**Author**: Mário Santiago de Carvalho**Part of**: Coimbra Between Science and Education (coord. by Mário Santiago de Carvalho)**Peer-Reviewed**: Yes**Published**: December 9th, 2022**DOI**: 10.5281/zenodo.7101873

The latest version of this entry may be cited as follows: Carvalho, Mário Santiago de, “Coimbra Jesuit Mathematicians. A Possible Reasoned Catalog”, *Conimbricenses.org Encyclopedia*, Mário Santiago de Carvalho, Simone Guidi (eds.), doi = “10.5281/zenodo.7101873”, URL = “https://www.conimbricenses.org/encyclopedia/coimbra-jesuit-mathematicians”, latest revision: December 9th, 2022.

Table of Contents

## Introduction

As the Portuguese historian of mathematics Henrique Leitão once wrote, “many of the essential works that could provide a sound basis for more analytical work on the development of the history of mathematics in Portugal are still not at our disposal” (Leitão 2004: 13). Unfortunately, the same may be said about the quite impressive number of very skilled and talented Jesuit mathematicians of Italian, French, German, Spanish, or Portuguese origin who “were central players in spreading mathematical culture by teaching and setting up schools, and in developing modern mathematics” (Gatto 2018). Despite this difficulty, any inventory of mathematicians working in the Portuguese assistancy may not ignore the teachers who spent some time of their life in Coimbra while waiting for the ship that would take them to the Eastern and Western missions. Smoothing some of these difficulties, Italian historian Ugo Baldini has done a great deal of work, allowing to advance one’s knowledge about those mathematicians and some of their books, and the present article is much in debt to his catalogs and several papers. Periodically, however, research must be resumed. Thanks to other recent contributions, notably by the Belgian historian Noël Golvers, the publication of the catalog below is on point. In fact, it is crucial to inspect non-European sources, as well as to go beyond the Portuguese and Roman Archives (Golvers 2017b, for Chinese Archives). Nevertheless, without the work of the three major historians mentioned so far, the present article would be nonexistent. The catalog below will show that the final word about the history of mathematics in Coimbra was not said yet. This is perhaps true of any sort of research, but I expect to show how some of the lacunae below may be seen at the image of Mendeleev’s periodic table. Allow me to explain this with just one case. Thanks to a testimony by Antoine Thomas (1644-1709) one knows that the chair of mathematical studies in Coimbra had been empty after the departure of Adam Aigenler (1633-1673) in the spring of 1673 (Golvers 2017b, and 2019b). There is no need, however, to jump to the conclusion that such vacancies were a rule, never filled in, or that mathematics was a regular void in the Jesuit Coimbra colleges. For instance, one knows now that the gap between Diego Seco (1575-1623) and Cristoforo Borri (1583-1632) is narrowed by Wenceslaus Pantaleão Kirwitzer’s (1588-1626) presence. Still, much remains to be done, and this type of work will never finish. Two caveats are nevertheless needed. First, a catalog related to Coimbra may not give an objective picture of mathematics in the Portuguese Jesuit colleges. It would be ideal to give a picture including all the colleges of the first historical phase of the Society of Jesus in Portugal (1542-1759), and to interpret all the data regarding them in a systematic way. Regrettably, this is impossible, given the state of the art. With his vast experience in this domain, Baldini acutely pointed out that “the Society’s presence in the country’s life was such that even minor document deposits in small towns could offer surprises” (Baldini 2008: 76). Second, I share the consensual idea that the Jesuit contribution to mathematics in Coimbra, or in Évora, for that matter, cannot be compared with the level of work actually done in the Lisbon college of Santo Antão/S. Anthony, notably its famous “Aula da Esfera” (class on the ‘Sphere’). Its historical unique role during almost one and a half century is unquestionable even in the wider context of the Iberian Peninsula (Baldini 2013: 74). However, I cannot accept Gatto’s (2018) assumption that to know the Portuguese assistancy’s contribution to mathematics one can rely exclusively on the names of the Jesuits who taught in the Lisbon college. For obvious reasons, the present catalog will stick to the mathematicians who had any relation with the two Jesuit colleges of Coimbra, the college of Jesus (from 1542 onward), and the college of Arts (from 1555 onward).

**Jesuit Mathematics Calendar and Outlook**

From their beginning, Jesuit Constitutions (1546) prescribed the teaching of mathematics along with the more philosophical matters (Lukács 1965: 283). For instance, in the Jesuit college of Padua the students were to study one year of logic and two and a half years of physics, metaphysics, mathematics, and ethics (Lukács 1965: 11). Despite the fact that mathematics became part of the curriculum, for the first time (1548), in the college of Messina, only in the Roman College (1553) did mathematics get the status of a permanent course within the *studia superiora *(Gatto 2018). Hieronymus Nadal (1507–1580), rector of the college of Messina, designed the first curriculum as follows: the first three books of Euclid’s *Elements*, practical arithmetic, *Sphaera, Astrolabium, *and planetary theories according to the Ptolemaic model. Nadal’s teaching in Messina was short-lived, as was the case in the six colleges of the German assistancy. The importance of mathematics for the formation of the Jesuits was explicitly recognized in Coimbra, and there is no reason to suppose that the general curriculum would then differ from the one in Messina. In 1557, i.e., only two years after Coimbra’s college had been entrusted to the Society of Jesus by king John III (1521-1557, in office), Luis Gonçalves da Câmara (1519-1575) requested general Diego Lainez (1558-1565, in office) to send a teacher of mathematics from Rome to Coimbra (Baldini 2004: 304). It seems that such request was never satisfied, and it cannot be known if the requested teacher was meant to teach in the college of Arts or in the college of Jesus — or even in both, likely. Another information is surely more important, and sound. Namely, the presence in Évora and Coimbra of some lecture notes similar to those from the Collegio Romano (Wallace 1990: 254) which evidence the trend eventually initiated by the *calculatores* and the *parisienses* dating from the 1570s and 1580s. Broadly, these manuscripts would be meaningless without already some kind of interest in mathematics in the Portuguese colleges, and there is some evidence of it already in the 50s. Those lecture notes could signal a Spanish input into the first generation of professors at the Roman college (Baldini 1992: 37), notably due to Francisco de Toledo’s (1532-1596) teacher, Domingos de Soto (1494-1560), whose history is also linked to that of the young Galileo (1564-1642), as discovered by historian William A. Wallace (1997). However, historians read the Portuguese situation of mathematics as backward and poor, to say the least. For instance, Baldini justifies “the scientific backwardness” of the Portuguese Province, as far as mathematics is concerned, both institutionally, and doctrinally. To be more precise, he mentions “the very small number of mathematics chairs and, consequently, (…) the equally low number of those trained in the discipline above an elementary level”, and also “the fact that mathematics was generally little esteemed and the disciplines considered more important – theology and philosophy – were cultivated by men on whose cultural mathematical methods and results exerted very little influence” (Baldini 2004: 311-12). This is true, and historian António Lopes (1993) recognizes that the first deficiency of the Portuguese Jesuits, in the teaching centers and from the very beginning, was their weak inclination towards mathematics and exact sciences. But I trust there is another reason for this situation, pointed out already by historian of science Luis Miguel Carolino (2000), and this has to do with the tense relationship between philosophers and mathematicians amid the Portuguese Province as well, and with the role played by the “philosophers’ party”, so to speak, in the relation between those two disciplines, considered the most important ones. Curiously, Christopher Clavius (1538-1612), who would later become the chief mathematician of the Jesuits in Rome, began his philosophical studies in Coimbra around the years preceding Câmara’s request. Between 1555 and 1560, Clavius took notice of the poor level of mathematics in the Coimbra Jesuit colleges, if any. Likely, this gave him the occasion to get acquainted with the work of one of the most important European mathematicians, Pedro Nunes (1502-1578), who taught at the university of Coimbra (1554/62). Even though Clavius was never a pupil of Nunes during his five years in Coimbra studying Arts, there are at least thirty-four references to Nunes in Clavius’ later published works (Knobloch 2004: 164). Oddly, in 1574, when Clavius moved to Messina, where he developed a partnership with Francesco Maurolico (1494–1675), and both established a pedagogical framework for mathematical studies, “which remains a masterpiece in its genre” (Gatto 2018), in Coimbra a professor of rhetoric, Cipriano Suárez (1524-1593), was being somehow regarded as a possible contributor to “mathematics”, as this discipline was understood by the “philosophers’ party”. I shall return to Clavius’ presence in Coimbra, as well as to Suárez, but for the moment let me continue to unfold the calendar regarding the entrance of mathematicians in the Portuguese Province, and in the city of Coimbra specially. Relations between the university dons and the Jesuit teachers were always tense and difficult. Established in the city of Coimbra in 1537 for good, the Portuguese university created a math chair one year later. This creation justifies the presence and teaching of Nunes, who eventually lectured it under two different institutional conditions, for according to the 1559 Statues the chair belonged already to the Arts course, and this decision was taken only four years after that course was entrusted to the Society of Jesus. But both the fact that we do not know of any existence of mathematics in the college of Arts, and the fact that mathematics was mentioned individually in an individualized form as a university chair only in 1653 (Queiró 1997: 771), are consistent with the information about what was happening in the Jesuit Coimbra colleges. In between, mathematics was taught at the university by André de Avelar (1546-1623), from 1592 to 1612 – in the medical course, to be precise (Fiolhais & Martins 2016: 713). It must be noted, however, that: (i) Cipriano Suárez was thought to act as a “mathematician”, as early as 1574; (ii) João Delgado (1553-1612) was teaching mathematics in 1586; and (iii) the Jesuit chair of mathematics was vacant in 1653. Furthermore, Baldini states that before 1590 mathematics was not taught publicly or by a special professor, and only some notions of elementary geometry and of the ‘sphere’ were imparted by the philosophy teacher before reading Aristotle’s *De Caelo* in the second or third class, a practice almost abandoned before 1600 (Baldini 1998). May it be inferred from this peculiar situation that the Jesuits could not attend mathematical lessons beyond the walls of their colleges? It is likely, but not impossible at all. Should one infer that the frequent vacancies shown by the catalog below were a sign that mathematics was totally ignored? Could it be a sign that the philosophers’ party was still dominating? As mentioned already, since Clavius never followed Nunes’ lessons, a negative answer to the two last questions might be given. Two things seem to point to a more complex situation: on the one hand, historian Golvers’ discoveries regarding the personal but international library of a fourteen years old Jesuit, Francisco Pereira de Lacerda (1637-1656), and the relations between Ferdinand Verbiest (1623-1688) and Francisco Rodriguez Cassão (1596–1666), a physician but also an active astronomer of the university of Coimbra; and on the other hand, Carolino’s pertinent interpretation of André de Almada’s (1570-1642) scientific academy in Coimbra (Carolino 2016) seem to point to a more complex situation. Jesuits like Cristoforo Borri, for instance, could indeed participate in the works of such an informal academy. It seems also highly unlikely that Lacerda’s mathematical library would pass unnoticed to “the Jesuits or other competent persons within the Coimbra context” in 1654 and 1655, and “the presence of (…) rather advanced mathematical books next to the ‘poor’ mathematical collection of the Jesuit college library shows that mathematics could, in particular conditions, be part of personal culture and practiced also by young novices, in the margin of the simultaneous official *curricula*.” (Golvers 2018: 73-74). It is not possible to get an exact picture of the kind of lessons partaken by the Jesuit colleges back then, and what this “personal culture” might have looked like, but it is more or less consensual that not only curricular lessons (meaning, courses for beginners), but also more advanced lessons were offered and attended in Coimbra. One problem remains, notwithstanding: how to discriminate between each one of the four possible sites where mathematics could be dealt with in the city of Coimbra, namely, the two Jesuit colleges, the university, or more or less informal sessions? As it will be seen ahead, Golvers reads one reference to the “nostros” by Verbiest as an indication that his “basic mathematics” courses were part of the “domestic teaching” reserved for Jesuit “*scholastici*” only, with exclusion of (lay) external students, as existed in the Coimbra college until 1692. To sum up, there are no indications whether, when, and to what extant mathematics was “taught” beyond the college walls, namely by Jesuits. It is likely that in general, with their own specificity – which was by no means a looser relation with Aristotle’s texts –, private lessons included more advanced or more specialized topics than those which were part of curricular courses. Given that these types of classes (as well as conferences, informal seminars and various other forms of teaching), which supplemented or replaced public courses, were not recorded in catalogs, there is no solid way to find out where and when they took place or who taught them. Eventually, the difference between classes depended upon the presence or lack of major experts, as well as the time they stayed in Coimbra. As will be seen ahead, the attribution to João Delgado of one manuscript on *Explanationes in sphaeram Ioannis de Sacrobosco… anno salutis 1587 (BGUC # 1184) *may be – assumed it is by Delgado – a testimony of those private lessons in mathematics at the college of Jesus, as an alternative to those given by the university of Coimbra (Gomes 2012: 237), as well as by the college of Arts. I mentioned the “time” the experts stayed in Coimbra because a huge number of known Coimbra teachers were foreign Jesuits waiting for the right wind to sail to the missions: they were known as “Indipetae”. This was also the case in all other colleges, as well as in the Évora university, but in Coimbra the last foreigners is Albert Eusebio Buchowski (1658-1717), around the second half of the 1690s. For better or worse, being the site of a higher, older, and prestigious Jesuit college, Coimbra welcomed those youngsters who aimed at going overseas. This had consequences, and the most important is contradictory in its terms. For if it could benefit from the mixing of several cultural traditions, and all sort of educations in mathematics – a kind of transculturality, as one would say nowadays –, it did not allow the newcomers to gain roots, thus preventing the creation of a true mathematical school, with a consolidate tradition and historical productivity. Nevertheless, the study of mathematics grew as its relevance for the missionary goals increase, in the beginning of the 17^{th} century in particular, when Matteo Ricci (1552-1610) persuaded Clavius and his superiors in Rome of the need of a sound formation in mathematics, especially envisaging Eastern audiences. And this had its consequences in Portugal, too. It is quite significant, says Lopes (1993), that among the twenty-six mathematics teachers who passed through these schools until 1680, seventeen were foreigners. In fact, Jesuits in Portugal always struggled to provide the colleges of Lisbon, Coimbra, Évora, Braga, Santarém, among others, with competent math teachers. It is usually said that only in 1692 did the teaching of mathematics become official and offered in a regular basis in Coimbra, as well as Évora. In 1685, general Charles de Noyelle (1682-1686, in office) admonished the provincial, insisting that the Portuguese Province should cultivate the study of mathematics, not only with the exercise of the “magisterium” in view, but above all to be able to send proficient mathematicians to China (see Lopes 1993). The year 1692 coincides with the presence of the Swiss John König’s (1639-1691) and the Portuguese Manuel do Amaral’s (1660-1689) in Coimbra. Also, the reason why 1692 may mark the beginning of a new period in the Jesuit history of mathematics in Portugal results from several instruments published by general Tirzo Gonzalez (1686-1705, in office). Between 1692 and 1702 he gives his personal appraisal of mathematics as pure scientific research, but this period of reformation continues with general Michelangelo Tamburini (1706-1730, in office) and his letter to the provincial of the Portuguese Province, on April 11, 1711 reinforcing mathematics (see Baldini 2004: 318-27; Rosendo 1996, and 1998). A later date could be highlighted in this calendar, 1754. This is the year of the publication of the *Elenchus Questionum, *or *List of Subject-Matters to be Taught *(Andrade 1973: 295-6). The *List *provided an updated orientation for the studies in the Portuguese Province, and mathematics appeared among the physical matters therein, based on the explanation of Peripatetic, Cartesian and Newtonian opinions. Broadly speaking, “restricted physics” should deal with the opinions of Aristotle, Descartes, Kepler, and Newton; the Torricelli tube, the Magdburg hemispheres, and the opinions of Descartes, Gassendi, and Borelli on several scientific topics should be discussed as well (Martins 1999: 20). This last scientific and philosophical horizon coincided more or less with the time of Inácio Monteiro’s (1724-1812) teaching, but other Jesuits, outside Coimbra, would witness it. The moment is, however, close to the time the Society of Jesus was expelled from Portugal (1759).

