By the end of the fourteenth century the Alfonsine tradition, which origi-nated in Castile a little more than 100 years earlier, had become the main computational tool for European astronomers. A great variety of astronomi-cal tables, often accompanied by texts, followed this tradition, using the same models to describe the motions of the celestial bodies and the same under-lying parameters, but differing in presentation. This “Alfonsine corpus”, as we have recently named it, dominated the scene of Western astronomy for several centuries.1
Within this corpus are the Parisian Alfonsine Tables of which hundreds of copies in manuscript are extant as well as two editions that appeared before 1500. Of particular interest are various methods and tables for finding the time from mean syzygy (i.e., conjunction or opposition of the Sun and the Moon) to true syzygy, starting with the method described in the canons by John of Saxony (1327).2 A specific approach to this problem, summarized below, was introduced by an otherwise almost unknown Nicholaus de Heybech of Erfurt (c. 1400).3 Heybech’s table was modified as it was transmitted from Erfurt to Poland, then to Salamanca, and finally to Jerusalem, and it is a remarkable example of the variety within the Alfonsine corpus that did not involve any
* Journal for the History of Astronomy 39 (2008), 345–355.
1 See José Chabás and Bernard R. Goldstein, The Alfonsine Tables of Toledo (Dordrecht and Boston, 2003).
2 José Chabás and Bernard R. Goldstein, “Computational astronomy: Five centuries of finding true syzygy”, Journal for the history of astronomy, xxviii (1997), 93–105.
3 José Chabás and Bernard R. Goldstein, “Nicholaus de Heybech and his table for finding true syzygy”, Historia mathematica, xix (1992), 265–289. We have seen a dozen manuscripts of Nicholaus de Heybech’s tables: Basel, Universitätsbibliothek, f.ii.7; Dijon, Bibliothèque Municipale, 447; Paris, Bibliothèque nationale de France, lat. 7287 and lat. 7290a; Cues, 211;
Vienna, Nationalbibliothek, 2440; Cracow, Biblioteka Jagiellońska, 609, 610, 613, 1852, and 1865 (twice); and Princeton, University Library, Grenville Kane Collection 51. Several authors have mentioned other manuscripts containing the same material: Bern, 454; Vatican, Pal. lat. 1376;
Vienna, Nationalbibliothek, 5245; and Munich, Clm 14111 and 26666.
changes in the theory or the parameters. Moreover, in his computations of the entries in his table, Heybech used a table for lunar velocity that first appears in Paris in the 1330s; this is part of the Alfonsine corpus and differs significantly from the tables for lunar velocity in the traditions of al-Battānī and of al-Khwārizmī.4 We thus offer an illustration of the transmission of a computational technique within the Alfonsine corpus that kept evolving during its long journey through a considerable part of Europe and beyond.
Heybech’s method was presented in the form of a single table in 5 columns and consisted in taking the time interval between mean syzygy and true syzygy as the difference between two independent terms, one for the Sun and one for the Moon. Each term is calculated separately, and both require the computa-tion of a set of minimum and maximum values and the use of an interpola-tion scheme for intermediate values. In Heybech’s table, this scheme is a list of interpolation coefficients, ranging from 0 to 1 (column iii, headed minuta proportionalia, where 60 minutes = 1), depending on the mean lunar anomaly, when computing the solar term, and on the mean solar anomaly, in the case of the lunar term. Besides column iii, the computation of the solar term requires columns i (headed equatio solis) and ii (headed diversitas equationis solis), both given in hours and minutes. As for the lunar term, besides column iii, its com-putation requires columns iv (headed equatio lune) and v (headed diversitas equationis lune), the former given in hours and minutes, and the latter in min-utes of an hour.
The solar term (Δts) can be obtained by means of the expression, Δts= c1(κ̄) – c2(κ̄) · c3(ᾱ),
where κ̄ is the mean solar anomaly and ᾱ the mean lunar anomaly, and c1, c2, and c3 represent entries in columns i, ii, and iii, respectively. The entries in col. i depend on κ̄ and assume that ᾱ = 0°; those in col. ii also depend on κ̄ and represent the differences between the values for ᾱ = 0° and ᾱ = 180°, for a given κ̄; and the entries in col. iii, ranging from 0 to 1, are for interpolation for other values of ᾱ between 0° and 180°.
