The practice of geometry in medieval astronomy

Symposium in the 9th ESHS Conference

31.08-03.09 2020

org. Samuel Gessner, Richard Kremer



Alfonsine astronomy, widely prevalent in Latin Europe from the late 13th through the early 16th centuries, is generally thought to have been based on a set of tables commissioned by King Alfonso X of Castile and to have consisted primarily of tables used for computing eclipses and planetary positions for astrological purposes.  ALFA, an ERC-funded, international, collaborative project launched in 2017, has been expanding this understanding of Alfonsine astronomy by finding, in the manuscript codices, scattered but richly diverse visual materials that supported astronomical practices extending beyond computation.  This session will explore this visual culture by analyzing the interdependence of computation and the mathematical understanding of astronomical models in their visual, geometrical, or material realizations. Addressing the role of diagrams in canon texts, the use of geometric constructions for problem solving, the spatialization of quantitative data, and mechanical (i.e., in brass) expressions of abstract computational models, the case studies presented in the session will illustrate diagrammatic and material practices within Alfonsine astronomy. Including two speakers from ALFA and two external collaborators, the session will feature presentations by early career investigators coming from a variety of backgrounds (history of science, astronomy, mathematics, cultural history, history of philosophy, philology, history of the book and libraries) as well as comments and questions by more senior scholars.


Program and abstracts

Nick Jacobson (Paris Observatory, ALFA project)

The justificatory role of diagrams in an anonymous set of fourteenth-century canons for the Alfonsine tables (Erfurt Q366 ff. 70v-73v)

In the middle of the fourteenth century, Johannes de Wasia (d. 1395), a master of arts and later a theologian at the Sorbonne, collected several sets of canons for the Alfonsine tables. One of these sets included diagrams designed to aid the user in finding the true positions of the planets. An extant witness to these canons has been preserved in a manuscript in Erfurt with the shelf mark UFB, Amplon. Q. 366, ff. 70v-73v. Scholars have thought it rare for canons in the Alfonsine tradition to contain diagrams treating planetary motion, as users could limit themselves to strictly arithmetic techniques when operating the tables to these ends. The Erfurt manuscript, however, provides clear examples of the integration of geometrical methods into the computational procedures of the canons. In this paper, I will present on a canon from this set that offers procedures for finding true solar positions. I argue that there are actually two parallel chains of exposition within the canon, the first of which instructs the user in purely arithmetical terms, while the second resorts to a geometrical demonstration with a strong reliance on the geometrical diagram. This format of presentation bears many similarities to that of the Theorica Planetarum of Campanus of Novara, which suggests that the diagram for the canon served a justificatory function of the efficacy of the canon’s procedures. If I am correct that the geometrical element of the canon offers – to use Campanus’s words – “the reasoning behind the mode of operation with tables,” then it is also possible that this second portion of the canon gives the reader glimpses, not only into how tables were used, but also into the methods by which a table-maker constructed valid tables in the first place.


Laure Miolo (PSL-IRIS-Scripta, Paris Observatory, ALFA project)

Figuring and calculating eclipses at the end of the 15th century: Lewis of Caerleon and his geometrical canons

Lewis of Caerleon (d. c. 1495) was a physician trained in Cambridge, and perhaps also in Oxford. His case presents a striking example of astronomical practices in a troubled context, that is the War of the Roses. A physician of the Lancastrians, he was incarcerated for conspiracy by Richard III in the Tower of London in 1484. Released when Henry VII became king of England, in August 1485, he was appointed as Royal physician. In parallel with his career as a physician, Lewis of Caerleon was an expert in astronomy. During the 1480s he composed eclipse and parallax tables, eclipse canons and other short texts related to eclipse phenomena. He also performed a series of eclipse calculations ranging from 1481 to 1485. His incarceration did not stop his activity as he continued to compose texts and table in the Tower of London. Amongst several eclipse canons he composed, there is a canon for calculating eclipses geometrically: De modo calculandi eclipses geometrice sine tabulis. This text is preserved in two manuscripts, displaying diagrams for illustrating it. Canons for figuring geometrically an eclipse are quite common in the Medieval and Early Modern Latin tradition. Although Lewis of Caerleon’s canons contains a part dedicated to this geometrical figuration, the main purpose is to calculate ‘geometrically’ the magnitude and duration of the eclipse in order to avoid the use of the tables, which is unusual. My paper will focus on this peculiar text. In that framework, I will explore Lewis of Caerleon’s sources, vocabulary, and diagrams in order to better understand the aim of these canons as well as to situate them in context.


Angela Axworthy (Max-Planck Institut für Wissenschaftsgeschichte, Berlin)

Geometrical models of the universe in the fifteenth century: the case of Prodoscimo de’ Beldomandi

As a means to provide an insight into the ontological and epistemic status of geometrical models of celestial motions in fifteenth-century Alfonsine astronomy, this paper will consider the cosmological teaching of Prosdocimo de’ Beldomandi (1370/80–1428), who was a professor of mathematics at the university of Padua and who contributed to the Alfonsine tradition by providing a new compilation of his predecessor Jacopo de Dondi’s tables for Padua in 1424. As others before and after him, Beldomandi tackled the issue of the geometrical representation of the universe within a commentary on the Tractatus de sphaera by John of Sacrobosco, which he commented in 1418 and which was printed in Venice in 1531. Beldomandi was then addressing an audience of students, and above all of beginners in the field of astronomy, as was Sacrobosco in his time. But the extensive discussion of the content of the Sphaera by this Paduan professor, who was influenced by the fourteenth-century Parisian school of natural philosophy as well as by the work of his predecessors of the university of Padua, leads the reader quite far from the rather schematic teaching of the Sphaera and from what students were expected to know about spherical astronomy in this context. An overview of the parts of this commentary on Sacrobosco pertaining to the number, form and mutual relation of celestial spheres, the possibility of contrary motions in the celestial realm, as well as the ontological status of partial orbs, will enable to present Beldomandi’s views on the role of geometry in the apprehension and the depiction of the celestial order and of planetary trajectories.


Samuel Gessner (Paris Observatory, ALFA project)

The transposition of numerical data from tables onto the scales of instruments: Theorice novelle (15th c.)

Alfonsine astronomy included a practice of reworking tabular source material into new and often subtly clever formats. Besides novel types of tables, astronomers have also attempted to recast the tabulated quantities into diagrams, among which planetary equatoria. Poulle distinguished between ‘geometrical’ and ‘mathematical’ equatoria based on whether their shape materializes eccentricities and epicycles of the planets, or whether their configuration rather allows to ‘seperately obtain the components familiar to table users’. Within this second group, different ways to ‘spatialize’ quantitative data make their appearance during the Alfonsine period, including the special syzygy instruments with oblique lines from the 1440s, horoscopic instruments with “curvy” lines, both studied by Kremer (2011), and Petrus Apianus’ and Philipp Imser’s latitude instruments of 1540 and 1550s. This paper will look into the small corpus of Teorice novelle instruments established by Poulle (1980). This type emerges within astronomical circles in Erfurt and Leipzig and often comes with mean motion tables and user instructions. Directly examining the material realization of some instruments in parchment or paper allows tracking the problems the maker confronts when pursuing the simple principle on which they are based. Scrutinizing manuscript instructions for the Teorice Novelle provides hints at the mindset behind their conception. Together both approaches allow going a step beyond Poulle in characterizing the instruments’ relationship with the tabular approach in astronomy.