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-makers 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.

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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 astronomical 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 planetary 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.

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tial objects relied on a compromise between previous approaches and it was based on round numerical data of no astronomical signifcance. For the apogees, radices, 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 parameters 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 approaches, 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).

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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 astronomers chose as epoch a date which did not previously have any astronomical signifcance, 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). The round number 9º differs from all parameters previously used for trepidation. The results, shown in Table 1, are in very good agreement with text, except in two cases.

Table 1: Motion of the eighth sphere and recomputation

Radix in Ratdolt «Eius motus» in Computation of Difference in Ratdolt the periodic term seconds Flood 3,19;41, 0 0,2;57,12 -3; 1,17 -245 Nabonassar 5,20;48, 0 0,5;40,27 -5;40,27 0 Philip 5,42;35,27 0,2;40,55 -2;40,57 -2 Alexander 5,43;12, 7 0,2;35,29 -2;35,28 +1 Cesar 5,57;15,18 0,0;25,45 -0;25,45 0 Incarnation 5,59;12,34 0,0; 7,25 -0; 7,25 0

4. Other authors take this epoch to be a day earlier, May 16, 16 ad: see, e.g., Mercier 1977, p. 59.

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Diocletian 0,13;47,51 0,2; 8,15 2; 8,17 -2 Hijra 0,31;10,26 0,4;38,42 4;38,41 +1 Yazdegard 0,31;41, 3 0,4;42,45 4;42,47 -2 Alfonso 1, 3;34, 4 0,8; 4, 1* 8; 3, 9 +52

* In all manuscripts we consulted the entry is 8;4,1º as well.

For precession, there is yet another parameter used in Alfonsine astronomy. In his Expositio intentionis regis Alfonsii circa tabulas eius (1321) John of Murs at- tributes to the Tables of Alfonso a rate of precession of 1° in 72 ½ years: Poulle 1980, p. 258. This constant differs from the values used by Ptolemy (1º/100y) and al-Battānī (1º/66y). It is equivalent to a rate of 0;0,49,39º/y, or 0;0,0,8,9,22º/d. The explanation of this «new» parameter was given by Goldstein (1994), who showed that it is the same as that used in the treatise On the solar year which is extant in both Arabic and Latin versions. This treatise has traditionally been ascribed to Thābit Ibn Qurra (d. 901) but was probably composed by the ninth- century scholars Banū Mūsā: Morelon 1987, p. lii.

2. Apogees

For the linear term in Alfonsine precession/trepidation, one has to refer to the tables for the «radices of the apogees». The editio princeps lists separately the radices for the apogees of the Sun and the planets for 10 eras, from the Flood to Alfonso (Ratdolt 1483, c8r - d1r). In all cases we are told that the entries do not include the motion of the 8th sphere, that is, they include the linear term for precession/trepidation but not the periodic term (trepidation). We also note that, following the standard tradition, the apogees of the Sun and Venus are considered to be the same; they are given here to seconds, whereas all other apogees are displayed to thirds. However, this is a false precision because for all planets the number of thirds is the same for each era, indicating that the entries were computed following the same procedure. Moreover, the differences between eras are the same for each planet. For example, between the Flood and the Incarnation, Jan 1, 1 ad (JDN 1721424), the apogees of the four planets (Saturn, Jupiter, Mars, and Mercury) have increased by 22;47,21,23º, whereas that of the Sun and Venus has moved 22;47,21º. We may also note that even though the apogees for the planets are given to thirds, for each of them the number of seconds and thirds at the Flood and Alfonso’s is the same, indicating an unlikely coincidence or an ap-

