Geometrical reasoning and visual culture: shaphing astronomy and other scientiae mediae

Workshop ALFA
Observatoire de Paris, Salle du Conseil
January 18-22, 2021

org. Samuel Gessner


The workshop will be organised online, link will be sent to participant shortly before the workshop sessions. If you want to participate please email



This workshop will focus on visuality and materiality in scientific knowledge in the late medieval and early modern periods by bringing to the fore questions of the relationship between visual culture and geometrical reasoning. Recent history of knowledge production and reproduction emphasized how intertwined they are with their visual and material representations and codification. This trend is a good opportunity to look back at the powerful overviews of medieval scientific illustrations, including diagrams and tables, by Evans (1980) and Murdoch (1984) that still provide important orientation into the variety of formats and uses of the visual component in medieval books. Since their publication, Saito (2006, 2012) and Sidoli (2009) have done groundbreaking work for various cases of mathematics in classical antiquity. Eastwood and Grasshof (2003) and Obrist (2012) have extensively worked on planetary and cosmological diagrams in early medieval Europe. Taking inspiration from such work, the workshop proposes to take a closer look at some cases from various mathematical sciences in medieval and early modern times, from and beyond Europe, and the multiple functions and uses of diagrams and models.

Within the Alfonsine tradition (end 13th c. to end 16th c.) astronomers not only mastered multiple practices depending on what context they deployed their astronomical knowledge in (from teaching, explaining theoretical choices, preparing new forms of tables, explaining the use of tables, helping with calendar reform, casting horoscopes, or producing ephemerides and annual prognostications), but also, in accordance with the purpose, their reasoning depended on knowledge conveyed in various tools: on calendrical rules for dating historical observations, on numerical tables, on conventional algorithms, on geometrical arguments produced as prose accounts or as visualizations by diagrams or by brass and wood models, and so forth. The ERC project ALFA has produced an overview of the variety of material the Alfonsine astronomical tradition has bequeathed on us in the form of manuscripts with texts, tables, diagrams and instruments. For ALFA it is crucial to examine how astronomical knowledge expressed in these different media (formats of expression) relates, whether different formats also represent different ways of thinking about astronomy (reasoning styles) and when they were used by mathematical astronomers. In particular, we ask ourselves to what extent expressing knowledge in different media shaped astronomical practice and triggered specific developments of astronomy in the Latin West.

Center stage in this workshop is the question of the diverse roles played by geometry and to what extent it depended on the form of visualizations (as diagrams, graduated instruments and artisanal models) in the complex web of Alfonsine astronomical practices. To do so much can be learnt about the role of geometry and its visualization practice by widening the scope to other mathematical sciences or Scientiae Mediae (for instance optics, statics to name just the most obvious). By crossing perspectives through various fields of knowledge (astronomy, mathematics, and Scientiae Mediae) this workshop will shed new light on the history of mathematical astronomy and other mathematical sciences within a large time frame (ca. 13th to 16th c). It revisits topics explored by recent projects like “Diagrams, figures, and the transformation of astronomy” (Jardine 2008-2014) or work that dwelled on the edition of diagrams like the conference “The problem of diagrams and drawings criticism in mathematical texts” (Napolitani, Mascellani et al. 2004) or the paper by Jardine & Jardine (2010). But in this workshop the quest remains more focused: the visual expression of geometrical reasoning. By closely looking at the vehicles of visualization (formats and materials) of geometrical concepts can one detect changing roles of geometry? Three levels of questioning will be addressed:

