Here are some informations and comments concerning the questions raised about Pedro Nunes.
The 1537 "Tratado da Sphera" by Pedro Nunes consists of several annotated translations (of works by Sacrobosco, Peuerbach, Ptolemy) plus two original treatises on navigation and charts.
These two treatises are the most important part of the book. In them, Nunes analyses rhumb-lines (that is, lines of constant bearing). He states as a desirable characteristic for charts that rhumb-lines appear as straight lines. He is perfectly conscious of the metric distortions this will introduce in charts, needing corrections by tables and instruments.
The two navigation treatises appeared expanded, in Latin, in 1566 (Basel) and again in 1573 (Coimbra).
Already in the 1537 book Nunes distinguished the rhumb-lines (later called loxodromes) from great circles, drawing several of the former spiraling round the north pole. This is before Mercator's globe.
He carries his analysis further in the 1566 book, describing in detail a procedure (involving spherical triangles) to calculate the latitude and the longitude of points in a rhumb-line from the equator. This is mathematically equivalent to constructing a chart by the so-called Mercator projection. Mercator's chart appeared in 1569, without full explanations as to its construction.
Extraordinarily, at the end of his detailed explanation, Nunes includes an empty table of rhumbs (for seven angles), leaving "diligent teenagers" to fill the empty columns "following the preceding demonstrations"!
The first person to actually publish tables allowing the construction of charts with the required property of straight rhumb-lines was Edward Wright, in "Certaine Errors in Navigation" (London, 1599). This book was reprinted in facsimile in 1974 (Walter J. Johnson, Inc.; no. 703 in the series "The English Experience").
In his Preface, Wright says this: " (...) it may be, I shall be blamed by some, as being to busie a fault-finder myself. For when they shall see their Charts and other instruments controlled, which so long time have gone for currant, some of them perhappes will scarcely with pacience endure it. But they may be pacified, if not by reason of the good that ensueth hereupon, yet towards me at the least, because the errors I poynt at in the chart, have been heretofore poynted out by others, especially by Petrus Nonius, out of whom most part of the first Chapter of the Treatise following is almost worde for worde translated (...)"
This part of Wright's book is the only English translation I am aware of concerning all or part of Nunes' "Tratado da Sphera" and its subsequent versions. The "French trans. prior to 1562, published in France" mentioned in the 'Dictionary' entry quoted by Janet Barnett seems quite mysterious. The closest possibility I see is the translation into Latin, by Elie Vinet, of one of Nunes' annotations (fol. 37-45), and included by Vinet in his own edition of Sacrobosco's "Sphaera", published in Paris in the 1550s.
Portuguese editions, naturally, are another matter. In particular, a full transcription of the 1537 original, with abundant and valuable notes, was published in Portugal in 1940.
I use this opportunity to point out a serious error in the article on the loxodrome appearing in the Encyclopaedia Britannica (1994-2000 electronic version; I haven't seen the paper edition). The full article reads:
"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."
I don't understand where the 1550 date comes from. But the serious error lies in the statement that Pedro Nunes "mistakenly believed [the rhumb-line] to be the shortest path joining two points on a sphere". It is abundantly clear from Nunes' writings and illustrations that this is not so, and that there was no confusion to him between rhumb-lines and great circles (see above). Already in the 1537 treatises he studies the question of how to navigate along a great circle by successive changes of bearing.
Still concerning translations of Nunes, there are references to contemporary versions of the "Libro de Algebra" (Antwerp, 1567) in Latin and French, existing only in manuscript form.
Finally, the year 2002 will mark the 500th anniversary of Pedro Nunes' death.
Joao Filipe Queiro Universidade de Coimbra Portugal