J. F. Queiró

University of Coimbra

The Encyclopaedia Britannica (online and DVD versions) contains serious mistakes of fact in two articles where the Portuguese mathematician Pedro Nunes is mentioned.

I wrote Britannica in April 2002 about this. They answered as follows: "Your comments have been forwarded to the appropriate department for consideration. Our editorial staff will review all of the information you have provided and will consider appropriate changes. We assure you that our editors will give your comments careful consideration."

A long time afterwards, the errors persist.

The errors appear in the articles "loxodrome" and "analytic geometry - curves of double curvature".

Here are the relevant passages:

**loxodrome**

**also called RHUMB LINE, OR SPHERICAL HELIX, curve cutting the
meridians of a sphere at a constant nonright angle. Thus, it may
be seen as the path of a ship sailing always oblique to the
meridian and directed always to the same point of the compass.
Pedro Nunes, who first conceived the curve (1550), mistakenly
believed it to be the shortest path joining two points on a
sphere (see great circle route). Any ship following such a
course would, because of convergence of meridians on the poles,
travel around the Earth on a spiral that approaches one of the
poles as a limit. On a Mercator projection such a line (rhumb
line) would be straight. Rhumb lines are used to simplify
small-scale charting.**

**from analytic geometry - curves of double curvature**

**Loxodrome, or rhumb line, or spherical helix (see 169) is
usually defined as the curve cutting the meridians of a sphere
at a constant angle. The curve was first conceived by Pedro
Nunes in 1550. Its equations may be written in terms of [beta],
the constant angle, and [phi] and [theta], the longitude and
colatitude, respectively, of a point on the loxodrome. Nunes
believed that a loxodrome joining two points on a sphere was the
shortest distance on the sphere between those points. But
19th-century mariners realized that great-circle sailing is
preferable for shortening distances.**

The same errors are made in the two articles:

1) The 1550 date is wrong: Pedro Nunes wrote about this subject
in books published in 1537 (Tratado
da Sphera, Lisbon) and 1566 (Petri Nonii Salaciensis Opera, Basel,
Switzerland). No book of Nunes was published in 1550.

2) Much more important, Pedro Nunes __did not__ believe the
rhumb line to be the shortest path between two points on the
sphere. Actually, his main point in the 1537 book (in Portuguese,
with two chapters – "Treatises" – on navigation) is precisely that
distinction. I quote (my translation):

"[In the art of navigation] there are two ways: the first is to take always the same route, with no change (...). The second way would be to follow a great circle (...)."

"(...) to follow a great circle (...) is to travel a shorter distance."

"If we want to follow a great circle, it is necessary to know the change in the position-angles, to change the route accordingly."

"(...) the path followed sailing by a route is not a great circle (...), because we always make with the new meridians an angle equal to the one at departure, and this would be impossible if we had followed a great circle; it is rather a curved and irregular line."

"(...) rhumbs [are not] circles, but curved irregular lines, which make equal angles with the meridians we sail through (...)."

In the book published in Basel (1566), written in Latin, Nunes goes much further, and gives a procedure, involving spherical trigonometry, to describe points on a rhumb line. This is sophisticated mathematics, much beyond the point of distinguishing between rhumb lines and great circles. In this book we find the following sentence:

"This curved line is different
[from a great circle] and is
similar to a helix (...)."

3) But this is not all: already in 1537, Nunes also analyses the
question of the maritime chart, and states the desired property of
rhumb lines on the globe being represented by straight lines on
the chart, usually attributed to Mercator. He had great influence
on Edward Wright (1558-1615), who gave the first rigourous
construction of the so-called Mercator chart in 1599, in a book, *Certaine
errors of navigation*, where Pedro Nunes is abundantly
quoted.

4) A final point: With all of the above in mind, it is astonishing to read in Encyclopaedia Britannica that only in the 19th-century did mariners realize that great-circle sailing is preferable for shortening distances.