  • The first level will look into the materiality of the visualizations, their constructive approaches and their formats and conventions. Geometrical aspects were visually expressed following variant conventions and in many formats: not only by diagrams on parchment or paper page, but also in the form of paper or parchment instruments that included graduated scales, sometimes moveable parts (volvelles) and silk threads (e.g. in the case of equatoria). In rare cases even three-dimensional models like armillary spheres and geared planetary models have survived until today. Depending on the techniques involved, different specialists were called upon to realize such visualizations. Indeed, often such collaborations of scholars and craftsmen yielded creative solutions and sometimes involved compromise. As Shank (2012) has shown for Regiomontanus’ diagrams, the material tools for producing or reproducing diagrams impacted their use and their connection to a related text. So adopting a similar level of analysis can we show to what extent visual and material expression of geometry shaped, pushed, drove or constrained the scientific arguments?
  • A second concern of the workshop is the specific topics that have seen geometrical developments within astronomy, and within other fields like mathematics and the Scientiae Mediae. Not every chapter of astronomical knowledge has equally seen its contents neither discussed in terms of geometry, nor were all geometrical aspects translated into any visual format. The distinction between matters that have been geometrized and others that were thought of in non-geometrical terms allows highlighting the actors’ expectations and understanding of the role of geometry. What weight did such geometrical notions have in astronomical thought in comparison with other ‘styles of reasoning’ (getting the chronology right, adjusting numbers in tables and algorithms)? The participants who investigate on fields other than astronomy will be challenged to evaluate the weight given to arguments referring geometrical properties as compared with arguments of another kind such as physical, numerical, chronological arguments. To what extent did historical scholars seek to coordinate the various dimensions of arguments? What were the available notions of ‘coherence’ among different levels of arguments?
  • Thirdly, the workshop will probe the diversity of roles of geometry and its representations. If Ptolemy in his Almagest famously used geometry to ground computational tables, this role is not necessarily the one that Alfonsine astronomers have assigned to geometry. To the contrary, some evidence indicates that astronomers – while working with conventional computations – searched for a way to make “geometrical sense” of them (as happened e.g. in the case of the movement of the 8th sphere). Likewise, in mathematics and the Scientiae Mediae both the function of geometry and also the means of its visualizations varied. The “moving mental images” described by Crowther & Barker (2013) are one important instance, while there are, however, many more variants: ranging from mnemotechnic figure, classificatory device, to lettered diagram supporting geometrical reasoning, and from nomogram-like instrument for graphical/spatialized computation to mechanized representation (as with geared equatoria and planetary clocks).

Participants: Agathe Keller, Henry Zpeda, Razieh Mousavi, Renée Raphael, Sabine Rommevaux, Samuel Gessner, Scott Trigg, Seb Falk, Matthieu Husson, Richard Kremer, José Chabas, Eleonora Andriani, Camille Bui, Ségolène Albouy, Tristan Dot, Leni Gauffier, Jean-Claude Penin




Monday, January 18th

2.30 – 5.15pm  (break 3.45 to 4pm)

Matthieu Husson (Observatoire de Paris-PSL, ALFA project), Samuel Gessner (Observatoire de Paris-PSL, ALFA project)


Sabine Rommevaux (CNRS, SPHERE, Université de Paris)

Rôle de la géométrie dans l’Arithmetica integra de Michael Stifel

Henry Zepeda (Assistant Professor, Wyoming Catholic College, Lander)

Geometry and geometrical reasoning in Peurbach and Regiomontanus’ Epitome Almagesti


Tuesday, January 19th

9.30 – 12.15pm (break 10.45 to 11pm)

Scott Trigg (University of Hong Kong)

Visualizing the configuration (hayʾa): Thinking with images in Islamic astronomical manuscripts

Razieh Mousavi (PhD student, PPIWG, Berlin)

Questioning the reliability of astronomical diagrams: al-Farghānī’s figure-free text on Ptolemaic astronomy


2.30 – 5.15pm  (break 3.45 to 4pm)

Agathe Keller (CNRS, SPHERE, université de Paris)

Uses of a trigonometric circle in relation to tables of sines and sine differences in some Sanskrit mathematical and astronomical commentaries

Samuel Gessner (Observatoire de Paris-PSL, ALFA project)

Visualizing the geometry of the 8th sphere in Alfonsine astronomy (15th and 16th c.)


