## The loxodrome and two projections of the sphere

### Pedro Nunes and the loxodrome

During the Discoveries period, trade routes, in particular, spice trade routes, had a crucial role on the economical development of countries. Driven by the search of alternatives to the known routes and due to the geographical location of their country, Portuguese navigators started a set of sea travels that gave rise to the European Age of Discoveries (from XV century to XVII century). It was in this context of huge commitment that the Portuguese gave a substancial contribution to the development of navigation to that point that, at the time, is acquired the status of science. Several Portuguese researchers dedicated their attention to Cartography, Astronomy and Nautical Science, being Pedro Nunes (1502-1578) one of the most important.

In 1537, Pedro Nunes published two works on some of the problems related with certain navigation routes that keep the course of the ship steady, intersecting all meridians with the same angle. These routes determined a new type of curve, known as a loxodrome, and not a great circle, which is the curve on the sphere that minimizes the distance between any two points.

Pedro Nunes suggested that the ship should try to follow the course of a great circle, making the necessary adjustment at periodic time intervals to counter the effect of the loxodrome. To this end, the mathematician proposed a method to solve this but received harsh criticism due to its difficult implementation in high sea, at the time.

Regarding this controversy, Nunes once wrote: "Dizem mal de meus tratados aproveitando-se deles e usando muitas vezes de minhas próprias palavras, e querendo falar em tudo danam tudo. Tenho determinado por esta razão, acabando de alimpar algumas obras que escrevi, passar meus estudos à filosofia, e a largar-lhes as matemáticas, no estudo das quais perdi a saúde irremediavelmente." (Randles, 1989 [1])