At Lisbon in the

The places of mathematical practice in seventeenth century Europe are scattered geographically and socially, yet, for reasons that shall become obvious, networks of vivid communication do exist. Hence, in many respects, European mathematics constitutes a specialized field, that develops according to a relatively autonomous dynamics, which also means that it distances itself from the values of other kinds of activities. Studying the role of mathematical instruments is promising an approach to measuring these distances or proximities (by gauging the degree of autonomy), and also to understanding the dynamics of the field at that time. Two main facets of the ‘sector’ can be considered: one is the mathematical facet, where mathematical reasoning internal to the field applies, while the other facet relates to other aspects of the instrument considered as an object material, symbolical, economical. From this vantage point, the ‘sector’ resembles a hinge where two realms – mathematics and the remainder of the world – can be observed in their relation to each other. This is why the examined text may reveal something about the impact of further or lesser distances of those two realms.

The Jesuit Ignace Stafford (alias Lee or Badduley) (

The treatise is divided into two main parts: first a set of classical definitions and axioms, followed then by a series of propositions (nearly always formulated as ‘problem’) of which the solution is usually given discursively in general terms (without demonstration though) and then illustrated by various numerical examples running over several paragraphs. In some instances there are references to demonstrated propositions in a work called

What then is special about Stafford’s

Curiously enough in Stafford’s text, the sector is called ‘el pantometra.’ As we will now show, this particular terminology points at the circle of readers the author had in mind when he wrote up his treatise. We know that ‘pantometrum’ is a generic name given to several instruments at the time that had the particularity – according to its etymology – of combining several modes and objects of measurement, usually angle measurements.

He worked as a private teacher to officers and merchants. Also, from

[…] the use of the logarithmic and geometric instruments that I wanted to describe, mainly [the use] of the gramelogia, of the

[…]

In his manuscript treatise about military architecture Stafford writes:

But when one executes [the solution] by the

His argument for using the term ‘el pantometra’ is ‘para no dejar la phrasi comun’ and because ‘vulgarmente se llaman’, which hints, as to the readership he had in mind, at a group of people, Jesuits or not, very probably with links to the Habsburg court or Iberian military circles. Those would be familiar with Coignet’s manuscripts or the French publication and terminology, and hence using ‘el pantometra’ to designate the sector-like instrument. One of the few things we know about Stafford’s adolescence, is that in

This means that Stafford wrote his

In these circles Coignet’s works could have circulated through the Spanish network as early as

One of the strongest factors of autonomy of the field of mathematical knowledge is the adherence to a traditional corpus of mathematical concepts, problems, and specialized methods. During the Renaissance, as is well-known, mathematical authors maintained a conscious reference to antique authors such as Euclid, Archimedes, Apollonius etc. Among the authors of the sixteenth century, there was a clear tendency of presenting even new developments as based on their venerable classical foundations. This allegiance to classical antiquity meant, on the formal level for example, that a mathematical text would start with a set of definitions and axioms and would then be divided in a series of propositions (a format alternative to dialogue,

These characteristics appear combined in Stafford’s

The author relies doubly on antique heritage: he uses a solution attributed to Apollonius for the geometrical construction of two mean proportional straight lines by

We know that, beyond the influences from present and past, mathematical work does not emerge completely isolated even from the world outside mathematics. This is why one needs to qualify the degree of autonomy of mathematical knowledge and its development. We shall now see in the case of Stafford’s

When commenting on the different currencies with their subdivisions of coins, Stafford gives preference to the Portuguese, as he says:

Mrs. ‘Money’ although admitting greater variety in her integer and fractional numbers [compared to units of weight and volume] […] the Portuguese is doubtlessly the best of all, not only because it is the simplest, but also because it divides up into tiny parts, and comforts the poor who may, with little substance, show off thousands of coins.

Another example, famous already in his time, is Stafford’s invective against the habit of using the bad ‘ordinary sea charts’ instead of one using the Mercator projection. He comments on this problem when introducing the meridian line on the sector.

[…] that’s why on the ordinary chart the ratio of longitude of Friesland is double with respect to the globe, and to the true [longitude]; on the isles of Groenland and Groeland it is four times the true one because of the meridian of Friesland being double of its parallel, and [the meridian] of Groeland, and Groenland four times its parallel.

