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.