**Mathematics According to the ****Coimbra Jesuit Course**

Inácio Monteiro published in Portuguese language a book for mathematical classes, the *Compendio dos Elementos de Mathematica* (Monteiro 1754-56), later resumed in Latin in the first part of his *Philosophia Libera* (Monteiro 1766). The two versions show how mathematics was integrated in a Jesuit course of philosophy in the 18^{th} century. If, as it will be seen below, Pedro da Fonseca’s (1527-1599) outlook was definitively surpassed, mathematics remained a part of philosophy. A trend like this began very soon in the history of the Portuguese Province, namely with the publication of the *Coimbra Jesuit Course* (1592-1606). It must be noticed that this *Course *embodied the spirit of what I have been calling the “philosophers’ party”. Jesuit discussions of the scientific status of mathematics are scarce (see Baldini 1992), but one can find a brief summary of the matter presented by the *Coimbra Jesuit Course* (Carvalho 2020a: 279-81; see also Coxito 2005, and Mota 2007). Two Jesuits were the major contributors for the *Course*, Manuel de Góis (1543-1597), and Sebastião do Couto (1567-1639). Both agreed that mathematics belongs to the speculative or contemplative sciences and that, despite being one science in its genus, it can be divided into two species, a division justified by the kind of abstraction in question, either from sensible or from intelligible matter. Exact or pure mathematics (*syncera puraque*) is thus divided into geometry, which abstracts from sensible matter and studies the continuous quantity, and arithmetic, which abstracts from both sensible and intelligible matter, and studies the discrete quantity, that is, the number. Due to its particular type of abstraction, mathematics is considered an intermediate science. Music, optics (*perspectiva*) and astronomy, in contrast, are mixed (*mixtae*) mathematics, as they occupy an intermediate place between physiology and pure mathematics, and their object is, accordingly, partly mathematical, partly physical (*naturalis*). Mathematical beings depend upon the abstraction of matter, and can be considered either in connection with the nature of things (*rerum natura*), or mathematically. This division is of paramount importance, since it explains in what sense Aristotelian physics differs so much from Galilean or Newtonian science (Baldini 1992: 36, and 52). It is not in the least due to “epistemological obstacles”, such as primary experience and substantialism, two psychological features (Bachelard 1938) broadly shared by the authors of that time as well. According to Góis and Couto, mathematicians study the planes, solids, distances, points in bodies, and the fundamentals of mathematical procedures (*more mathematicorum*), namely, the elements, definitions, axioms, hypotheses and postulates. Among mathematical forms there are such items as the even, the odd, the straight, the curve, the number or the figure. In spite of being aware that many authors consider mathematics to be a true science, Couto will sustain the opposite view. For this, he ends up being attacked by true mathematicians of the Society, as it will be seen ahead. Couto reasons in line with an absolute Aristotelian sense, and says that mathematics cannot be truly a science because in their demonstrations mathematicians do not and cannot proceed by “a priori” causes (Carolino 2000: 79-80). The degree of evidence and certainty of mathematics in relation to physics is also discussed by Couto, and he admits that mathematics surpasses physics, in the degree of certainty. As regards demonstration, Couto continues, a mathematician differs from a metaphysician because the former considers only the formal cause, and the latter three, the final, the formal and the efficient. Mathematicians also differ from natural philosophers, who consider the four causes, according to Góis. As regards the order of learning, mathematics predates natural philosophy, and helps in the learning of other sciences. Despite its importance and relevance for the Society of Jesus, the doctrine just presented by the *Coimbra Jesuit Course* marks a step back in the epistemology of mathematics. Not the least because ignores the important discussion around the certainty of mathematics (*de certitudine mathematicarum*), which in Rome justified the opposition between Giuseppe Biancani (1566-1624) and Benet Perera (1536-1610; Baldini 1992: 51; see also Gatto 2018). Before it was resumed in Coimbra by João Delgado and Cristophoro Borri, the issue was the object of Clavius’ opposition to the paradigm championed by Pedro da Fonseca and Sebastião do Couto. A testimony by Borri reads like this: “in my readings and in the preamble to mathematics, I [sc. Borri] efficaciously refuted that which [Couto] had stated in his *Logica Conimbricense*, in which he attempted to fallaciously demonstrate that mathematics is not science” (see Carolino 2007: 191; Santos 1951: 145). Contemporary and eventually mixed with the work Clavius was doing in Rome, a different trend was also stemming from an emphasis on the “physical sciences, and particularly those parts that made use of mathematical reasoning” in Coimbra (Wallace 1995: 698). This “new direction” cannot be found in the *Coimbra Jesuit Course*, and that is why “natural philosophy in Portugal became less technical and mathematical from the end of the sixteenth century onward”, thus possibly explaining “why there is no conspicuous use of calculatory terminology in the famous *Cursus*.” (Wallace 1990: 254). Prior to the publication of the *Coimbra Jesuit Course*, it was possible to comment on Aristotle’s oeuvres, like the *De Generatione*, *Physica* VII, and the *Meteororum*, without leaving untouched several topics of mathematics, some of them related to Galileo’s early topics (Wallace 1995). If the strictly philosophical relevance of the *Coimbra Jesuit Course* prevented the continuity and the deepening of anything closer to a “mathematical physics”, it gave an argument for increasing the autonomy of mathematics as such – at least to the Jesuits most involved with the future of mathematics as such, meaning, to some of those mentioned in the catalog below. Mostly for institutional as well as more prosaic historical reasons or circumstances, those mathematicians could not attain the prestige that their fellow philosophers gained. Nonetheless, and according to Ugo Baldini, the former may have contributed to a new epoch, for “already during the years 1620-1640 the evidence provided by astronomical observations had persuaded some philosophers in S. Antão and Coimbra, informed by the mathematicians, to give up classic Aristotelian theses…” (Baldini 2004: 342-43).

