**Organisation:**

**Matthieu Husson** (CNRS, Observatoire de Paris), **Clemency Montelle** (Canterbury University, New Zealand), **Benno van Dalen** (Bavarian Academy of Sciences and Humanities, Germany)

During the third workshop the team will gather for one week and work toward two complementary objectives. Morning will be dedicated to a thorough discussion of the content of the collective book and each contribution to it. Afternoon will be devoted to collective discussion and definition of the TAMAS database tool. This workshop will be also the occasion to organise future research in the project as two new full time members have now joined us: Galla Topallian TAMAS IT and Anuj Misra TAMAS postdoc

**Monday January 30, 2017**

**9h 9h15 ***General introduction of the workshop*

**Matthieu Husson **(CNRS, Observatoire de Paris)

**9h15-10h45 ***The Almanac of Jacob ben Makhir, *

**José Chabás **(Pompeu Fabra University Barcelona, Spain)** and Bernard R. Goldstein **(Pittsburgh University, USA)

Jacob ben Makhir Ibn Tibbon (ca. 1236 – ca. 1305) was a member of the prominent Ibn Tibbon family established in Languedoc, in southern France. Jacob was also known by his Provençal name, Profeit Tibbon, later rendered in Latin as Profatius. Although probably born in Marseille, he was active in Montpellier both as an author and as a translator. Besides his translations, almost exclusively on mathematical and astronomical matters, Jacob is also the author of two original works, a text on a new version of an astronomical instrument, called the quadrant, and a set of tables entitled *Almanac*.

Jacob’s *Almanac* consists of a text that includes a prologue and canons, and a number of tables to compute the true positions of the planets and the two luminaries. It was originally written in Hebrew, and diffused mainly in Latin. The main part of Jacob’s *Almanac* consists of tables for the true positions of the celestial bodies at intervals of a few days, from which their true positions at any time in between can easily be determined, following the pattern set in other almanacs such as that ascribed to Azarquiel (d. 1100). It also contains a series of tables for computing the circumstances of solar and lunar eclipses. In this paper we analyze all tables in the *Almanac* and put them in their proper context.

**10h45-11h ***Break*

**11h-12h30 ***Computer-aided analysis of astronomical tables in Yuan and Ming China (A.D. 1271-1644): A case study on the sunrise and sunset tables*

**Li Liang **(Beijing Institute for the History of Natural Science, Chinese Academy of Science, China)

This article introduces two types of time tables in the Chinese calendrical systems during the Yuan and Ming period. The first type was used for calendrical calculation by experts, while the other one mainly for laymans. With the help of computer-aided analysis and case studies, two methods which were used to calculate these tables are explained. The analysis shows that the first method named arc and sagitta method is accurate and more coincident with the theory, but too complicated and could bring heavy calculation burdens. The second method named nine domains method is simple and easy, but not accurate enough. The official astronomers in the Yuan and Ming dynasty all followed the first method to calculate the sunrise and sunset tables used for their capital Beijing and Nanjing respectively. Moreover, how these tables were transmitted to Korea and how they were modified and used in Korea capital by Korean astronomers are also discussed in the article.

**12h30-14h30 ***Lunch*

**14h30-16h ***Database discussion*

**16h-16h15 ***break*

**16h15-17h30 ***Database discussion*

**Tuesday January 31, 2017**

**9h-10h30 ***Editing Dinakara’s Candrārkī*

**Clemency Montelle **(Canterbury University, New Zealand)

The *Candrārkī*, `Moon and Sun’, was one of several table-texts compiled in Sanskrit by Indian astronomer Dinakara. The work contains around 26 verses which describe how to use the accompanying tables for computations related to solar and lunar phenomena. The epoch of the *Candrārkī *is 31 March 1578 and it survives in at least a dozen manuscripts. Some manuscripts contain the text only, some only the tables. Yet further copies contain both the tables and the text. We explore some of the challenges related to editing and analysing these tables.

**10h30-11h ***Break*

**11h-12h30 ***Editing and analysing the Tabule magne of John of Lignères (Paris c. 1325)*

**Matthieu** **Husson **(CNRS, Observatoire de Paris)

John of Lignères is, with John of Murs, John Vimond and John of Saxony, among the most important astronomers of the Parisian milieu which formed the Parisian Alfonsine Tables between 1321 and 1327. These tables formed the core of the Alfonsine tables which in various formats were the main computational tool of astronomy in Europe well into the 16th century.

Among his various astronomical works ranging from the design of planetary instrument to the redaction of a Theorica planetarum, the Tabule magne are a striking example of John of Lignères ability to explore new layout and were particularly important in the reception of the Alfonsine tables in England. The tables and canons are known to be, extent or in part, in seven manuscripts which testify the reception of the work in different milieus and present multiple variations, in the order and content of the canons, in the selection and format of tables or in the type of numbers used.