[…] I will note an example of this effect by Pedro Nunes. The distance between Lisbon and Ilha Terceira is estimated at

Stafford then moves on to give an ‘historical’ account of how the technique of realizing a Mercator projection came about. One sees in this set of citations how, while introducing contemporary elements, the author also includes moral judgement, and criticizes realizations and practices in his present time world, practices contrary to mathematically guided reason. The following are the issues he apparently points out: the erroneous way of navigation (repeating Nunes’ example), the making of private profit by exploiting the market price instead of selling at the adequate price (the servant of the cloth merchant), the cheating on the product to resist the obscure forces of offer and demand (the inn keeper), the spending of life years and money on war (the knight), the misery of the poor (showing off their little substance) etc.

In addition to this non-mathematical matter intruding into Stafford’s text from the historical context one should underscore that a couple of very recent mathematical innovations make their appearance, too. There is the case, as we have seen, of the demonstration borrowed from Apollonius. Although inherited from antiquity, this and other methods for finding two mean proportionals are included in Eutocius’ commentary of Archimedes’

Finally, but very significantly, Stafford’s presence in the town of Lisbon was itself a product of the historical circumstances: As an effect of the confrontational policy of the English crown and the Roman church hierarchy, many English Catholics were driven into exile to catholic countries.

At some point, Stafford seems to acknowledge the geographically distant traditions his

Stafford attributes the invention of the sector to Christophorus Clavius (

A text about the design and use of a sector implies the bridging between the spheres of ‘mathematical concepts’ and ‘materially crafted’ instruments. Stafford’s text gives a few hints about the difficulties of bridging this gap. Admittedly, there are very few of such passages where we could speak of a conscious perception of the distance between abstract reasoning and instrumental operation.

The first I could locate is rather trivial. It is included in problem number

Proposition

[…] of the number that corresponds in the tangent table to the required arc, five figures on the right hand side are removed, given that the radius is

[…]

A few pages further in the text, the author draws the attention to another condition that depends on the actual size of the material instrument. If the latter is of medium size, acknowledges Stafford, it is in fact materially impossible to engrave distinctly the thousandth in the scale of solids:

Thus, the author is well aware of the distance lying between his theoretical explanation of how to design an instrument and the actual material manufacture.

Eventually, Stafford manifests awareness of still another kind of distance: the one left behind by travelling knowledge while it is handed down from the first text that would describe the instrument (the source) down to his own text. He writes at the end of his ‘Appendix’:

Up to this point the authors [Gunter and Oughtred] of these instruments teach us their design and construction. If [they did it] better than [myself in] the preceding propositions, then I have struggled in vain: and if not, I wish to heal the illusion of those curious people who are only satisfied with what they expect to be better at the source, than in the river: and it is not always like this because often the water at the source is crude and too thin, and flowing down it acquires perfections, like the Tajo river by the gold sands with which it becomes loaded.

He compares the knowledge about the sector to a river and states the problem first in the form of a paradox: People would usually think that it is better to drink water at the source rather than from the river, i.e. better to read Gunter than Stafford. But, in this case – meaning his own work where, on the construction of the scales of the sector, he had added detailed accounts, which were actually missing in Gunter’s account – he thinks that the knowledge about the instrument has grown better and richer in his own text. This phenomenon compares to the Tajo river, that is known to carry gold sand at the end of its meandering when it reaches the estuary at Lisbon.

From this we may conclude, that Stafford acknowledges that circulation does not leave knowledge

By way of conclusion, one could ask: how do the described kinds of distance affect the circulation of knowledge. Stafford uses the mathematical knowledge and the know-how to design the mathematical instrument coming from the writings of their inventors, makers, publishers. As far as the sector is concerned, he refers to Edmund Gunter’s

Stafford’s

This increased distance between two traditions – the one of ‘De usu & fabrica’ and the one of arithmetic textbooks – nevertheless contributes to the circulation of instrument knowledge in a context where it would not necessarily be expected. This double move of bringing two genres together but immediately enhancing their distance again, clearly ensures a wider circulation if not of the instruments themselves so at least of the knowledge about them.

