Presentation of “Petri Nonii Salaciensis Opera” by
Pedro Nunes (this text in
Portuguese)
João Filipe Queiró
Departamento
de
Matemática,
Universidade de Coimbra, 2002.
Pedro Nunes (Alcácer do Sal, 1502 –
Many aspects of his life remain obscure to this day. We know
the place and year of his birth, as well as the place and date
of his death. We have some information on his activity, as a
professor – in the university and at the royal court – and as
a cosmographer. A few details concerning positions held and
salaries received have also come to us, through documents
written by others. And we have his works, from which we can
extract an intellectual profile and draw some additional
biographical conclusions.
All of this, which is much, is not
enough to obtain a picture of the man. Information about his
family, his youth and his formative years is lacking.[1]
We hardly know of any European travels by him. There are practically no letters, to and
from Pedro Nunes, and they have certainly existed and would
today be documents of the utmost interest for the better
understanding of the Portuguese mathematician’s life and work.[2]
Even a portrait is lacking: there is no contemporary painting
or engraving representing him.
In the absence of this information, we
are left – and from a scientific perspective it is the most
important thing – with Pedro Nunes’ writings. The works
printed during his life have been identified,
and it is unlikely, although not impossible, that others might
surface. Also, no unpublished manuscripts have been found,
with the exception of a single one, more than fifty years ago.
But even about these works there is much we don’t know, in
general, with respect to the circumstances surrounding
publication: the choice of printer, the funding, the print
run, distribution and sales, the reading and use by men of
science, teachers, students.
The publication of Pedro Nunes’ printed
works during his life is concentrated in two periods:
1537-1546 and 1566-
More than before,
Pedro Nunes addresses himself, with these two deeply meditated
works, to the European scientific community. He chose Spanish
for the Libro de Algebra, and published it in
The object of the present facsimile
edition is this latter book. Of all of Pedro Nunes’ works,
this one is possibly the least studied by historians of
science. But it probably is one of the most interesting.
*
The Petri Nonii
Salaciensis Opera consist of four texts, well
identified in the front page, although not so well in the
contents pages compiled by the author.
The first two of these texts may be considered
Latin versions of the navigation treatises included in the
1537 Tratado da Sphera. But while the first,
De duobus problematis circa nauigandi artem, is
indeed, in large measure, a translation of the Tratado sobre certas dúvidas da navegação, in
which, for the first time, constant-bearing navigation and
great-circle navigation were clearly distinguished, the second
one is a very different work, much longer and developed than
the Tratado em defensam da
carta de marear.
This extensive text,[3]
entitled De regulis & instrumentis, ad uarias rerum
tam maritimarum quam & coelestium apparentias
deprehendendas, ex Mathematicis disciplinis, deals
successively with the analysis and design of navigation
charts, astronomical navigation and the laying out of
rhumb-lines on a globe.
On the first subject, Pedro Nunes again discusses
at length, as in 1537, the constant-latitude rectangular
chart, in particular against the requisite that rhumb-lines be
represented by straight lines. This is a typical contribution
originating from navigational needs, which inform Nunes’
discussion more than the convenience of faithfully
representing the globe. After pointing out the inevitable
errors in a constant-latitude chart, with respect to rhumbs,
coordinates of points on the globe, and distances between
them, Pedro Nunes suggests the use of a solution of the type
found in Ptolemy’s Geography (in turn inspired by
Marinos of Tyre), where there appear several rectangular
regional maps with varying proportions in their grids,
according to the central parallel in each.[4]
Pedro Nunes recommends that different charts should be made
according to latitude, and adds that the extremal parallels in
each chart should not be too distant from one another
(“Extremos autem parallelos non admodum à se inuicem distare
oportet”, pág. 25), to avoid excessive distortion.
Furthermore, he says that in all charts the whole longitude of
the globe should be included. (“Et ponenda est in omni tabula
universa orbis longitudo”). Pedro Nunes thus proposes,
differently from Ptolemy, a collection of horizontal strips
covering the surface of the globe. Here we practically have
already the so-called Mercator projection. It remains only, as
Gomes Teixeira points out,[5] to
join the narrow partial maps in a single chart, keeping their
inner proportions and using the same scale everywhere for
longitudes. According to Nordenskiöld,[6] it
was in this way that Mercator proceeded in the design of his
map, using ten-degree intervals between the parallels.
The first author to publish tables
allowing the construction of charts – with “increasing
latitudes” – in which rhumb-lines are represented by straight
lines was Edward Wright,[7]
who says that the errors in the chart with constant latitudes
were pointed out by several authors before him, “especially by Petrus Nonius, out of
whom most part of the first Chapter of the Treatise
following is almost worde for worde translated”. Wright
used the Latin treatises of Pedro Nunes, and not the 1537
texts, as we can see from the beginning of his chapter II, in
which he mentions “Petrus Nonius in his second booke of
Geometricall observations, rules, and instruments”, which
alludes to the title of the second treatise.
The next chapters of the De regulis
& instrumentis deal with astronomical navigation,
including several procedures to find geographical coordinates.
In these chapters we find Pedro Nunes’ interesting
commentaries to Copernicus, which were analysed in detail by
Henrique Leitão.[8]
The last seven chapters of this second
treatise are dedicated to the problem of laying out
rhumb-lines on a globe. This problem is harder than the
construction of the increasing-latitudes chart, although in
modern mathematical terms one can see that the two are
equivalent. Pedro Nunes presents a complicated procedure,
consisting of the sequential resolution of several spherical
triangles, to obtain points which are approximately on
rhumb-lines on the sphere. This remarkable work by Pedro Nunes
was studied in detail by Raymond d’Hollander.[9]
At the end of his construction, on page 172, Pedro Nunes
includes a table to present the results of the calculations
for several rhumbs. This table is empty, and Pedro Nunes says
on the previous page that its filling can be carried out by “studiosi adolescenti”. Here we see
the mathematical spirit, after devising a procedure to solve
a problem, showing no interest in the computations necessary
to the application to practical situations. It should be
recalled that both Gomes Teixeira and Raymond
d’Hollander point out the inaccuracy of a criticism to Pedro
Nunes’ construction by Simon Stevin, which unfortunately is
repeated still today by those who have not read the original
text of Pedro Nunes.
