**Monteiro da Rocha**'s parents were João Teixeira de Sousa and Maria Pereira who were farmers. He was the eldest of his parents four children, having younger brothers Germano Manuel Pereira, João Pereira and Goalter Pereira Rocha. Monteiro was born in Canaveses, a small town in the Vila Real district about 100 km east of Porto. We know nothing about the first eighteen years of his life up to 15 October 1752 when he joined the Society of Jesus, known as the Jesuits. Let us look briefly at the situation in Portugal at this time.

The King of Portugal, John V, died in 1750 and was succeeded by Joseph I who preferred enjoying himself to ruling the country so delegated power to Sebastião José de Carvalho e Melo, later to become the Marquis of Pombal. Melo sought to modernise Portugal and decided that the Jesuits, with their influence on science and education, were a major problem. He acted against the Jesuits making various accusations concerning the Treaty of Madrid which had been drawn up to settle disputes between Spain and Portugal over their territories in South America. The modern boundaries of Brazil were essentially set by the Treaty of Madrid, but the Jesuits with many missions in that part of the world resisted giving these up.

With this background, it is not too surprising that, on joining the Jesuits in 1752, da Rocha was sent to the Jesuit College of the Bay of All Saints in the city of São Salvador da Bahia in Brazil. It was at this College that da Rocha obtained a solid foundation to his education. Although we have only a little specific information about the topics studied by da Rocha at the College, we know that the standard education provided there was in three areas, Letters, Philosophy and Theology. Letters involved the study of Latin, Greek, History, geography, poetry and oratory. Philosophy involved the study of mathematics, physics and astronomy while Theology involved the study of sacred scriptures, theology and Hebrew. Da Rocha was taught science by the German Johannes Josephus Breuer (1718-1789). A college record concerning da Rocha stated (see for example [1]):-

On 28 June 1759 a royal charter was issued which prevented Jesuits in Portugal from teaching and forced the closure of all their colleges. This led to the expulsion of the Jesuits from Portugal and all their overseas territories. Gomes Teixeira writes [5]:-... his ability is good, judgement and understanding are sufficient; progresses well in Latin; in Philosophy, which he has only just begun, he makes good progress; not yet much experience; all goes well; melancholic.

Returning to the secular life, da Rocha remained in São Salvador where he applied for positions teaching Latin Grammar and Rhetoric. After being approved as a teacher he taught the children of the State Governor. He returned to Portugal in 1766 and, in March of the following year, he enrolled as a student at the University of Coimbra. After three years of study of Canon Law, he was awarded a bachelor's degree on 25 June 1770.Monteiro da Rocha ... left[the Jesuits]in1759, with other members of the same order, young people like him, at a time when the houses that the Society owned in that Brazilian city[São Salvador]were surrounded by military forces.

Melo, the Marquis of Pombal, initiated university reform in 1770 and created the Literary Board which was to write new Statutes for the University of Coimbra. Francisco de Lemos, who, along with Melo, was in charge of reforming the University of Coimbra, was well aware of Monteiro da Rocha's talents and invited him to participate in the process of drawing up the Statutes. On 16 August 1771, da Rocha travelled to Lisbon where, along with seven other members of the Board, he began the process of deciding on the reforms and drawing up the Statutes. Many years later he wrote a letter to Francisco de Lemos saying [1]:-

The reform of the University of Coimbra established scientific education there, created a Faculty of Mathematics and proposed the setting up of a University Observatory. The new Statutes were approved and put into effect on 28 August 1772. Before teaching began, Monteiro da Rocha, who had drawn up the mathematics syllabus as part of the new Statutes, was appointed as a professor in the chair of Physics and Mathematics of the new Faculty of Mathematics at Coimbra. Since one of the requirements was that professors should have doctorates, the University held a ceremony on Friday 9 October 1772 at which da Rocha, and other newly appointed staff, were awarded doctorates. He was given the honour of delivering the inaugural address to the new Faculty of Mathematics on the following day, Saturday 10 October 1772.It is now thirty years since I left for Lisbon on the suggestion of Your Excellency; a time that will never cease to be present vividly in my memory.

Our description of da Rocha's career to this point seems a little strange. In 1772 he is appointed as a professor of mathematics and physics (probably applied mathematics would be most accurate in today's terminology) yet he seems not to have studied much mathematics up to that point. Gomes Teixeira makes the following educated guess [5]:-

Da Rocha held the Chair of Applied Mathematics at the University of Coimbra from 1772 until 1783 when he was appointed to the Chair of Astronomy. He continued to hold this Chair until 1804 when, on reaching the age of seventy, he retired. While at Coimbra he also served as Dean of the Faculty of Mathematics and as Director of the University Observatory. He also served as Vice-Rector of the University from 1786 until he retired in 1804. In this role he set up certain regulations which played an important role in the development of scientific research. For example he drew up regulations for the Astronomical Observatory which required those working at the Observatory to make scientific foreign trips to update their knowledge.We do not know how he learnt mathematics; probably he studied Arithmetic, Elementary Geometry and the principles of Astronomy at the College of Baía, where he was educated, and then continued progress without a teacher in the study of the other branches of those sciences and in the improvement of the knowledge that he had received in that College. Men of unusual talent, such as he was, quickly leave behind their teachers and continue alone on their path.

In 1779 da Rocha was appointed as Principal of the Royal College of Nobles of the three Provinces of the North. On 16 January 1780, he was elected a member of the Academy of Sciences of Lisbon, and a few years later he became the Director of the Academy's Class of Exact Sciences. In 1798 he was elected a member of the Royal Maritime, Military and Geographical Society which had been founded on 30 June 1798, and in the following year he was elected a member of the Board of the General Directorate of Studies and Schools of the Kingdom of Portugal.