**A Catalog of mathematicians in Coimbra**

The present catalog will not follow the historic division proposed by Baldini (2004: 301) for the history of mathematics – namely, prehistory (1590-1692), history (1692-1759), and post-history. His proposal pays attention to the fact that, in Rome, the study of mathematics only began with Clavius’ lessons, i.e. from 1562/3 onward. Local conditions, however, and the catalog’s first intention behind the present publication, may explain the different approach, more akin to prosopography than to epistemology.

**From Christoph Clavius (1556) to João Delgado (1586)**

Christoph Clavius introduced mathematics to the Roman college but, as mentioned before, it would be be wrong to think that the study of mathematics was simply nonexistent there prior to Clavius’ breakthrough. The same situation would likely be happening in Coimbra, as time went by. Indeed, historian Wallace (1995: 697) sees no difference between what was going on in Rome and in Coimbra during the 80’s, the decade that would finally see a true Jesuit mathematician teaching in Coimbra, João Delgado. The program designed by Clavius for the course surpassed the one that Baltasar de Torres (1518-1561) had taught between 1557 and 1560 in the Roman college: Euclid’s *Elements *(first six books), arithmetic, cosmography, astrology, planetary theory, the Alfonsine tables, perspective, and gnomonics (Lukács 1965: 179). Earlier, when Clavius was studying philosophy in Coimbra, Jerome Nadal and Pedro da Fonseca were exchanging views about philosophy and both agreed that a philosophy course should make room for mathematics. Gatto (2018) mentions Clavius’ saying to Bernardino Baldi (1533-1617) that his interest in mathematics had been aroused by Fonseca’s lessons on Aristotle’s *Second Analytics *and by the question: “why is the sum of the internal angles of a triangle always equal to two right angles?”. These lessons by Fonseca did not come to our days, but Clavius’ saying shows what kind of “mathematics” could be treated from a logical point of view. Moreover, faced with the need to design the scope of mathematics in a philosophical course Fonseca explicitly called for the contribution of a rhetoric, Cipriano Suárez, as already said here. Marcos Jorge (1525-1571) and Pedro Gómez (1534-1600) are the two other names mentioned by Fonseca (Gomes 2012: 164), the latter of which would even have the merit of being closer to Toledo. The “mathematical” task attributed to Suárez was very well defined by Fonseca, and differed considerably from the seminal word that allegedly explained Clavius’ “conversion” to mathematics. It should be “Aristotelian mathematics”, meaning, geometry, demonstration, cosmography, astrology, perspective (i.e. mostly, *De Caelo* and *Meteororum*), along with the theory of the planets following the fourth book of Sacrobosco’s *Sphaera*, and Plinius’ teachings about other issues related to meteors, like winds, water springs, and so on (Carvalho 2010: 28). Arguably, Fonseca’s notion of mathematics was not particularly narrow or outmoded, although it certainly clashed with Nunes’ teaching contents at the university, eventually diverged from Câmara’s best dreams, clashed with Clavius’ future perspective, and could have no relation at all to what would later be Delgado’s paradigm. A picturesque account by Christoph Grienberger (1564-1636), regarding a consultation of Clavius to Fonseca in connection with that Aristotelian issue is quite telling, as the latter was allegedly unable to satisfy the former’s curiosity. Baldini (2004: 312) situates this episode around 1558 and, just two years earlier, the Jesuit Iberian philosophers were raising objections to the significance and role of mathematics in the proposed version of the *Ratio Studiorum* (Carolino 2000: 76). Since Fonseca failed to give Clavius a satisfactory answer, the latter started studying Euclid’s *Elements *by himself and going beyond it. “Undoubtedly”, Gatto (2018) underlines, “in that period, Clavius also had to study astronomy, as, in 1557, he was able to predict the solar eclipse he described in detail in his *Commentarius *on Sacrobosco’s *Sphaera*.” According to Stephenson et al. (1996), Clavius’ first observation at Coimbra allowed him to describe “the effects of the fall of darkness which accompanied the full disappearance of the Sun”. Clavius’ manuals and his commentaries on classical mathematicians – especially Sacrobosco’s *Sphaera* and Euclid’s *Elements* – remained basic references in Jesuit schools (and several non-Jesuit schools) in Europe and other continents until after the mid-17th century. There is probably no comparable case in pre-contemporary history in terms of diffusion and influence (Baldini 2013: 74). Considering the clash between Clavius and Fonseca, and the real influence of the latter in the Portuguese assistancy (Carvalho 2020b), this first period may be considered as a pre-scientific (meaning “philosophical”) one as regards the epistemological status and teaching of mathematics. But if Clavius’ interest in astronomy dis somehow rise in Coimbra, Fonseca cannot be the sole explanation. From several manuscripts discovered by Wallace concerning “physical mathematics”, so to speak, three at least were in Coimbra, and belonged to the Jesuit milieu. They pertain, of course, to a later period, but it is not impossible that they signal the presence of an older interest. In fact, they represent three different cultural circumstances of the Portuguese Province. One of the manuscripts, tentatively attributed to Luis de Cerqueira (1552-1614) by Stegmüller (1959: 255), questions how the velocity of motion is to be judged “from an effect”, a “common practice in treatises composed at the university of Paris, where Soto had studied” (Wallace 1995: 681). If the manuscript is in fact his, one must remember that Cerqueira taught an entire philosophical course, and hence “mathematics”, in the Coimbra college of Arts during the academic year 1581/85. The second manuscript, dating 1588/89, reports Manuel de Lima’s (1554-1620) teaching at the northern college of Braga, not Coimbra. The last one, by Stephanus del Bufalo (dated 1596), a teacher of philosophy at Rome from 1595 to 1598, is just one of several extant manuscripts by Bufalo in Portugal reporting his Roman lessons. Nobody knows when they actually appeared in Portugal, but their presence might be related to the interest and to the practice of the kind of “physical mathematics” that distanced itself from Fonseca’s backward perspective. Broadly, they invoke and sometimes put together the two traditions already mentioned, viz., the *calculatores* and the *parisienses. *Bufalo deals with issues such as alteration, degrees of qualities, intension and remission of forms, and action and reaction. In the last of these he mentions the teaching of the *calculator* Richard Swineshead (*fl*. 1340/44) and contrasts it with those of Pietro Pomponazzi (1462-1525), Ludovico Buccaferri (1482-1545), Flaminio Nobili (1530-1590), and Jacopo Zabarella (1533-1589). Moreover, in his discussions of *gravitas* and *levitas* he cites the *doctores Parisienses* and compares their teachings with those of Girolamo Borro (1512-1592) and Francesco Buonamici (1533-1603), both of whom were among Galileo’s teachers at the university of Pisa” (Wallace 2006: 326). A comparison between these teachings and the doctrine actually published by Manuel de Góis in the *Physica* was never done, but thanks to Carvalho’s (2020a: 221, 261 e.g.) instrument of research, it is possible to have a quick access to some of the mathematical topics just mentioned, and confirm that the *calculator* Swineshead, as well as the *parisienses*, are mentioned by the *Jesuit Coimbra Course.* If until today no one has compared Góis’ mathematical allusions with Bufalo’s text, it is at least possible to recognize that the Roman manuscripts are almost contemporary with the preparatory work for the composition of the *Course*.

**From João Delgado (1586) to Cristoforo Borri (1626)**

Arguably, a new paradigm might have started with João Delgado, a former pupil of Clavius in Rome (1580/5). A specific token of the novelty he brings is connected with the fact that he introduced the discussion of an issue that Borri would push further, the *quaestio de certitudine mathematicarum. According to Carolino (2007), Delgado was one of the first Jesuit mathematicians to argue that “mathematics fit all the main features of Aristotelian science, and therefore it should be considered as a true science (…) within a strict Aristotelian framework”. *It is usually claimed that Delgado addressed this issue only in Lisbon, but this is an argument “a silentio”. In theory, nothing would prevent him to deal with it already in Coimbra, for he first lectured a full course on mathematics at the college of Jesus (1586/89), likely the right place to reinforce Clavius’ refutation of Couto’s idiosyncratic ideas, endorsed by Fonseca. Nevertheless, Clavius’ Academia was “fundamental in sustaining and guiding the teaching of mathematics in Lisbon for half a century (1590-1594, 1596-1597, 1605-1608), having exerted an enormous (if not exclusive) and long-lasting influence on the culture and practice of mathematics in Portugal.” Such an impact is not recognizable in Coimbra, but Delgado’s acquaintance with applied mathematics allowed him to take charge of the construction of the new College of Santo-Antão-o-Novo (1591). Delgado’s lessons and mathematical contributions can only be accessed through his students’ notes and the manuscripts he left (Leitão 2004: 747-8, Silva & Ferreira 2008: 103-08). However, based on Rodrigues (1931b), Carvalho (2021) recalls that Delgado’s complete course on mathematics, written in Portuguese language, did exist, and that his superiors welcomed its translation into Latin. Assuming it is by Delgado, the extant manuscript in Coimbra mentioned above, the *Explanationes in sphaeram Ioannis de Sacrobosco*, breaks with the custom of teaching the *De Sphaera *“before reading Aristotle’s *De coelo*, in the second or third class [of philosophy].” (Baldini 2004: 301). Therefore, together with Delgado’s presence in Coimbra, those lessons might signal the new paradigm of mathematics being taught in Coimbra. Although Delgado was unable to create a solid and lasting school, some of his disciples became math teachers and helped the Society to fulfill its mission (Baldini 2013: 76). Martim Soares (1561-?), João Pinto (1557-1613) and Diogo Seco (1575-1623) belong to this second generation of Delgado’s pupils who have taught in Coimbra, in 1591/93, 1593/94, and 1605/06 respectively (Baldini 2013: 72). It seems that none of Delgado’s most prominent students at Lisbon attained the Coimbra chair, namely Antonio Leitão (1580–1611), Francisco da Costa (1567–1604) and Francisco Machado (?–1659), but it is likely that the author of the astronomical and cosmographical *Tianwnlüe*, Manuel Dias (1574-1659), met João Pinto and Christopher Grienberger in his early years as a student of Jerónimo Barradas (1561-1646) in Coimbra (on Dias, see Leitão 2008). The teaching activities in Coimbra by Soares, Pinto and Seco were preceded by the English Richard Gibbons (1546/9-1632), who taught in 1590/92, and who had been a pupil of Clavius in Rome (1574/6) as well. There is testimony to the presence of the already mentioned German Grienberger – he too a pupil of Clavius in Rome (1591ff.) – in Coimbra in 1599, but Grienberger did spend more time teaching mathematics in Lisbon (1599/02) than in Coimbra. Grienberger was well acquainted with the French mathematician François Viète’s (1540–1603) new or symbolic algebra, which represents with alphabetic letters the coefficients and the unknowns of equations, and which is a precondition to understand René Descartes’s (1596-1650) *Géométrie.* To Grienberger is also attributed the diffusion of some of Galileo’s discoveries in Portugal (Fiolhais & Martins 2016: 716). With only one year in the activity of teaching, it is possible that the Bohemian Venceslau Pantaleão Kirwitzer was in Coimbra, teaching during the academic year of 1616/17 (Golvers 2010). To sum up, this second period may have known for the first time a different approach to mathematics, thanks to the input of Clavius and the massive presence of some of his Portuguese and foreign pupils, notably Delgado, who could have left their marks in some of his students. The lack of manuscripts belonging to these Portuguese Jesuits or related to their lessons prevents us from taking any other step, and it is also evident that one – if not the sole reason – why this new paradigm did not go further is because most of the teachers only remained in Coimbra for one year. I stress that Grienberbg’s testimony about Fonseca’s lack of expertise in mathematics indicates that the two contemporary paradigms were likely colliding. This clash could also explain why mathematics was so scarcely taught during this period. Indeed, between Seco’s and Kirwitzer’s time, the champion of the logic paradigm Couto managed to publish his volume on the *Dialectics* to integrate the *Coimbra Jesuit Course.*