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proximation somewhere in the computation of these values. It follows that the underlying rate for the motion of the apogees is about 0;0,0,4,20,41,17,19º/d, very nearly the same as the value found in the editio princeps 0;0,0,4,20,41,17,12º/d, for the mean motion of the eighth sphere, that is the linear term in Alfonsine precession. This unique value is equivalent to 1 revolution in about 49000 years and differs from any analogous value in previous zijes. Indeed, the identifcation of these two quantities is clearly stated in the heading of another table, entitled Ta- bula medii motus augium et stellarum fxarum (Ratdolt 1483, d4v). The coincidence of these two values can be considered a characteristic feature of the Alfon- sine Tables that were reworked in Paris in the early fourteenth century. By identifying one of the terms of precession/trepidation with the motion of the apogees, the fact that these apogees are sidereally fxed is somewhat obscured. For background, note that in the Almagest the planetary apogees are sidereally fxed and the solar apogee is tropically fxed (Neugebauer 1975, pp. 58 and 150). But in the above-mentioned treatise, On the solar year, it is argued that the solar apogee is also sidereally fxed (Morelon 1987, p. 66; Neugebauer 1962b, p. 289, § 146). In any case, the new parameter 0;0,0,4,20,41,17,12º/d (or about 0;0,26,27°/y) for both the motion of the apogees and linear precession, corresponding to a period of 49000 years, in turn depends very strongly on a calendar year of 365;15d, as explained below. The round number 49000 was not based on any observation; rather, it was derived from an approximation of the length of the tropical year. The initial value taken for the tropical year was 365 ¼ - 1/136 days, which is ascribed by Ibn Ezra (d. 1167) to several astronomers including Azarquiel (d. 1100). The data given in this text is that the tropical year is 365 days plus an excess of revolution of 87;21°, where a day is taken to be 360°: Millás 1947, p. 83:14. Now, 87;21°/360° = 0;14,33,30d. The difference between a Julian year of 365;15d and a tropical year of 365;14,33,30d is 0;0,26,30d, and 0;0,26,30 = 1/135;51 ≈ 1/136. Hence, the implied length of the tropical year is 365 ¼ –1/136 days. In some manuscripts of Ibn Ezra’s text this defcit is given as 1/106, but in one manuscript it is given correctly as 1/136: Millás 1947, p. 95:13 - 14; cf. Toomer 1969, pp. 318 - 319. At a rate of 0;0,26,30º/y, the time needed to progress 360º is 360/0;0,26,30≈360 · 136 = 48960y, which was rounded to 49000y. The defcit from a Julian year of 365;15d is obtained by dividing the time needed to progress 1º (360/49000) by the daily velocity of the Sun (360 / T), where T is the length of the tropical year. Thus,

T = 365;15 – (360/49000) / (360/T).

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This expression can be reduced to

T = 365;15 – T/49000.

Hence, 365;15 = T + T/49000, which is equivalent to

T = 365;15 · (49000/49001).

The result for T is 365;14,33,9,57,4,26,... d. The length of the tropical year in John of Murs’s Expositio (Paris, Bibliothèque Nationale de France, MS lat. 7281, 157r; cf. Poulle 1981, p.253), and in the editio princeps of the Parisian Alfonsine Tables (Ratdolt 1483, d3r) is given as 365d 5;49,15,59,34,3h. This value, where the excess over 365 days is expressed in hours, is equivalent to 365;14,33,9,57,20,7,30d, which only differs in the ffth sexagesimal place from the accurately computed value for T in days shown above. As North noted (1996, p. 463), it is nonsensical to make a «real» motion dependent on an arbitrary value for the number of days in a calendar year, and there is no precedent for such dependence in earlier dis- cussions of precession and trepidation. We also note the false precision of the length of the tropical year — to seven places — because we start with a round number 49000, and then divide it by 49001, which will yield as many places as one is patient enough to compute. The motion of the apogees is taken to be uniform, and the distance of the apogee of each planet from the solar apogee is kept constant in each of the 10 eras, indicating that the entries for the apogees given in the tables were systematically computed. Comparison with previous sets of tables is only possible for the Hijra, because this is the only epoch for which values of the planetary apogees are generally given. The following entries for the apogees at the Hijra are displayed in the editio princeps of the Parisian Alfonsine Tables:

Sun and Venus 75;59,21º Saturn 237;57,40,58º Jupiter 158;10,58,58º Mars 119;46,11,58º Mercury 195;13,31,58º

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For astronomical purposes the epoch of the Hijra is noon on Wednesday, July 14, 622 (JDN 1948438), where the civil epoch of the Hijra is 6 hours later — at sunset, considered the beginning of Thursday: Neugebauer 1962a, pp. 9 - 11, and Sachau 1879, p. 34. These values may be compared with those for the Hijra found in previous zijes and computed for different meridians. See Table 2, where we have also included al-Battānī’s apogees for 880 ad.

