# Alfonsine Astronomy: mathematical practices

28.09. 2020-02.10.2020

Organisation: José Chabás, Richard Kremer, Matthieu Husson

This conference will be held in a virtual room. The conference is open to the public upon registration. To register simply send an email to pascale.baudoin@obspm.fr at the latest on Septembre 24. The appropriate contact information will be sent to participants via email.

## Rationale

The central scientific aim of ALFA’s second phase is to “analyse the Alfonsine astronomers’ practices, their relations to mathematics, to the natural world, to proofs and justification, their intellectual context and the audiences they addressed”. To tackle this aim, we will focus more specifically on the mathematical practices attested in the Alfonsine corpus. The kernel of the astronomers’ practices is mathematical: the kind of numbers and quantities they shaped, their relation to computations, rounding and interpolation, their use of diagrams and instruments, their use of standardized procedures, or their attitude toward proofs and justification.

It will be important to analyse each of these different layers of mathematical practices, ranging from numbers and quantities to procedures and argumentation, on its own sake by asking specific questions about the ways Alfonsine astronomers engaged with them in order to address astronomical problems and produce various types of solutions, be they computational, graphical, tabular, or most often a mix thereof. It will be also important to consider, from the manuscript evidence, how these different layers of the Alfonsine mathematical practices are articulated in order not only to provide solutions to astronomical problems but also to reveal self-reflexive understandings and evaluations of these astronomical problems by the historical actors.

Thus, we seek to come closer to the Alfonsine astronomers’ practices and to assess more precisely what kind of mathematical astronomy they fostered and what were its main lines of tension and development over the course of the 14^{th} and 15^{th} centuries. This, in turn, will enrich our understanding of the attitudes of Alfonsine astronomers on broader issues like empirical observation, relation to natural philosophy, or links to astrology or medicine.

This conference is the first of two scheduled for this set of issues (the second will occur in 2021). In this first meeting, our aim is to discuss possible case studies in order to explore the issues briefly sketched above. Alfonsine astronomy was not an isolated nor an entirely autonomous set of practices. It has profound roots in the mathematical astronomy developed earlier in *al-Andalus* and shares space with other astronomical traditions, for example, that found in Hebrew or Byzantine Greek materials.

## Participants

Albouy Ségolène, Andriani Eleonora, Avelar Helena, Bui Camille,Glen Van Brummelen; Chabás José, Covanov Svyat, Dot Tristan, Gessner Samuel, Hadrava Petr, Hadravova Alena, Husson Matthieu, Jacobson Nick, Kremer Richard, Lanunza Tayra, Leone Isabella, Levy Tony, Miolo Laure, Mozzafari Mohamed, Penin Jean-Claude, Saby Marie-Madeleine, Samso Julio, Tur Alexandre, Zieme Stefan

## Program

NB: the hours indicated are for Paris civil time

**Session 1: new sources**

**Monday 28, September 2020**

**16.30-19.30**

**Alena Hadravova**, On *Canons to the* Tabule resolute

**José Chabás**, *Computational practices in mathematical astronomy: Hermann of Saxony’s version of the Parisian Alfonsine Tables*

**Laure Miolo**, *Lewis of Caerleon (d. 1495), a fifteenth-century physician and astronomer working on eclipses*

**Session 2: Computing (with) tables**

**Tuesday 29, September 2020**

**16.30-19.30**

**Mohammed Mozzafari**, *Observational basis for planetary parameters in the* Alfonsine Tables

**Petr Hadrava**, *Mathematical treatment of Alfonsine trepidation*

**Stefan Zieme**, *The tables in Gerard of Cremona’s Latin Translation of the *Almagest

**Session 3: Computing (with) tables**

**Wednesday 30, September 2020**

**16.30-19.30**

**Jean Claude Penin**, *A 15th century craftsman, astronomer and mathematician : Jean Fusoris*

**Camille Bui**, *The evolution of computation of planetary latitudes throughout the first half of the 14th century in Eastern Europe.*