**From Cristoforo Borri (1626) to Albert Buchowski (1692)**

The well-known scholar Italian Cristoforo Borri taught at the Coimbra Jesuit college in 1626/27, that is, almost twenty-years after the publication of the last volume of the *Coimbra Jesuit Course* by Couto. Meanwhile, the appearance of comets in Europe in 1618 caused an intellectual turmoil in Portugal, hence the amount of manuscripts and publications on these phenomena appearing in Coimbra as well (Carolino 2003: 167-98, and 239-87). Borri’s testimony about André de Almada’s academy at the university of Coimbra, “a community whose members shared information, letters and research interests”, must not be omitted (Carolino 2016: 126). In his capacity as a member of Almada’s academy, Borri mentions the instruments for astronomical purposes owned by this professor of theology, and praises him for being “the most distinguished amateur in mathematical sciences” (Carolino 2016: 125). It is worth noticing that Borri’s *Collecta Astronomica* describes the author’s observation of the moon in Coimbra with the telescope he borrowed from Almada, eventually after having read Tycho Brahe’s *Astronomica instauratae mechanica* (1602), according to Carvalho (1944; see also Albuquerque 1965). Carolino is adamant that Borri’s scientific program is linked to the academy: “…certainly instilled by the gregarious character of André de Almada and by the facilities that the would-be rector of the University of Coimbra provided, [Borri] engaged in a plan of astronomical observations. These observations covered the common topics of astronomy at the time, including celestial coordinates of several stars, the surface of the Moon, and Mars and its motion” (Carolino 2016: 127). To be more precise: “Basically, based upon astronomical observations and other theoretical reasoning, Borri put forward theories such as the tripartite division of the universe (distinguishing an airy heaven, *caelum aereum*; ethereal heaven, *caelum aethereum*; and the Empyrean heaven, *caelum empyreum*), the fluidity and corruptibility of heavens, and the celestial nature and location of comets.” (Carolino 2016: 128; Carolino 2008). Also, Borri’s name is associated with another mathematical issue. In a country whose early modern mathematical tradition has been described as being mainly confined to nautical needs, Borri’s explicit insistence on the theoretical question of the certainty of mathematics (*de certitudine mathematicarum*) “reemerged as a bone of contention, with important consequences at the cosmological level”, against Couto’s backward position (Carolino 2007). Likely, forty-three-year-old Borri expanded the route opened by Delgado in that respect, and his move made it possible to integrate mathematical data into the philosophical debate, particularly with regard to the new cosmology. This would be of significance, considering the missionary goal of the Society, and the importance these topics could have in the Eastern areas. The case of Antoine Thomas below, notably his references to China in his own textbook confirms this, for it clearly confirms the existence of eager *Indipetae *among the pupils in Coimbra who demanded a more suitable and deeper approach to mathematics, namely to astronomy. Golvers (2017a) interprets the emphasis put on astronomy by Thomas’s teachings as the reflection of the reputed importance of astronomy within the China Mission, since Matteo Ricci. It seems that Borri’s views received “the general approval of the whole Coimbra college, not only from the mathematicians, but also from the philosophers and the theologians” (Santos 1951: 145). This is a crucial note. Literally taken, the approval is a considerable argument for the idea that mathematicians were imposing themselves, that the recognition of their approach to mathematical science was increasing, or that the philosophers’ party (meaning, Fonseca, Couto and so on) was losing some of its influence. While Almada’s informal and conventional “scientific academy” was stimulating research on astronomy and stimulating links of patronage (Carolino 2016), the published *Coimbra Course* was being subject to constant revisions by several other Jesuit philosophers – Baltasar Teles (1596-1675), Francisco Soares Lusitano (1605-1659), António Cordeiro (1640-1722), and so on; one must not forget that as soon as it was published, the *Course* immediately received bad reviews inside and outside the country, but the reasons for that can only be hypothesized. Lastly, it was in Coimbra that Borri (1631) wrote his oeuvre, previously conceived in China, where he put forth the location and celestial nature of comets (Carolino 2007). Santos (1951: 136) mentions that Borri did publish his critical thesis (*conclusiones*) on Ptolomeus’, Copernicus’, and Brahe’s cosmological doctrines, which he had taught before a Coimbra audience. Contrary to the fate of the *Collecta Astronomica, *this scholar booklet did not survive. In this second period, Borri opened the door to many foreign mathematicians, and in some cases the teaching periods of some of them at Coimbra increased. For instance, the Irish Simon Fallon (c.1604-1642) taught in Coimbra for two periods, 1627/28, and 1630/33, which ranks him just after Delgado, but he too ends his life in St. Antão College, arguably teaching mathematics. Till Manuel do Amaral and Eusebio Buchovski, Delgado’s record was never beaten. The English John Rishton/Farrington (c.1615-1656), who had studied theology in Liège, might have succeeded Fallon in the chair, for he is said to have taught in Portugal Hebrew, Greek and mathematics between 1648/49, although it is not sure if he did it in Coimbra or in Lisbon. Nevertheless, in Liège Fallon seems to have kept in touch with Francis Line (1595-1675), “the best English Jesuit mathematician in the middle of the century” (Baldini 2004: 386). After a possible period of vacancy after Rishton’s teaching, Ferdinand Verbiest entered in Coimbra in the early summer of 1656. His eight months stay in Coimbra (1656/57), attested by only one autograph letter from Verbiest to Athanasius Kircher (1602-1680), dated 18th December 1656, might have been very fruitful, for when he was sent to Coimbra he had been already trained at the Jesuit college in Leuven during the winter semester of 1644-45, as a disciple of André Tacquet (1612–1660) – a young talented and already reputed Jesuit mathematician, author of one textbook on arithmetic, and another on elementary geometry, both reprinted many times and translated into Italian and English (Gatto 2018). To this historical period, installed sometime between Verbiest and Buchowski’s teaching periods, belong also the teaching titles – on mathematics, astronomy, geography, perhaps also on hydraulics – found in archaeological excavations in the site of the Coimbra colleges (Duarte et al. 2020). In the article Golvers wrote for this Encyclopedia, the Flemish-Belgian historian gives three crucial indications, two of them already touched upon before. They are all external but are connected with the Coimbra scholar atmosphere, namely: (i) the Flemish Ignatius Hartoghvelt (1628-1658) describes a severe material life in the college, negatively qualifies the condition of its library, and connects the philosophical perspective of the courses (which he dubs the “Spanish influence”) with a complete rejection of any Cartesian theme (on Hartoghvelt, see Golvers & Simões 2022); (ii) Francisco Pereira de Lacerda owned a library with advanced mathematical books of a rather up-to-date personal, mathematical culture published in Switzerland, Italy, Holland and England, along with a copy of Clavius’ algebra (on Lacerda, see Golvers 2018); (iii) Verbiest had access to the well-provided private library of a prominent member of the Coimbra university, Francisco Cassão (Golvers 2020c). Contrasting (iii) with (I), Golvers’ words are clear, but puzzling: with the same background as Verbiest, and having Jesuit libraries of the Flemish-Belgian Jesuit colleges as a standard, Hartoghvelt’s negative appraisal of the Coimbra library could confirm one year earlier the bad situation of the study of mathematics in Coimbra (see also Rodrigues 1950a: 213). However, it remains to be explained the exceptional case of (ii): how did a young Jesuit in his second year as a humanities student have a “personal” library? And why could Hartoghvelt not notice it? Hartoghvelt was at least not very well informed, a conjecture that may be confirmed by the mistake he made concerning the alleged “Spanish influence” on the philosophical course. Furthermore, Hartoghvelt could be mistaken about the college members’ alleged ignorance of Descartes. Another testimony by Golvers (2020b), this time regarding the French traveler, physician, and diplomat Balthasar de Monconys (1611-1665), former student of the Jesuits in Lyon and Salamanca, enables the following conjecture. Despite being in Coimbra for only three days (from 11 to 14 December 1645), Monconys visited not only the Benedictine mathematician, Pedro de Meneses (dates unknown), but also the Jesuits, among whom he found Rishton explaining Descartes’ theory on the tides and on gravity (Golvers 2020b: 489). What is significant here is that Monconys opposes, without explicitly admitting it, the mathematical updating of the Jesuits to the outdated mathematical knowledge of a Benedictine colleague (see Boletim 1879: 196). Indeed, “both the topic of tides and gravity were treated by Descartes