Table 2: Apogees for the Hijra, as well as those in al-Battānī’s zij for 880 ad

al-Khwārizmī a al-Battānī b Toledan Tables Ibn al-Kammād Ibn al-Bannāʼ (Arin) (Raqqa) (Toledo) (Córdoba) (Marrakesh) Sidereal Tropical Sidereal Sidereal Sidereal Sun 77;55 82;14 77;50 76;45,21 76;44,17 Saturn 244;55 244;28 240; 5 238;38,30 239;42,45 Jupiter 172;32 164;28 164;30* 158;21, 0 159;42,45 Mars 128;24 126;58* 121;50 119;41, 0 122;12,45 Venus 81;15 82;14 77;50 76;45,21 76;44,17 Mercury 224;54 201;28 197;30 198;21, 0 198;24,17

a. Note that for al-Khwārizmī the positions of the apogees of the Sun and Venus differ (which is not the case in the other zijes): Neugebauer 1962a, p. 99, and Pedersen 2002, p. 1228. b. The values given for al-Battānī are tropical and correspond to year 1191 of the Seleucid Era (879/880 ad): Nallino 1903 - 1907,1: 239 - 241, and 2: 108, 114, 120, 126, and 132. * For discussion of these problematic values, see below.

In Table 2 for Mars the text of al-Battānī has 126;58º (Nallino 1903 - 1907, 2: 120). But the differences displayed in Table 3 suggest that the value for the apogee of Mars attributed to al-Battānī by subsequent astronomers was 126;18º, for they yield differences similar to those for the other planets. Likewise, the value 164;30° for the apogee of Jupiter seems to be a «faulty» value used by the author(s) of the Toledan Tables, rather than 160;30° which would conform with the values in other zijes: see Table 3. Moreover, as shown in Table 2, it is clear that the apogees for each planet — including Jupiter (as corrected to 160;30º) — in the Toledan Tables, Ibn al-Kammād, and Ibn al-Bannāʼ are quite close to one another and probably have a common source, suggesting that they all refect a tradition going back to Azarquiel: cf. Samsó and Millás 1998, p. 286. The best results in comparing the values of the apogees in PAT are obtained with those in al-Battānī’s zij and the Toledan Tables (see Table 3, where the two prob-

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lematic entries for Jupiter in the Toledan Tables and for Mars in al-Battānī’s zij are included).

Table 3: Comparison of the apogees for the Hijra in the Parisian Alfonsine Tables and the Toledan Tables with those in al-Battānī’s zij for 880 ad

al-Battānī Toledan Parisian al-Battānī Todedan al-Battānī Tables Alfonsine – Toledan Tables – PAT Tables Tables – PAT Sun 82;14° 77;50° 75;59,21º 4;24° 1;50,39° 6;14,39° Saturn 244;28 240; 5 237;57,41 4;23 2; 7,19 6;30,19 160;30 3;58 2;19, 1 Jupiter 164;28 158;10,59 6;17, 1 164;30 -0; 2 6;19, 1 126;58 5; 8 7;11,48 Mars 121;50 119;46,12 2; 3,48 126;18 4;28 6;31,48 Venus 82;14 77;50 75;59,21 4;24 1;50,39 6;14,39 Mercury 201;28 197;30 195;13,32 3;58 2;16,28 6;14,28

We can also offer an explanation for the difference of about 4;23° between al- Battānī’s values for the apogees and those in the Toledan Tables. According to many sources, the sidereal coordinates were equal to the tropical coordinates about 40 years before the Hijra, that is, about 582: Samsó 1997, p. 108; cf. Goldstein 2011, pp. 78 - 79. At a rate of 1° in 66 years for precession (al-Battānī’s value: see Nallino 1903 - 1907, 1:128), in 297 years (= 879 - 582) the apogees would have moved 4;30°, which is close to the differences in the apogees that we found. How- ever, 4;23° in about 297 years corresponds to about 1° in 68 years (more precisely, 297/4;23 = 67;45,52), which is within the range of values for precession in the Mid- dle Ages. It should be noted that this epoch, about 582AD, is unrelated to the epoch of the periodic term in PAT, which has been shown to be 16 ad. In sum, despite the variation in the readings of some entries, al-Battānī seems to be the source of the apogees for the Hijra in the Parisian Alfonsine Tables as well as in the Toledan Tables.

3. Radices

The editio princeps of the Parisian Alfonsine Tables also lists the radices for the mean motions of the 8th sphere and the celestial bodies for the same 10 eras used

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for the apogees (Ratdolt 1483, c8r - d1r). The entries are given to fourths for the two luminaries and to thirds for the rest. Table 4 displays the radices for the mean motions at the Hijra in the editio princeps of PAT and in previous zijes, as noted in Table 2.

