Introduction 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 . 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 , 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 . 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 . 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 , 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" , 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" . 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 , 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 , 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" . 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 , 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 which he considers, is the inverse of the eccentricity of the conic section. He justifies, through general considerations, that, if p > q , the conic section is an ellipse, if p = q , a parabola and if p < q , a hyperbole, indicating in this latter case the existence of two branches of the curve. He expresses regret that he is unable to give detailed demonstrations, limiting his treatment to general considerations and writes: "but the brevity which I have imposed upon myself in this work does not permit a more detailed description and I reserve this for another publication". (I suppose that we find here a reference to the Princípios Mathemáticos in which a study of conic sections is also included). Thus Anastácio da Cunha assumes that his readers know at least the general properties of an ellipse, in which , and writes the equation in the greek manner, corresponding to that given by Apollonius . It is from this equation that he obtains, from the variation of p , the other equations of the conic sections. First of all he obtains the equation of a circumference, a particular case of an ellipse, considering that the directrix is at an infinite distance, which he indicates by ; then he brings the directrix closer to the vertice, to , obtaining the equation of a parabola; he follows this with a consideration of the situation in which , which allows him to obtain the equation of a hyperbole; finally, he considers the limiting case in which p = 0 , which allows him to obtain the equation corresponding to a cone degenerated into two coincident lines. The equations are skillfully transformed by allowing the ratio to vary; the cases which correspond to the limiting values, zero and infinity, are dealt with by operations corresponding to the determination of their limits. This section of the manuscript seems to me to be of sufficient merit that it warrants a special and separate study. We can detect here the capacity of the author to synthesize and also his skill as a "lucid organizer of Mathematics" which later were revealed throughout the Princípios Mathemáticos. This organisational capacity was appropriately stressed by João Filipe Queiró . It is only after completing this Preparação that Anastácio da Cunha begins the first part of his work which he calls Nova theórica das minas. He starts by describing the general principles of combustion of gunpowder and referring the reader to the explanations already given in Theorica da pólvora which we suppose are included in the "Carta Físico-Mathemática". We note the importance which Anastácio da Cunha attributes to experiments, in the principles which he assumes. An example may be found in the suggestion that the excavation produced in the surface by the explosion of a mine has a parabolic form, as confirmed by experiment. The hypothesis which Anastácio da Cunha presented has a geometric basis, bearing in mind the images of transformation of conic sections, presented in Preparação. More specifically, the process which involves the passage of a circumference, through an ellipse to a parabola. Anastácio da Cunha transfers this process into three dimensional space. He supposes that gunpowder, located below the surface, has a spherical shape and that it maintains this form at the beginning of combustion. After the combustion initiates, the sphere expands, expelling the earth which compresses it and extending to the upper surface through the path of least resistance and altering its shape to that of an ellipsoid. Finally, after the expulsion of all the earth above the mine the ellipsoid is transformed to a paraboloid. This is his conjecture which he presents as such. In this part of the work he also enunciates general principles which relate the dimensions of paraboloides with the corresponding charges of the mines and also with the velocity at which the earth is expelled by the explosion. The second part of Ensaio sobre as Minas is constituted by the mine theory of the mathematician John Muller, whose work José Anastácio da Cunha, faithfully translates from the original text in english, but with a re-ordering consistent with his personal view of the subject. The text referred to is "The Attac [sic] and Defence of Fortified Places" three editions of which have been published in London: the first edition in 1747, the second in 1757 and the third in 1770. After an extended period of work on the third edition, the most easily obtained, only recently have I been able to obtain a copy of the second edition, of 1757, which was that which José Anastácio da Cunha used. I am certain of this because occasionally, in a gesture in recognition of his sources, he mentions the corresponding pages of the original text. When he mentions page 218 of Muller''s text the text of the manuscript corresponds effectively to that of page 218 of Muller''s second edition, while in the first edition this text is to be found on page 233 and in the third edition on page 196. It was important to identify the correct edition used by José Anastácio da Cunha because only by directly comparing the text with that of the manuscript found in the District Archive of Braga was I able to correct some of the mistakes in the latter document. Reference to the second edition of Muller''s