in his *Principia Philosophiae*, of which the first, Latin edition appeared in Amsterdam in 1644, i.e. just one year before Monconys’s quotation from Rishton in Coimbra.” Golvers continues: “Apparently Harthogvelt did not know, or was not informed on the ideas Rishton already a decade earlier expressed on the spot. When Harthogvelt – following his self-presentation – once tried to introduce some questions from Descartes’s philosophy in the discourse of the local professors, this was not received well, because – in their opinion – Descartes was not ‘subtle’ enough ‘in philosophia naturali’” (Golvers 2020b: 490; see also Golvers 2020c: 160-62). Historian Joaquim de Carvalho (1939) had already detected the first echoes of Descartes in Portugal in Jean Gillot (ca. 1613–1657), and in João Pascasio Cosmander (1602– 1648), and historian Santos (1937) had discovered some Cartesian echoes in the work of Francisco Soares Lusitano, professor of philosophy in Coimbra between 1636 and 1654. So, despite what he himself says, Hartoghvelt was not the ‘first’ to introduce Cartesian themes (whether of philosophical or geometrical in nature) in Coimbra. Let me return to the presentation of the teachers. Apparently, the French Jacques Cocle (1628-1710) succeeded Verbiest. Cocle arrived in Portugal to go to Brazil as a missionary, but for one year he too taught mathematics in 1659/60. Having known of his expertise, the Provincial requested his assistance, at least for that short period. After Verbiest and Cocle, the presence of *Indipetae* continues. Educated mostly in Ingolstadt, as far as mathematics is concerned, the Bavarian Adam Aingenler was sent to Coimbra after April 11, 1672, when he was still in Lisbon. Once again, Aigenler’s presence in Coimbra certainly filled a vacancy, as in the preceding years no courses of mathematics had been offered, due to a lack of appropriate candidates (Golvers 2019b). As usual, Aigenler’s courses were too limited in time, spanned between the second half of 1672 and the beginning of 1673, two academic semesters to be precise. If his name is not mentioned in the *Catalogi Triennales *of 1649-1675, it is probably – Golvers explains – for the same reason as in the case of Verbiest some seventeen years earlier, namely because he taught a special class for Jesuits only («*nostros*»). The possibility of Aigenler having taught beyond Jesuit walls cannot be put aside, as the same historian later recognizes (Golvers 2020c: 165). On one hand, Aigenler is known for having prepared didactical instruments in Coimbra for his classes, notably one “rota astronomica”, a “wheel” built to demonstrate the mutual position of the main planets. On the other hand, it is now known that, besides classes of mathematics and Hebrew for the novices, Aigenler also offered open courses for other interested Jesuits, as well as special Sunday courses likely of an inclusive nature, i.e. to Jesuits, non-Jesuits, and secular students (Golvers 2020c: 165). Yet once again, after a five year period of vacancy, the French Antoine Thomas took the chair over, apparently only during one year, 1678/79. Together Aigenler and Thomas could not but leave some sort of mathematical fingerprint in Coimbra. The latter’s presence was originally meant to last one year, but in fact it lasted until March 1680, likely with no other extra-curricular occupations. Again, his presence had been requested by the Portuguese Provincial, Luis Álvares (1675-1678, in office), certainly motivated by Thomas’ mathematical reputation. While in Coimbra, he managed to write a book, the *Synopsis mathematica*, later published circa 1685, thanks to the financial support of Maria de Guadalupe, Duchess of Aveiro (1630-1715) when he was already in China (Golvers 2017a: 213). When Thomas arrived in Coimbra, he too found a poor (scholarly? mathematical?) library, confirming a complaint already heard from Hartoghvelt and Verbiest. At the end of March 1678, Thomas started his courses in Latin, since he had not yet had the occasion to learn Portuguese, as Golvers (2020a) observes. The Portuguese language in mathematics was of common use in Lisbon, but it is known that that feature was much older, and was in Coimbra as well. The situation with Delgado was mentioned before, but even Pedro Nunes defended the use of Portuguese in scientific texts, and contributed to it, particularly with the translations of the *Treaty of the Sphaere* as well as of Ptolemy’s *Geography* (Leitão 2002). Eventually, following Aingenler’s path, Thomas’ public consisted not only of Jesuit novices (*“scholastici*”) but also of lay students. Among the former, there were several potential candidates for the Chinese mission, such as José Soares (1656-1736) and probably Adam Weidenfeld (1645-1680). As a matter of fact, the *Synopsis* was explicitly written for the use of Jesuit candidates for the China mission, and describes in detail the minimum level of mathematical, and especially astronomical, knowledge and skills that were expected from them. We are rather well informed about Thomas’ mathematical courses, thanks to a series of short references in his correspondence of this period – mostly letters to his patroness, Duchess of Aveiro –, but especially by the *Synopsis*. Intended for all those wishing to be introduced (*tyrones*) to mathematics – that is, both Jesuits and non-Jesuits –, Thomas’ lessons offered “an average instruction in fifteen different mathematical disciplines, excluding only algebra. It was all expressed in a ‘simple’, that means non-specialized (Latin) discourse, and interspersed with references to more specialized books for ‘further readings’ for those who were interested in pursing their study…” (Golvers 2020c: 168). During his two year stay in Coimbra, Thomas also made a series of astronomical observations (on 15 July 1678 and continued during the years 1678-1680). The place of observation seems to have been* the lateral terrace or platform between each bell tower and the façade of the Jesuit Coimbra church (see Lobo 2019), but many more references to Coimbra were detected in the more than one thousand Latin pages that compose the **Synopsis (Golvers 2020c: 169). *The instruments Thomas used were “*tubi Belgici”*, telescopes consisting of tubes which he probably had brought from Douai, says Golvers, without taking into consideration, however, that already on July 18, 1627, Borri had used a telescope (*tubospicilum*) to watch the moon from the sky of Coimbra (Simões et al. 2020: 16). As a mathematical textbook, the *Synopsis* was still recommended to math beginners by three Jesuit mathematicians, two of them being Portuguese, Diogo Soares (1684-1748) and Inácio Monteiro (see Leitão 2007: 232), the third one being the Italian mathematician Niccolo Giampriamo (1686-1759). John König/João dos Reis might have been ordered to go to Coimbra to fill another vacancy, after Thomas’ departure. But, contrary to what was usually happening there, he taught mathematics during a larger period, 1682/86, and only to Jesuit students, according to Baldini (2004: 394). If there really was such restriction, by contrast with Aigenler’s and Thomas’ inclusive teaching experiences, the reason lying behind it is not known. König had a previous teaching experience in Dillingen (1675/6) but his presence in Coimbra was certainly an advantageous one. For instance, he asked permission to publish the thesis he presided to, he was authorized to publish a *Commentarium seu lucubratio mathematica*, which unfortunately never happened, and several other manuscripts of his, whose content is unknown to us, are also referred to in the correspondence. Overall, that amount of work could be a sequel to his earlier writings on cosmography and geography, published in Germany before 1680, subject matters that he could arguably have deepened during the rest of his whole life in Portugal. König gave lessons to the Portuguese José de Sequeira (c.1656-1690), and Manuel do Amaral, the former having received a negative appreciation in mathematics by the superiors, the latter having been teacher of mathematics at the Arts as well as at the Jesuit college of Coimbra during 1686/89, before going to the missions in Brazil. While attending his second year of theology, the Portuguese Francisco Barbosa (1658-?), replaced (*agit magistrum substitutum matthematices*) Amaral during 1689/90, but he had probably also been a student of König. Altogether, König, Amaral, and Barbosa were able to teach for nine years without interruptions. It seems that the German Philippe Bourel (1659-1709), even though he had no special training in mathematics, taught in 1691/92, but this period ends with the activity of the Bohemian Albert Buchowski, trained in Breslau (1691/2). There is no evidence that the new paradigm put forward in the second period came to a halt in the third. On the contrary, the period started with Borri insisting and enlarging Delgado’s view on the certainty of mathematics. Of course, one may only be cautious here, for again the lack of material information is considerable. There is, finally, a predominance of foreign Jesuits owning the chair of mathematics and mostly for very short periods of time. Surely, this must be related with the Roman and Portuguese superiors, the former more concerned with the Eastern missions, the latter seeking to ensure the level of mathematics in the Province. Whether the latter’s wish was ever materialized and to what extant is still not known.

