*History of Mathematics in Portugal*, which was published in 1934, the year after he died, Gomes Teixeira wrote about Monteiro da Rocha. We give below an English version of Teixeira's text:

Monteiro da Rocha was the subject of a conference at the Lisbon Academy of Sciences where we spoke about his life, the talk later being published in our

*Panegyrics and Conferences*. Let us reproduce here the essential part of this conference talk, with some modifications from time to time.

Monteiro da Rocha started to write mathematical research papers late in life. He was 48 years old when he introduced the first. Before that he had spent a great deal of time in theological studies, then in the organization of university studies and in the translation of books for the use of the students of the new Faculty of Mathematics. In addition he had entered the world of mathematics self-taught, provided only with a few texts. He had to find his own way. He came so late to the heights of this world, but not so late that he could not enrich it with his own precious works, as we shall see.

After the discovery of the law of universal gravitation, some astronomers, taking for themselves the geometric point of view, continued to study the movements of the stars by means of regularly continued observations; others, taking for themselves the mechanical point of view, occupied themselves with the deduction through mathematical analysis of the consequences of the application of the law of gravitation to the various planets, satellites and comets. Monteiro da Rocha is in the first group of astronomers we have just mentioned: he was a practical astronomer.

The first memoir he wrote was devoted to the determination of the parabolic orbits of comets, and was presented to the Academy of Sciences of Lisbon in 1782, shortly after its foundation. He was one of the founding members of the Academy.

It is well known that the first geometer who dealt with this problem was Isaac Newton, who gave two geometrical methods for solving it, which are masterpieces of invention, but lead to too approximate results. Other more exact methods were later given by Leonhard Euler, Johann Lambert and Joseph-Louis Lagrange in extremely remarkable memoirs, but these methods, being in fact models of analytical elegance and theoretical interest, are astronomically imperfect because of the difficulty of their application. The first practical process to solve this problem was published in 1787 by Heinrich Wilhelm Olbers.

This process, which has become a classic, does not differ essentially from that which is contained the Memoir of Monteiro da Rocha, which had been presented to the Academy of Sciences of Lisbon in 1782 before Olbers' work appeared, but whose publication had been delayed until 1799, the year in which the first volume of the collection of Memoirs of this Academy came out. This coincidence in the methods used by the two astronomers was noted by Professor Duarte Leite in an article about the work of the Portuguese astronomer published in the *Scientific Proceedings* of the Polytechnical Academy of Porto, where it is also shown that the two methods are linked to that of Lambert by the beautiful theorem discovered by Euler in 1743 which connects the time taken by a comet to describe an arc of a parabola to the length of its chord and to the vectors of the extreme points, the theorem used by Lambert, Olbers and Monteiro 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.

Let us note that the Portuguese astronomer applied his method to Halley's Comet.

Another important memoir of Monteiro da Rocha that we believe should be considered here is devoted principally to the prediction of eclipses of the Sun.

For this prediction easy graphical methods were employed in the eighteenth century which gave the times of the contact of the Sun and the Moon in a given place on the earth with an error less than one minute and analytical methods that gave these times with an error less than one second. The methods of Philippe de La Hire, Nicolas Louis de La Caille, Dionis du Séjour and Jérôme Lalande, the most used at that time, were of the latter case.

Séjour's method, the more analytical and the one that leads to the most accurate results, was the starting point of Monteiro da Rocha's investigations. But the method given by the Portuguese astronomer is simpler than that of the French astronomer, and this simplicity results from the way he considered the parallaxes, and from the way of referring the Sun and the Moon to the Equator, while Séjour refers one of these bodies to the Equator and the other to the Ecliptic.

But in matters of this nature, in order to judge the value of a method, it is not enough to recognize it theoretically, it is still necessary to have it applied. A Portuguese scholar, who was Director of the Astronomical Observatory of Coimbra, Dr Rodrigo Ribeiro de Sousa Pinto, who applied for many years the method of Monteiro da Rocha for calculating the eclipses that appear in the *Ephemerides* published by this Observatory, says in a booklet on the calculation of these Ephemerides, that the formulas given by Monteiro da Rocha are the simplest and most elegant of all the ones he knew.

Likewise, Jean Baptiste Delambre, who analyzed the work of Monteiro da Rocha and compared it to that of Séjour in a long report from the *Connaissance des temps* for 1807, says Monteiro da Rocha's formulas are simpler than those of Séjour and that the Portuguese astronomer, applying his method to the eclipses considered by the French astronomer, obtained the same results by much shorter routes.

The work of Monteiro da Rocha, which we have just discussed, was published by the astronomer in Supplements to the volumes of the Ephemerides of the Astronomical Observatory of the University of Coimbra, for the years 1804 and 1807 and was then linked to other astronomical works by the same author, and notes of the translator, published in 1808 in the French language by Manuel Pedro de Melo in a volume entitled *Mémoires sur l'Astronomie pratique*.

