The first fact that we know about Delgado (other than his place of birth) is that he entered the Society of Jesus (commonly known as the Jesuits) in 1574. We, therefore, know nothing about the first 21 years of his life. In 1576 he went to Rome to study with one of the most eminent scholars of mathematics of the day, namely the Jesuit Christopher Clavius. Clavius, from a region which is now Germany, was well-known to the Jesuits of Portugal since after entering the Jesuit order in 1555 in Rome, he had studied at the University of Coimbra in Portugal in the following year. Let us note that the University of Coimbra was founded in Lisbon in 1290 but moved permanently to Coimbra in 1537. Portugal had a second university, the University of Évora which existed between 1559 and 1779, but after the University of Évora was closed the University of Coimbra was the only Portuguese university until the University of Lisbon was founded in 1911.
Clavius spent four years in Coimbra studying at the university before returning to Rome where, after studying theology, he taught mathematics at the Jesuit Collegio Romano in Rome from 1564. He had been teaching there for twelve years when Delgrado arrived in 1576 to begin his nine years studying at the Collegio Romano. This was an extremely important time in Delgado's mathematical development and Clavius's influence would be the basis of the contribution he made after returning to Portugal.
In 1585 Delgado returned to Portugal with the intention of going to Brazil but, for some reason that is not known, the trip never happened. It has been conjectured that may have taught mathematics in 1585-86 but this is not known for certain. What we do know for certain is that in 1586-87 he taught in Coimbra, but not giving courses open to all but just teaching members of the Jesuit Order at the Jesuit residence in the city. In the two years 1587-89 he continued to teach in Coimbra but during these years his courses were at the College of Jesus which was open to both Jesuit and lay students. This Jesuit College was initially built in 1542 and is the oldest Jesuit College in the world. It remained an independent college until 1759 when, after the Jesuits were expelled from Portugal, the College became part of the University of Coimbra. In 1589-90, Delgado was sent to Évora where the Jesuit University of Évora was operating. During this year he did not teach but spent it in priestly duties.
Although in Delgado's time there were only two universities in Portugal, there were other Jesuit colleges where elementary mathematics was taught. The most important was the College of Santo Antão in Lisbon which started to give courses from 18 October 1553. These were held in a house used by the Jesuits since 1542. Portugal, a small country with a population of around one million at this time, was ambitious in building a large overseas empire and an important requirement to achieve this was to have men with great navigation skills. Mathematics and astronomy were the foundation of these skills but Portuguese universities concentrated on teaching theology so the King of Portugal appealed for instructional courses to be set up. Teaching of Esfera, meaning the sphere, was set up in the College of Santo Antão and from the beginnings this course was given only to Jesuit students. However, there were no dedicated mathematics teachers in Portugal so still only very basic mathematics (beginnings of Euclid's Elements and some use of navigational instruments) was being taught by professors of philosophy. Only the first part of Sacrobosco's 1220 text Tractatus de Sphaera Ⓣ was used for this teaching. The first chapter of this book deals with the shape and place of the Earth within a spherical universe. The second chapter deals with various circles on the sky. The third chapter describes rising and setting of heavenly bodies from different geographical locations while the fourth chapter gives a brief introduction to Ptolemy's theory of the planets and of eclipses. This mathematics teaching did not accomplish the need for many skilled navigators so, in 1590, Delgado was appointed to teach the Aula de Esfera, the Course of the Sphere, at the College of Santo Antão in a course available to both Jesuits and lay students. This appointment made him the first professor of mathematics in Portugal. Delgado held this appointment until his death 22 years later, but he did not teach the Aula de Esfera continuously at the College of Santo Antão during these years.
Delgado had a second position as the chief architect for Portugal. Among the works he undertook as an architect we mention those for the College of Santo Antão, the Jesuit Novitiate at Cotovia, and the College of Arts in Coimbra. When Delgado was occupied with these architect's duties, the Aula de Esfera was delivered by Francisco da Costa (1567-1604) who was his assistant.