Wednesday, January 20th

2.30pm – 5.30pm  (break 4.30 to 5pm)

Seb Falk (University of Cambridge)

Geometric diagrams and geared clockwork in the collected works of Richard of Wallingford

Renée Raphael (University of California)

Visualizing mines and claiming proprietorship: Geometrical reasoning in Luis Capoche’s 1585 Relación General del Asiento y Villa Imperial de Potosí

Wrap up discussion and conclusion



Sabine Rommevaux

Rôle de la géométrie dans l’Arithmetica integrade Michael Stifel

Michael Stifel fait paraître en 1544, à Nuremberg, une Arithmetica integra en trois livres, qui contient une arithmétique des nombres entiers et fractionnaires, une classification des nombres irrationnels associée à une lecture arithmétique du livre X des Éléments d’Euclide et une algèbre. Il s’agit donc fondamentalement d’un livre d’arithmétique dans lequel sont traités, de manière parallèle, les nombres entiers et fractionnaires, les nombres radicaux ou irrationnels et les nombres cossiques, soit les nombres de l’algèbre. Dans cette Arithmetica integra, la géométrie n’est toutefois pas absente. Nous nous interrogerons sur son rôle et nous verrons comment la notion médiévale de “contractio” peut nous éclairer à ce sujet.

Michael Stifel publishes in Nuremberg the three books of his Arithmetica integra, in 1544. It contains an arithmetic of integer numbers and fractions, a classification of irrational numbers associated with an arithmetic reading of book X of Euclid’s Elements, and an algebra. Therefore, basically, the book is about arithmetics and is treating integer and fractional numbers, the radicals, irrationals and cossic numbers, i.e. the numbers of algebra, in parallel. In this Arithmetica integra geometry, nevertheless, is not absent. We are going to question its role and will see how the medieval notion of “contractio” helps elucidate this affair.


Henry Zepeda

Geometry and geometrical reasoning in Peurbach and Regiomontanus’ Epitome Almagesti

The Epitome Almagesti of Regiomontanus, unsurprisingly, falls within the Ptolemaic tradition of astronomy, and thus it contains many geometrical figures, some copied directly from the Almagest. I will thus examine some of the uses and features of geometrical figures in the medieval Ptolemaic corpus. The Epitome also contains many figures that are not in the Almagest, and Regiomontanus frequently modifies its figures in significant ways. Of particular interest are the ways in which non-geometrical passages and drawings from the Almagest, e.g. material on instruments and methods of observing, are made geometrical by Regiomontanus and the ways in which he adds geometrical figures for material that he has taken from sources that are not geometrical. I will also briefly address the strange phenomenon, not unique to the Epitome, of manuscripts that lack the figures that are so essential to understanding the text.


Scott Trigg

Visualizing the configuration (hayʾa): Thinking with images in Islamic astronomical manuscripts

Ulugh Beg’s 15th c. Samarqand observatory and madrasa is one of the most famous Islamic scientific institutions, producing astronomical observations that were not equaled until Tycho Brahe. Less is known about the processes of research and education at Samarqand, but a number of commentaries produced by Samarqand scholars promise to shed light on the intellectual life of the classroom. Instruction in the sciences was primarily an aural process, however, as anyone who has perused the Almagest can attest, it is quite difficult to visualize the complex and dynamic motions produced by the mathematical machinery of eccentrics and epicycles without the aid of images.

In this paper, I explore the relationship between image and text in Islamic manuscripts from the 13th-15th centuries, particularly commentaries written by the director of the Samarqand observatory Qāḍīzāde al-Rūmī, and one of Qāḍīzāde’s prominent students Fatḥallāh al-Shirwānī. Using examples of how these commentators portrayed the components and motions of complex models, as well as more basic celestial phenomena, I describe how students could begin to put together an impression of how they combined to produce the apparently irregular motions we observe in the heavens, as well as the outstanding problems of Ptolemaic planetary theory that these astronomers were attempting to explain and resolve. I analyze the extent to which images may have helped the reader visualize different kinds of motion and extrapolate from two-dimensional static “mathematical” images to the three-dimensional dynamic “physical” motions of the heavens. I suggest that differences in the kinds of images each commentator chose to add to their texts, the manner in which they portrayed the models and particular motions, and the ways in which they highlighted and analyzed different issues reflect differences in the intended audiences for the two commentaries.