Table 4: Radices for the Hijra

al- al-Battānī5 Toledan Ibn Ibn-Isḥāq PAT, 1483 Khwārizmī (Raqqa) Tables al-Kammād (Toledo) (Toledo) (Arin) Tropical (Toledo) (Córdoba) Sidereal Tropical Sidereal Sidereal Sidereal Sun 113;25,48 113;58, 4 113;41,11 113;20,30* 113;21,24 114;52, 2, 0,50 Moon, λ 117;45,17 119;43,16 120;58,18 120;34,42 120;32,32 122; 1,16,23,53 Moon, α 104;36, 1 106;30,40 108; 8,39** 108;11 108; 8,39** 107;21,27,42,28 Moon, 233;47,38 233;45,18 234; 9,55 233;30 234;19,37 233;20,35,51 node*** Saturn 117;58,37 116;15 115;51,15 115;30,30* 116;17,15 118;21, 0, 3 Jupiter 330;16,49 332; 3 331;39,37 330;19* 330;11 331;43, 9,52 Mars 210;25,15 211;48 211;24,59 211;26* 210;37,25 212;42, 3,32 Venus, α 46; 0, 3 45; 7 45;28,37 45;21 44;30 47;41,17,16 Mercury, α 68;53,10 73;23 73;46,18 74; 1 73;25,29 73;26,14,31

* The sidereal longitudes were obtained by adding the corresponding radix for the center of the Sun and the planets to their respective apogees: see Chabás and Goldstein 1994, p. 33. ** Ibn-Isḥāq has exactly the same value for the lunar anomaly as in the Toledan Tables. *** The entries in the tables are 360 – n, where n refers to the longitude of the ascending node.

A comparison of the radices in the various sets yield interesting results. As can be seen from Table 5, the radices for the 5 planets and the lunar node in the To- ledan Tables are clearly related to those in the zij of al-Battānī, for they differ by a constant of about 0;23°. This amount is mainly due to the difference between sidereal and tropical coordinates. As mentioned above, the date when tropical and sidereal coordinates agreed with one another was taken to be about 40 years before the Hijra and, with a rate of precession of 1° in 100 years (Ptolemy’s value), in 40 years precession would amount to 0;24°. However, this correction for pre-

5. In the zij of al-Battānī, the radices are for the Hijra, although the apogees are for 880 ad (see Table 2).

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cession should only apply to planetary longitudes, not to the anomalies of Venus and Mercury (despite the textual evidence displayed in Table 5).

Table 5: Comparison of the radices for the Hijra in al-Battānī’s zij and the Toledan Tables

al-Battānī TT Batt. – TT (Raqqa) (Toledo) Tropical Sidereal Sun 113;58, 4 113;41,11 +0;16,53 Moon, λ 119;43,16 120;58,18 –1;15, 2 Moon, α 106;30,40 108; 8,39 –1;37,59 Moon, node 233;45,18 234; 9,55 –0;24,37 Saturn 116;15 115;51,15 +0;24 Jupiter 332; 3 331;39,37 +0;23 Mars 211;48 211;24,59 +0;23 Venus, α 45; 7 45;28,37 +0;22 Mercury, α 73;23 73;46,18 +0;23

For the recomputation of the longitudes we assume that the difference in time from Raqqa to Toledo is about 3 hours, which we obtain considering the longitude of Toledo to be about 28° (counted from the «meridian of water»: cf. Comes 1994), rather than the value of 11° (counted from the Fortunate Islands) that occurs in the Toledan Tables (Toomer 1968, p. 134; Pedersen 2002, pp. 1512 and 1516; cf. Kennedy and Kennedy 1987, p. 357) and the longitude of Raqqa to be 73;15° (Nallino 1903 - 1907, 2:41). Then the difference between these two cities is about 45° (≈ 73;15° –28°). In this time the mean Sun moves 0;7,24°, and 0;24° – 0;7,24° = 0;16,36° (in contrast to 0;16,53° in Table 5). Anal- ogously, the lunar mean motion in longitude in 3 hours is 1;38,49°, and 0;24° – 1;38,49° = –1;14,49° (in contrast to –1;15,2° in Table 4). And the mean lunar motion in anomaly in 3 hours is 1;37,59° (exactly as in Table 5): in this case there is no correction for precession which only applies to longitudes. The difference for the lunar node is about –0;24° (rather than +0;24°) because the nod- al motion takes place in the retrograde direction. The agreement is much better than for the apogees, where the differences range between 3;58º and 4;28º (see Table 3, above). We conclude that the radices for the Hijra in the Toledan Tables were all derived from al-Battānī’s zij and did not depend on any new observations. We have

