ESHS 2022 symposium: Approximation, Precision and Error as Actors’ Categories in the Alfonsine Astronomical Tradition

Organisation: Sophie Serra, Samuel Gessner


Most historiography of ancient astronomers’ achievements has pinpointed errors and approximations as a way to characterize them with respect to corrections and improved precision in the later course of history. In explicit contrast to this, the aim of this symposium is to ask how those astronomers themselves handled precision, approximation and error and how they thought about it. The case studies included in the symposium will address these problems on three levels.

The first looks at the Alfonsine astronomers’ awareness of problems related to manuscript transmission. The historical actors knew that variability, ambiguities and information loss are inevitable in the context of a manuscript culture. Among the various means to mitigate these problems are, for example, the careful checking or recomputing of specific quantities (e.g. radices, differences).

The second level concerns the computational procedures. For one and the same object (e.g. eclipse), depending on their goals and on the context, astronomers would recur to different procedures with different intrinsic precision. Also, their use of what we name today “significant digit” in computation often shows good control of precision, but still awaits a convincing explanation for the modern historian. Diagrams and geometrical instruments may also be interpreted as a means of error control. They simultaneously come, due to the imperfection of any material instrument, with a related vocabulary of approximation, precision and errors inherent in vision and sense perception.

Finally, the symposium will discuss Alfonsine authors’ epistemological stances on approximation, precision and errors. To what extent were they seen as inseparable from any human pursuit of truth? As a Scientia Media intermediate between mathematics and natural philosophy, was astronomy considered peculiar, regarding the approximation of truth? What meaning did numerical precision have for the Alfonsine astronomers ? What kind of reference to truth and its approximation were they interested in ?



Parisian pre-Alfonsine astronomy: changing times and changing values

Sophie Serra

One of the most popular of the astronomical texts that flourished in Paris at the turn of the 13th and 14 th century is the commentary on the Canon of Arzachel by John of Sicily, known by its incipit « Inter cetera veritatis philosophicae documenta ». This text was composed during the 1290s in Paris, and we don’t know much about its author. But this very lack of information, coupled with the compilation nature of the text), confers to it an important place in the study of the history of late medieval astronomy, since a lot of changes happened in Paris a few years later, giving birth to the « Parisian Alfonsine Astronomy ». As F.S. Pedersen has noted in his edition, the Inter cetera veritatis is « polished to an extent rarely seen in such works », thus offering an entry point into the pre-Alfonsine astronomers theoretical conception of astronomy and how they actually practised it at this turning point. It opens with a lengthy prologue where John paraphrases Ptolemy and Boethius in a vibrant defence of the preeminence of mathematical astronomy over the other theoretical sciences because of its unalterable objects, the precision of its results and the unmistakable nature of its conclusions. But at the same time, later in the text, John incorporates up-to-date values in computations, and often fails to discuss changes and revisions. Is it because he was not competent to assess the significance of such changes ? Because, as a compiler, he prefered to stay silent about numerical changes in order not to be forced to intervene on the textual parts ? Or because he did not want to address complex issues about truthfulness ?


Computing lattitudes as a medieval astronomer

Camille Bui

With the Toledan Tables, two traditions for computing planetary latitudes were spread across western Europe from the 11th century onwards. They either derived from tables from al-Khwarizmi or from the zij of al-Battani, hence following the Ptolemaic model. With the emergence of the Alfonsine Tables, some scholars such as John Vimond (1320) or John of Murs (1321) reworked the latter type of tables, although they did not change its parameters. This idea of “revising” latitude tables was pursued later on by other astronomers, in particular by William Batecombe who wrote the Oxford Tables (1348), Prosdocimo de’ Beldomandi (1424) and Giovanni Bianchini (1442). This has resulted in a variety of latitude tables whether they are more user-friendly, “extended” or use a double-entry system. Medieval astronomer were aware that different types of tables existed and sometimes had several at their disposal. By putting ourselves in the shoes of these historical actors and adopting their method of calculation, we will compute latitudes using different tables. It will be interesting to compare the results obtained and to try to deduce the practises of a medieval astronomer. What did he expect when choosing one table over the other? Was he aware of the differences that could be generated? Was he choosing to be faster in his calculations at the expense of accuracy or was it the other way round? Did he take into account other elements that we might have disregarded?


The challenge of precision measurements using astronomical instruments: the case of the directorium

Nick Jacobson

Sometime in the 1320s, a short treatise attributed to John of Lignères was written on an astronomical instrument called the directorium (preserved in BnF Latin 7295, ff. 78r-78v and elsewhere). The directorium enabled users to project a planet’s ecliptic longitude onto the celestial equator without recourse to ascension tables. This was necessary for finding what were called directions between different planets or planetary aspects, which was an essential technique in astrological medicine during the late Middle Ages. What is more, the text specifies how the instrument was designed to measure a planet from its location in latitude as well as longitude. In this paper I will explain how the instrument was designed such that it could be sensitive to the precise measurements of a planet in latitude. I will also discuss criticisms of the instrument found in the commentary on al-Qabisi’s Introduction to Astrology by John of Saxony who doubted its efficacy, and preferred using tables instead. I will consider under what conditions the instrument could have been useful to a practitioner, despite John of Saxony’s objections. In doing so, I hope to think more broadly about the inclusion of astronomical instruments in canon texts, and under what circumstances the approximative results of an instrument could be preferred to the more precise results attainable with astronomical tables.

Geometrical approximation and the correct planetary positions on early astronomical clocks

Samuel Gessner

In his unsurpassed study of planetary mechanisms from 1980 E. Poulle notoriously scolded historical authors when their mechanical constructions deviated from  the Ptolemaic models, or praised them when these perfectly complied. In contrast, in this paper I seek to adopt an agnostic position with regard to the makers’ choices. What did they consider to be the true reference for the mechanical imitation of planetary motion? Was it the planetary positions observed in the sky? Was it Ptolemy’s (or a different) geometrical constructions? Was it the numerical data within planetary tables? Documents addressing the makers’ stances and the criteria for their choices are rare. A few writings relating to planetary clocks, however, show that approximation was seen as legitimate, not only by the commissioners and mechanics but also by the astronomers involved in their elaboration. By examining the various mechanical approaches to planetary motion together with some rare documents the paper tries to reconstruct a historical understanding of an “adequate” planetary mechanism.