Pedro Nunes (1502 - 1578)

Mathematics, Cosmography and Nautical Science in the 16th century.


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Quadrant with a nonius scale.


The Nautical Chart



The nautical chart was a very important instrument used in navigation. It was a natural evolution of the portolan chart, used mainly in Mediterranean navigations. This "evolution" introduced many geometrical problems, unnoticed by the pilots of the ships. Pedro Nunes identified the main geometrical problems of the square nautical chart and provided solutions. His work was a precursor of Mercator's.


See his work in: Obras, Vol. I; Obras, Vol. IV.




The Loxodromic Curve



A loxodrome is a curve in the surface of a globe, made by following a constant bearing course, so it cuts all meridians with the same angle. Pedro Nunes referred to this curve as a rhumb line but it was the Dutch mathematician Willebrord Snell who named it loxodromic curve.

The origin of the loxodromic curve is deeply related with the problems of the nautical chart, principally with the confusion between what was a great circle and a constant bearing course. The rhumb line was misinterpreted as a straight line in the common nautical chart.

The loxodromic curve is a straight line in a Wright-Mercator chart.



See his work in: Obras, Vol. I; [Manuscrito de Florença]; Obras, Vol. IV (Chapter. 1).





Cosmographic and astronomical topics



The width of the "clime" zones.


The notion of klimata was introduce in classical antiquity (as far as we now by Aristotle). It was used to divide the Earth in parallels (usually seven). Two consecutive parallels had a difference of half hour for the length of the longest day of the year.

As far as we know, no classical, Arabic or medieval author left a mathematical method to find the length of the climes. Pedro Nunes was the first to present a geometrical method to find out this values. He included it in a commentary to the climes' chapter of Sacrobosco's Sphaera, published in his Tratado da sphera (1537). His Annotatio in extrema verba capitis de climatibus is certainly his most published text since it was included in many editions of Élie Vinet's editions of Sacrobosco's Sphaera.



The minimum twilight.


Nunes adressed the question in the book De crepusculis (1542). In it Nunes answered a question by a pupil -- Prince Henrique, one of the king’s brothers and future king -- about the problem concerning the length of twilights for different regions. In this book, Nunes showed how an atmospheric phenomenon could be explained using the “most certain and evident mathematical principles”.



The height of the atmosphere.


Extra meridian methods for latitude determination.



The dial of Achaz.


A geometric solution of a biblical miracle. Nunes dealt with the problem in his Tratado da sphera (1537) and later in his Opera (1566).



"Annotation on the Moon".







The Nonius scale.


Nunes devised this graphical procedure as a solution to enhance instruments’ precision and presented it for the first time in De crepusculis (propositio III).



The shadow instrument.


The instrument lying on the plane.


A simple solution that enabled a observer to obtain the height of the sun. Due to the needs of a steady plane in which the instrument should lie it was never much used aboard.



The nautical ring.


Another interesting and simple proposal of an instrument destined to obtain the height of the sun.



Mathematical diagrams/graphic solutions.




Other Mathematical Topics



Archimedes' calculation of pi.



Copernicus errors in trigonometry.


Pedro Nunes was an early reader of Copernicus' De revolutionibus (1543). In his Opera (1566) Nunes presents some mathematical reflections on Copernicus' text. Nunes was aware of the physical implications of Copernicus theory but he left that discussion to the “philosophers”. His main concern was the mathematical consistency of Copernicus' work: Nunes pointed some trigonometrical errors in it.



Solution of higher order equations.


Nunes dealt with this question in his Libro de Algebra (1567).



The contact angle.


Nunes dealt with this problem in his Libro de Algebra. The debate was originated in the III book of Euclides' Elements, and registed many commentaries until the 16th century.

Nunes refuted Jacques Peletier demonstrations and concluded that the contact (or contingence) angle is a quantity.

He also used some of Jordanus arguments (proposition II, De ponderibus) establishing an interesting use of an example from "natural philosophy" to sustain mathematical ideas.


Libro de Algebra: Letter to the Reader.


This important text included in the Libro de Algebra is an important example of the well known disputes around the resolution of third degree equations. It was also important to legitimate algebra as a mathematical field. 



Rowing: Annotation to Aristotle's Mechanica.



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Bruno Almeida and Henrique Leitão (Centre for the History of Sciences, Lisbon University)