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also detected an inconsistency in the Toledan Tables: there should be no correction for precession in the radix of anomaly. But, while the lunar anomaly is not corrected for precession, the anomalies of Venus and Mercury include this correction. It follows that the author(s) of the Toledan Tables did not compute the radices correctly for Venus and Mercury. We are aware that Ptolemy’s rate of precession of 1° in 100 years, which seems to have been used in computing the radices, is inconsistent with the rate of 1° in 66 years that we found in Section 2 for the apogees. And the Toledan Tables as well as PAT invoke models of trepidation rather than precession. The inconsistencies suggest that the authors of these sets of tables accepted material from a variety of sources. In contrast, the radices for the Hijra in PAT show no affnity with those in al- Battānī’s zij or those in the Toledan Tables (see Table 6).

Table 6: Comparison of the radices for the Hijra in al-Battānī’s zij and the Parisian Alfonsine Tables

al-Battānī PAT, 1483 (Raqqa) (Toledo) Batt –PAT Tropical Tropical Sun 113;58, 4 114;52, 2 –0;53,58 Moon, λ 119;43,16 122; 1,16 –2;18, 0 Moon, α 106;30,40 107;21,28 –0;50,48 Moon, node 233;45,18 233;20,36 +0;24,42 Saturn 116;15 118;21 –2;5 Jupiter 332; 3 331;43 –0;20 Mars 211;48 212;42 –0;54 Venus, α 45; 7 47;41 –2;34 Mercury, α 73;23 73;26 –0;3

4. Mean motions

The editio princeps of the Parisian Alfonsine Tables displays tables for the daily mean motions of (1) access and recess of the 8th sphere (d4r); (2) apogees and the fxed stars (d4v); (3) Sun, Venus, and Mercury (d5r); (4) Moon in longitude, anom-

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aly, and argument of latitude (d5v - d6v); (5) the lunar elongation from the Sun (d7r); (6) lunar node (d7v); (7) Saturn, Jupiter, and Mars in longitude (d8r - e1r); and (8) Venus and Mercury in anomaly (d1v - d2r). Of special interest is the value of the mean motion for the Sun, Venus, and Mer- cury, 0;59,8,19,37,19,13,56º/d, already reported by John of Murs in his Expositio intentionis regis Alfonsii as belonging to the «Tables of Alfonso» (Poulle 1981, p. 253). This value results from the length of the tropical year, 365;14,33,9,57,4,26... d which, in turn, derived from a period of revolution of 49000 years (see section 2, above).6 A potential source has been suggested for the Alfonsine parameters of the planetary mean motions. Chabás and Goldstein (2012, pp. 57 - 59) listed values for the mean motions used by different pre-Alfonsine authors and concluded: «Compari- son of Ibn al-Raqqām’s values for Venus and Mercury and those given in the Parisian Alfonsine Tables show very close agreement (the difference is about 0;0,0,0,0,0,30º/d), another clear indication of the Andalusian character of the Pa- risian Alfonsine Tables that has not previously been recognized»: see Chabás and Goldstein 2012, pp. 57 - 59. Based on a comparison between the mean motions in PAT and those in the zij of Ibn Isḥāq, Samsó (2019, p. 363) concluded: «The fact that most Alfonsine mean motions derive clearly from Ibn Isḥāq’s parameters shows the Toledan origin of the Parisian Alfonsine mean motion tables», and he later added that «Ibn Isḥāq’s mean motion parameters seem to derive from those of the Toledan Tables» (Samsó 2020, ch. 7, section 7.6.2.2.2.2). Thus, the Arabic origin of the Alfonsine mean motions of the planets seems to be frmly established. In order to compare systematically the Alfonsine (tropical) values with those used by Andalusian and Maghribi astronomers (sidereal), we start with those quantities not affected by trepidation, namely, lunar anomaly and the anomalies of Ve- nus and Mercury. The results are shown in Tables 7 and 8.7

6. The value for the daily solar mean motion confrms that of the tropical year, 365;14,33,9, 57,4,26... d, and shows that the value for the tropical year reported by John of Murs in the Expo- sitio (Poulle 1981, p. 251), 365;14,33,9,59,20,... d, has to be understood as an error in computing by John himself or an error that was already in the source on which he depended. Cf. North 1996, pp. 458 - 460. 7. The following values are taken from Chabás and Goldstein 2012, except those for Ibn Isḥāq and Ibn al-Bannāʼ, which were taken from Samsó and Millás 1998, p. 262.