This paper is based on research financed by the

Henrique Leitão & Lígia Martins (eds.),

The full title of the work is ‘Arithmetica practica geometrica logarithmica’, in

This work could be spotted in a manuscript copy as Ignace Stafford,

The following works of the

Ad Meskens, ‘Michiel Coignet’s contribution to the development of the sector’,

For a list of Coignet manuscripts relating to all kinds of instruments, see: Meskens, ‘Coignet’s contribution’ (n.

Edmund Gunter,

‘[e]l uso de los instrumentos logarithmicos, y geometricos que pretende siempre apuntar, principalmente los de la gramelogia, del pantometra, y del radio, por ser los mas insignes, y expeditos, que hasta a ora se han inventado. Y porque son los de que se sirven las personas, por cuya contemplacion he tomado el presente assumpto entre manos’. Cf. Stafford, ‘Arithmetica’ (n.

‘El Autor del

‘Pero quando se executa por

João Pereira Gomes,

This designation appears in the subtitle of Stafford’s only printed work

Carlos Sommervogel (ed.),

Stafford, ‘Arithmetica’ (n.

The ‘real de prata’ was a Portuguese silver coin. Around

‘Proposicion ^{a} Como se reconoce el quarto numero proporcional a otros tres dados en proporcion direct; continua o discreta […]’; ‘

‘

‘

‘

‘[Prop.

‘[Prop.

‘[Prop. ^{a} ] Como se reconoce el valor, y proporcion de qualquier numero quebrado […] La señora moneta aun admitte mayor variedad en sus numeros enteros, y quebrados [than weight and volume measures] [...] y sin duda la Portuguesa es la mejor de todas; no solo por ser la mas facil; sino tambien porque ordinariamente se haze en partes menudas, consuela a los pobres, que con poca substancia pueden hazer ostentacion de millares de monedas’. Stafford, ‘Arithmetica’ (n.

One understands that there is a sense of a lack of mathematical training in the kingdom at that time from a short remark by the Jesuit Padre Provincial, Luis Lobo, on the back of the cover page to the printed elementary mathematics treatise by Stafford: ‘Licencia de Luis Lobo, P. Provincial da Comp[anhia] em Portug[al]

On Nunes’ work on theory of navigation, see commentary and notes in: Henrique Leitão (ed.),

‘[Prop.

‘

Paradoxically, in several instances, it is precisely the mathematics that will help to find out the best way to cheat.

An ‘analogy’ is the equation of two similar ratios. It is moulded after the Greek word for ‘proportionality’.

For instance the Act of Supremacy (

Henrique Leitão, ‘A periphery between two centres? Portugal on the scientific route from Europe to China (sixteenth and seventeenth centuries)’, in: Ana Simões, Ana Carneiro, Maria Paula Diogo (eds.), ^{4}

‘

The best account of the early history of the development of the sector starting from the second part of the sixteenth century, and many elements about Coignet, is given by Filippo Camerota,

‘Propos. ^{a} ; Probl. ^{o}: Como se executa la fabrica de las escalas de tangentes en el Pantometra, y las escalas de secantes. […] del numero, que corresponde en la tabla de tangentes al arco pedido, se quitan sinco notas a la mano derecha, siendo el radio

‘

‘[…] Hasta aqui los Autores [Gunter and Oughtred] destos instrumentos nos enseñan sus descripsiones y construciones. Si mejor que las precedentes proposiciones me ha cançado en valde: y si no, deseo que se desengañe la curiosidad de los que no se contentan sino con la expectacion de lo que siempre imaginan mejor en la fuente, que en el rio: y no es assi siempre porque muchas veses el agua en la fuente es cruda y demasiada mente delgada y corriendo gana perfeciones, como el Tajo por los arenas de oro, con que se enrriqueze’. Stafford, ‘Arithmetica’ (n.

Mario Biagioli, ‘From print to patents: Living on instruments in early modern Europe’,

Engraving of two sides of the sector with the text: ‘These instruments are wrought in brass by Elias Allen [...]’. From: Gunter,

Coignet’s sector as depicted in Hulpeau,

Sir Antony van Dyck,

A brass sector with scales labelled in Portuguese, unsigned and undated, inv. T.

Sector as depicted in Christophorus Clavius,