The two Latin treatises by Pedro Nunes
on navigation were translated into Spanish by Juan Cedillo
Diaz, in a manuscript kept at the Madrid National Library.[10]
The third text in the
Opera is a short essay, entitled In Problema mechanicum
Aristotelis de Motu nauigij
ex remis, on the movement of boats with oars in the Mechanica
of Aristotle. This essay
was studied by Henrique Leitão.[11]
The fourth and last, In Theoricas Planetarum Georgii
Purbachii annotationes aliquot, is an extended
collection of annotations to the Theoricae Novae Planetarum by the 14th century Austrian
mathematician Georg Peurbach.[12]
These have been described as being among the most important
annotations to the influential work of Peurbach.[13]
*
The present publication is being carried
out for several reasons. Of course there is a wish to mark the
500th anniversary of Pedro Nunes’ birth. But there are deeper
reasons. As is known, the content of Petri
Nonii Salaciensis Opera was reproduced in a
volume published in
*
The reasons that led Pedro Nunes to
choose the Officina Henricpetrina, in
During the 16th
century, the
The Petri Nonii Salaciensis
Opera is a 304×205mm volume, with 308 pages plus 16
unnumbered pages. In the title-page we see the well-known
emblem of the publisher, still in use today, illustrating a
passage from the Book of Jeremiah.[15]
The colophon is on page 307:
“Basileae, Ex Officina Henricpetrina, Anno M. D. LXVI,
Mense Septembri.”
It is to be remarked that the extant copies of
this 1566 edition are not all identical. Some of them include
four pages of errata compiled by the author to point out
mistakes in the text and in the illustration captions.[16]
This shows that Pedro Nunes had no opportunity to correct
proofs of the text set by the Basel printer, but he had access
to the printed pages in time to organize the errata, which
were included only in part of the print run, since there exist
copies of the Opera
without them.[17]
This interesting fact helps us understand what led to the
The copy of the Petri Nonii Salaciensis Opera used for the present facsimile edition
belongs to the Mathematics Library of Coimbra University. As
can be seen from an inscription on the title page, it was once
part of the great library of the Santa Cruz Monastery in
The volume is in good condition, with the original parchment binding, and contains the above mentioned errata, which shows that it was part of the “complete” print run. It contains several handwritten notes and illustrations on the margins, presumably from the 16th century. There are handwritten additions to the errata, which may be from the author himself, in the same writing as the marginal notes. The corrections indicated in the errata are introduced throughout the text in blue ink. These corrections seem to date from the 20th century.
[1] The little we know can be found in
Inquisition depositions by two grandchildren of Pedro Nunes
in the 17th century.
[2] An explanation for this lack of documents lies in the sad fate of Pedro Nunes’ papers, lost by his uninterested heirs. In this respect see Joaquim de Carvalho, preface to Defensão do tratado da rumação do globo para a arte de navegar, Coimbra, 1952.
[3] It occupies pages 13 to 189 of Petri
Nonii
Salaciensis Opera.
[4] Mathematically, this corresponds to
saying that, if the central parallel has latitude j, the longitude degrees on the grid
should be multiplied by cos j, or, which is the same thing, the latitude
degrees should be multiplied by sec j. Pedro
Nunes gives several examples of situations of this type.
[5] Francisco Gomes Teixeira, Elogio Histórico de Pedro Nunes, Panegíricos e Conferências, Coimbra, 1925. The text is reproduced in his História das Matemáticas em Portugal, Lisbon, 1934.
[6] A. E. Nordenskiöld, Facsimile
Atlas to the Early History of Cartography,
[7] Edward
Wright, Certaine Errors in Navigation,
[8] Henrique Leitão, Uma nota sobre Pedro Nunes e Copérnico, Gazeta de Matemática, 143, p. 60-78, 2002.
[9] Raymond d’Hollander, Historique de
la loxodromie, Mare Liberum, 1,
p. 29-69, 1990.
[10] Los dos libros de la arte de nabegar, de Pº Núñez de Saá, traducidos de latin en Castellano, por el doctor Sñ. Cedillo Diaz.
[11] Henrique Leitão, O Comentário de Pedro Nunes à Navegação a Remos, Lisbon, Comissão Cultural de Marinha, 2002.
[12] Recall that a Portuguese translation of part of
that work was included in the 1537 Tratado da Sphera.
[13] See, e. g., C. D. Hellman and N. M.
Swerdlow, Georg Peurbach, Biographical
Dictionary of Mathematicians, vol. 4,
[14] Henrique Leitão, Uma nota sobre Pedro Nunes e Copérnico.
[15] “Is not my word
like as a fire? and like a hammer that breaketh the rock in
pieces?” (Jeremiah, 23:29)
[16] It is through an indication of Pedro Nunes in the errata that we
know the year of his birth. On the first line of page
[17] The inicial capital letter in the errata, the body of the four pages as well as the type in which they were set, are different from those used in the main text. The paper too is slightly different.
[18] António Mariz, the
[19]
See J. M. Teixeira de Carvalho, A Livraria do Mosteiro de
Santa Cruz de Coimbra, Coimbra, 1921.