book also allows a doubtful point about the library of José Anastácio da Cunha to be clarified. The fifteenth item in the list of the books which were confiscated by the Inquisition is described as "6 volumes of Muller''s vorks [sic] in the english language". However, in the known bibliography of J. Muller there are no 6 volume works. What then was this collection of volumes? It happens that in the second edition mentioned above, there is inserted a list of publications, which were at the time recently published by the author. Six different books are referred to under the title: "A system of Mathematics, Fortification and Artillery. In 6 volumes". The discrimination of the content of the volumes is also given. The following list is transcribed in from the english text: "Vol. I. Algebra, Geometry, and Conic Sections. II. Trigonometry, Surveying, Levelling, Mensuration, Laws of motion, Mechanics, Projectiles, Gunnery, &c. Hydrostatics, Hydrawlics [sic], Pneumatics, and Theory of Pumps. III. Fortification, Regular and Irregular. IV. Practical Fortification in four Parts. V. Artillery in six parts. VI. Attack and Defence of Fortified Places, Mines, &c. Three Parts". I should add that, in the catalogue of the library of the Academia Militar in Lisbon, there exists this same set of six works of Muller cited here, under the designation: "Elements of Mathematics" (6 volumes). It seems quite probable that it is this set, which is referred to as the 15th item of the list of José Anastácio da Cunha''s books which were confiscated by the Inquisition. In the second part of Ensaio sobre as Minas it is worth noting that José Anastácio da Cunha disagrees with some of the affirmations of Muller, even though in the section designated by "Instrucção" he has presented this author as a Master. It is Anastácio da Cunha''s rigor and critical sense which prevents him from simply giving a direct translation, and leads him to insert personal notes of his disagreement, support or explanation of the original text. It seems pertinent to make an observation regarding the omission of the use of the symbol in the calculation of the volume of a truncated paraboloide. In the english text J. Muller uses the letter r to indicate and José Anastácio da Cunha follows faithfully the original text in this respect. As is well known, the symbol was used for the first time by an amateur mathematician William Jones (1675 - 1749), in 1706, but the definitive use of this symbol is due to Euler. Euler adopted the symbol in 1737 and since then its use has gradually become universal . José Anastácio da Cunha already used the symbol in Princípios Matemáticos, although with certain inconsistencies, as noted by Tiago de Oliveira . It is also worthwhile observing that the second part of Ensaio sobre as Minas is enriched by the introduction of the method of Daniel Bernoulli for the determination of the largest and smallest roots of a rational equation. José Anastácio da Cunha takes this section from another book by Muller, the "Traité Analytique des sections coniques, fluxions et fluentes", published in Paris, in 1760. This is a french version, significantly enlarged, of the english text . As is already known, this method of Daniel Bernoulli was published for the first time in the records of the Academy of Sciences of S. Petersburg, Tome III, in 1728 . Later the method was explained and developed by Euler in his "Introducio in analysin infinitorum" of 1748, translated into french in 1786, 1796 and 1835 under the title of "Introduction à L'Ánalyse Infinitésimale" . However it should be noted that José Anastácio da Cunha did not limit himself to a translation of the french text of Muller and to giving examples of applications in relation to mines. The author was careful in the rigor of his demonstrations, as has been emphasized by E. Giusti , and went beyond the translation of Muller. In a personal note he refers the reader to Newton''s book "Analisis per Quantitatem Series Fluxiones ac Differentias, cum enumeratio linearum tertie ordinis" published in London in 1711, where the justification of the calculations presented is to be found . It is curious to note that José Anastácio da Cunha refers to Newton as "Sir Newton", the same expression used in his Carta Físico-Mathemática. The third part of Ensaio sobre as Minas is also interesting and rich, though in a different way. This is the section which the author designates as the "Prática das Minas". To a great extent this continues as a translation of Muller''s book . However the order of the subjects is different and a preoccupation with pedagogical aspects is noted. He directs the text to beginners, which do not know, as he says, more than the four operations and the extraction of the square and cube root. He himself advises those who have not studied Algebra, to consider only the third part. In this manner the problems of the original text are dispersed and systematized into ten rules, and the application of each one of these rules is separated into various steps. For each rule there is a concrete example, for which the data are to be found in Muller''s book, as is also stressed in the notes inserted in the text. When the problem is more complex, as in the tenth rule, he constructs tables which immediately provide the answer from the data, and whose use he teaches carefully. The preoccupation with pedagogical aspects throughout the third part is clearly expressed by José Anastácio da Cunha who ends with the affirmation: "my intention is only to be useful to the young officers, my comrades". It is the same sense of service which also seems to have been present in the elaboration of Princípios Mathemáticos, as can be read in the statements made by Anastácio da Cunha in his interrogation during his trial: "to be useful to the public and to the state" . This is the attitude of someone who wishes to share with others the talents he has received more abundantly: wisdom, intelligence, discernment and even linguistic ability. I believe that the greatest tribute that can be paid to an author is the publication of his works. My work on this edition has as its primary objective to offer to scholars a text which I suppose to be very much closer to the original than the manuscript which I found; this is my way of paying homage to José Anastácio da Cunha. 