**From Albert Buchowski (1692) to Inácio Martins (1725)**

Albert Buchowski was the last foreign Jesuit to teach mathematics in Coimbra, which he did for three years, 1692/95. All the subsequent teachers of mathematics were of Portuguese origin. Almost at its end, this last period witnessed the presence of two relevant mathematicians, Eusébio da Veiga (1718-1798) and Inácio Monteiro. Both ended their careers in Italy, but while the latter was sent from Coimbra to Santarém, the former was sent to Lisbon, becoming the last professor of the “Aula da Esfera” (Trigueiros 2015). Another feature of this period has to do with the higher number of years during which teachers were able to give their lessons or contributions. Some of them stand out for this particular reason, like Inácio Correia (3 years), Filipe Pereira (3 years), António Monteiro (3 years), Bernardo de Oliveira (3 years), Luis Gonzaga (4 years), Inácio Vieira (4 years), João de Faro (4 years), Inácio Monteiro (4 years), Paulo de Mesquita (6 years), and Inácio Martins (14 years). They also trained several “students” whose importance could be signaled, the following turning to be the most successful teachers: Filipe Pereira (with 6 known students), Bernardo de Oliveira (6 students), Inácio Monteiro (6 students), António Martins or Lourenço Rodrigues (7 students), Paulo de Mesquita (7 students), Eusébio da Veiga (7 students), and Inácio Martins (14 students). When Buchowski requested to go on the missions, the general fixed as a condition that he should spend “some years” in Portugal teaching mathematics (Baldini 2004: 401). For unknown reasons he sailed to Colombia and Ecuador instead of China, as he had initially requested, sailing not from Lisbon but from Seville. It is said that this third period finally saw the opening to the public of the home lessons or instead two alternative classes, one exclusively for the Jesuits, the other to anyone interested. On March 29, 1692, the general Gonzalez ordered Bernardo de Abreu (1665-1733), to whom I shall return, to be trained by Buchowski, given Abreu’s proficiency in mathematics, but the catalogs explicitly limited his lessons to Jesuits (*domi nostratibus explicat*). It is not known if this is to be read as if no lessons of mathematics were being offered at the college of Arts, or if Buchowski lectured two levels of mathematics; if this is the case, it is likely that the higher levels were offered at the college of Jesus, not at the Arts. Luis Gonzaga (1666-1747) succeeded Buchowski, and taught in Coimbra, 1695/98, just after the sixth recommendation by general Gonzalez to improve the level of mathematics. Indeed, Gonzaga’s teaching coincides with the arrival of Caspar Castner (1655-1709) and Giovanni Francesco Musarra (1649-1718), sent by the general to Portugal to contribute to the growth of mathematics. However, neither of them ever entered Coimbra, but the latter taught in Évora for a short period (1696/97), and exchanged letters with Gonzaga (see Baldini 2004: 407), where he describes the state of mathematics in the country after the 1692 reform, and points out many lingering problems. To be sure, in July 1723, Paulo de Mesquita (1693-1729) as well as Xavier Francisco (1695-1731) once again informed the general complaining that the Coimbra superiors were not following all the rules the students were supposed to follow (Baldini 2004: 432). Earlier, in a letter to the new general Tamburini, replied to on August 7, 1706, Inácio Vieira had complained about problems regarding the teaching of mathematics in Coimbra (Baldini 2004: 414). Gonzaga’s life is also related to Évora (1695/99) and Lisbon (1700/05), where he spent much more time teaching mathematics, as usual. Due to bad health, José Botelho (1661-1699) spent only one year, 1695/96, teaching mathematics, and Gonzaga took up the the chair again for a last year in Coimbra. Buchowski’s influence may be also detected in two students of his, António de Barros (1664-1708) and Bernardo de Abreu, but the former became a botanist in China, so that only the latter could teach in Coimbra, 1699/02 (?). The growing number of known Portuguese mathematics students reveals a new and last effort by the Province to send missionaries to the East. For instance, Francisco Cardoso (1673-1723), who taught in Coimbra in 1699/00, sailed to Asia, traveled across China, and became a geographer. Data on the early years of the 18^{th} century are uncertain. Between 1702/05, Inácio Correia (1673-?) gave lessons to António Martins (1677-1754), João Mendes (1676/78-1752), and Alexandre Duarte (1679-1741), but only the first one is recognized has having proficiency in mathematics, and his name was already mentioned in connection with the numerous “students” he might have taught. In addition, in the aftermath of the 1692 reform, eventually as a distant echo of Borri’s ideals (Carolino 2000: 84), general Gonzalez asked Inácio Correia to give António Martins a prize for his brilliant mathematical discussion. Inácio Vieira (1678?-1739) succeeded Martins in the academic years 1705/08. Historian Baldini (2004: 414) refers that records of Vieira’s lessons are extant, and that he taught mathematics for twelve years (Baldini 2004: 430), but only four of them in Coimbra, as suggested in the catalog below. To Inácio Vieira are linked the names of a few students who also played the role as teachers in Coimbra, such as Lourenço Rodrigues (1682-1743), who taught in 1711(?)/14, and Diogo Soares, in 1714/15, who taught the same subject matter later in Lisbon (Baldini 2004: 429-30). A manuscript by Soares, entitled *New Atlas* (*Novo Athlas Lusitano ou Theatro Universal do Mundo todo*, 1721), is extant (Leitão 2007: 232; Gomes 1960: 518), but Soares’ most relevant work is his cartographic work carried out in Brazil, from 1729 to his death (Leite 1947), first in collaboration with Domenico Capacci (1694-1736). Having in mind that Soares learned mathematics in Coimbra, underlining the importance of his cartographic work, historian Gomes makes two notes that are worth reflecting on: (i) Soares’ proficiency was common to other Jesuits of that period, and it (ii) it was probably a result of the matters taught in the Portuguese colleges of Coimbra, Lisbon, and Évora during the first half of the century (Gomes 1960:518). One of the thesis presided to by Soares, and published in Coimbra (1715), is registered by Gomes (1960: 518), and repeated by Baldini (2004: 732). Jerónimo de Carvalhal (1684-?) gave a two-year course, 1708/10, but his presence in Lisbon as a teacher is surer. Because of this lack of information Baldini (2004: 417) suggests that the possibility of Bartolomeu de Gusmão (1685-1724), best known for his aerostatic flight (*Passarola*), had some “connection with the college’s mathematicians” in that period. Little is known also about the years immediately following Carvalhal’s teaching, but the aforementioned António Martins and Lourenço Rodrigues, as well as Anton Stieff (1660-1729) and Joseph Spindler (1675-1730) are teaching candidates (see Baldini 2004: 416), even though the last two are more unlikely. There is more certainty with the names of a few students who could have taught in Coimbra for more than a decade, after 1710. Besides the already mentioned João Mendes, Jerónimo de Carvalhal, António Martins, and Lourenço Rodrigues, the names of João Pitta (1685/87-1741), Inácio da Silveira (dates unknown), José Fróis (1692-?), and Francisco Custódio Correia (1686-?) stand out. Ugo Baldini mentions that Silveira taught for two years, and that in 1710/11 he was in his third probation year (Baldini 2004: 421). After studying mathematics for five years, Filipe Pereira (1687-1757) taught the discipline, in 1716/19, for Jesuits only. As it is easy to see, so far, several references have been made in connection with the periodic opening and closing of Jesuit cloisters to outsiders. It is not clear why Jesuit doors seemed to close at times, but whatever the explanation might be, it must at least take into consideration the following: (i) Jesuit inner politics or situation in a particular time; (ii) Jesuit relations with the university at that time; (iii) the seeming invisibility of mathematics amid the university itself (Queiró 1997: 775). Unfortunately, less is known about the history of mathematics in the university of Coimbra than about that of the Jesuit colleges of the same city. Among Filipe Pereira’s students were the aforementioned João Pitta and José Fróis, as well as Jacinto de Almeida (1692-?), Manuel de Torres (1712-1748), Paulo de Mesquita, who succeeded Pereira in the chair in 1719/23, and Inácio Martins. In 1718, Mesquita’s *Theses Mathematicas*, defended under Pereira, were published in Coimbra (Baldini 2004: 733). Just like Pereira, Mesquita also had numerous students. In addition to Francisco, the following are known: Manuel da Silva (1697-1771), Inácio Vieira, João Mendes (1676/8-1752), Pedro Ferreira (1687-1744), Inácio Martins, and Francisco Ribeiro (1702-1761/6), who had Manuel de Carvalho (1710-1734) as his student. Carvalho defended the *Demostrationes mathematicae* published in Coimbra, in 1733 (Baldini 2004: 739). Allow me to adduce the case of Pedro Ferreira just to show the difficulties of working with the sources available: according to Baldini (2004: 430), who limited himself to Roman catalogs, Ferreira, a Jesuit from 1703, studied mathematics for two years (sometime between 1711 and 1717), and taught it for one year, according to one catalog. However, according to later catalogs, Pedro Ferreira was credited with a three-year teaching (1718/19, 1726, and 1743).