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Table 7: Mean motions in anomaly in the zijes of Ibn al-Raqqām and Ibn Isḥāq and comparisons with PAT (º/d)

PAT– PAT Ibn al-Raqqām Ibn Isḥāq PAT – Isḥāq Raqqām Moon 13;3,53,57,30,21,4,13 13;3,53,56,17,51,25,27 0;0,0,1,12,30 13;3,53,56,17,52,4 0;0,0,1,12,29 Venus 0;36,59,27,23,59,31 0;36,59,27,23,58,51 0;0,0,0,0,0,40 0;36,59,27,23,59,25 0;0,0,0,0,0,6 Mercury 3;6,24,7,42,40,52 3;6,24,7,42,40,23,56 0;0,0,0,0,0,28 3;6,24,7,42,40,49 0;0,0,0,0,0,3

Table 8: Mean motions in anomaly in the zij of Ibn al-Bannāʼ and the Toledan Tables and comparisons with PAT (º/d)

PAT Ibn al-Bannāʼ PAT – Bannāʼ Toledan Tables PAT – TT Moon 13;3,53,57,30,21,4,13 13;3,53,56,17,52,7 0;0,0,1,12,29 13;3,53,56,17,57 0;0,0,1,12,24

Venus 0;36,59,27,23,59,31 0;36,59,27,23,59,32 –0;0,0,0,0,0,1 0;36,59,29,27,29 –0;0,0,4,3

Mercury 3;6,24,7,42,40,52 3;6,24,7,42,41,5 –0;0,0,0,0,0,13 3;6,24,7,39,31 0;0,0,0,3,10

In the three cases, the values in Ibn al-Bannāʼ and Ibn Isḥāq are the same, with differences of 3, 7, and 16 sixths. The differences amount to about 30 sixths between Ibn al-Raqqām and Ibn Isḥāq. Since these are values not explic- itly stated in the zijes but are derived from entries suffciently far apart in time in the corresponding tables, we consider these values to be essentially the same in all three zijes, despite the minor differences among them displayed here. All three zijes compare well with the values in PAT for Venus and Mercury. In contrast, those in the Toledan Tables do not seem to have served as the basis for PAT. For the lunar anomaly, all three Arabic zijes and the Toledan Tables give ap- proximately the same value, which is different from what we fnd for the anomalies of the two inferior planets; thus, all differ from PAT by the same amount. It turns out that this is the same value used in al-Battānī’s zij as well as in Ptolemy’s Almagest (13;3,53,56,17,51,59°). The reason for the small discrep- ancy in PAT of about 0;0,0,1,12,24° from this «long-standing consensus» is not clear to us. We now turn to the mean motions in longitude. The results of the comparisons with PAT are shown in Tables 9 and 10.

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Table 9: Mean motions in longitude in the zijes of Ibn al-Raqqām and Ibn Isḥāq and comparisons with PAT8 (º/d)

PAT Ibn al-Raqqām PAT – Raqqām Ibn Isḥāq PAT – Isḥāq Sun 0;59,8,19,37,19,13,56 0;59,8,11,28,26,22,5 0;0,0,8,8,52 0;59, 8,11,28,26,22 0;0,0,8,8,52 Moon 13;10,35,1,15,11,4,35 13;10,34,52,46,51,19,31 0;0,0,8,28,20 13;10,34,52,46,53 0;0,0,8,28,18 Node –0;3,10,38,7,14,49,10 –0;3,10,46,40,59,49,30 0;0,0,8,33,45 –0;3,10,46,40,58,42 0;0,0,8,33,45 Saturn 0;2,0,35,17,40,21 0; 2, 0,27,46,42,52 0;0,0,7,30,57 0; 2, 0,27,46,44,53 0;0,0,7,30,55 Jupiter 0;4,59,15,27,7,23,50 0; 4,59,7,36,24,31,1 0;0,0,7,50,43 0; 4,59,7,36,25,41 0;0,0,7,50,42 Mars 0;31,26,38,40,5,0 0;31,26,31,9,4,52 0;0,0,7,31,0 0;31,26,31,9,5,59 0;0,0,7,30,59