The wealth and depth of these variations concentrated in a relatively small number of manuscripts for a work of a significant historical importance make the Tabule magne a particularly interesting case study in the perspective of exploring how a database combining edition and mathematical analysis tools may open new perspectives in the inquiry of ancient astronomical tables.

**12h30-14h30 ***Lunch*

**14h30-16h ***Database discussion*

**16h-16h15 ***break*

**16h15-17h30 ***Database discussion*

**Wenesday February 1, 2017**

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**9h-10h30 ***Computing and copying Tables in medieval monastic contexts*

**Seb Falk **(Postdoc, Cambridge)

Medieval Christian monasteries were an unusual context for astronomy. Monastic houses were organised in close networks that were in constant communication, exchanging texts and ideas with each other and (in the later medieval period) with the new universities. Yet monks themselves took a vow of stability, committing to remain in one community for life. It is therefore interesting to examine how astronomical tables were copied and adapted for use in different monastic locations.

This chapter will examine a set of tables made c. 1380 at the Benedictine house of St Albans (England), for use 200 miles further north, at the priory of Tynemouth. The tables are mostly a close copy of a set accompanying a popular instrument treatise, but an additional table of oblique ascensions was computed for use at the new latitude; this computation was carried out using different parameters and techniques. The copyist added a commentary explaining his intentions for the use of the tables and instrument at – and only at – the new latitude. Moreover, a further set of tables of astrological houses was added that demonstrates the breadth of the monks’ astronomical interests.

Through a combination of mathematical, statistical and codicological methods, this chapter will reconstruct the copying and computational procedures used by the Benedictine producers of this manuscript. It will draw new conclusions about monastic mathematicians: their learning, practices and motivations for the study of astronomy. It will also discuss the possibilities and limitations of the use of simple statistical techniques to analyse medieval tables.

**10h30-11h ***Break*

**11h-12h30 ***Analysing Islamic Geographical Tables*

**Benno Van Dalen **(Bavarian Academy of Sciences and Humanities, Germany)

In their book Geographical Coordinates of Localities from Islamic Sources (Frankfurt am Main, 1987), E.S. and M.-H. Kennedy collected more than 10.000 pairs of geographical lon-gitudes and latitudes that are found in over 80 Arabic and Persian sources. These include sev-eral geographical treatises (most importantly, Yāqūt’s Muʿjam al-buldān and Abu ’l-Fīdā’s Taqwīm al-buldān), a large number of zījes (astronomical handbooks with tables), and several instruments. Islamic geography and its influence in Europe has been studied intensively since at least two centuries, with detailed investigations by Lelewel, Nallino, King and Sezgin as some of the landmarks. Also a first analysis of the data in Geographical Coordinates outlines four main mathematical-geographical traditions in the Islamic world, namely Ptolemy, the geography of al-Maʾmūn as laid down in al-Khwārizmī’s Kitāb Ṣūrat al-arḍ (ca. 825), the otherwise unknown Kitāb al-aṭwāl wa-l-ʿurūḍ li-l-Furs cited by Abu ’l-Fīdā, and the geo-graphical table in al-Qānūn al-Masʿūdī, the monumental zīj by al-Bīrūnī (ca. 1030). However, no systematic investigation of the more than 10.000 coordinates collected by the Kennedy’s and the exact relations between the sources has been carried out yet.

The proposed contribution will contain an edition of one of the geographical tables included by the Kennedy’s, namely that in the anonymous Shāmil Zīj (ca. 1240), on the basis of four available manuscripts (the Kennedy’s used only one). This table will be compared in detail with two related tables, namely that from Ibn al-Fahhād’s ʿAlāʾī Zīj (ca. 1175) as con-tained in the hitherto unstudied manuscript from the Salar Jung Museum in Hyderabad and that in the Mulakhkhaṣ Zīj by al-Abharī, extant in manuscripts from Utrecht and Kolkata. The edition and comparison will show that the transmission of geographical tables yields a much more complicated picture than that of most other types of mathematical and astronomical ta-bles (only star tables present a somewhat similar range of textual and numerical problems). First of all, many of the localities in geographical tables were unknown to the scribes, so that we find scribal variations in the names that go far beyond a simple confusion due to the simi-larity of letters of the Arabic alphabet. Secondly, except if the localities are systematically arranged by increasing longitude, by climates or in regions, the coordinates in a table do not bear any relation to another. Whereas tabular values of mathematical functions can often be corrected on the basis of the assumption that their tabular differences change in a smooth and regular way, this is not the case for a geographical table. Also so-called slides, in which parts of certain columns of digits or numbers appear some rows too high or too low, can be more easily recognized and corrected in a mathematical table. If a locality whose coordinates are part of a slide is copied into a different table with a different order, it will be almost impossi-ble to restore the correct coordinates. A glance at the coordinates found in Geographical Co-ordinates confirms that a wide range of variants appear in the coordinates for well-known localities (for which new observations may be expected to have been made in the course of time) as well as for traditional localities (often ancient, frequently in ruins), whose coordinates could only have been copied from one source into the next.