**From Inácio Martins (1725) to José Teixeira (1759)**

Inácio Martins (1693-1738) is a case of his own because the fourteen years he spent in Coimbra were dedicated to mathematics, first as a student, then as a teacher. The obituary by Manuel da Veiga (1566-1647) registered in Carlos Sommervogel’s *Bibliothèque* (VIII, 530 B), a manuscript found in Coimbra and written in Portuguese, entitled *Relação da morte do P. Ignacio Martins com testemunhos que delle e de suas cousas se deram*, cannot be related to our mathematician but rather to the famous homonym Jesuit who lived in 1531-1598, contrary to what Baldini (2004: 438) asserts. Martins assumed the chair in 1723/37, after having defended the *Theses cosmographicas Terraqueae* already in 1719. In 1726 is published in Coimbra the *Labyrinthum Mathematicum*, a thesis discussed by him with José Teixeira followed two-years latter by João Evangelista’s *Conclusiones Astronomicas*, and the *Fulgentissimos Mathematicae Splendores*. Later in 1736, and still in Coimbra, were published the *Conclusiones Mathematicas de Astronomia* *Sphaera, Geographia & Optica, *defended by Inácio Coutinho but presided to by Martins (Baldini 2004: 734ff). In this way, it would be impossible to downplay Martins’ considerable influence among several students, although our knowledge of their increasing number comes only from the data that have survived and does not mean by itself that by then classes were knowing more graduations (*matheseos deputati*). Among those linked with Martins, are also Francisco Ribeiro, Xavier Francisco, Bento Peixoto (1702-?), João Inácio (1704-1749/53), José Carneiro (1694-1727), Manuel de Torres (dates unknown), Pedro Ferreira, Manuel da Silva (dates unknown), Luís da Silva (1703-?), Paulo Ferreira (dates unknown), Domingos José (1707-1774), José Leonardo (1705-?), José de Mesquita (1705-), Luís da Silva (1703-?), João de Loureiro (1717-1791), Felix da Rocha (1713-1781), Inácio Coutinho (1710-1738), Caetano Moniz (1711-?), and finally Francisco de Albuquerque (1713-?). Of the names just mentioned, it must be said that Felix da Rocha became president of the Astronomical Tribunal in Beijing (Baldini 2004: 443). Marcelo Leitão (dates unknown) might have succeeded Inácio Martins, in 1737/39, but the catalogs do not register an impressive number of students’ new names, like those before. Nevertheless, in 1739/40, 1741/42, and again in 1743/44, the figure of João de Faro (1705-1790) emerges, and two of his lessons on Aristotle’s *De Caelo* and on *Physica particularis* are extant (Baldini 2004: 451). Furthermore, if he was still teaching mathematics in 1743/44, for one year he gathered it with the teaching of the course of Arts inaugurated in 1742. Faro is mentioned as teacher of Moniz, for instance, but Manuel de Sousa (dates unknown) and Inácio Soares (1712-?) were also among his math students. It is just a curiosity, but it could be mentioned that the Lisbon newspaper *A Gazeta*, in its issue of June 13, 1754, announced the philosophical and mathematical scope of Inácio Soares’s *Philosophia Vniversa Eclectica*, in northern Braga. The name of António Soares (dates unknown) appeared for only one year, 1740/41, in the chair of Coimbra. He would be succeeded by an impressive teacher, Bernardo de Oliveira (1714-1796), who began teaching mathematics in 1742/43, and who would resume the chair left by Inácio Monteiro, due to the latter’s departure to the college of Santarém. The period of Portuguese mathematical presence in the East is now over, and the period of an eventual influence in Italy begins, due to the Marquis of Pombal’s (1756-1777, in office) pogrom. Much less in known about this last period. After Oliveira’s deportation to Italy, he continued to teach mathematics in Ferrara (1754/61), first residing at the college of Cotignola in 1770, and later living in Cento and Bagnacavallo, where he died. When Oliveira was send back to Coimbra for the second time, he might have taken with him José Teixeira (1729-1799), his fellow mathematician sent to Coimbra to become a theology student. Either Oliveira or Teixeira were the last teachers of mathematics at the Coimbra college, and both cooperated in the observation of the eclipse in Coimbra on July 30, 1757 (Sommervogel V 1896). While in Coimbra, Oliveira taught Eleuterio de Sousa (1717-1768) and João de Sá (1720-1789), but the names of José Leonardo (1705-1772), João de Faro again, Manuel de Sousa and Inácio Soares reappear in 1742/43, i.e. during the academic year credited to Oliveira. António Monteiro (1715-1773) held the chair of mathematics in Coimbra for three years, 1744/47, and later that of rhetoric (four years), before leaving Coimbra to Lisbon, at whose college he lectured in philosophy. His presence is relevant, for during his three-year period in Coimbra Monteiro taught Eusébio da Veiga, to whom I shall return, and Victorino Teles (1718-1798). After being deported to Italy, the latter became a renowned astronomer, and published the *Effemeridi Romane calcolate pel mezzo di vero del meridiano di Roma*, the first volume of which gives a series of meteorological observations made in Rome by the author and his team (Baldini 2004: 452). But Veiga was not António Monteiro’s sole student. He trained (*matheseos deputati*) several other Jesuits in Coimbra, like Manuel dos Santos (1719-1768), Inácio de Carvalho (1719-1793), José de Espinha (1722-1788), João de Faro, and Bernardino Correia (1708-1798), who in 1746 were both likely the subject of a decree issued by the Rector of the Coimbra college reacting against an excessive amount of liberty “perceived as threatening the original aim of philosophy teaching in the Society: namely that of supporting theological discourse” but at the same time “permitting the teaching of many new doctrines and works” (Baldini 2004: 378) as those established by the *List or Elenchus Quaestionum* referred above. Two of these Jesuits must be evoked now. Inácio de Carvalho was one of the names criticized by the famous anti-Jesuit Luis Antonio Verney (1713-1792), and José de Espinha became President of the Astronomical Tribunal of Beijing. The swan song is heard as the first years of Eusébio da Veiga’s entry into the Society of Jesus coincide with an effort on the part of the Ignatians to revive the studies of mathematics in the province. Considered “an astronomer in a strict technical sense, with qualified mathematical knowledge” (Baldini 2004: 452), Veiga was one of the Jesuits who most promoted the science of mathematics in Portugal (Rodrigues 1950b: 420ff), and in Italy in its full extent. He started lecturing mathematics in Coimbra, 1747/49, according to Baldini (2004), or 1743/49, according to Trigueiros (2015), and finished his career in Lisbon (1750/59), with the full and explicit support of general Tamburini. Such type of divergences occurs frequently among historians interpreting historic data differently. I believe, and this is a rule of interpretation I always stick to in these matters, that one of the following may explain the divergence of the two possible teaching periods mentioned: the presence of experts in mathematics in the catalogs, apparently sharing academic years or teaching responsibilities with a tutor (*matheseos deputat/magistri mathematici et revisores*), is an indication that either they really acted as substitute professors (for a short or a long period of time), teaching in other classes under supervision, or were just advanced students periodically replacing a teacher in his absences. Historian Trigueiros authored the most up-to-date biography of Veiga, where he establishes the first day of his novitiate in Coimbra in September 21, 1731, and says that in 1743 he combined already the teaching of mathematics (*magistri mathematici et revisores*) with that of humanities, with 1748 being the year he taught mathematics while he was a third-year student of theology (Trigueiros 2015: 156). Veiga presents himself (1758) as “mathematics public professor” (*Professor publico que foi de Mathematicas, e agora Filosofia no Real Colegio dos Estudos geraes de Santo Antão na Cidade de Lisboa*), which has been interpreted as a reference to his classes in Lisbon, not in Coimbra, which were, some interpreters say, of a private nature. However, there is no reason to assume it, since at least from the time of Inácio Monteiro, the Coimbra King’s college of Arts (*Regale Artium Collegium Conimbricensis*), run by the Jesuits, would be a pragmatic alternative for those affected by the frequent vacancies in the mathematics chair at the university, the last of them occurring between 1726 and 1772, according to historian Queiró (1997: 774). Veiga published the *Eclipsis partialis lunae observata Ulyssipone die 27 Martii anno 1755*, and the *Planetario Lusitano*, “the first modern work of its kind by a Portuguese” (Baldini 2004: 463). While in Coimbra, Veiga gave lessons to Francisco Taveira (1725-1770), Agostinho de Araújo (1723/25-1756), Filipe de Gamboa (1722-?), Manuel Mendes (1716-1782), João Inácio, João Pinheiro (dates unknown), José Bernardo de Almeida (1729-?), and André Rodrigues (1728-?), or had at least the assistance of some of these younger Jesuits. Rodrigues and Almeida are interesting cases because they, too, ended their career as the last Portuguese in the Presidency of the Astronomical Tribunal in Beijing. We only know that Filipe de Gamboa taught in Coimbra, 1751/52, thanks to his 1752 letter to general Visconti informing that his students had abandoned his classes, and that an advanced student of him, Cristóvão Ribeiro (1725-1795), was not able to discuss his theses in public (Baldini 2004: 458), the reason for these tribulations being ignored. Baldini also reports that on January 2, 1753, Gamboa received an answer from the general related to his transference to Évora “to refine his competences”, which the Italian historian interprets as a lower level of the teaching of mathematics in Coimbra (Baldini 2004: 459). This is a curious interpretation, for this last period gave a considerable number of teachers to Lisbon, where existed a more demanding chair. As is usually the case, since we do not know the date of Gamboa’s death, it is possible that he left the Society. Considering the relevance of his published work, as well as its recognition by European circles, Inácio Monteiro might be the most important mathematician of the Portuguese province, a rank he might share with Eusébio da Veiga. Nevertheless, it is worth noting that Monteiro’s and Veiga’s mathematical careers were viewed by Baldini (2004: 362-3) as a sound argument to reinforce the idea that Portugal gave no special recognition to mathematics, for both Jesuits, despite being the two most relevant mathematicians of their time, could not finish their career teaching that discipline. Monteiro taught in Coimbra between 1753/56, and the appreciation of his oeuvre, the *Compendio*, and the first part of the *Philosophia Libera* (see Monteiro 1754 and 1766, respectively) will be the subject of an entry in this encyclopedia. There is, therefore, no need to go deeper into his mathematical contributions. When he was sent to Coimbra to study theology, Monteiro complained to father general Ignacio Visconti (1751-1755, in office) about not being appointed teacher of mathematics (*mathematicae/mathesis magisterio*), which indicates a common custom. António de Brito (1726-?), António do Vale (1726-?), João da Cunha (1716-1790), Tomás de Campos (?-1766), Jerónimo Mendes (1721-?), and Francisco Gião (1699-1761) were among Monteiro’s students. Bernardo de Oliveira and José Teixeira are the last mathematicians connected to Coimbra: the former was already mentioned, and the latter, being a student of theology between 1755 and 1756, was still in Coimbra in the year 1759, which is the last year registered by Baldini’s Table, the historical source I am much indebted to. The same historian refers that the Portuguese catalogs in Rome mention Tomás de Campos teaching mathematics for five years, and Jerónimo Mendes holding the chair in 1749/53 (Baldini 2004: 462), which surely happened not in Coimbra but in Évora. At the university of Évora, before João de Borja (1711-1767), Tomás de Campos sustained a thesis dealing with the following subject matters: speculative geometry, Archimedes’ selected theorems, the sphere, geography, practical geometry, rectilinear trigonometry, astronomy, mechanics, statics, hydrostatics, optics, catoptric, dioptrics, gnomics and the circles revolutions (Gomes 1960: 572). Just like Bernardo de Oliveira, José Teixeira was the other name mentioned by Veiga as his Coimbra sources regarding the observations collected by the *Planetario Lusitano*.

**Final Remarks**

The list below gives the broadest and most complete appraisal to date of the role played by Coimbra and its contribution to Portuguese Jesuit mathematics. A period of two hundred years was covered, and all the teachers and “mathematical experts” known were included. I hope this list will grow and become more and more accurate as historical research advances. The division into four periods proposed above is conjectural, but justifiable, notably by local historical reasons. Since “mathematics” is also an historical word – not to say “inaugural”, or metaphysical (Heidegger 1962: 53-59) –, there are no reasons to exclude “mathematicians” belonging to the period before 1682 or 1692. I trust João Delgado’s presence may mark an epistemological breakthrough, and one may easily assume that after Delgado it would be difficult to continue teaching mathematics from a strict philosophical point of view, though not impossible. This must be ascertained, since the content of the lessons is mostly ignored, but it is most likely that Aristotle’s philosophical texts were not anymore the subject of material reading. It remains impossible to situate or distribute the presence of the teachers between the two colleges (or other “private” rooms), or to understand the nature of their lessons in accordance with the precise site of the colleges. The catalog still registers several lacunae, allegedly due to periodic chair vacancies (1560/74, 1574/86, 1594/99, 1599/05, 1606/16, 1617/26, 1628/30, 1633/48, 1649/56, 1657/59, 1660/72, 1673/78, 1679/1682), but the last thirty-three years show almost no absences in the chair holders or in “students” who could fill the lacunae. Even this last phase must be interpreted in the light of the Society’s global policy.

**Table of Mathematics Teachers in Coimbra Colleges of Jesus and Arts**

Academic years | Teachers [and Students, or Mathematical Experts] | Nationalities |
---|---|---|

