Born: 1744 in Lisbon, Portugal
Died: 1 January 1787 in Lisbon, Portugal

Anastácio da Cunha's father was Lorenzo da Cunha, who was a painter, and his mother was Jacinta Ignes. Da Cunha was educated in the town of his birth, studying at the Congregation of the Oratory in Lisbon. However, he did not learn much of mathematics and physics during his formal education and it was these topics which he studied on his own. It is certainly fair to say that in these areas he was self-educated, a fact which may have led to his later work being particularly innovative. During the period of his education Lisbon was hit by an earthquake on 1 November 1755 and two-thirds of the city was reduced to rubble. However rebuilding the old town soon created one of the most beautiful of European cities.

When da Cunha was nineteen years old, in 1763, he volunteered for military service and he became a lieutenant of artillery giving ten years' of service. He spent most of his years in the army at Valença do Minho. Subsequent events were greatly influenced by the politics of Portugal during the period. In 1750 Joseph I had been crowned king of Portugal. Joseph appointed Sebastiao de Carvalho marques de Pombal as a minister and soon de Pombal came to have such an influence over the running of Portuguese affairs that his powers became almost absolute. De Pombal put through a series of major reforms, and around 1758-59 he moved against the Jesuits and the Society of Jesus. Da Cunha supported these religious reforms by de Pombal and also the later reforms he put in place such as the reform of university education, the beginning of commercial education, the creation of trading companies, and the reorganisation of the army. Da Cunha [1]:-

... became known as a progressive thinker, talented poet, and author of an original memoir on ballistics.

As part of his university reforms de Pombal appointed da Cunha professor of geometry in 1773. His period in the Faculty of Mathematics at Coimbra University was, however, short. King Joseph I died on 24 February 1777 and suddenly de Pombal lost all his powers. Queen Maria I assumed the throne and she set political prisoners free. De Pombal was accused of having abused his powers, was found guilty by a judicial tribunal, and then banished from Lisbon. Da Cunha was arrested and imprisoned by the Inquisition. In October 1778 he was sentenced by the General Council of the Inquisition in Lisbon to three years in prison for being a follower of Voltaire and supporting heretical views. He was freed in 1781 but prison had ruined his health. Although he was appointed as Professor of Mathematics in the College of São Lucas and took up his mathematical research again, he died a few years later.

Da Cunha wrote a 21 part encyclopaedia of mathematics Principios Mathemáticos which he began to publish in parts from 1782 (it was published as a complete work in 1790) which contained a rigorous exposition of mathematics, in particular a rigorous exposition of the calculus. The book contained the elements of geometry and algebra in addition to the calculus. In all areas da Cunha paid unusual attention to methodology as well as rigour. Struik, reviewing [11] writes:-

His importance for the history of mathematics is due to his "Principios Mathemáticos", published posthumously in 1790 and translated into French by J M d'Abreu [Racle, Bordeaux, 1811]. This book is characterised by the attempts at rigor, especially in the calculus. Da Cunha develops a criterion for the convergence of a series which he uses to define the exponential function in a rather modern way, and from these develops the binomial series. His definition of the differential of a function y = f (x) anticipates that of Cauchy, and, written in our present notation, amounts to this:

if Dy = f (x+Dx) - f (x) can be represented in the form | Dy| = A Dx + e Dx, where A is independent of Dx and e → 0 when Dx → 0, then A Dx is called the differential of y = f (x).

In Principios Matemáticos da Cunha also gave a definition of the convergence of a series which is equivalent to Cauchy's convergence criterion. However, this was not realised until comparatively recently since most historians of mathematics studied the French translation of 1811 which is inaccurate at the crucial place where this definition is given.

There is a second publication of a mathematical work by da Cunha discussed by Bogolyubov in [3] where he studies:-

... the Portuguese mathematician J A da Cunha's work ["Essay on the principles of mechanics"(Spanish), London, 1807; Amsterdam 1808], which contains interesting approaches to the foundations of mechanics, similar in many respects to contemporary ones.

1. A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York 1970-1990). 
   http://www.encyclopedia.com/doc/1G2-2830904859.html

Articles:

1. R L S Baroni and I F Balieiro, The concept of infinitesimal in J A de Cunha's 'Mathematical principles' (Portuguese), in II Portuguese-Brazilian Conference on the History of Mathematics & II National Seminar on the History of Mathematics, São Paulo, 1997 (Rio Claro, 1997), 361-368.
2. A N Bogolyubov, The views of J A da Cunha in the domain of mechanics (Russian), in Studies in the history of mathematics 19 'Nauka' (Moscow, 1974), 177-187.
3. A L Duarte and J C e Silva, Some comments on the 'Mathematical principles' of José Anastácio da Cunha (Portuguese), Proceedings of the XIIth Portuguese-Spanish Conference on Mathematics II (Braga, 1987), 274-289.
4. A J Franco de Oliveira, Anastácio da Cunha and the Concept of Convergent Series, Archive for History of Exact Science 39 (1988), 1-12.
5. J F Queiró, José Anastácio da Cunha : A forgotten forerunner, The Mathematical Intelligencer 10 (1) (1988), 38-43.
6. A P Yushkevich, J A da Cunha et les fondements de l'analyse infinitésimale, Revue d'histoire des sciences et leur applications XXVI (1973), 3-22.
7. A P Yushkevich, C F Gauss et J A da Cunha (French), Rev. Histoire Sci. Appl. 31 (4) (1978), 327-332.
8. A P Yushkevich, C F Gauss and J A da Cunha (Russian), Istor.-Mat. Issled. 24 (1979), 186-190, 387-388.
9. A P Yushkevich, J A da Cunha and problems on the foundations of mathematical analysis (Russian), in Studies in the history of mathematics 18 'Nauka' (Moscow, 1973), 157-175, 337.
10. A P Yushkevich, J A da Cunha et les fondements de l'analyse infinitésimale, Rev. Histoire Sci. Appl. 26 (1) (1973), 3-22.