Introduction During the Middle Ages and early modern times tables were a most success- ful and economical way to present mathematical procedures and astronomical models, facilitating computations based on them. One reason for depending on these techniques was the absence of modern mathematical notation to represent the algorithms that astronomers were using. Indeed, some major sets of tables were the direct result of the development of new astronomical approaches. In the second century ad Ptolemy composed the Almagest, a com- prehensive handbook of astronomy and related mathematical procedures: in it he presented a set of observations on the basis of which he determined the parameters of his geometrical models for planetary motion. He then compiled tables which were also included in this treatise. At a later date he revised these tables, making them easier to use, in a work called the Handy Tables where, for example, Ptolemy displayed entries at 1°-intervals, rather than at 3°- and 6°- intervals as he had done previously. Nicolaus Copernicus (1473–1543) provides a somewhat different example of this pattern. In 1543 his magnum opus, De revolutionibus, was published, in which he described a set of planetary mod- els to replace those based on the Almagest, and included tables for computing planetary positions with these new models. In 1551 Erasmus Reinhold (1511– 1553) compiled the Prutenic Tables, based on the models of Copernicus’s De revolutionibus, presented in ways that facilitate computation. Most of the time, however, the compilation of complete sets of tables—or of individual tables— just reflected partial changes in the parameters and the models underlying a particular theory, or new methods to compute the positions of the celestial bodies without changing the underlying models. This is particularly true in the Middle Ages, when the Ptolemaic models were rarely challenged. Astronomical tables are a basic component of astronomy, although they are frequently neglected (or considered an unintelligible sequence of numbers) because, more often than not, it is no simple matter to establish the structure of such tables, and sometimes it is even difficult to identify the problem which a particular table addresses. Yet, to understand astronomical tables and to fully tackle the procedures used to compile them, various skills, both linguistic and mathematical, are required at the same time. The papers included in this volume give many examples where the meaning and purpose of such tables has been determined by careful analysis. As we have shown in our recent monograph, A Survey of European Astronom- icalTablesintheLateMiddleAges (Brill, 2012), astronomical tables are a primary source of historical information. Through their analysis it is possible to insert © koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_002 2 introduction them and their compilers in an astronomical tradition, thus displaying unex- pected links between authors. These analyses also allowed us to discover com- putational techniques, interpolation methods, and approximation procedures, as well as to identify changes in the standard parameters and the geometrical models. It should be stressed that an analyst of a medieval table should only appeal to modern mathematics very selectively and judiciously, given the risk of falling into anachronisms that may lead to incorrect results. In particular, modern mathematical functions are generally to be avoided although medieval mathematical procedures expressed verbally may legitimately be translated into algebraic notation. In recomputing a table consistency requires making use only of computational techniques, concepts, and strategies that were avail- able at the time. For example, in computing entries in a table for finding the length of daylight on a given day, one needs to use trigonometrical tables that can be found in the same set of tables or in a previous set; indeed, this is how it was done by medieval astronomers. In other words, we have to evaluate medieval scholars according to criteria consistent with their own time, which may be different from those appropriate for other periods. The pioneering work on this subject goes back to the beginning of the twen- tieth century. In the years between 1899 and 1907 C.A. Nallino published an edition, translation, and commentary on the zij (i.e., a set of astronomical tables) of al-Battānī (ca. 900). This work was extended in 1956 by E.S. Kennedy in his survey of Islamic astronomical tables. Moreover, in 1962 O. Neugebauer provided an extensive commentary on the zij of al-Khwārizmī, originally com- posed in Baghdad in the ninth century that is only extant in a Latin version by Adelard of Bath (1075–1160), which was published by H. Suter in 1914. While the publication and analysis of Islamic astronomical tables have continued to this day, considerable attention has also been devoted to astronomical tables in Latin, Byzantine Greek, Hebrew, and the vernacular languages in Europe, including our own contributions to this field which have been built on those of our distinguished predecessors. As a result of these scholarly efforts a critical mass of studies is available which provides a good understanding of the overall framework for the transmission of astronomical ideas and procedures through tables in the period from Ptolemy to the early sixteenth century. The study of many sets of tables in a variety of languages still to be examined in European libraries (and others in different parts of the world) will surely lead to signif- icant modifications of this framework in addition to an expansion of its con- tents. In particular, there are at least three major topics concerning medieval astronomy in the Latin West, where the analysis of tables is likely to play a crit- ical role in enhancing our understanding of developments in this domain: the early stages of astronomy in Latin Europe, based on Arabic materials in the.