
Bernoulli’s enthusiasm is easy to understand and to share. The logarithmic spiral is found in nature and art. The Golden spiral, whose growth factor b is the golden ratio, is a special case, and so is the circle as b tends to 0.
At the same time, it is difficult not to laugh at the manner in which Bernoulli’s wish was finally granted. Perhaps confused by the reference to Archimedes, the appointed mason cut an Archimedean spiral at the bottom of the tombstone, which has none of the properties Bernoulli so admired in the logarithmic spiral. And, to add insult to injury, he missed the word ‘numero’ from the motto. Bloody builders – always the same…
Bernoulli’s considerations made an impression on me when I first read Maor’s book. The spira mirabilis as a symbol of fortitude and constancy in adversity, or of the human body restored to its perfect self even after death. But after reading the passage in its entirety, I find it even more beautiful and inspiring. And how about taking a picture of my son – simillimus filius patri – next to the spiral, before any of this had come back to my mind?
By the way, my son’s name is Maurits, like (but not named after) M. C. Escher.