1555/60 (?) | Christopher Clavius | German |

1574 | Cipriano Suárez | Spanish |

1581/85 | Luís de Cerqueira | Portuguese |

1586/89 | João Delgado | Portuguese |

1590/92 | Richard Gibbons | English |

1591/93 | Martim Soares | Portuguese |

1593/94 | João Pinto | Portuguese |

1599 | Christopher Grienberger | German |

1605/06 | Diogo Seco | Portuguese |

1616/17 | Venceslau Pantaleão Kirwitzer | Bohemian |

1626/27 | Cristoforo Borri | Italian |

1627/28 | Simon Fallon | Irish |

1630/33 | Simon Fallon | Irish |

1648/49 (?) | John Rishton/Farrington | English |

(vacant chair (?), according to the testimony by I. Hartoghvelt) | ||

(1654/55) | (Francisco Pereira de Lacerda's mathematical library) | Portuguese |

1656/57 | Ferdinand Verbiest | Flemish |

1659/60 (?) | Jacques Cocle | French |

1672/73 | Adam Aigenler | Bavarian |

1673/78 | (vacant chair, according to A. Thomas's testimony) | |

1678/79 | Antoine Thomas | French |

[José Soares; Adam Weidenfeld] | ||

1682/83 | João König | Swiss |

1683/84 | João König | Swiss |

[Manuel do Amaral; Francisco Barbosa; Rodrigo da Costa] | ||

1684/85 | João König | Swiss |

[Manuel do Amaral; André Mendes] | ||

1685/86 | João König | Swiss |

[José de Sequeira] | ||

1686/89 | Manuel do Amaral | Portuguese |

1689/91 | Francisco Barbosa | Portuguese |

1691/92 | Philippe Bourel | French |

1692/93 | Albert Eusebio Buchowski | Bohemian |

[António de Barros; Bernardo de Abreu] | ||

1693/94 | Albert Eusebio Buchowski | Bohemian |

[Bernardo de Abreu] | ||

1694/95 | Albert Eusebio Buchowski | Bohemian |

1695/98 | Luiz Gonzaga | Portuguese |

1695/96 | José Botelho | Portuguese |

1696/99 | Luiz Gonzaga | Portuguese |

1699/02 (?) | Bernardo de Abreu | Portuguese |

1699/00 | Francisco Cardoso | Portuguese |

1702/03 | Inácio Correia | Portuguese |

[António Martins] | ||

1703/04 | Inácio Correia | Portuguese |

[João Mendes; António Martins] | ||

1704/05 | Inácio Correia | Portuguese |

[Alexandre Duarte] | ||

1705/06 | Inácio Vieira | Portuguese |

1706/07 | Inácio Vieira | Portuguese |

[Lourenço Rodrigues; Diogo Soares] | Portuguese | |

1707/08 | Inácio Vieira | Portuguese |

[Diogo Soares] | Portuguese | |

1708/10 | Jerónimo de Carvalhal | Portuguese |

1710/11 | António Martins (?)/ Lourenço Rodrigues (?) | Portuguese |

[João Pitta; J. Mendes; J. de Carvalhal; A. Martins; L. Rodrigues; Inácio da Silveira] | Portuguese | |

1711/12 | [João Pitta] | Portuguese |

1713/14 | Lourenço Rodrigues | Portuguese |

[José Fróis; Francisco Custódio Correia; Pedro Ferreira (?)] | Portuguese | |

1714/15 | Diogo Soares | Portuguese |

[José Fróis] | Portuguese | |

1716/17 | Filipe Pereira | Portuguese |

[Jacinto de Almeida; João Pitta; José Fróis] | Portuguese | |

1717/18 | Filipe Pereira | Portuguese |

[Paulo de Mesquita; Inácio Martins; José Fróis] | Portuguese | |

1718/19 | Filipe Pereira; Pedro Ferreira (?) | Portuguese |

[Inácio Martins; Paulo de Mesquita; João Garção; Manuel de Torres] | ||

1719/20 | Paulo de Mesquita | Portuguese |

[Xavier Francisco; Manuel da Silva; Inácio Vieira; João Mendes; F. Pereira; I. Martins] | ||

Paulo de Mesquita | Portuguese | |

1720/21 | [Xavier Francisco] | |

Paulo de Mesquita (?) | Portuguese | |

1721/22 | Paulo de Mesquita (?)/ Xavier Francisco (?) | Portuguese |

1722/23 | Inácio Martins (?) | Portuguese |

1723/24 | Paulo de Mesquita/Inácio Martins | Portuguese |

1724/25 | [Francisco Ribeiro] | |

Inácio Martins | Portuguese | |

1725/26 | [Bento Peixoto; João Inácio; Francisco Ribeiro; José Teixeira; Xavier Francisco; José Carneiro; Manuel de Torres; Pedro Ferreira; Manuel da Silva] | |

Inácio Martins | ||

Portuguese | ||

1726/28 | [João Inácio] | |

Inácio Martins | Portuguese | |

1728/29 | Inácio Martins | Portuguese |

1729/30 | [Luís da Silva] | |

Inácio Martins | Portuguese | |

1730/32 | Francisco Ribeiro | Portuguese |

1732/33 | [Manuel de Carvalho] | |

Inácio Martins | Portuguese | |

1733/34 | [Pedro Ferreira; Domingos José; José Leonardo; José de Mesquita; Luís da Silva; João de Loureiro] | |

Inácio Martins | ||

Portuguese | ||

1734/35 | [Paulo Ferreira; Felix da Rocha] | |

Inácio Martins/Inácio Coutinho | ||

1735/36 | Inácio Martins | Portuguese |

1736/37 | [Caetano Moniz; Francisco de Albuquerque] | |

Marcelo Leitão (?) | Portuguese | |

1737/39 | João de Faro | Portuguese |

1739/40 | [Manuel de Sousa; Inácio Soares; Caetano Moniz] | |

António Soares | Portuguese | |

1740/41 | João de Faro | Portuguese |

1741/42 | Bernardo de Oliveira | Portuguese |

1742/43 | [Eleuterio de Sousa; João de Sá; José Leonardo; João de Faro; Manuel de Sousa; Inácio Soares] | |

João de Faro; Pedro Ferreira (?) | ||

Portuguese | ||

1743/44 | António Monteiro | Portuguese |

1744/45 | [Eusébio da Veiga; Victorino Teles] | |

António Monteiro | Portuguese | |

1745/46 | [Manuel dos Santos; Inácio de Carvalho; José de Espinha] | |

António Monteiro | Portuguese | |

1746/47 | [Manuel dos Santos; Inácio de Carvalho] | |

Eusébio da Veiga | Portuguese | |

1747/48 | [Francisco Taveira; Agostinho de Araújo] | |

Eusébio da Veiga | Portuguese | |

1748/49 | [Filipe de Gamboa; Agostinho de Araújo; Manuel Mendes; João Inácio; João Pinheiro; José Bernardo de Almeida; André Rodrigues] | |

1751/52 | Filipe de Gamboa | Portuguese |

[Cristóvão Ribeiro] | ||

1752/53 | Inácio Monteiro | Portuguese |

1753/54 | Inácio Monteiro | Portuguese |

[António de Brito; António do Vale; João da Cunha; Tomás de Campos; Jerónimo Mendes; Francisco Gião | ||

1754/55 | Inácio Monteiro | Portuguese |

1755/56 | Inácio Monteiro/Bernardo de Oliveira | Portuguese |

1756/57 | Bernardo de Oliveira/José Teixeira | Portuguese |

**Bibliographical References**

**Primary Sources**

- Borri (1631), Christophorus.
*Collecta astronomica ex doctrina… De tribus caelis, Aereo, Sydereo, Empyreo,*Ulysipone: apud Matthiam Rodrigues. - Martins (1719), Inácio.
*Theses cosmographicas Terraqueae.**Ac Caelestis Sphaerae Naturam, & affectiones Mathematic disquisitione …Praeside P.M. Philippo Pereira…**Demonstraturus consecrat Ignatius Martins,*Coimbra: Ex Typ. In Regali Artium Collegio Soc. Jesu. - Martins (1726), Inácio.
*Labyrinthum Mathematicum Variis…explicandum A R. P.Ignatio Martins Soc. Jesu, Matheseos Professore, Extricaturus offert Josephus Teixeira*, Coimbra: Ex Typ. In Regali Artium Collegio Soc. Jesu. - Martins (1728), Inácio.
*Conclusiones Astronomicas D. Joanni Evangelistae Dicatas Praeside R. P. ac S. M. Ignatio Martins Societatis Jesu Matheseos Professore…*, Coimbra: Ex Typ. In Regali Artium Collegio Soc. Jesu. - Martins (1730), Inácio.
*Fulgentissimos Mathematicae Splendores… Praeside R. P. Ignatio Martins Soc. Jesu, Matheseos Professore, Expositurus offert Aloysius da Sylva…,*Coimbra: Ex Typ. In Regali Artium Collegio Soc. Jesu. - Martins (1736), Inácio.
*Conclusiones Mathematicas de Astronomia*,*Sphaera, Geographia & Optica… Praeside P. M. Ignatio Martins e Societ. Jesu Matheseos Professore, defensurus offert Ignatius Coutinho*…, Coimbra: Ex Typ. In Regali Artium Collegio Soc. Jesu. - Monteiro (1754-56), Inácio.
*Compendio dos Elementos de Mathematica*, Coimbra: Real Collegio das Artes, 2 vols. - Monteiro (1766), Inácio.
*Philosophia Libera seu Eclectica Rationalis, et Mechanica Sensuum ad studiosae juventutis institutionem accomodata, ac per lectiones digesta*, 7 vols., Venice: Antonii Zatta, 1766. - Thomas (1685), Antoine.
*Synopsis Mathematica complectens varios Tractatus*, Douai: Typis Michaelis Mairesse. - Veiga (1758), Eusébio da.
*Planetario Lusitano, explicado com problemas, e exemplos praticos para melhor intelligencia do uso das efemerides, que para os annos futuros se publicão no planetario calculado: e com as regras necessarias para se poder usar delle não só em Lisboa, mas em qualquer Meridiano (…) Para uso da Nautica, e Astronomia em Portugal, e suas Conquistas*, Lisbon: na officina de Miguel Manescal da Costa.

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*Vértice*, vol. XXV: 14-27. - Andrade (1973), A. A. Banha de. “Inácio Monteiro e a evolução dos estudos nas aulas dos Jesuítas de Setecentos”,
*Revista Portuguesa de Filosofia*29: 289-304. - Bachelard (1938), Gaston.
*La formation de l’esprit scientifique*, Paris: Vrin. - Baldini (1992), Ugo.
*‘Legem impone subactis’. Studi su Filosofia e Scienza dei Gesuiti in Italia, 1540-1632*, Roma: Bulzoni Editore. - Baldini (1998), Ugo. “As Assistências Ibéricas da Companhia de Jesus e a Actividade ceintífica nas Missões asiáticas (1578-1640). Alguns aspectos culturais e institucionais”,
*Revista Portuguesa de Filosofia*54: 195-245. - Baldini (2004), Ugo. “The Teaching of Mathematics in the Jesuit Colleges of Portugal, from 1640 to Pombal”, in
*The Practice of Mathematics in Portugal*, edited by Luis Saraiva, and Henrique Leitão, Coimbra: Universidade de Coimbra, 293-465. - Baldini (2008), Ugo. “The Jesuit College in Macao as a meeting point of the European, Chinese and Japanese mathematical traditions. Some remarks on the present state of research, mainly concerning sources (16
^{th}-17^{th}centuries)”, in Luís Saraiva and Catherine Jami (ed.),*The Jesuits, the Padroado and the East Asian Science (1552-1773)*, New Jersey/London: World Scientific, 33-79. - Baldini (2013), Ugo. “A Escola de Christoph Clavius: um agente essencial na primeira globalização da matemática europeia”, in
*História da Ciência Luso-Brasileira. Coimbra entre Portugal e o Brasil*, ed. Carlos Fiolhais, Carlota Simões e Décio Martins, Coimbra: Imprensa da Universidade de Coimbra, 51-76. - Boletim (1879).
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