In order to understand the significance of Delgado's contributions, we need to examine a little of the background to the position of mathematics at this time. The main problem posed was whether mathematics met the requirements of a science as set out by Aristotle in the Analytica posteriora. This revolved around what became known as the 'Quaestio de certitudine mathematicarum', namely the question of the certitude of mathematics. The debate arose from the writings of Alessandro Piccolomini (1508-1578) who was born in Sienna, studied in Padua, Bologna and Rome, and taught in Sienna and Rome. He raised serious doubts about the scientific value of mathematics, stating that the discipline did not meet the most important scientific requirement, that of presenting the causes of the conclusions. He published Commentarium de certitudinem mathematicarum disciplinarum Ⓣ (1547) which :-
... contains the provocative thesis that certitude in mathematics does not rely upon proofs. Piccolomini argues that, since a mathematical proof cannot meet the requirements for the most certain kind of proof, the so-called demonstratio potissima, mathematics does not derive its certainty from its proofs; instead, it derives its certainty from its content.Francesco Barozzi (better known by the Latin form of his name Franciscus Barocius) published Opusculum, in quo una Oratio, et duae Quaestiones: altera de certitudine, et altera de medietate Mathematicarum continentur Ⓣ in 1560. This was written to argue against the views of Piccolomini and argues that mathematical certainty derives from the nature of its proofs:-
... the certitude of mathematics is contained in the syntactic rigour of demonstrations.Christopher Clavius produced a version of Euclid's Elements in 1574 which defended the certainty of mathematical proofs. However, Benito Pereira (1536-1610), a Spanish Jesuit who taught in Rome, endorsed Piccolomini's views in De communibus omnium rerum naturalium Ⓣ (1576). In fact Pereira went further than Piccolomini denying any scientific value to mathematics.
We explained above that Delgado was in Rome, studying with Clavius, from 1576 to 1585 so we see that he was in Rome at just the time when this argument was raging. Jesuits around the world took sides in the argument, the Jesuits in Coimbra, for example, taking the side of Benito Pereira. We should understand that, in many ways, the debate on the certitude of mathematics was between philosophers who sided with Piccolomini and Pereira, and mathematicians who took the opposing view. This, of course, illustrates the importance in the Portuguese context of the mathematician Delgado opposing the philosophers of Coimbra.
We know in detail what Delgado taught in his 1605-06 Aula de Esfera since the text of that course survives. We also have details of his 1606-07 course. A manuscript containing material thought to be from his 1598 course also exists. The book  contains a collection of fascinating articles about these courses of Delgado, although we should point out that it is 45 years old and some of its conclusions have been superceded by more recent research. Delgado begins by citing the theses and arguments of the detractors of mathematics. Some, he explains, argue that mathematics does not proceed by means of proper causes, but only by the formal arguments which is a sign of inferiority in relation to physics. Others argue, says Delgado, that mathematics does not proceed by any cause since mathematical causes are only metaphorical, and others do not accept that mathematical proofs proceed from their own principles. Although he is clearly referring to the arguments by Piccolomini and Pereira, he does not mention them by name. In particular, he explains that some reject Euclid's proof that the sum of the angles of a triangle add to two right angles because it uses the external angle in the proof and this is not a part of the triangle.
Then Delgado explains that his view of mathematics is completely the opposite of that presented by these detractors of mathematics. He ends his introductory chapter giving four theses which give his view of mathematics :-
Delgado's thinking can be summarized as follows: Mathematics fulfils the Aristotelian requirements so that one can and should consider it a perfect science; mathematics is a superior discipline, whose results influence the production of theories of other scientific disciplines, such as physics. João Delgado is an example of the Collegio Romano's role in the formation of specialized core groups in the teaching and promotion of mathematics. What stands out in the introductory chapter of his course is the breadth of topics covered, the extent of his defence of mathematics, his deep knowledge of the technicalities involved, and the force of his attack on his opponents, which goes well beyond the brief arguments presented in the prolegomena of works of Clavius.Luís Miguel Carolino looks at this aspect of Delgado's contribution in . He writes in summary:-
Challenging the views traditionally endorsed by philosophers on this matter, Delgado argued that mathematics fit all the main features of Aristotelian science, and therefore it should be considered as a true science. By doing so, Delgado was one of the first Jesuit mathematicians to defend the scientificness of mathematics within a strict Aristotelian framework.A more detailed summary of Luís Miguel Carolino's excellent paper  is given at THIS LINK.
The Jesuits of Coimbra reacted strongly to Delgado's strongly worded attack. Sebastião do Couto (1554-1638) produced the University of Coimbra response and the argument would rumble on for many years, particularly in Italy, France, Germany and Portugal. Delgado, however, played an important role in this argument and deserves considerable praise for his defence of mathematics and its importance to other disciples, something which today is completely taken for granted.
One other aspect of Delgado's contribution is important for the development of mathematics in Portugal, namely the small group of fellow mathematicians he built at the College of Santo Antão in Lisbon. Most of these mathematicians, such as Francisco da Costa, taught at the College of Santo Antão and helped to build Portuguese mathematics.
No book by Delgado was ever published but it is known that he intended to write one. His aim was to produce a text to teach mathematics in a three year course and he sent a request to Rome in 1606 for an assistant to assist him in producing this work. Clavius would have had to approve the final version of such a publication but, as far as we know, no text from Delgado ever reached him. It is not impossible, although very unlikely, that one day a draft of this book might turn up in some library or private collection. More probably, however, the book was never written.
Article by: J J O'Connor and E F Robertson