José Anastácio da Cunha (1744-1787)

Mathematician and poet, born in Lisbon on 11 May 1744. Of humble descent, his father, Lourenço da Cunha, was a painter of some fame, who spent a short period in Rome in order to improve his technique. His mother, Jacinta Inês was a servant and received elementary education. José Anastácio da Cunha was educated in Lisbon, at the Convento de Nossa Senhora das Necessidades, which belonged to the Congregation of the Oratorians. According to Anastácio da Cunha’s own testimony to the Inquisition, the Oratorians taught him Grammar, Rhetoric and Logic until the age of 19. As far as Physics and Mathematics were concerned, he taught himself.

In 1764, he was nominated Lieutenant of the Artillery Regiment of Oporto and was placed in the parade grounds of Valença do Minho. The Artillery Regiment of Oporto was made up of a majority of foreign officials, many of them protestant, who influenced Anastácio da Cunha. He is believed to have adhered to ideals such as tolerance, deism and rationalism, which integrated his scientific and poetic production.

Due to his friendship with the British officials, he learnt to speak English fluently which, together with his knowledge of other languages such as French, Latin, Greek and Italian, enabled him to translated authors like Voltaire, Pope, Otway, Horace, Rousseau, Holbach, Helvetius and others. It is thought that it was during this period that he joined the masonry, by influence of his foreign friends.

In 1769, at the request of Major Simon Frazer, he wrote a memoir on ballistics, titled Carta Fisico-Mathematica sobre a Tehoria da Polvora em geral e a determinação do melhor comprimento das peças em particular, where he pointed out some errors and lack of precision he had found in some works on artillery. In 1773, the Marquis of Pombal nominated him lector of Geometry at the University of Coimbra, following the reform of the University carried out in 1772. Anastácio da Cunha did not find a favourable environment to the development and application of his skills and after the death of D. José I in 1777, he was denounced to the Inquisition, imprisoned on 1 July 1778 and accused of being involved with the British protestants in Valença, of reading Voltaire, Rousseau, Hobbes and other dangerous authors and of corrupting the younger generations through his eloquence. He was considered guilty of the accusations and excommunicated, drawn away from this position at the University, from his titles and had his possessions confiscated. He was also condemned to participate in an act of faith and was shut up in the Congregation of the Oratorians in Lisbon, after which he was deported to the city of Évora for four years. He was also forbidden to return to Valença and to Coimbra. After two years in the Oratorians, his sentence was reduced to compulsory residence in the Oratorians.

Between 1778 and 1781, he was forced to dedicate himself to private teaching in order to survive and in 1783 he was hired by Superintendent Pina Manique for the Colégio de S. Lucas of the Casa Pia as a Mathematics teacher, having drawn up the pedagogical programme of the Casa Pia. While at the Casa Pia, he concluded his work Princípios Mathematicos. Around 1785-86, he lost his position at the Casa Pia for unknown reasons. He died on 1 January 1787.


Principios mathemáticos para instrucção dos alunos do Colégio de S. Lucas da Real Casa Pia do Castello de S. Jorge, finished in 1786 a few days before his death and published in 1790 in Lisbon. The writing of this work started in 1786, with a first version having been concluded in 1771. This version was presented to the Board of the Faculty of Mathematics, with no response having been obtained on its value. There is information that the first chapters printed were already circulating in Lisbon in 1782, however the final proofreading was only concluded on the eve of his death by Anastácio da Cunha himself. It is considered to be his most important work and although it was published as a didactic book, it started being written while Cunha was a lector in Coimbra. Its importance far exceeds many of the didactical aspects.

Carta Fisico-Mathematica sobre a Theoria da Polvora em geral e a determinação do melhor comprimento das peças em particular, drawn up in 1769, but only published in 1838, in Oporto.

Ensaio sobre os Principios de Mechanica, of which there are no known copies, but which is believed to have been published by Domingos de Sousa Coutinho in London in 1807. There is only one published version in the magazine O Instituto, of Coimbra, in 1856.

Ensaio sobre as Minas, an unpublished manuscript found in the District Archive of Braga and published in 1994 in a joint initiative of the District Archive of Braga and the Department of Mathematics of the University of Minho. This manuscript was aimed at the instruction of officials of the Artillery Regiment of Oporto, quartered in the Square of Valença do Minho.

As a poet, he did not publish anything while still alive; his poems were only published in 1839 by Inocêncio da Silva under the title: Composições poéticas do Doutor José Anastácio da Cunha. For this edition, Inocêncio da Silva was accused of “abusing the freedom of the press in religious matters” and the copies on sale were confiscated by the judicial power. Only in 1930 were his poems re-edited, along with others discovered in the meantime by Hernâni Cidade under the title A obra poética do Dr. José Anastácio da Cunha: com um estudo sobre o anglo-germanismo nos proto-românticos portugueses. His poetic works are considered as the precursors to the romanticism.

