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 :-
... 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  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: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.
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).
There is a second publication of a mathematical work by da Cunha discussed by Bogolyubov in  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.Many historians discuss the influence of da Cunha's remarkable, yet little known, work. Yushkevich, in , claims that da Cunha should rank with Bolzano, Cauchy, Abel and others for his contributions to the principles of the calculus. In  Yushkevich notes that da Cunha was not quite as unknown as had previously been thought since an anonymous, but unfavourable, review of Principios Matemáticos Ⓣ appeared in 1811 in the Gottingische gelehrte Anzeigen Ⓣ. In the same year, Gauss wrote a letter to Bessel in which he commented positively on da Cunha's definitions of the exponential and logarithmic functions. Despite this da Cunha's work was not widely known and, sadly, had little influence on the development of mathematics.
Article by: J J O'Connor and E F Robertson