Pedro Nunes

Pedro Nunes was a Portuguese mathematician of the sixteenth century, perhaps the greatest mathematician of his time in Europe.

Yesterday I was treated to a very informative short presentation about Nunes and his work by the historian of science Henrique Leitão. Here are three things I learned.

First, one of Nunes’ five books was a book on Algebra. What is remarkable about it for its time is the philosophy. Nunes believed that algebra is not just a growth from the root of geometry, but an entirely new subject. His proof was that some results in geometry are more easily proved by means of algebra than by geometric methods.

Second was his discussion of the rhumb line (now called the loxodromic curve), the line traced by a ship which sails always on a bearing making a constant angle with the meridian. Such a line is not a great circle, since it spirals in to the north and south poles. (This fact was already a great novelty at the time, a curve having a finite limit point.) The mathematical tools of the time did not permit finding its equation, but Nunes proposed a “finite difference method”. The navigator sets a bearing making the given constant angle with the meridian, and sails straight (i.e. alon a great circle) until his bearing deviates from the required value by more than a given fixed amount (say one degree); then he corrects the bearing and continues. This gives a practical method for calculating rhumb lines. Nunes’ method was used by many others, and tables were produced.

There has been a lot of interest in the question of how Mercator calculated his map projection. Leitão and a colleague propose a new answer to this. Since rhumb lines appear as straight lines in Mercator’s projection, he could simply use existing tables based on Nunes’ method to derive the spacings of the parallels. This hypothesis appears to fit the data better than any other suggestion.

Nunes’ third remarkable achievement was the following. Suppose that you place a vertical stick in the ground, and watch the movement of its shadow as the day progresses. Almost everyone would say that the movement of the shadow was monotonic. However, Nunes did the calculations and showed that retrograde motion of the shadow was possible under some conditions, and worked out exactly what the conditions were. He admitted that he had never seen the phenomenon, despite knowing what to look for, and nobody he had spoken to had seen it either; yet he had sufficient confidence in his mathematics that he could confidently assert its existence. This caused a certain amount of religious controversy; the fact of a shadow standing still is described in the Bible as a miracle, and yet Nunes was proposing that standing still or even reversing can happen strictly in accordance with natural laws. (I believe that this phenomenon, though small, has now been observed.)

Anyway, the reason for this was an extraordinary event yesterday. I mentioned in March that I was teaching a group theory course to PhD students in compuational algebra at the Universidade Aberta (the Portuguese Open University). There was a celebration of the successful completion of the first year of the course, at which two Pedro Nunes awards (voted by the students on the course) were presented, to Michael Kinyon and me, by the Chancellor of the University. The ceremony began with a short presentation by João Araújo (the driving force behind the course) of how the computational algebra course was set up, and how it had run.

All this in a morning out from the Portuguese Mathematical Society summer meeting, at which I lectured. I will probably say something about this meeting later.