Main scientific contributions

Anastácio da Cunha’s most relevant work is the Principios Matemathicos, where he reveals remarkable creativity added to his great knowledge. Here he reformulates the Geometry of Euclides and was one of the first to note that the axiom of Euclides on the equality of all the right angles is not independent of the other axioms.

Cunha preceded Cauchy in the clear presentation of the notion of infinite and of infinitesimal quantity: “The variable that can always take on a value larger than any greatness that is proposed shall be called infinite; and the variable that can always take on a value smaller than any greatness that is proposed shall be called infinitesimal” Definition II, Book XV). Unfortunately his manuscript titled “Sobre o infinito matemático” is not known, as it was not published and was therefore lost. This notion, however, seems to correspond to the classical notion of infinite potential that is translated by the possibility of always going further without ever reaching a final element, however, and is translated in the work by the &c., which always indicates a finite expression, but which may continue “for as long as necessary” or “as much as one wanted”.

As far as the theory of the series is concerned, it precedes the works of Augustin Louis Cauchy (1789-1857) and Bernard Bolzano (1781-1848). The general principle of convergence of numerical series is taken up for the first time as a fundamental issue and claimed very precisely.

“Mathematicians call a convergent series one whose terms are determined in a similar manner, each by the number of preceding terms, in such a way that the series may continue forever, and finally it becomes indifferent whether to continue it or not, as the sum of all the terms one decided to add to those already written or indicated can be disregarded with no significant error: the last ones are represented by &c. after the first two, or three, or however one wishes: it is however necessary that the terms written show how they could continue the series, or that it be shown in another way”. (Definition I, Book IX).

His innovative contributions regarding differential calculus should also be highlighted, as for the first time he formulated a precise analytical definition of differential, thus called fluxion. Cunha further presents a definition of exponential that reveals his merit and originality for that time period.

Also worthy of mention is his "Ensaio sobre os princípios de Mechanica" where he defends that mechanics, although using mathematical methods, is physics and not simply mathematics. In the words of José Francisco Rodrigues, "Cultura e Ciência em Portugal no Século das Luzes", Colóquio/Ciências, 1, Feb. 1988, p. 84: "It is in Portuguese, once again to the dismay of José Anastácio da Cunha, that for the first time such a clear and up-to-date distinction is made of Rational Mechanics and Physical Mechanics and that the axiomatic separation of cinematics from dynamics is announced, a key aspect in the future development of the physical and mathematical sciences”.

Fernando Reis


José Anastácio da Cunha uma tragédia eterna
História das Matemáticas em Portugal - Anastácio da Cunha
José Anastácio da Cunha - Biography
Projecto Vercial - Anastácio da Cunha
A História da Matemática no Ensino da Matemática
João Filipe Queiró - Minas: da prática à teoria
João Filipe Queiró - José Anastácio da Cunha: Um Matemático a Recordar, 200 Anos Depois


ALBUQUERQUE, Luís de, "Cunha, José Anastácio da", in Dicionário de História de Portugal, Porto, Figueirinhas, 1981, vol. II, p. 256.
Subcomissão de Lisboa para a Exposição de Homenagem a José Anastácio da Cunha, José Anastácio da Cunha: 1744-1787: matemático e poeta, Catálogo da exposição, Lisboa, Bib. Nacional, 1987.
Departamento de Matemática da Faculdade de Ciências e Tecnologia da Univ. de Coimbra, Em homenagem a José Anastácio da Cunha, Coimbra, F.C.T.D.M., 1989.
FERRAZ, Maria de Lurdes, RODRIGUES, José Francisco, SARAIVA, Luís, coord. Anastácio da Cunha: 1744-1787: o matemático e o poeta, actas / Colóquio Internacional, Lisboa: Imprensa Nacional-Casa da Moeda, 1990.
FERRO, Joäo Pedro, introdução, transcrição e notas de, O processo de José Anastácio da Cunha na Inquisição de Coimbra, 1778, Lisboa, Palas, 1987.
QUEIRÓ, Joäo Filipe, "José Anastácio da Cunha: um matemático a recordar, 200 anos depois", Boletim da SPM, n.º 29, Setembro 1994, pp. 1-18.
QUEIRÓ, Joäo Filipe, "Minas: da prática à teoria", Boletim da SPM, n.º 30, Dezembro 1994, pp. 79-80.
RODRIGUES, José Francisco, "Cultura e ciência em Portugal no século das luzes: a obra matemática de José Anastácio da Cunha", Colóquio/Ciências, nº 1, Fev. 1988, pp. 74-86.
SILVA, Inocencio da, ARANHA, Brito, Diccionario Bibliographico Portuguez, Lisboa, Imprensa Nacional, T. IV, pp. 221-231.
Universidade de Évora, Bicentenário da morte de Anastácio da Cunha: matemático e poeta, Évora, Universidade, 1988.

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