Razieh Mousavi

Questioning the reliability of astronomical diagrams: al-Farghānī’s figure-free text on Ptolemaic astronomy

No doubt, the Elements of Astronomy by al-Farghānī (fl. c. 850 CE) is among the most read astronomical texts in the premodern world in both Arabic and Latin. That is not only confirmed by the number of surviving manuscripts, but also by the wide range of references to this book by medieval scholars. In spite of this large popularity, the purely descriptive format of the text singularizes it among many technical astronomical works. Although the Ptolemaic model of the universe is faithfully followed, al-Farghānī’s text abstains from any illustration, let alone any geometric diagram. In contrast, in his extended book on astrolabes, he adopts a different attitude and begins his work with a discussion on the significance of geometrical reasoning for affirming the reliability and functionality of astronomical instruments. This dual attitude might at first appear contradictory if one fails to discern between distinct formats of astronomical exposition in the early Islamic time. A further apparent contradiction lies in that al-Farghānī emphasizes the “principles of the celestial motions” as a goal for compiling the Elements of Astronomy, which signals the incongruence between the material (physical) representation of the cosmos and geometrized modelling in his viewpoint.

In this paper, I compare al-Farghānī’s writing style in his two surviving books and explore the arguments of the ninth-century Muslim astronomer for explaining Ptolemaic astronomy without the recourse of diagrams. To do so, I will also examine early planetary diagrams as a part of the visual culture of scientific works in Islamic astronomical texts.


Agathe Keller

Uses of a trigonometric circle in relation to tables of sines and sine differences in some Sanskrit mathematical and astronomical commentaries

In early theoretical Sanskrit astronomical texts (siddhānta), tables of sines (ardha-jyā) and sine differences (khaṇda-jyā) often come with rules to explain how the values they contain can been derived. The earliest commentaries on these rules rest on the construction of a diagram, a trigonometrical circle, in which these geometrical derivations can be visualized. In this paper, I will argue that this diagram is a mnemonic device with a subsequent long presence in Sanskrit astronomical and mathematical literature. I will also explore how the first prose text describing the construction of this diagram, Bhāskara’s (629) commentary on the Āryabhaṭīya (499), was used and transformed by subsequent commentators. I will thus contrast Pṛthūdhaka’s ( 850) construction and use of this diagram in his commentary on Bhramagupta’s derivation of a Sine table in the Theoretical Astronomical Treatise of the True Brāhma School (Brāhmasphuṭasiddhānta 628) with the one prescribed by Bhāskara showing how the same diagram is constructed differently and especially how differently the diagram, the geometrical reasoning and the numerical computations are articulated in both commentaries. The paper will also try to account, more tentatively, for the reasons of such differences.


Samuel Gessner,

Visualizing the geometry of the eighth sphere in Alfonsine astronomy (15th and 16th c.)

Combining a linear with an oscillating term (“access and recess”, later “trepidation”) to describe the movement of the 8th sphere was understood, e.g. by John of Murs in his Expositio (1320s), as a way to account for the shift in longitude of the planetary apogees and the fixed stars, typical of Parisian Alfonsine astronomy. Historians have focused on astronomers’ arguments against and for trepidation but rarely questioned the tools available to the historical actors in these discussions like geometrical concepts, diagrammatic practice and other visual tools. The geometry of both motional components was clear from historical texts: Ptolemy’s Almagest and Pseudo-Thabit’s De motu octave spere, respectively. The earliest text known to provide diagrams in an attempt at a geometrical interpretation of the combined movement is by Nicolaus Comes de Comitibus (1450) followed soon by Peurbach’s Theoricae novae planetarum (1453). In this talk I look into the way these accounts resort to discourse combined with diagrams. Moreover I compare and contrast them with a few examples of later three-dimensional constructions, instruments traditionally called “trepidation spheres”, that show the 8th sphere. I will seek to clarify what the medium – using words, diagrams or brass models – implies, in terms of what geometrical aspects become explicit or remain hidden.