King John VI of Portugal came to the throne in 1816 on the death of his mother Maria I (she was the daughter of King Joseph). However, before that, because his mother Maria was mentally ill, he had served as Prince Regent from 1799. On 21 March 1800, da Rocha had been appointed as Councillor to the Prince Regent and, in recognition of his services, he received the Commendation of the Order of Christ of Portalegre. On 7 July 1801, a Royal Charter granted him the Commendation:-

In March of 1802, dressed in the robes of the Order of Christ, da Rocha was received into the Order:-Having regard, and having respect to his known merit; and to the great progress which has been made by his studies, to the teaching in the same Faculty by Doctor José Monteiro da Rocha, I would like to make him the recipient of the aforesaid vacant Commendation of Portalegre.

In 1804 da Rocha retired from the University of Coimbra when he was appointed as preceptor to the crown prince Peter (Pedro), the second son of King John VI. Peter at this time was six years old. From that time on Da Rocha lived on his estate at São José de Ribamar, near Lisbon.On2March1802, by decree of20October and24December1801, and Order of the Ministry and State Services,31August of the same year, D. João VI, Prince Regent of Portugal, informed the Rector of the College of Tomar of the Order of Christ that Doctor José Monteiro da Rocha ... wished and had devotion to serve ... and receive the habit of the said Order ... in the Church of this College.

Our description of da Rocha's life has shown his importance in the progress of science in Portugal through his administrative skills. However we must explain a little about his contributions to mathematics and astronomy. Fernando Figueiredo writes [9]:-

Da Rocha was very unlucky with his work on orbits of comets. He gave a simple method of calculating a parabolic orbit given three observations which he presented to the Academy of Sciences of Lisbon in 1782. However, at this stage the Academy was newly founded and was not publishing the papers which were read to it. They did get round to producing a journal and, in the first two volumes, published papers which had been presented to the Academy since its foundation in 1780. Da Rocha's paperMonteiro da Rocha was a key figure in eighteenth-century Portuguese science. His scientific work covered very diverse mathematical and astronomical domains. At a mathematical level his studies on integral and numerical calculus are the most relevant. He proposed a method to accelerate the convergence of numerical series, similar to the one that would be formulated by Lewis Richardson in the beginning of the twentieth century. Da Rocha was responsible too for the translations into Portuguese(carried out between1773and1775)of a set of fundamental French textbooks(Bézout, Marie and Bossut)to be used at lessons. However Monteiro da Rocha's main scientific work was in the field of astronomy. This work spans from theoretical to practical astronomy, the most significant elements being the following: a work on the determination of comets' orbits; several papers on the calculation of eclipses; a work on longitudes; astronomical tables of the Sun, Moon, and planets and charts of Jupiter's satellites; a work on the use of the small rhomboidal net; and a work on the use and calibration of the transit instrument.

*Determinação das órbitas dos Cometas*Ⓣ was only published in 1799, two years after Wilhelm Olbers published essentially the same method. Gomes Teixeira writes [5]:-

The following four astronomy papers by da Rocha,Monteiro da Rocha and Olbers must therefore figure together in the history of astronomy, as the first inventors of a practical method for the determination of parabolic orbits of comets.

*Cálculo dos Eclipses*Ⓣ (1803),

*Uso do retículo Rhomboidal*Ⓣ (1805),

*Uso do Instrumento de Passagens*Ⓣ (1805), and

*Exposição dos methodos particulares de que se faz uso no calculo destas Ephemerides*Ⓣ (1807) were translated into French and appeared as

*Mémoires sur l'Astronomie Pratique*Ⓣ by M J Monteiro da Rocha (Paris, 1808). The French titles are:

*Mémoire sur l'usage du Réticule Rhomboïdal*Ⓣ (pages 1-16);

*Mémoire sur l'usage de l'Instrument des Passages*Ⓣ (pages 17-29);

*Nouvelle méthode sur le calcul des Éclipses sujettes aux effets des parallaxes*Ⓣ (pages 30-120); and

*Exposition des méthodes particulières employées dans les calculs des Éphémérides de Coimbra*Ⓣ (pages 121-164).

The contributions we have just mentioned are in the area of astronomy but da Rocha also made contributions to mathematics. As noted above, he undertook the translation of a number of major French texts into Portuguese. For example *Elementos de Arithmetica* Ⓣ (1773), *Elementos de Trigonometria Plana* Ⓣ (1774) and *Tratado de Hidrodynamica* Ⓣ (1775) were all translations into Portuguese of texts by Étienne Bézout.

Interesting works on mathematics by da Rocha include *Additamentos à regra de M Fontaine para resolver por approximação os Problemas que se reduzem às Quadraturas* Ⓣ (1797) and *Solução Geral do problema de Kepler sobre a medição das Pipas e Tonéis* Ⓣ (1797). These two papers are discussed in detail by Gomes Teixeira in [5]. For Teixeira's text describing these papers and other information about da Rocha's work, see THIS LINK.

Let us end this biography by quoting from [27]:-

Monteiro da Rocha did not compete in an effective way in the progress of the world of mathematics. His talent was mainly of a practical nature. He did not create theories, but he solved more or less difficult problems. Whenever he had to solve an question, he pondered it deeply until he found the easiest solution and took his study to the last numerical details. Thus he dealt with the problem of parabolic orbits of comets and gave the first practical solution to this problem; he took up the problem of predicting eclipses and gave an easier method to solve it than the other processes employed in his time; he took care of the measure of casks and gave a solution that exceeds in its approach, and is not inferior in its simplicity, to the best than had been given previously; he dealt with Fontaine's quadrature rule and gave, for the first time, conditions that could be applied with confidence.

**Article by:** *J J O'Connor* and *E F Robertson*