In 1987 the bicentenary of the death of José Anastácio da Cunha
was celebrated. Perhaps because of the injustices which he suffered
during his lifetime, the Portuguese as a nation, have felt a need to
recognize the contributions which have been made by this man. This was
not however the only motivation for the celebrations: the many events
hosted by the Universities of Évora, Coimbra and Lisbon, with the
participation of Portuguese and foreign mathematicians and historians,
had the principal objective of paying homage to the achievements of this
remarkable portuguese mathematician, poet and innovator in the field of
thought which prevailed during his lifetime. Much has been written and
much has been said about his work. Two books
were published
containing the communications of the participants in the conferences
which have taken place Nota29.
We followed with emotion the path which he chose during his life:
his birth into humble surroundings in Lisbon in 1744; his education in the
Congregation of Oratório; his enlistment at the age of 19 into the
Artillery Regiment of Oporto stationed in Valença do Minho; the high
and passionate lifestyle which he lived during these years in Valença; his
contacts within the military environment with foreign army officers with
liberal ideas, many of them of protestant religion; the love for the young
Margarida, described and sublimed into many of his verses; the fame of
his mathematical culture which, through the Conde de Lippe, was
transmitted to the Marquis of Pombal; the invitation which he received
from the Marquis of Pombal to lecture in the recently created Faculty of
Mathematics in the University of Coimbra; his life in Coimbra and, after
the fall from power of the Marquis, the libelous accusations of free-
thinker, heretic and libertine and his trial, condemnation and
imprisonment by the Inquisition in 1778. The last years of his life were
lived in Lisbon, as a professor of the pupils of the Real Casa Pia, after a
part of the sentence in prison had been pardoned.
As we began to know José Anastácio da Cunha better we couldn''t
help admiring him. His main mathematical works were re-printed and
also several critical reviews were published. We registered with pride the
praise of Gauss Nota30, one of the greatest mathematicians of all time; his
poetic works have also attracted many critical comments; the attention of
the reader was drawn, in particular, to the critical appreciation of
Fernando Pessoa Nota31.
In the field of philosophy, Anastácio da Cunha had a more far-
seeing vision than that of many of his contemporaries; perhaps he
considered them with a certain disenchantment, but in spite of this, or
perhaps because of this, he didn''t cease to love them Nota32. The celebrations
of 1987 were the beginning of a more profound study of José Anastácio
da Cunha, a more enthusiastic and renewed study, more informed and
based on firmer foundations. The Portuguese Group of Mathematical
Historians, Seminário Nacional de História da Matemática, decided that
"the life and works of José Anastácio da Cunha would be a priority theme
for 1988". Thus, in the context of the discipline of History of Mathematics
in the course of Licenciatura em Ensino da Matemática in the
University of Minho, with a group of students of the 1987-88 course, it
was decided to research further the life of José Anastácio da Cunha. My
young and enthusiastic students went to Valença do Minho to look for
new details of the period in which Anastácio da Cunha lived there.
Unfortunately, apart from the kindness of the people who helped them
with their research, their efforts were fruitless and they found no new
facts. My share of the project research took me to the Library and
District Archive of Braga. It was in this Archive that I found, beside a
manuscript of the poetic works of José Anastácio da Cunha, another
manuscript entitled Ensaio sobre as Minas, previously unpublished.
In a corner of the first page of the manuscript, in small letters in
red ink, the following note is written: "Unpublished, regarding the most
notable author of this book see Inocêncio, Tome 4, page 221 and
following. Inocêncio did not know of the existence of this unpublished
edition". In fact this manuscript is not cited in the Portuguese
Bibliographical Dictionary of Inocêncio Francisco da Silva Nota33, under the
list of works published by José Anastácio da Cunha. However there are
at least three references to this "Ensaio" which we will now cite, in
chronological order.
The first of these citations is of José Anastácio da Cunha himself in
the "Carta Físico-Mathemática" Nota34, completed on the 5th of November of
1769 and later published in Oporto, in 1838. On page 29 there is a note
inserted within the parenthesis, "although with respect to mines nothing
can be precisely determined, as I have shown in a previous treatise". It
seems reasonable to conclude that this previous treatise was Ensaio
sobre as Minas. A second reference is made in the "O Processo de José
Anastácio da Cunha" Nota35. On page 73, in a part of a letter written by João
Baptista Vieira Godinho to José Anastácio da Cunha, in 1771, we can
read: "by return of mail send back to me an extract of your work entitled
Arithemética - Ensaio das Minas or your Dissertation - Ensaios sobre a
Pyrrotecnia - etc., with all the motivation which stimulates us to study
each work".