In the first five volumes of those Ephemerides are some other works of Monteiro da Rocha of great interest for the practice of this science. We will not mention them here in order not to tire the reader. We will add, however, that Monteiro da Rocha was the founder of these *Ephemerides* which have continued to be published up to the present time, and that the volumes that appeared in his time, very well made, as Delambre said, are remarkable not only for the value of the memoirs that our astronomer published in them, but also for containing some very useful Tables, which did not then appear in the analogous publications of other countries.

Monteiro da Rocha's purely mathematical works are less important than his astronomical works.

The first of these works deals with the problem of measuring the volume of liquid contained in a barrel, full or not, without emptying it; an industrially useful problem proposed by Johannes Kepler in his *Stereometry*.

This problem can only be solved approximately, and among the solutions that were given before Monteiro da Rocha considered it, the best is one that the Jesuit astronomer Esprit Pezenas published in the *Memoirs* of the Academy of Sciences of Paris.

The barrel can be considered as a geometrically indefinable solid of revolution, and to measure its capacity, it is replaced by a geometrically defined solid of revolution approximately equal in volume. Monteiro da Rocha enumerates and examines the several ways that this substitution has been made. In all of the solids replacing the barrel, the end sections and the middle section are in common with it. Our mathematician employs a new solid, which has in common with the barrel not only the sections mentioned, but also two new sections equidistant from those. The solution thus obtained is more closely approximate than those given previously, but less simple. For this reason Monteiro da Rocha, in order to facilitate its application, considered it necessary to calculate a table which makes it very practical, both in the case that one wants to measure the total capacity of the barrel and that of only a part of it.

Another noteworthy work on pure mathematics by Monteiro da Rocha is titled *Additions to Fontaine's rule to solve by approximation the problems that are reduced to quadratures*. It was published in 1797 in Volume II of the *Memoirs* of the Academy of Sciences of Lisbon.

In this work our mathematician once again revealed the fineness of his mind and his practical skill, giving a remarkable way of calculating the convergence of Alexis Fontaine's formula, clarifying it with well-chosen examples, and taking from his doctrine new rules, more convergent than that of Fontaine, for the solution of the problem considered. It is a memoir full of sound doctrine concerning the convergence of the expressions in the infinite limit, which is astonishing to see written in the days when these questions were treated with little care and which can still be read with profit today.

It is convenient to devote here a few words, as in parentheses, to a memoir which, despite its small merit, led Monteiro da Rocha to compose the one of which we have just spoken.

Analysis and geometry help each other, but there are questions in the domain of the latter science to which the mathematician, without consideration, throws himself on the wings of the first, and, by flying, seeks to find by complicated formulas results to which the latter leads by a simple path.

This observation applies to a memoir of Dr Coelho da Maia, professor at the University of Coimbra, which was awarded a prize by the Academy of Sciences of Lisbon and published in its collections (Volume 1, 1797) under the title *Method of approximation of Fontaine*.

We will not mention this Memoir for what it is worth, but because of a polemic that gave rise to it. Coelho da Maia obtained, by means of horrendous calculations, the extension of the formula of Fontaine we mentioned, full of series expansions lacking rigour, but added nothing remarkable about their convergence. Anastácio da Cunha gave two very simple geometric demonstrations of this formula, and he censured the Academy of Sciences not only for posing such a simple question, but also for having rewarded such a mediocre memoir. Monteiro da Rocha replied indirectly to the author of the topic in the previous Memoir put forward for the contest, defending the proposed theme which required the study of the conditions of convergence of the formula. The subject was thus well defended, but not the society that rewarded the Memoir, because the author of it had not sufficiently studied this difficult part of the question. As we have said, Monteiro da Rocha, who had the burden of seeing a prize awarded to someone who did not deserve it, studied it but, on the other hand, he composed another Memoir on the subject worthy of a prize.

After analyzing the scientific writings of Monteiro da Rocha, let us devote a few words to the general appreciation of his scientific work, reproducing what we said in our Panegyrics and Conferences:

"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."

We have finished our description and critique of the main works of Monteiro da Rocha; let us now say a few words about his life.

Monteiro da Rocha was born in 1734 in Canavezes, a village situated between Douro and Minho, and belonged to the Society of Jesus, in which he entered in Brazil, in Bahia, in 1752, but left in 1759, 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 were surrounded by military forces.

His talent was revealed first in his occupancy of the chair of Philosophy in the College of Baía and later in his occupancy of the chairs of Canon Law at the University of Coimbra.

Later, the Marquis of Pombal, informed of his value by D Francisco de Lemos, Bishop of Coimbra, called him to collaborate in the reform of the University, and, when this reform was put into execution, he was in charge of administering the Chair of Mechanics.

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 the composition of the Statutes of the University he proved himself as a great organiser, in teaching at that University he proved himself as a great teacher, in astronomical observations and calculations he proved himself as an astronomer and, in the Memoirs he published, revealed himself as a sage of high merit. It was a great example of intense activity, broad and fruitful.

He taught until 1804, when he retired, and died in Ribamar, near Lisbon, in 1819.