Table 10: Mean motions in longitude in the zij of Ibn al-Bannāʼ and the ToledanTables and comparisons with PAT (º/d)

PAT Ibn al-Bannāʼ PAT – Bannāʼ Toledan Tables PAT – TT Sun 0;59,8,19,37,19,13,56 0;59, 8,11,28,26,24 0;0,0,8,8,50 0;59,8,11,28,27 0;0,0,8,8,52 Moon 13;10,35,1,15,11,4,35 13;10,34,52,46,51,20 0;0,0,8,28,20 13;10,34,52,48,47 0;0,0,8,26,24 Node –0;3,10,38,7,14,49,10 –0;3,10,46,40,59,50 0;0,0,8,33,45 –0;3,10,46,42,33 0;0,0,8,35,18 Saturn 0;2,0,35,17,40,21 0; 2, 0,27,46,42,52 0;0,0,7,30,57 0;2,0,26,35,17 0;0,0,8,42,23 Jupiter 0;4,59,15,27,7,23,50 0; 4,59,7,36,24,34,54 0;0,0,7,50,43 0;4,59,7,37,19 0;0,0,7,49,48 Mars 0;31,26,38,40,5,0 0;31,26,31,9,5,32 0;0,0,7,31,1 0;31,26,32,15,17 0:0.0,6,24,48

As was the case for the lunar anomaly, the zijes of Ibn al-Bannāʼ, Ibn Isḥāq, and Ibn al-Raqqām have almost the same values for the six parameters considered here; clearly, they all belong to the same tradition. The values given in the Toledan Tables are generally higher in the fourths, except for the Sun, where the value differs from those in the three zijes by less than 1 sixth, and for Saturn and Mars, where the values differ from them by about 1 third. The comparisons in Tables 9 and 10 show that grosso modo the three zijes (those of Ibn al-Bannāʼ, Ibn Isḥāq, and Ibn al-Raqqām) share the same differences with PAT, concentrated around, say, 0;0,0,8,20º (for the Sun and the Moon) and 0;0,0,7,40º (for the superior planets). The entries in theToledan Tables also share these differences, but for Saturn and Mars. We conclude that the values for the

8. All values for Ibn al-Raqqām are slightly different in Samsó and Millás 1998 (reproduced here) from those in Samsó 1997.

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Sun, the Moon and Jupiter in PAT agree with the three zijes and the Toledan Ta- bles (adjusting for the differences in coordinates), whereas the values for Saturn and Mars agree with the three zijes (adjusting for the differences in coordinates), but not the Toledan Tables. The differences between the entries in PAT and in the Arabic zijes is due to precession. Their value is compatible with 0;0,0,8,9,22º/d, which is the rate of precession equivalent to 1º in 72 ½ years ascribed to Alfonso by John of Murs in his Expositio. A few conclusions can be drawn for the mean motions. First, the parameters in the zijes of Ibn al-Bannāʼ, Ibn Isḥāq, and Ibn al-Raqqām are the same, despite their differences in the sixths, and produce the same results. It should be recalled that 1 sixth is about 2 · 10 - 11, a tiny amount that, even when multiplied by a huge number such as the number of days in, say, 1000 years, it does not affect the result in the seconds, which is the precision usually found in astronomical tables. Sec- ond, comparisons of the mean motions in anomaly show that PAT borrowed the values used in these, or similar, zijes for the superior planets, indicating that An- dalusian/Maghribi astronomy is at the origin of the corresponding Alfonsine parameters. Third, as regards the quantities subject to precession, the daily mean motions used by Alfonsine astronomers were taken from then same source, to which a fxed quantity was added to convert sidereal coordinates into tropical coordinates. The two authors of this paper, in agreement with Julio Samsó, have repeatedly characterized the Castilian Alfonsine Tables, and thus the Parisian Alfonsine Ta- bles, as a zij, as Arabic astronomers had been compiling for centuries. And this so not only because of the structure and contents of this set of tables, but also for the approaches and parameters that have been identifed in them. The birthplace of these tables is undoubtedly Toledo — not any other place — as we have been argu- ing for a long time.9

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9. In particular, see Chabás and Goldstein 2003, pp. 243 - 244.

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