It seems indisputable that this "Ensaio das Minas" is the Ensaio
sobre as Minas which I have found in the Archive in Braga.
The third reference is also made by José Anastácio da Cunha
himself, in one of his own letters written in his defence against the
accusations to which he was subjected in the Qestão entre José
Anastácio da Cunha e José Monteiro da Rocha Nota36, commented on by
António José Teixeira: "The captain of mines of my regiment asked me
my opinion about the various authors which had published work related
to mines: I gave him a hand-written copy, almost without intending to,
leaving myself without a copy. Amongst other aspects I showed in this
work some of the mistakes of Mr._Dulacq, author which the marshal had
recommended to the artillery and engineer officers, who neither I, nor
any other member of my regiment, knew at the time. Afterwards the
marshal passed through Almeida, someone innocently, thinking he was
doing me a favour, offered a copy of my dissertation to the Conde de
Lippe, who naturally thought the offer an insult. Leaving for
Buckembourg, still with doubts regarding my innocence, he
recommended that my salary be doubled and that I be promoted".
From this passage it would seem reasonable to conclude that it
was the Ensaio sobre as Minas which so much impressed the Conde de
Lippe in the favour of José Anastácio da Cunha that he praised him so
highly to the Marquis of Pombal.
The fact that the Marquis nominated José Anastácio da Cunha for
the position of professor of Geometry at the University of Coimbra on the
5.th of October of 1773 is sufficient proof of the high regard which the
Marquis had for him. If further evidence of the high esteem which the
Marquis had for José Anastácio da Cunha is necessary it may be found in
the flattering letters of recommendation for José Anastácio da Cunha
which the Marquis also wrote to D. Francisco de Lemos Nota37, rector of the
University of Coimbra. As José Anastácio da Cunha affirmed in the letter
which has been transcribed above, he himself didn''t even keep a personal
copy of the manuscript Ensaio sobre as Minas. Presumably, at a later date,
copies of the manuscript must have been made. The route by which the
copy which was found in the District Archive arrived in the library of the
Conde da Barca, from where it was transferred to Braga, is not known.
The discovery of the manuscript was immediately communicated to
the members of the Seminário Nacional de História da Matemática where
the news was received with great pleasure. The participants in the
meeting of this group, which took place in the University of Minho in
1988, with the guest participant Professor Ubiratan d'Ámbrósio, through
the kindness of the staff of the District Archive of Braga, had the
opportunity to see the manuscript. From this time the publishing of the
manuscript was considered, however the process has taken longer than
expected. Firstly because at the time I had other urgent matters to attend
to, and secondly because the printing of the manuscript required the
collaboration of various people and Institutions.
From the first contacts with the manuscript I was sure that the
version which I had found was not the original. There were several errors
in the writing of words of french origin and in mathematical expressions
which could not have been made by José Anastácio da Cunha. On this
basis it was decided that a transcription, and not a facsimile edition, would
be made. Only the figures, tables and face-plate have been maintained as
found in the manuscript. Before proceeding to the transcription, it seems
pertinent to present the reader with some comments about this work.
1. General characteristics
José Anastácio da Cunha began the Ensaio sobre as Minas with
a section which he designates by Instrução, a sort of preface, in which he
affirms the inexistence of publications in portuguese on the subject and
refers to the work of various foreign authors, some of these he criticises,
others he praises. He makes firm recommendations to the readers of his
work that they be wary of being easily impressed by such words as
"demonstration", "evidence", and phrases such as "I will prove", etc.,
which often only serve to hide the errors and mask the ignorance of the
author. The attention of the reader is called to similar warnings given by
José Anastácio da Cunha on page 25 of his "Carta Físico-Mathemática" Nota38.
Still in the introductory Instrução he affirms that the work is
divided into three parts.
The first part consists of the theory of mines of José Anastácio da
Cunha; as he says "that which I can call entirely my own in this work".
However, as he comments, in order to make this Theory intelligible it
was necessary to precede it by a study of conic sections, which he
designates by "Preparação". In this section we see the preference of José
Anastácio da Cunha for a general approach rather than a specific
approach to particular examples. He describes a general treatment of
conic sections rather than dealing individually with each one of them.
In the section designated by Preparação he characterises the
conic sections in the manner chosen by Pappus Nota39, as sets of points such
that the ratio of distance from a fixed line (directrix) and a fixed point
(focus) is constant. He uses a system of orthogonal axes in which one of
the axes contains the major axis of the conic section, and the other the
tangent in one of the vertices. The directrix is then parallel to this latter
axis. Anastácio da Cunha designates by
p
and
q
the distances from the
vertice, which coincides with the origin, to the directrix and to the focus,
respectively. From this, it is immediately recognisable that the ratio