Abstract
Tables form one of the largest corpora of ancient and medieval astronomical texts. Unlike astronomical textbooks and treatises, which predominantly introduce astronomical concepts and set out astronomical theories, the production and use of tables formed part of the working practice of astronomy and astrology. Broadly speaking, tables can be divided into two types: (i) tables containing numerical values that are used in making calculations of astronomical positions or phenomena, such as mean motion tables, tables of anomaly and parallax tables, and tables of ascension; and (ii) tables that contain the results of astronomical calculation for a given span of time, such as ephemerides, almanacs and eclipse canons. The study of tables has and continues to make an important contribution to the history of astronomy. Not only do tables represent the instantiation of astronomical theories, they often provide our only evidence for theories which were either never formally written down in texts or for which such texts no longer exist. Furthermore, tables provide an important insight into how astronomical calculation was performed in practice – how geometrical models were transformed into mathematical functions, what strategies were used to simplify calculation, what roundings were made during the process of both producing a table which represents a theory and when using that table to calculate a specific position or phenomena, how such roundings affected the precision and accuracy of the overall calculation, etc.
The collection under review addresses two central methodological problems in the study of numerical astronomical tables: how to edit tables and how to analyse them. The two problems are related – analysis of a table requires a text to work with and editing a table requires choices to be made about the fidelity of the numbers in the text which is usually informed in part by a preliminary analysis of the table’s numerical content – although the former is given most weight in the volume under review.
The first step in the analysis of an astronomical table is to identify what it is, a process known colloquially as ‘table cracking’. In essence, table cracking involves analysing an unidentified (and sometimes only partially preserved) table to identify what astronomical function the numerical data it contains represents and to try to identify the source of that data. For example, faced with an unidentified medieval Latin table, a table cracker may be able to identify first that is a table for the solar mean motion and second that it is based upon the Toledan Tables. In doing so, the table cracker will often have to have made allowances for scribal errors in the table, the rounding of values, and, perhaps, small variations from what we would expect from any previously known set of tables. These small variations may themselves prove crucial in showing that the table’s author had slightly varied the underlying parameters of the table, perhaps in order to improve upon an earlier table that it is based upon. An excellent methodological introduction to various approaches to table cracking is provided in the contribution by Glen van Brummelen, Matthieu Husson and Clemency Montelle. In addition to outlining the main tools that have been applied to table cracking, the authors clearly discuss how we can assess the level of confidence with which the table has been cracked and how we can combine the results of table analysis with the study of astronomical treatises and theories in order to get a more detailed understanding of the history of astronomical calculation. Several papers in the volume provide concrete examples of table cracking.
Once a table has been cracked, further analysis of its contents can provide many insights into both astronomical practice and astronomical methods. This analysis may take several forms. For example, Richard L. Kremer applies the technique of exploratory data analysis, which looks at the residuals between different known tables (e.g. Ptolemy, the Toledan Tables, Ibn al-Kammād) and the table being analysed, in order to identify the most likely foundation of the table and, crucially, any regular or periodic differences between the table and its foundation which must be the result of either rounding or the use of a slightly different parameter or set of parameters or of a modified theory. In some cases, detailed analysis of preserved tables allows whole planetary models to be recovered. For example, in his chapter, Glen van Brummelen is able to reconstruct Jamshīd al-Kāshī’s planatery latitude model from a set of double argument tables for planetary latitude from his Khāqānī Zīj. Johannes Thomann attempts something similar in his contribution, attempting to reverse-engineer the astronomical models which were used to produce ephemerides which contain calculated planetary positions.
Although the volume is entitled Editing and Analysing Numerical Tables, the process of editing tables receives significantly less discussion than table cracking and analysis. In particular, it is a shame that there is no methodological overview of table editing comparable to van Brummelen, Husson and Montelle’s chapter on table cracking. Having said that, many of the methodological questions around editing tables, albeit with a focus very definitely on questions relevant to dealing with Sanskrit sources, are brought up by Montelle in her chapter on editing Sanskrit tables. The editing of any text, especially one preserved in multiple copies, requires the editor to make decisions and impose what may be an artificial structure and ‘correctness’ on the text. As Montelle makes clear, in editing tables these problems are greater still: copies of tables may have different layouts, especially with regard to the splitting of a table onto multiple pages, they will often have many scribal errors, some sources may include visual information (the use of colour, spacing, ruling style, etc) that are not found in other sources, and some sources may include ‘corrections’ made by the scribe. Thus, the editor is forced to choose whether or not to follow the layout of one particular source, whether to correct what seem to be obvious mistakes present in all manuscripts, whether to give priority to what seems to be the best sources from an astronomical or a manuscript perspective, whether to provide an idealised Ur-text or to provide an accurate rendition of the preserved sources even when it is clear that they are corrupt, etc. In the end, the choices made will depend upon the purpose of the edition and the author’s preference and there is no single right or wrong way to edit a table.
This volume makes a significant contribution to the study of astronomical tables and, given the role of tables in early astronomical practice, to the study of the history of astronomy itself. I particularly commend the editors for their intentional inclusion of discussion of methodological questions, which will be particularly useful in training future scholars. And with that in mind, the publication of the book as an open access e-book is particularly welcome.