Seb Falk

Geometric diagrams and geared clockwork in the collected works of Richard of Wallingford

Oxford, Bodleian Library MS Ashmole 1796 is the most complete collection of the astronomical works of Richard of Wallingford (1292-1336). Produced c. 1350 at St Albans Abbey, where Wallingford was abbot from 1327 until his death, it contains the only copy of his treatise on the astronomical clock he built for the abbey church. This incomplete treatise contains five pages of illustrations showing the gearing for its complex astronomical functions. The illustrations, probably added after the treatise was written, combine multiple perspectives and display both practical and decorative elements. This presentation will discuss their purpose, and will compare them with the diagrams of the Albion treatise in the same manuscript, which vary from sparsely geometric to evocatively tangible.  The diagrams have the potential to tell us much about how Richard and his later readers saw the purposes of such instruments: as three-dimensional diagrams, teaching tools, practical devices or even historical artefacts.


Renée Raphael

Visualizing mines and claiming proprietorship: Geometrical reasoning in Luis Capoche’s 1585 Relación General del Asiento y Villa Imperial de Potosí

Previous scholarship has explored the topic of subterranean geometry associated with treatises on mining. However, a closer examination reveals a distinction in this sub-genre of mining texts: geometrical reasoning only comes up when authors are interested in claims of ownership or describing geography. This pattern suggests that subterranean geometry was more closely associated in the early modern period with the practices of surveying, not mining. After a survey of classic texts on Renaissance mining drawn from Italian, Latin, German, and Spanish printed sources, I will illustrate the practical use of subterranean geometry in several archival documents related to mining in the Spanish Americas, focusing specifically on Luis Capoche’s 1585 Relación General de la Villa Imperial de Potosí.


Selected bibliography

Crowther, Kathleen M. & Peter Barker, “Training the Intelligent Eye. Understanding Illustrations in Early Modern Astronomy Texts”, Isis, vol. 104, n. 3, 2013, p. 429-470.

Eastwood, Bruce & Gerd Graßhoff, “Planetary Diagrams – Descriptions, Models, Theories. From Carolingian Deployments to Copernican Debates”, in The Power of Images in Modern Science. J. Renn W. Lefèvre, U. Schoepflin eds. (Basel: Birkhäuser, 2003), p. 197-226.

Evans, Michael W., “The Geometry of the Mind”, Architectural Association Quarterly, vol. 12, no. 4, 1980, p. 32–55.

Jardine, Boris & Nicholas Jardine, “Critical editing of early modern astronomical diagrams”, Journal for the History of Astronomy, vol. 41, 2010, p. 393-414.

Jardine, Nicholas (PI), Project “Diagrams, figures, and the transformation of astronomy”, 2008-2014, AHRC, AH/F011024/1.

Murdoch, John E., Album of Science: Antiquity and the Middle Ages (New York: Scribner’s Sons, 1984).

Napolitani, Pier Daniele, Ottavio Besomi, Enrico Giusti, Carlo Maccagni, “The problem of diagrams and drawings criticism in mathematical texts”, 2nd Bricks focused workshop, University of Pisa, 25-27 November 2004.

Obrist, Barbara, “Visual Representation and Science: Visual Figures of the Universe between Antiquity and the Early Thirteenth Century”, Spontaneous Generations – Journal for the History and Philosophy of Science, vol. 6, n. 1, 2012, p. 15-23.

Saito, Ken & Nathan Sidoli, “Diagrams and arguments in ancient Greek Mathematics: Lessons drawn from comparisons of the manuscript diagrams with those in modern critical editions”, in Karine Chemla ed., History and historiography of mathematical proof in ancient traditions (Cambridge: Cambridge University Press, 2012), p. 135-162.

Saito, Ken, “A preliminary study in the critical assessment of diagrams in Greek mathematical works”, SCIAMVS vol. 7, 2006, p. 81-144.

Shank, Michael H., “The geometrical diagrams in Regiomontanus’s edition of his own Disputationes (c. 1475): background, production, and diffusion”,  Journal for the History of Astronomy, vol. 43, 2012, p. 27-55.

Sidoli, Nathan & Ken Saito, “The Role of Geometrical Construction in Theodosius’s Spherics”, Archive for History of Exact Sciences, vol. 63, 2009, p. 581-609.