**Glen van Brummelen**, *The Emergence of Auxiliary Astronomical Tables in Medieval Europe*

**Session 4: working with texts, tables and diagrams**

**Thursday 1, October 2020**

**16.30-19.30**

**Nick Jacobson**, *The Role of Planetary Diagrams in a Fourteenth-Century Canon Set to the Alfonsine Tables (Erfurt Q366 ff. 70v-73v)*

**Samuel Gessner**, *Uneven scales on instruments: using tables and geometry in Alfonsine astronomy*

**Matthieu Husson**, *Interpolations in alfonsine astronomy: canons, tables and diagrams*

**Session 5: Mathematical practices, Ephemerides and astrology**

**Friday 2, October 2020**

**16.30-19.30**

**Richard Kremer**, *What can we learn about computational practices from a study of medieval ephemerides?*

**Alexandre Tur **Figurae caeli* as a source for the history of astronomical practices*

**Helena Avelar ***John of Saxony 1327 canons and astrology*

**Session 6: General discussion**

**Friday 2, october 2020**

**19.30-20.00**

## Abstracts

**Alena Hadravova** (Academy of Sciences of the Czech Republic)

*On canons to the Tabule resolute*

“Tabule resolute” have been explained in several sets of canons identified already in the secondary literature. In the last year, I devoted myself to the canons “Mirabilis in altis Dominus”. Now, I would like to add several notes on two newly identified copies of the canons “Mirabilis” in the manuscript preserved in the library of Regional Museum at the castle Mikulov, and then to continue with chosen remarks on the canons “Girum recensendo” by Andreas Grzymała from Poznań, “Nemo integre sapit”, calculated to the year 1388, and on very detailed and exact canons “Pro introduccione generali” by John of Głogów.

**José Chabás** (Pompeu Fabra University Barcelona, Spain)

*Computational practices in mathematical astronomy: Hermann of Saxony’s version of the Parisian Alfonsine Tables*

Hermann of Saxony is the author of a set of tables accompanied by a text, which seems to be his only extant work. Hitherto, this astronomer and his work have been given little scholar attention. He compiled tables for 1359 for the meridian of Paris, making Hermann of Saxony a member of the second generation of Parisian Alfonsine astronomers, after that of the Johns (Vimond, John of Murs, John of Lignères, and John of Saxony, among others). The text that goes with the tables display detailed examples of computations based on them, revealing the mathematical practices of an Alfonsine astronomer at work.

**Laure Miolo** (EPHE-PSL, ALFA, France)

*Lewis of Caerleon’s geometrical explorations*

In the troubled context of the War of the Roses, Lewis of Caerleon, physician of Elizabeth Woodville, Margaret of Beaufort and her son Henry (the future king Henry VII), produced canons and tables devoted to eclipse computations. Trained in Cambridge (BMed in 1465-1466) and doctor of Medicine by 1481, he was a fervent supporter of the Lancastrian faction. This political involvement led him to be incarcerated in the Tower of London by Richard III in 1485. From 1481 to 1485, Lewis of Caerleon wrote several canons and composed set of parallax and eclipse tables. In this range of time, three different moments – during which he revised his works – can be considered as defining in his activity. In that framework, the evolution of his astronomical practices is legible for every canons and tables he produced. Although he innovated in creating new works, the physician relied on important sources such as al-Battānī, Richard of Wallingford, Simon Bredon, Jean de Lignères or the Cantabrigian astronomer John Holbroke. Thanks to his extant notebook which contains a part of his sources and some of the primitive versions of his astronomical works we are able to have a better understanding of his working methods and practices. Furthermore, the fact that other of his works are compiled in manuscripts commissioned by him for wealthy patrons also questions the intended audience of his treatises and tables. Lewis of Caerleon’s main concern was the eclipse computation and more specifically the calculation of the magnitude and the duration of the eclipse. For that purpose, he composed sets of eclipse and parallax tables as well as canons related to them. However, he also provided a peculiar canon allowing to avoid the use of the tables in the eclipse computation. It is said to be entirely based on geometry. This paper will explore Lewis of Caerleon’s geometrical practices and adaptations through an analysis of this specific canon preserved in two witnesses and his sets of eclipse and parallax tables.