4 Bernard R. Goldstein, “Lunar velocity in the Ptolemaic tradition”, in The investigation of difficult things: Essays on Newton and the history of the exact sciences, ed. by P.M. Harman and A.E. Shapiro (Cambridge, 1992), 3–17; idem, “Lunar velocity in the Middle Ages: A comparative study”, in From Baghdad to Barcelona: Studies in the Islamic exact sciences in honour of Prof.
Juan Vernet, ed. by J. Casulleras and J. Samsó (2 vols, Barcelona, 1996), i, 181–194.
figure 3.1 Facsimile of Nicholaus de Heybech’s table (excerpt): Vienna, Nationalbibliothek, ms 2440, f. 74v
Similarly, the lunar term (Δtm) can be obtained by means of the expression, Δtm= c4 (ᾱ) – c5 (ᾱ) · c3 (κ̄),
where c4 and c5 represent entries in columns iv and v, respectively. The entries in col. iv depend on ᾱ and assume that κ̄ = 0°, and the entries in col. v also depend on ᾱ and represent the differences between the values for κ̄ = 0° and κ̄ = 180°, for a given ᾱ. Again, the entries in col. iii are for interpolation. Thus, according to Heybech’s table, the time from mean syzygy to true syzygy is given by
(1) Δt = Δts– Δtm= c1(κ̄) – c2(κ̄) · c3(ᾱ) – c4(ᾱ) + c5(ᾱ) · c3(κ̄).
Heybech’s method for determining Δt was appreciated by many medieval astronomers, for it simplified their computations without sacrificing accu-racy.5 In the manuscripts that preserve this table there is usually a short canon explaining its use, but not the method for computing the entries. In all cases the tables are identical, but for copyist’s errors.
Madrid, Biblioteca Nacional, ms 3385, containing a set of tables in Latin which we call the Tabule Verificate for Salamanca (ff. 104r–113r), has much the same material as in Heybech’s table, but with a different presentation.6 In these tables, fully part of the Alfonsine corpus, the epoch is 1 Jan. 1461. The name of the author of the Tabule Verificate (henceforth tv) is unfortunately not known, although we have identified Nicholaus Polonius as the most likely candidate among the few astronomers working in the Castilian city of Sala-manca. Polonius was a Polish scholar who came to Salamanca no later than 1460 and held the newly established chair in astronomy/ astrology at the uni-versity there until 1464. It is reasonably clear that he brought the Tabule Res-olute (a form of the Parisian Alfonsine Tables) with him from Poland.7 The
5 See Chabás and Goldstein, “Computational astronomy” (ref. 2). The conventions for the algebraic signs in Eq. 1 are not well described in Heybech’s canons, whereas the versions in tv and Zacut are unambiguous because the headings tell the user when to add and when to subtract.
6 José Chabás and Bernard R. Goldstein, Astronomy in the Iberian Peninsula: Abraham Zacut and the transition from manuscript to print (Philadelphia, 2000), especially pp. 23−36.
7 On this set of tables, see Jerzy Dobrzycki, “The Tabulae Resolutae”, in De astronomia Alphonsis Regis, ed. by M. Comes, R. Puig, and J. Samsó (Barcelona, 1987), 71–77; José Chabás, “Astronomy at Salamanca in the mid-fifteenth century: The Tabulae Resolutae”, Journal for the history of astronomy, xxix (1999), 167–175.
tables we have labelled tv 7 (f. 106r–v) and tv 8 (f. 107r–v) in a previous pub-lication represent Heybech’s columns for the lunar correction and the solar correction, respectively, but they are organized in a different way: column 2 in tv 7 displays c4(ᾱ), that is, column iv in Heybech’s table; column 4 in tv 7 is equivalent to 1 – c3(ᾱ), that is, the complement in 1 to column iii in Hey-bech’s table; column 2 in tv 8 represents c1(κ̄) – c2(κ̄), that is, the difference between columns i and ii in Heybech’s table. However, there is no longer a col-umn equivalent to Heybech’s colcol-umn v. The rest of the colcol-umns in tv 7 and tv 8 contain the arguments and line-by-line differences of the entries of other columns.
With the entries in tv 7 and tv 8,
(2) Δt = [c1(κ̄) – c2(κ̄)] – c4(ᾱ) + [1 – c3(ᾱ)] · c2(κ̄).
This expression is equivalent to (3) Δt = c1 (κ̄) – c2 (κ̄) · c3 (ᾱ) – c4 (ᾱ),
and it agrees with the first three terms in Eq. (1). The suppression of the fourth term, c5 (ᾱ) · c3 (κ̄), is indeed an acceptable approximation because in Eq. (1) its contribution is at most 0;04h, a small amount compared with the maximum values of the first, second, and third terms (4;47h, 1;01h, and 9;40h, respectively).