2. Notes on the transcription: In the transcription the general criteria established by A. H. de Oliveira Marques, João José Alves Dias and Teresa Ferreira Rodrigues were used. Regarding the omission of symbols or words which are repeated at the beginning of various pages, the criteria of João Pedro Ferro were adopted. As in this text there are extensive translations from english and some from french, which are indicated in italic, we have opted to use a bold font to indicate the text which was underlined in the original manuscript. 3. Acknowledgments A large number of people and Institutions contributed in various ways to make the edition of this book possible. I begin with a reference to Doctor Norberto Cunha, a scholar and admirer of José Anastácio da Cunha, who I met at the beginning of my research in the Public Library of Braga. I am pleased to express my thanks to Doctor Norberto Cunha for the suggestions provided at the beginning of my search and at later stages in the process. I also thank Dr._Armando Malheiro da Silva for his efforts in verifying that the volume was indeed an unpublished work. I direct a few words of thanks to the students of History of Mathematics of the academic year 1987-88, especially to Dr._Victor Neves, Dr._David Paiva and Dr. Francisco Assis, who with their enthusiasm accompanied the beginning of my research. I should mention the unconditional support provided, from the beginning of this project, by the Department of Mathematics of the University of Minho, through the person who was at that time the director of the Department, Professor Raquel Valença. This initiative has always been given her stimulus and all the necessary financial support. Apart from this contribution Professor Raquel Valença has provided most useful information with regard to the method of Daniel Bernoulli, referred to by José Anastácio da Cunha. I record here my sincere gratitude. All the texts in english of J. Muller which I have used were obtained through Dr._Stella Mills, and it was with her help that the second edition of Muller''s work was identified as the one used by Anastácio da Cunha. I appreciate that this process took much effort and time. In addition, it was with Dr._Stella Mills that I discussed various aspects during the project, receiving from her many valuable suggestions; I am most grateful for her support. I wish to thank Dr._Luísa Lapa de Souza for the kindness she showed in obtaining the memoirs of Belidor from the Royal Academy of Sciences in Paris. Many other people have kindly helped me with their useful and opportune suggestions. I thank in particular Dr. Grattan-Guinness, with whom I spoke about the manuscript during his visit to the Department of Mathematics of the University of Minho, in April of 1990, and who provided many valuable suggestions. I extend my thanks to my colleagues from the University of Coimbra, Doctors António Leal Duarte, Jaime Carvalho e Silva and João Filipe Queiró, who have already produced interesting work on José Anastácio da Cunha ; their suggestions and support were extremely useful. To Professor Jean Dhombres I am also very grateful for having provided a rapid access to the Bibliothèque Nationale of Paris, the only viable option, given the little time which was available. I must also mention here the precious support that was given by the Estado Maior do Exército through the personal influence of General Guilherme Belchior Vieira, who I met at the 1.o Encontro Luso Brasileiro de História da Matemática, which took place in Coimbra between the 31.st of August and the 3.rd of September of 1993, and where I presented the Ensaio sobre as Minas. From the beginning, General Guilherme Belchior Vieira offered enthusiastic support which was provided in two complementary manners. On one hand he made available the books of the Military Academy and of the Army, including other elements of the Historical military archive, which were extraordinarily useful in the search for the authors mentioned by José Anastácio da Cunha. On the other hand he obtained financial support for the publication of the book through the Estado Maior do Exército. It is due to the funds available from the Mathematical Center of the Department of Mathematics of the University of Minho, from the Estado Maior do Exército and the District Archive of Braga, that the publication of this edition has been made possible. I would also like to thank the director of the District Archive of Braga, Dr._Maria Assunção Vasconcelos, for all the stimulus and support which this project has received. In particular I thank the computer science team of the District Archive of Braga, Dr._Clara Sofia Moreira, who carried out the final processing, and Professor José Nuno de Oliveira, as the person responsible for the software with which the text was processed. To Professor Michael Smith who, kindly, accepted the task of translating this introduction into english, I express my sincere thanks. I also thank the Department of Computer Science of the University of Minho for the use of the facilities provided in the final phase of the electronic edition of this book. Finally, I express my gratitude to all those who have supported me in the search, which has not yet yielded results, for other copies of the manuscript Ensaio sobre as Minas. In particular I would make special reference to the efforts of Dr._Fernanda Maria Campos (Portuguese National Library); Dr._Luís Cabral (Municipal Library of Oporto); and Dr._Maria Teresa Mendes (Main Library of the University of Coimbra).