New Approaches and Parameters in the Parisian Alfonsine Tables* José Chabás Universitat Pompeu Fabra Barcelona (Spain) [email protected] Bernard R. Goldstein Dietrich School of Arts and Sciences University of Pittsburgh [email protected] Abstract: In this paper we examine several aspects of the Parisian Alfonsine Tables [henceforth PAT] that appeared in Latin around 1320, frst in Paris and then diffused throughout Europe. Our focus is on the tables for precession/trepidation, the motion of the planetary apogees, the radices for mean motions, and the mean motions. The goal has been to identify sources for the parameters, and the result is that, for the most part, Anda- lusian zijes provided the required information, indicating continuity from the table-mak- ers in al-Andalus to Latin Europe. We derive the parameter for the Alfonsine motion in precession from the length of the tropical year ascribed to Azarquiel by Abraham Ibn Ezra. Although the sources for some parameters in PAT have not been identifed, there is no evidence that new observations played any role. Among the sources for PAT were the Toledan Tables and al-Battānī’s zij. In the case of the planetary apogees it is shown that the values in PAT were probably derived from those in al-Battānī’s zij, whereas those in the Toledan Tables were almost certainly derived from that zij, despite the fact that the Toledan Tables use sidereal coordinates and al-Battānī used tropical coordinates. In the case of the radices, again values in the Toledan Tables were derived from those in al- Battānī’s zij, but the values in PAT show no such affnity. The mean motions in PAT are closely related to those in several Maghribi zijes, which supports the suggestion that a lost work by Azarquiel may be the common source. * The contribution of one of us (JC) was written in the framework of the European Research Council project ALFA, Shaping a European Scientifc Scene: Alfonsine Astronomy, under the EU program Horizon 2020 (Grant Agreement 723085) Chabás, José; Goldstein, Bernard R. (2020-2021). «New Approaches and Parameters in the Parisian Alfonsine Tables». Suhayl 18, pp. 51-68. ISSN: 1576-9372. DOI: 10.1344/SUHAYL2020.18.3. 18108_Suhayl18_2020.indd 51 11/12/20 10:40 José Chabás, Bernard R. Goldstein Keywords: Alfonsine Tables, trepidation, apogees, radices, mean motions, al-Battānī, To- ledan Tables, Azarquiel, John of Murs Introduction The Parisian Alfonsine Tables were the most widely diffused set of astronomical tables in Europe in the fourteenth and ffteenth centuries. They originated in Cas- tile in about 1272, but only the canons (or instructions) in Castilian survive, that is, the earliest version of the tables appeared, in Latin, in Paris around 1320. There are several sets of canons to PAT in Latin, but none of them addresses the sources for the parameters underlying the tables, and little information can be gleaned from the Castilian canons either. In general, each manuscript of a set of medieval astro- nomical tables tends to have a slightly different collection of tables. Fritz S. Ped- ersen’s edition of the Toledan Tables (2002) provides the clearest example of the variation in the manuscripts of the same set of tables. No comparable study has been made of the many manuscripts of PAT, and so we have to rely in part on the early printed editions (Ratdolt 1483 and Santritter 1492). The goal of this paper is to investigate the sources of the underlying parameters in PAT, where we focus on precession and trepidation (§ 1), planetary apogees (§ 2), radices (§ 3), and mean motions (§ 4), based on the zijes of al-Khwārizmī (f. 830), al-Battānī (d. 929), Ibn al-Kammād (f. 1100), Ibn Isḥāq al-Tūnisī (f. 1193 - 1222), Ibn al-Raqqām (d. 1315), and Ibn al-Bannāʼ (d. 1321), as well as on the extant Latin version of the Toledan Tables.1 In previous studies we have addressed the plane- tary equations and the velocities of the Sun, the Moon, and the planets, and the results of this new study are consistent with them, that is, most of the parameters in PAT originated in the Iberian peninsula.2 As will be shown below, the approach to the general problem of the frame of reference for the coordinates of the celes- 1. For the zij of al-Khwārizmī, we consulted Suter 1914 and Neugebauer 1962a; for the zij of al-Battānī, we consulted Nallino 1903 - 1907; for the Toledan Tables, we consulted Toomer 1968 and Pedersen 2002; for the zij of Ibn al-Kammād, we consulted Chabás and Goldstein 1994; for the zij of Ibn Isḥāq we consulted Samsó and Millás 1998 and Samsó 2019; for that of Ibn al-Raqqām, we consulted Samsó 1997 and Samsó and Millás 1998; and for that of Ibn al-Bannāʼ, we consulted Samsó and Millás 1998. 