The story of a problem

Some days ago mathematician John D Cook of the Endeavour blog posted the following mathematical problem from the 15^th century.

In 1471, Johannes Müller asked where you should stand so that a vertical bar appears longest.

To be more precise, suppose a vertical bar is hanging so that the top of the bar is a distance a above your eye level and the bottom is a distance b above your eye level. Let x be the horizontal distance to the bar. For what value of x does the bar appear longest?

Note that the apparent length of the bar is determined by the size of the angle between your lines of sight to the top and bottom of the bar.

 At the same time he sent me an email asking if I knew whether Müller had solved the problem himself and if so what his solution was. Now Johannes Müller is nowadays much better known as Regiomontanus and although I am considered something of an expert for the life and work of Franconia’s greatest 15^th century mathematicus I had to honestly answer John’s enquiry with a resounding no, I don’t know. Although well aware of Regiomontanus’ prowess as a mathematician I have always been more interested in his activities as an astrologer, his achievements with his teacher Peuerbach in setting the astronomical revolution in motion and his activities as the world’s first scientific printer publisher. However I was intrigued by John’s question, which as he pointed out seemed to require a solution using calculus a couple of hundred years before it was invented, and set about finding the answer. My first step was to order all the relevant literature that I could find from the university library and through inter-library loan and as I type there is a literally one metre high stack of books next to my computer all of which are dedicated to the life and work of Herr Müller or contain papers on the same. Where to look? (Any reader wishing to know more details about Regiomontanus’ life should follow the links to earlier posts where they will find most of the details)

Now the only mathematical, as opposed to astronomical, book that Regiomontanus wrote was his trigonometry textbook, which was published posthumously about sixty years after his death,[1] so where to look for mathematical problems of the type posted by John Cook? The answer is to be found in his correspondence. During the four years that he spent in Italy, 1461 – 1465 in the entourage of Cardinal Bessarion Regiomontanus made the acquaintance of many of Italy’s leading mathematicians with whom he then corresponded when he moved on. Parts of his correspondence with Paolo dal Pozzo Toscanelli (1397 – 1482) doctor of medicine and Florence’s leading mathematician, the Ferrara professor of mathematics and court astrologer to Leonello d’Este, Marquis of Ferrara, Giovanni Bianchinni, and Jacob of Speier court astrologer to the Duke of Urbino still exist. Toscanelli and Bianchini had also been friends of Peuerbach. In these letters Regiomontanus challenges his correspondents to proffer solutions to long lists of complex and challenging mathematical problems. Such challenges were fairly common throughout the Early Modern Period. The most well know examples being the various challenges involving Tartaglia and the solutions of the cubic and bi-quadratic algebraic equations. Even at the end of the 17^th century the various mathematicians involved in the invention of the calculus challenged each other to solve difficult problems to prove their prowess in the new discipline. Most famous is the so-called brachistochrone problem posed by Johann Bernoulli, which according to his niece, Catherine Barton, Isaac Newton solved between supper and going to bed. Although his solution was submitted anonymously Bernoulli recognised it as Newton’s remarking, “as the lion is recognised from his print, but back to Johannes Müller. John’s problem is not in the surviving Italian letters so what now.

There is one more letter written by Regiomontanus containing a list of thirty-eight problems and it is here that the original of John’s problem can be found. After several years in Hungary Regiomontanus settled in Nürnberg in 1471 with the intention of reforming the whole of astronomy. Now even he realised that he would not be able to achieve this task alone and so he cast around for potential partners. His first choice fell on the Erfurt University professor of mathematics Christian Roder and it is the letter that he wrote in 1471 trying to win Roder for his project that contains the problem. The original problem is slightly different to the one John posted and reads as follows:

A ten-foot pole hangs vertically so that its lower end is four feet off the floor. Find the point on the floor from which the stick appears longest, or, as there are infinitely many such points that lie on a circle, find the diameter of the circle.[2]

Apart from being a somewhat tricky mathematical problem it is also interesting in that it is actually a problem out of the geometrical optic. The solution invokes Euclid’s fourth postulate of geometrical optics:

That things seen under a larger angle appear larger, those under a smaller angle smaller, and those under equal angles equal.[3]

The first lectures that Regiomontanus held as a freshly baked Magister Artium at the University of Vienna at the tender age of 21 were on geometrical optics a standard course in the scholastic university since Robert Grosseteste had made the metaphysics of light the central pillar of his natural philosophy in the 13^th century.

Unfortunately Regiomontanus does not offer us a solution to his problem but the addition of the information that the point lies on a circle, which must be tangential to the floor lead almost automatically to the solution offered by Pat Bellew and lvps1000vm at Endeavour, which is also the solution that was reconstructed for the problem at the end of the nineteenth century independently by both of the German historians of mathematics Moritz Cantor[4] and Sigmund Günther.[5] All of the surviving letters between Regiomontanus and Jacob of Speier, Bianchini and Roder were edited and published in the original Latin by Maximilian Curtze.[6]

 

---

[1] Ioanis de Regio Monte, De triangulus omnimodis, ed. Johannes Schöner, Johannes Petreius, Nürnberg, 1533

[2] Quoted from Ernst Zinner, Regiomontanus: His Life and Work, trans. Ezra Brown, North-Holland, Amsterdam etc., 1990 p. 106

[3] Quoted from David C. Lindberg, Theories of Vision: From al-Kindi to Kepler, University of Chicago Press,  Chicago, 1976, p. 12

[4] Moritz Cantor, Vorlesung über Geschichte der Mathematik, Zweite Band Von Jahre 1200 bis zum Jahre 1668, 2 Aufl. 1900, Johnson Reprint Corp, New York, 1965, p. 283

[5] Sigmund Günther, Geschichte der Mathematik, 1. Teil, Leipzig, 1908, pp. 302f.

[6] Maximilian Curtze, Urkunden zur Geschichte der Mathematik in Mittelalter und der Renaissance, B.G. Teubner, Leipzig, 1902, pp 185 – 336.