Pedro Nunes takes Coimbra's chair of mathematics, 16 October 1544

On 16 October 1544 a royal appointment named a doctor of medicine, then about 42, to teach mathematics at the University of Coimbra. Pedro Nunes had not lectured at a university for about twelve years. University records do not mention him at all between 1537, when the university left Lisbon, and 1544, when the king's order brought him back, now to Coimbra.3 sources

He was already the king's cosmographer, the official expert on the shape of the Earth, the heavens and the charts that Portuguese pilots carried. He had published a book on astronomy and the sea chart in 1537 and a Latin treatise on twilight in 1542. Henrique Leitão, the historian who curated the National Library's 2002 exhibition on Nunes, writes that in national terms the Coimbra chair was the highest distinction that could be given to a mathematician.3 sources

Nunes never sailed to Africa, Brazil or India. His contribution to the age of the Portuguese voyages was made at a desk, with geometry. Two of his ideas outlived him: the curve a ship traces when it keeps to one compass bearing, and a way of reading an angle more finely than the degree marks on an instrument allow.3 sources

Nunes was born in 1502 in Alcácer do Sal, a river town south of Lisbon, which is why his Latin name was Petrus Nonius Salaciensis. His parents are believed to have been Jewish converts to Christianity, and he was registered as a New Christian, the label Portugal gave to converts from Judaism and their descendants.3 sources

He began university studies around 1517 and studied at Salamanca in Spain, where in 1523 he married Guiomar Áreas. They had two sons and four daughters. Sources disagree on whether he studied first in Lisbon or in Salamanca. Back in Portugal he took a degree in medicine, which then included astronomy and mathematics, because doctors were expected to practise astrological medicine.3 sources

On 16 November 1529 King João III appointed him cosmographer of the kingdom, on a salary of 20,000 reais. Weeks later he began teaching moral philosophy and then logic at the university, which was still in Lisbon. He became a doctor of medicine in 1532.3 sources

The court also made him a tutor. He taught the king's brother Luís until July 1531 and, from October 1531, his brother Henrique, the future cardinal and, in 1578, king. Decades later the young King Sebastião, João III's grandson, was also counted among his pupils. He also taught navigation to two men who would command fleets: Martim Afonso de Sousa, who led an expedition to Brazil in 1530, and João de Castro, later viceroy of India.3 sources

Martim Afonso de Sousa came home from his Brazil expedition with a puzzle. Sailing east by the compass, he had expected that, as on a globe, he would eventually cross the equator. He never did. He put the question to his old teacher, and Nunes answered in a short treatise, probably written around 1534 and printed in 1537. The Dictionary of Scientific Biography dates the conversation to 1533.3 sources

Nunes recalled the episode in a second treatise, addressed to the king's brother Luís. Some time ago, he wrote, he had produced a little work on certain doubts that Martim Afonso had brought back from Brazil, and answering them had needed points of geometry and theory as well as the practical rules of navigation.2 sources

Pilots of the time treated a course on a single compass bearing and a great circle, the shortest route between two points on a globe, as the same thing. Nunes showed they were different. A great circle cuts each meridian it crosses at a different angle. A ship that holds one bearing cuts every meridian at the same angle, and so its track bends round the globe.3 sources

In the Portuguese texts of 1537 Nunes thought the curve would eventually reach the pole. In his Latin works of 1566 he corrected this: the ship comes ever closer to the pole but never touches it. He had no word for the line. He wrote of a curved and irregular line. The Greek-derived name loxodrome, meaning slanting course, was coined in the Low Countries early in the next century.2 sources

The 1537 book also contains a drawing of curves on a polar plan, which Nunes meant to show rhumb lines at bearings of 45° and 67.5°. Modern analysis has found that the drawing is made of joined arcs of circles, not true spirals. It was a first sketch, and the 1566 Latin version is the fuller treatment.3 sources

Prince Henrique, his former pupil, asked Nunes which day of the year has the shortest twilight. The answer became De crepusculis, printed in Lisbon in January 1542. It is usually considered his most sophisticated work, and it impressed the leading mathematicians of the day.3 sources

Buried in the book was a practical idea. Astronomers and navigators measured the height of the sun or a star with quadrants and astrolabes marked in whole degrees, and the scales were too small to subdivide further. Nunes proposed engraving extra scales. Inside the main arc of 90 degrees he drew 44 more arcs, each divided into one part fewer than the arc outside it: 89, 88, 87 and so on down to 46.3 sources

The reading works like this. If the sighting line falls between two degree marks, it will pass almost exactly through a mark on one of the inner arcs. Suppose it crosses the eighth mark of the arc divided into 63 parts. The angle is then 8/63 of 90 degrees, about 11.43°, which a simple calculation or a table turns into degrees and minutes.3 sources

In theory the method was very precise, but it was hard to engrave 45 different scales accurately on a small instrument, and the Danish astronomer Tycho Brahe remarked that it did not live up to its promise. Clavius and his associate Jacob Curtius simplified the engraving, and in 1631 Pierre Vernier replaced the fixed inner arcs with a single sliding scale.3 sources