2. See Goldstein and Chabás 2001; Chabás and Goldstein 2012, pp. 63 - 81, 95 - 101, and the literature cited there. 52 18108_Suhayl18_2020.indd 52 11/12/20 10:40 New Approaches and Parameters in the Parisian Alfonsine Tables tial objects relied on a compromise between previous approaches and it was based on round numerical data of no astronomical signifcance. For the apogees, radi- ces, and mean motions, early Alfonsine astronomers borrowed material from their Arabic predecessors and adjusted them to their needs in much the same way as the author(s) of the Toledan Tables treated al-Battānī’s material, in particular for the planetary radices and apogees at the Hijra. Although new approaches and param- eters are found in PAT, essentially there is continuity between the work done by table-makers in al-Andalus and Latin Europe. Among the common sources for PAT and zijes in al-Andalus and the Maghrib, it is likely that one of them was a lost work by Azarquiel. 1. The eighth sphere and “eius motus” Probably the most notorious change introduced by early Alfonsine astronomers concerns the treatment of precession and trepidation. They combined both ap- proaches, introducing a linear term for precession to be added to a periodic term for trepidation (see Chabás and Goldstein 2012, pp. 48 - 52; cf. Mercier 1977, pp. 58 - 59). Although original, this method for determining the position of the eighth sphere reproduces the traditional procedure used by all astronomers to locate any celestial body, consisting of a frst approximation using a mean motion to which an additional term, or correction, is applied. The editio princeps of PAT (Ratdolt 1483) has a table entitled «Radices motus octave sphere ad eras subscriptas» (c8r). It displays the radices of the motion of the eighth sphere (also called motion of access and recess of the eighth sphere) together with an entry labeled «Eius motus est», for 10 eras from the Flood to Alfonso (see Table 1). In this edition, all entries are given to seconds in signs of 60º. The values of the radices range from 3,19;41,0º for the Flood to 1,3;34,4º for Alfonso’s era. Since the Flood corresponds to February 17, –3101 (JDN 588465) and Alfonso’s era in PAT to June 1, 1252 (JDN 2178503), the difference, 223;53,4º, in 1590038 days yields a mean motion of the eighth sphere of 0;0,0,30,24,49,33º/d, corresponding to one revolution in very nearly 7000 years.3 This rounded number 3. According to chapter 1 of the Castilian canons, the era of Alfonso began at noon, Sunday, Dec. 31, 1251, and the frst day of the era, Jan 1, 1252 (JDN 2178351), is considered to be the epoch of these tables. However, the printed edition of PAT uses a different epoch, June 1, 1252, the date of the beginning of Alfonso’s reign (Chabás and Goldstein 2003, 21 - 22, 144). 53 18108_Suhayl18_2020.indd 53 11/12/20 10:40 José Chabás, Bernard R. Goldstein is astronomically meaningless. The entries for the radices can be computed by means of the expression 360 t / 7000, where t is the number of years since the epoch. The precise date of the epoch is not given, and one would think of the In- carnation as a possibility, but the corresponding entry is 5,59;12,34º, which is shortly before completion of a full revolution. Rather, early Alfonsine astrono- mers chose as epoch a date which did not previously have any astronomical sig- nifcance, May 17, 16 ad (JDN 1727039), the time when sidereal coordinates were considered to agree with tropical coordinates and thus trepidation was zero.4 This epoch in PAT is unprecedented. Indeed, consider, for example, the case of the Hijra, where the radix in PAT is 31;10,26º. The corresponding time t derived from expression 360 t / 7000 is about 606 years from epoch. Since the date of the Hijra is 622 ad, the epoch was taken to be 16 ad. The other cases yield the same result. This precise date was already identifed as the origin of trepidation by Giovanni Bianchini in his tables for the planets (c. 1442): Chabás and Goldstein 2009, pp. 28, 32. As explained below, the values for the radices listed in this table were used in the computation of the corresponding entries labeled «Eius motus est». They rep- resent the periodic component (trepidation) and can be recomputed by means of the expression arcsin (sin 9º · sin (360 t / 7000)), where 9º is the maximum value of the equation of access and recess in Alfonsine astronomy (see Ratdolt 1483, d3v). This expression involving sin 9º, rather than just 9º, as a coeffcient was already proposed by Delambre (1819, p. 251).