At the end of my contribution I intend to outline a program by means of which geo-graphical tables in a database may be tackled automatically. This includes, in particular, com-parisons of tables that take into account: the order of the localities (with the possibility of in-cidental omissions or additions of entries), common scribal mistakes in place names, common scribal mistakes in numbers, and the possibility of slides of parts of columns with numbers. To all possible scribal errors a frequency or probability of their occurrence may be assigned, which will make it possible to design a statistically based measure for the similarity of two given tables. The probabilities for the scribal errors may, for example, be derived from all Arabic and Persian tables for which multiple copies will be available in the database.

**12h30-14h30 ***Lunch*

**14h30-17h30 ***Alhambra *(to be confirmed)

**Thursday February 2, 2017**

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**9h-10h30 ***Cracking the Tabulae permanentes of John of Murs, Paris, c. 1321*

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

In their classic study of “Five centuries of finding true syzygy,” J. Chabás and B.R. Goldstein identified as the earliest tabular solution to this difficult computational problem a double-entry table prepared in the twelfth century by Ibn al-Kammad. But al-Kammad’s table requires users, before entering the table, to first compute the solar and lunar corrections as well as the solar and lunar velocities at the time of mean syzygy. In the early 1320s, John of Murs, possibly assisted by Firmin of Beauval, constructed a more “user friendly” (Chabás and Goldstein) double-entry table that requires knowing only the solar and lunar anomalies at the time of mean syzygy. Versions of this table were incorporated (without attribution) into the subsequent eclipse tables of Immanuel ben Jacob Bonfils (1350s), John of Gmunden (1420s), and Georg Peurbach (1450s); the latter were printed in 1514 and widely used throughout the sixteenth century. In 2001, B. Porres and J. Chabás published an edition, with English translation, of the canons to the Tabulae permanentes but they did not ask how the tables were computed. In this paper, I shall offer a proposal for that algorithm and will discuss the “table-cracking” procedures I employed to find that algorithm.

**10h30-11h ***Break*

**11h-12h30**

*Al-Kashi’s Double-Argument Tables of Planetary Latitudes in the Khaqani Zij Strengths and weaknesses of standardized statistical tools for analyzing historical astronomical tables*

**Glen van Brummelen **(Quest University, Canada)

We present an unusual document from early 15th-century Iran: a set of double-argument tables for determining planetary latitudes from Jamshid al-Kashi’s Khaqani Zij. Although Islamic tables of planetary latitudes generally simply copied from Ptolemy or some other source, these tables exhibit a unique structure and give rise to several possibilities for their computation. Intriguingly, the tables are only partly completed in all available manuscripts, providing some insight into the patterns of computation. However, many mysteries remain; the tables have not been fully analyzed. To approach them, we also present several standardized statistical procedures that have been developed to analyze pre-modern astronomical tables: Benno van Dalen’s method of parameter estimation (and a proposed improvement by David Bellhouse), as well as my table dependence test. This leads to an important methodological question: is it preferred to apply quantitative methods designed for specific tabular situations, or standardized tools that work with a wide variety of situations?

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**12h30-14h30 ***Lunch*

**14h30-16h ***Database discussion*

**16h-16h15 ***break*

**16h15-17h30 ***Database discussion*

**Friday February 3, 2017**

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**9h-10h30 (by visio conference)**

*The Am**ṛ**talahar**ī** of Nity**ā**nanda.*

**Anuj Misra **(Postdoc TAMAS, Observatoire de Paris)

The *Am**ṛ**talahar**ī* (or perhaps, the *Khe**ṭ**ak**ṛ**ti*) of Nityānanda is an undated Sanskrit manuscript found in the collection of University of Tokyo discovered by Prof. Pingree. This manuscript contains numerical tables for computing calendrical elements (like *tithis, nak**ṣ**atras, yogas*) along with other astronomical tables like the right and oblique ascensions tables of the zodiacal signs, the planetary equation tables and mean motion tables. It also includes a combined Sine table, solar declination table, lunar latitude table, and various gnomonic shadow-length table for arguments extending from one to ninety degrees or arc.

This work, along with the Siddhāntasindhu (a Sanskrit translation of the Zīj-i Shāh Jahānī, c. 1630s) and the Sarvasiddhāntarāja (a siddhāntic text, 1639) represents the three works know to have been authored by Nityānanda. We attempt to mathematically analyse a selection of the tables of the *Am**ṛ**talahar**ī* in an effort to understand the challenges in reconstructing numerical tables from seventeenth century Mughal India.

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**10h30-11h ***Break*

**11h-12h30 ***Project general discussion*