The Nonius of Pedro Nunes
António Estácio dos Reis, Academia de Marinha

This article was published in the magazine Oceanos - number 38 April / June 1999, under the title “Navios e navegações - Portugal e o Mar”
Comissão Nacional para as Comemorações dos Descobrimentos Portugueses

The nonius concept

Tycho Brahe turns to the nonius of Pedro Nunes

An unforeseen event makes history

Vernier’s solution

 Estácio dos Reis

(click on the picture above to see an enlarged version)

Vernier’s nonius. João Maria Pedroso, 1800. Museu da Marinha, Lisbon

(click on the picture above to see an enlarged version)

Rapid reading system used in sextants for at least half a century and which make it possible to read angles with an approximation to the arc minute.

Vernier’s solution

The nonius, as it was designed by Pedro Nunes, was very difficult to render concrete, which led some of the Portuguese mathematician’s peers to try to improve the original design. One of them was the Jesuit, Christopher Clavius, originally called Schlussel (1537-1612), born in Bamberg, in Bavaria, who took his place in the History of Science for being the main person responsible for the reform of the calendar, a mission entrusted to him by Pope Gregory XIII. Another such person was Jacob Curtius (J. Kurtz or Curz), also of German nationality, chancellor of the emperor Rudolf II and who, like him, dedicated himself to the study of astronomy and mathematics. Unfortunately, given the dimensions of this text, we will not be able to devote ourselves to the contributions of these two scientists. The more interested reader though may consult the work “O único exemplar vivo do nónio de Pedro Nunes?”, where this issue is developed.

It was, however, Pierre Vernier who ended up finding a practical solution. Pierre Vernier (1584-1638) was born in Ornans, close to Besançon, in western France, and inherited the passion for mathematics from his father, who also initiated him in the exact sciences and their practical applications, especially as regards instruments used in cartography. He read the works of Tycho Brahe and Clavius. He was captain of the castle of Besançon, which he controlled until his death, having been made an honorary citizen of this city, as gratitude for having organised the military defence when the region was threatened by the legions of the German Ernest de Mansfeld.

Having dedicated himself to cartography, he became aware of the imperfection of the instruments used on the field and in an attempt to apply the methods recommend by Father Clavius to measure angles using a compass and a special scale, he designed the movable sector, a discovery which he presented in the work La construction l'usage et les proprietés du quadrant nouveau de mathématiques, published in Brussels in the year 1631. This nonius, also known as “vernier” in some countries, is therefore a graduated cursor placed next to the scale of the instrument and which slides along it. It was designed in such a way that the n divisions of the scale correspond to n-1 or n+1 divisions of the cursor, the former being called a retrograde nonius and the latter a direct nonius. As an example we chose a nonius of the second type (to this effect we chose a straight scale) where it is easy to see that each division of the CD nonius corresponds to 9/10 of each one of the divisions of the AB scale.


Diagram illustrating how Vernier’s nonius works, as indicated in the text.

If we wish to measure the length of X, we must look for the graduation on the nonius which coincides or is closest to a division on the AB scale. In the case of the diagram, it is the sixth division, therefore the length of X is exactly 2.6. The 8 divisions of the AB scale correspond in fact to the sum of the length of X and the six divisions of the nonius which, as we saw, are equal to 9/10 of these. Therefore, 8=6x9/10+X, where X=2.6.

The Vernier nonius was without a doubt an ingenious solution as it was easy to make and immediately supplied the value of a fraction of the smallest division of the scale.

Pierre Vernier ended his days in the city where he was born, at the age of 57, without his invention reaching the dissemination it deserved and even passing unnoticed for a long time. The non-existence of scientific Academies and societies, which played a decisive role in the dissemination of Science as from the subsequent century, was perhaps the main reason for this oversight.

This type of nonius underwent great expansion as from the second half of the eighteenth century, not only in the octants and other nautical instruments of the same family of double reflection, but also in many other measuring instruments. However, Vernier’s invention ended up being dethroned in this century when the so-called rapid reading sextants began to be manufactured in which the fractions of a degree are read on a drum, graduated from zero to 60 minutes. Using these sextants, the height of the heavenly body is easily read at the moment of collimation.

We have finished expounding the methods imagined by Pedro Nunes and Pierre Vernier, yet two controversial issues still remain to be clarified:

1. Who should the real invention of the nonius, to which Vernier gave a practical form, be attributed to?

2. Which of the two terms should be used: nonius or vernier?

This discussion was probably started by Lalande, when in 1771 he published his Astronomia, where he states that "The division that is mostly used today in several authors is the division of Nonnius, although Nonnius is not actually the name of the author; yet he imagined another that was very successful, & that could lead to the one we have today. See his treaty De Crepusculis, printed in 1542. The true author of ours in its current state was Pierre Vernier,...", and, therefore, “I believe it is fair to re-establish the true author in his own right, & to call the piece that forms the division in question Vernier instead of nonius”.

The first reaction to this statement was by João Jacinto de Magalhães, referred to above, who said the following in one of his works published in 1775:

"1. I maintain the old name of Nonius of this piece, which the “Small Masters” of instrumental Literature started calling Vernier some years ago, with a success equal to that of women’s hairstyles which, despite all the ridicule of a useless and uncomfortable novelty, do not cease to be immediately imitated by some women of good sense for fear of being placed aside for their bad taste.

2. It is with true joy that I recommend to these “Small Masters” another far more beautiful name for the same piece, that of Clavius. It has a pleasant pronunciation and will not cease to make a fortune among all the Astronomers & Instrumentistas of good taste.

3. My guarantee for this novely is Père Pezenas, in chapter, page 83 of his Astronomia dos Marinheiros, printed in Avignon in 1766, in-8º where he remarks that his colleague Père Clauvius had already spoken of this division of the Nonius, twenty years before Pierre Vernier. It is remarkable that after this joke, this same Author preserves the name Nonius. Apparently it is because he has no more good taste than I. I lament it much for the both of us.”

There have been many opinions on this fascinating subject and it is perfectly natural that different standpoints be taken. Portuguese authors choose to call it the nonius, while the French prefer the expression vernier. However, this rule was not always adopted by everyone, as Delambre stated in 1818 on quoting Bailly: "the vernier is merely a perfected instrument; let the name Nonius be given to it with the traces of his genius". As far as we are concerned, if we may present our opinion, we opted for the designation nonius, as we feel that Vernier’s solution is no more than a phase in the evolution of a process began by Pedro Nunes, perfected by Curtius and Clavius, which was on the verge of success, but ended up being outdated when the sextants and other measuring instruments started using the already mentioned rapid reading system.

© Instituto Camões 2003-2005