**Mohamed Mozzafari** (Maragha Observatory, Iran)

The previous studies have already confirmed that the mean motions in the Parisian Alfonsine Tables are closely related to those found in the Western Islamic astronomical tradition, except for the mean lunar anomalistic motion; J. Chabás and B.R. Goldstein reasonably suggested that a lost work by Ibn al-Zarqālluh may be the common source. Nevertheless, there are some pieces of evidence, indicating that the role of observations in the development of astronomical theories in Toledo in Alfonso’s day should not be underestimated. First, the reports of three lunar eclipses occurring on 24 December 1265, 19 June 1266, and 13 December 1266 have survived from Isaac b. Sid, one of the authors of The Castilian Alfonsine Tables: the mid-eclipse times given by Isaac have a systematic negative error, not more than –20 minutes, which evidently show that he was a good observer in the context of medieval astronomy. Second, the solar theory provides a good criterion for the assessment of medieval astronomical tables with regard to their dependence on new observations: for the 1260s, the errors in the true longitude of the Sun according to the Alfonsine theory varies between –10′ and +40′, whereas the errors in Ibn al-Zarqālluh’s theory (underlying the Western Islamic zījes at the time) changes from –39′ to –61′. The error in the Alfonsine theory reaches zero around the autumnal equinox, and for about six months––the second month of spring through the second month of autumn––the errors are below ±10′. This significant accuracy is mainly because of Alfonso’s astronomers putting aside Ibn al-Zarqālluh’s trepidation theory as well as, to a lesser extent, presenting new epoch value for the mean longitude of the Sun. Third, the trio of lunar eclipses meets all critical conditions needed for the derivation of the radius of the epicycle, according to Almagest IV.6 and IV.11, and the mean motion of the Moon in longitude and in anomaly. Alfonsine value for the mean lunar motion in anomaly is superior to all values determined for it in Islamic astronomy. Fourth, the Alfonsine Tables has a new value of 3.12 for the eccentricity of Jupiter.

**Alexandre Tur** (BnF, ALFA, France)

Figurae caeli* as a source for the history of astronomical practices*

Astrological predictions, as well as annotated almanachs and ephemerides, often display a schematic representation of an horoscope in the form of an astrological square, known under the Latin name “figura caeli” (shape of the sky). These diagrams appear as being the astronomical ground for astrological predictions, as if they were a fronteer between astronomical computation and astrological interpretation. Lots of clues point however towards a more complex nature. In astrological prognostications, for instance, astrological squares are often traced but left blank. In other cases, they do not coincide with the textual analysis of the astronomical situation. Although their template seem to remain unchanged during the Alfonsine era, several variations can be identified : some astrologers, such as the 15th century Londoner Richard Trewythian seem for instance to have a preference for “doubble” horoscopes, aggregating in the same astrological square the astronomical positions of an event and the previous syzygy. Knowing more about how they were made and why would be a progress towards a better understanding of the relation between astronomical knowledge and calculation and their astrological application for alphonsine practitionners.

**Helena Avelar** (CIUHCT Lisbon, Portugal)

This presentation addresses several practical uses of the Parisian Alfonsine Tables by exploring some of the canons of 1327 by John of Saxony. These were widely copied together with the tables in the late Middle Ages and were included in the editio princeps of the Parisian Alfonsine Tables. This circulation provides an excellent basis to understand some major uses of the tables and to explore their application in astrology, namely in the specific calculations required to produce astrological charts. These could be used in any of the four main applications of astrology: Revolutions, that is, the study of the cyclical movements of the stars and their effects in kingdoms and regions; Nativities, the study of the horoscopes of individuals; Interrogations, the charts of specific questions, and Elections (sometimes called Inceptions), the selection of the best moment to begin an action or endeavour.

