I’m an alien

I’m a legal alien

I’m an Englishman in Nürnberg^{1}

As an English historian of mathematics living in Germany another question that I have had put to me several times by those with somewhat more knowledge of the history of mathematics is, “who invented logarithms the Scottish aristocrat John Napier or the Swiss instrument maker Jost Bürgi?” This question would appear to provoke at least as much nationalist sentiment if not even more than the Newton Leibniz dispute as older English language histories name Napier as the inventor and ignore Bürgi whereas similar German publications claim that Bürgi was the inventor. Before I go on to the main point of this post I think I should explain something for any possibly younger readers of this post. What I am talking about here is not the logarithmic functions you learnt about in that elementary calculus course but logarithmic tables, i.e. tables that enable the user to determine the value of the logarithm to a given base of any number in order to facilitate complex calculations. These tables, an essential part of my school education, are now only found in mathematical museums, as they have been made redundant by the ubiquitous pocket calculator, destroyer of the numerical abilities of modern school children.

The first log tables where published by Napier in 1614 followed closely by those of Bürgi in 1620. Bürgi’s supporters however claim that he started work on his tables in 1588 making him the real inventor. Modern research (by a German thank god!) has however shown that Bürgi didn’t start work on his tables until much later and the priority really does go to Napier. As with the calculus the concept on which logarithms are based has a long pre-history that goes back at least to Archimedes and involves many others before Napier and Bürgi independently brought the idea to fruition. Now multiple inventions or discoveries are actually very common in the history of science and technology and this phenomenon is often explained with the cliché “the ideas time was ripe”. Other examples of this are of course Leibniz and Newton with the calculus, Darwin and Wallace with the principle of natural selection or Harriot, Snel, Descartes and James Gregory who all independently discovered the sine law of refraction in the 17^{th} century. In many cases the time being ripe actually means that the co-discoverers were inspired by the same source or as I have chosen to express it in my title they drank at the same well. Darwin and Wallace were both inspired by the same passage in *An Essay on the Principles of Population* by Thomas Malthus. Newton and Leibniz both built their mathematical structures on the same works by James Gregory, Fermat, John Wallace and Isaac Barrow. Napier and Bürgi also both drank from the same well in their case one with the rather strange Greek name prosthaphaeresis.

Prosthaphaeresis means adding and subtracting and refers to the process in trigonometry where the multiplication of trigonometrical functions can be converted in the sum and or difference of trigonometrical functions to simplify calculation.

e.g. sin *a* sin *b* = ½[cos(*a* − *b*) − cos(*a* + *b*)]

The first known prosthaphaeresis formula can be found in the work of the Nürnberger mathematician Johannes Werner (1468 – 1522) who however never published it and so it disappeared again for more than fifty years. Prosthaphaeresis resurfaced in the hands of the itinerant German-Polish mathematician Paul Wittich (1546 – 1586). We don’t know if he learned the procedure from a manuscript of Werner’s work or rediscovered it for himself but having it, he proceeded to spread it around Europe. He taught the method to the physician, mathematician and astronomer John Craig (died 1620) at the University of Frankfurt on the Oder who on his return to Scotland taught it to John Napier. Wittich also taught the method to Tycho Brahe and his co-workers on the island of Hven, as it could be used to simplify complex trigonometrical astronomical calculations. When he left Hven Wittich journeyed to Kassel where he taught the method to William IV and his group of astronomers including Jost Bürgi. Bürgi would go on to teach the method to Nicolaus Reimers Bär, better known as Ursus, in exchange for Ursus translating Copernicus’ *De revolutionibus* into German for him, as he couldn’t read Latin. This was the earliest translation of *De revolutionibus* and the manuscript still exists. Ursus was the first to publish the method explaining as he did so who had taught it to him and from whom Bürgi had learnt it. This was to score points off Tycho with whom he was having a plagiarism dispute about the helio-geocentric world system, as Tycho was claiming incorrectly that he had discovered the prosthaphaeresis formulae. Ironically it was probably also Wittich’s work, which inspired Tycho to develop his world system.

Having both received the method of prosthaphaeresis, directly or indirectly from Wittich both Napier and Bürgi conceived the idea of turning multiplications into additions by using the exponents of powers and set out to calculate, construct and publish their logarithmic tables. It should be pointed out that another source for the appearance of logarithms at this point in history was the necessity for mathematical astronomers to perform complex trigonometrical calculation in order to determine the orbits of planets; this necessity also led Kepler, who actually published Bürgi’s tables, to produce an improved set of log tables of his own.

Even if you have learnt nothing else reading this post you can dazzle your dinning companions at your next social engagement by casually dropping the word prosthaphaeresis but should you be tempted to do so then I would recommend practicing a bit in advance as it’s rather difficult to say casually.

1) With apologies to Sting.