In 1537 King João III moved the university from Lisbon to Coimbra. The professorship of mathematics went with it. Nunes was named to that chair on 16 October 1544 and held it until he retired in 1562. The sources disagree on the exact day of his retirement: 4 February according to the Dictionary of Scientific Biography, 4 November according to the National Library's chronology.3 sources

Possibly to live up to the post, Leitão suggests, he soon published De erratis Orontii Finaei (1546), a book he had been planning for more than a decade: a demolition of the claims by Oronce Finé, professor at the royal college in Paris, to have squared the circle, trisected the angle and doubled the cube. University council minutes from 1546 to 1556 carry his signature beside the titles doctor in medicine, mathematics lecturer and cosmógrafo-mor.3 sources

On 22 December 1547 he was promoted to chief royal cosmographer, a post he kept until his death. From then on his life was split between Coimbra and the court in Lisbon. He spent most of 1557 to 1561 away in Lisbon, and Leitão notes that he published no new book until 1566, after he had left university teaching.3 sources

Nunes insisted that the voyages had been a work of science. In the 1537 treatise addressed to Prince Luís he wrote that the discoveries were not made by chance. In the language of his time, astrology meant what we call astronomy.2 sources

Working seamen were less sure. António Costa Canas, a naval historian and director of the Museu de Marinha, writes that Nunes's relations with seafarers were never peaceful, mainly because they found his mathematics hard to follow. As cosmographer he also had a hand in certifying instrument makers, chart makers and pilots.1 source

His method for steering along a great circle, with hourly corrections worked out by spherical trigonometry, was never adopted by Portuguese sailors. The instruments of the day were not precise enough, and the calculations were too laborious for the deck of a ship. Randles calls him a savant de cabinet, a scholar of the study.1 source

In 1549 Diogo de Sá, a writer of conservative Aristotelian views and a strong dislike of mathematics, attacked Nunes in De navigatione, a Latin book printed in Paris and partly written as a dialogue between a mathematician and a philosopher. Randles notes that Sá was no more a navigator than Nunes. His objection was practical: a pilot at sea has only a compass and a ship that goes where the wind pushes it.2 sources

Training was the other front. In 1559, under Nunes's auspices, a course known as the Aula do Cosmógrafo-Mor was set up to teach pilots. The Escola Naval, Portugal's naval academy, counts it as its forerunner, but notes that pilots still came to their examinations with a record of voyages more than with mathematics. Nunes's aim of pilots with real scientific training was met only in the eighteenth century.1 source

Free of teaching, Nunes gathered several of his main works, in Latin, into the Opera, printed in Basel in 1566. It reworked the 1537 treatises into a longer and more developed text. Unhappy with misprints in that edition, he reissued it at Coimbra in 1573 as De arte atque ratione navigandi.4 sources

On 11 September 1572 his former pupil, King Sebastião, called him to Lisbon to advise on the reform of weights and measures, which was promulgated in 1575. In 1577 Pope Gregory XIII consulted him on the calendar reform introduced in 1582. He died in Coimbra on 11 August 1578.3 sources

In his 1566 text Nunes asked for charts on which rhumb lines appear as straight lines, and suggested building them from narrow strips, each covering all 360 degrees of longitude. He never joined the strips into a single chart.2 sources

Gerardus Mercator's world map of 1569 was the first to show rhumb lines as straight lines. Leitão and the cartographic historian Joaquim Alves Gaspar argue that Mercator most likely drew it with a table of rhumbs, and they trace that path from Nunes's first account of the rhumb line in 1537 through a Portuguese text of about 1540 in which such tables first appear.3 sources

The mathematics of the new map was first published in 1599 by the Englishman Edward Wright, whose Certaine Errors in Navigation translated parts of Nunes word for word. Mercator's projection became standard at sea only in the mid-eighteenth century, once longitude could be found reliably. It is still the usual projection of nautical charts.4 sources

His peers abroad rated him highly. In 1558 the English mathematician John Dee named Nunes as the man to finish his work if he died, calling him that most learned and grave man who is the sole relic and ornament and prop of the mathematical arts among us. Leitão thinks the two almost certainly never met. Whether the German Jesuit Christopher Clavius studied under him at Coimbra is disputed: the Dictionary of Scientific Biography and MacTutor say so, while the Biographical Encyclopedia of Astronomers calls the claim erroneous.4 sources

His Latin name is on the Moon. The Jesuit astronomers Giovanni Battista Riccioli and Francesco Grimaldi named a crater Petrus Nonius, and the International Astronomical Union adopted the name Nonius in 1935. The crater measures about 70.6 km across.2 sources

In Coimbra, the university set up the Instituto Pedro Nunes in 1991, a non-profit body that links university research with industry. The Escola Naval traces its own lineage to the pilot course founded under his auspices in 1559.2 sources