**Petr Hadrava** (Academy of Sciences of the Czech Republic)

*Mathematical treatment of Alfonsine trepidation*

The trepidation, i.e. the supposed nonuniformity of precession of ecliptic, was assumed to determine not only the longitudes of fixed stars but also the positions of planetary apogeas. The tables of mean motion and equation of the ‘accessus and recessus octave spere’ were thus a necessary part of Alfonsine tables of planetary motions. Their values, however, do not correspond to the (pseudo-) Thebit’s treatise which is usually referred to in this context. An analysis of numerical values in these tables enables us to find the algorithm used for the construction as well as the errors in the computations of these tables.

**Stefan Zieme** (Humboldt University Berlin, Germany)

*The tables in Gerard of Cremona’s Latin Translation of the Almagest*

Until the 15th century, knowledge of the Almagest in the Latin West was constituted by Gerard of Cremona’s translation from Arabic into Latin. The text of Gerard’s translation has been examined carefully and its dependence on two different Arabic variants is well studied. However, the tables have not been scrutinized, and the relation to their Arabic or Greek counterparts is unstudied. In this talk, I will analyze the historical mathematical structure of tables in Gerard’s translation of the Almagest in comparison to their Arabic and Greek precursors. While Gerard’s text has proved to be a slavish translation from Arabic templates, the tables will turn out to be significantly different. The tables for e.g. the chord, declination, rising times, and solar equation appear to be recomputed in order to match Ptolemy’s textual explanations, which, in contrast, generally diverge in both Greek and Arabic tradition. The analysis of the tables offers a rare opportunity to scrutinize mathematical practices of early Latin astronomical material.

**Jean Claude Penin** (Independant researcher, ALFA, France)

*A 15th century craftsman, astronomer and mathematician : Jean Fusoris.*

To build his instruments of astronomy and calculate planetary tables, Jean Fusoris needed accurate trigonometric tables which gave him the chords of fractions of degrees. Whereas he did not own the tables that he would like, he built his own chords trigonometric table by steps of 15 minutes at a higher accuracy than the existing tables. After giving a short biography of Fusoris and a brief account of his works, I will analyse his method, pretty original it seems, to build his tables and I will put this work in the context of this time.

**Camille Bui** (CNRS-Observatoire de Paris-PSL, ALFA, France)

*The evolution of computation of planetary latitudes throughout the first half of the 14th century in Eastern Europe.*

Planetary latitudes are one of the last steps to accomplish in computing planetary positions. But in medieval astronomy, the tables used to compute planetary latitudes haven’t changed a lot since Ptolemy’s Almagest. However, during the fourteenth century in Europe, many modifications are brought to the latitude tables : the model underlying is left unchanged but the layout varies from author to author. Starting with John Vimond to William Batecombe, we will see that the different arrangements of the tables are probably conceived to ease the user’s computations. It will also allow us to understand how these kind of tables are made, as they are likely derived from more classical latitude tables such as the Toledan tables.

**Glen van Brummelen** (Trinity Western University, Canada)

*The Emergence of Auxiliary Astronomical Tables in Medieval Europe*

Auxiliary astronomical tables were a substantial and extensive tradition in medieval Islam, beginning as early as the 9^{th} century. These tables, computing functions that are more complicated than primitive trigonometric quantities but with no direct astronomical application, arise naturally in the context of spherical astronomy where solutions to different problems often share mathematical elements. We are fortunate to have two treatises with the same title ― the *Tabulae primi mobilis *― that allow us to trace the gradual birth of the idea of auxiliary tables in the works of their European inventor, the Italian astronomer Giovanni Bianchini, leading to their fullest realization in his *Tabulae magistrales*. Repeating the evolution in medieval Islam, one of these original auxiliary tables evolved into what we now call the tangent function. Regiomontanus copied Bianchini’s idea in his *Tabulae directionum* but took the notion much further in his single giant auxiliary table, his *Tabula primi mobilis*, a table whose idea would be rediscovered several times in following centuries. We shall trace the development of auxiliary tables from its European origin in the 15^{th} century through the end of the 16^{th} century

**Richard Kremer** (Darmouth College, USA)

*What can we learn about computational practices from a study of medieval ephemerides?*

Computation of long series of true planetary positions at regular time intervals (e.g., days) or consecutive eclipses has long been a central task of mathematical astronomy, from the Babylonian goal-year texts through Greek, Arabic and Latin astronomy and continuing into the modern period with the national editions of astronomical almanacs prepared by the Nautical Almanac Office of Great Britain, the US Naval Observatory, or the French Bureau des longitudes. Computing ephemerides necessitated a series of choices: which astronomical tables, local meridians, equation of time, parallax corrections, levels of precision, computational short-cuts, and output formats. During the Alfonsine period, daily ephemerides and eclipse lists were prepared not infrequently, especially in university milieus. The Alfonsine practice peaked with the printing of Regiomontanus’s massive ephemerides (1475-1506) in 1474. In this paper, I will explore what we can learn from these finished ephemerides about the computational practices of their makers. Can recomputation and statistical methods enable us to reconstruct how they computed positions, i.e., how they used the Parisian Alfonsine Tables and related eclipse tables in what J.D. North has called “feverish computational activity” (1977, p. 290)?

**Nick Jacobson** (CNRS-Observatoire de Paris-PSL, ALFA, France)

*The Role of Planetary Diagrams in a Fourteenth-Century Canon Set to the Alfonsine Tables (Erfurt Q366 ff. 70v-73v)*

Abstract: In the middle of the fourteenth century, Johannes 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 language into the computational procedures of the canons. In this paper, I will treat the canons in this set that offer procedures for finding true planetary positions. I argue that there are actually two parallel chains of exposition within the canons, the first of which instructs the user in purely arithmetical terms, while the second resorts to a geometrical descriptions with a strong reliance on the diagrams. This format of presentation bears many similarities to that of the Theorica Planetarum of Campanus of Novara, which may suggest that the diagram for the canon served a sort of justificatory function, revealing the cause for the efficacy of the canon’s procedures.

**Samuel Gesnner** (CNRS-Observatoire de Paris-PSL, ALFA, France)

*Uneven scales on instruments: using tables and geometry in Alfonsine astronomy*

This study aims at probing the Alfonsine astronomers’ understanding of the subtle interplay between tabulated values and geometrical constructions, numerical computation and analogue measurement using scales. It shall highlight the mathematical understanding of the various materials that astronomers were handling in the 15th century. To do that it proposes to browse a variety of instruments and their canons then common to Alfonsine astronomical practice: astrolabes, *teorice novelle equatoria*, syzygy instruments, and planetary latitude dials. Some of the scales on these instruments were unevenly divided. These uneven divisions, however, were arrived at by a diversity of procedures: geometrical construction, interpolation of tabulated values, and point by point construction from tabulated values of curved lines. The practices of this scale division reveal some aspects of the actors understanding of numerical and geometrical magnitudes and their correspondence, the practice of using them in analogous ways. This essay will examine the implicit notions of accuracy, approximation, ease of execution, notion of correspondence of numerical and geometrical magnitude that come to bear in the use or conception of such scales.

**Matthieu Husson** (CNRS-Observatoire de Paris-PSL, ALFA, France)

*Interpolations in alfonsine astronomy: canons, tables and diagrams*

Quantities manipulated in mathematical astronomy are in some cases represented as continuous magnitudes (e.g. in diagrams and to some extent in instruments) and in others as discrete numbers (e.g. in numerical tables). Practices related to interpolation, ubiquitous in astronomical computation, are an essential context where this mathematical tension is articulated and a locus of negotiation with respect to acceptable approximations of various kinds more or less deeply embedded in the tabular expressions of astronomical models. Rooted in an analysis of canons instructions with respect to simple or double arguments interpolations this study will also takes stock on the different kinds of tables and diagrams related to interpolation and map the surprising scope and diversity of this type of practices in mathematical astronomy of the Alfonsine tradition.