It should be noted that the changes introduced in these two tables by the unknown author of the Tabule Verificate are not mere variations in the positions of the columns, but imply a different approach from that in Heybech’s table, among other things because attention shifts from lunar apogee (ᾱ = 0°), which is assumed for col. i in Heybech’s table, to lunar perigee (ᾱ = 180°), which is assumed for col. 2 in tv 8.
The next step in the transformation of this table took place in 1513 in Jerusa-lem where Abraham Zacut (1452–1515) had recently arrived.8 Zacut’s best known astronomical work was composed in Hebrew in Salamanca (1478), and entitled ha-Ḥibbur ha-Gadol (The great composition). It was later published (with a number of modifications) in Latin and Castilian in Leiria, Portugal (1496), and entitled Almanach perpetuum. This work, in turn, was later
trans-8 For biographical details, see Chabás and Goldstein, Abraham Zacut (ref. 6), 6–15.
figure 3.2 Facsimile of tv 7 (excerpt): Madrid, Biblioteca Nacional, ms 3385, f. 106r
figure 3.3 Facsimile of tv 8 (excerpt): Madrid, Biblioteca Nacional, ms 3385, f. 107r
lated into Arabic and diffused in the Islamic world.9 Zacut’s tables are mainly based on the Parisian Alfonsine Tables, but some depend on astronomical traditions in Hebrew that began in Provence in the fourteenth century. For his method of finding the time from mean to true syzygy in the Almanach perpetuum Zacut depended on this Hebrew tradition that derived from the work of Levi ben Gerson (1288–1344) and was transmitted to him through Jacob ben David Bonjorn’s tables (c. 1360), and it was quite distinct from the Alfonsine tradition. But in his new set of tables of 1513 for Jerusalem Zacut included tables for finding Δt that represent a modified version of Heybech’s table. Ironically, in 1478 Zacut used the Christian calendar for mean motions, whereas in 1513 he used the Jewish calendar for this purpose. Zacut’s tables of 1513 in Hebrew for Jerusalem are extant only in fragments:10 we consulted New York, Jewish Theological Seminary of America [jtsa], ms 2574 (not dated), 15 folios (containing only Zacut’s canons and tables).11 This manuscript contains three tables for computing Δt by a method that is similar to the one in the Tabule Verificate.
Table 1 (f. 8b) is for the solar correction when the Moon is at perigee on its epicycle. The argument is the solar longitude, that is, the solar anomaly increased by 90° (under the assumption that the solar apogee is at 90°, which is an adequate value at the time), and it is given in degrees at intervals of 1°.
The entries are displayed in hours and minutes. They derive from Heybech’s table, and correspond to the difference between the entries in columns i and ii, c1(κ̄) – c2(κ̄), as was the case in the Tabule Verificate for Salamanca (tv 8, col. 2). Note however that, contrary to the Tabule Verificate, Zacut’s table for the solar correction is presented as a single table, not as a column of a table, as in tv 8.
Table 2 (f. 9a) is for the lunar correction when the Moon is at perigee on its epicycle. The argument is the mean lunar anomaly and it is given in degrees at intervals of 1°. The entries are displayed in hours and minutes. The entries
9 For Zacut’s tables in the Islamic world, see Julio Samsó, “Abraham Zacut and Joseph Vizinho’s Almanach perpetuum in Arabic”, Centaurus, xlvi (2004), 82–97; idem, “In pursuit of Zacut’s Almanach perpetuum in the eastern Islamic world”, Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, xv (2002–2003), 67–93.
10 See Bernard R. Goldstein, “The Hebrew astronomical tradition: New sources”, Isis, lxxii (1981), 237–251, p. 248.
11 Another fragment containing Zacut’s tables of 1513 is extant in NewYork, jtsa, ms 2567.
There is no hint in Zacut’s canons that he was aware of Heybech or that he had direct access to his table.
table 1 An excerpt of the table for the first correction (New York, jtsa, ms 2574, f. 8b). Table for the correction of the Sun when the Moon is at the perigee of its epicycle, in hours and minutes.
3s 4s 5s 6s 7s 8s
subtract
1 0; 4h 1;53h 3;16h 3;47h 3;18h 1;54h
2 0; 8 1;57 3;18 3;47 3;16 1;50
3 0;12 2; 0 3;19 3;47 3;14 1;47
4 0;16 2; 4 3;21 3;47 3;12 1;43
5 0;19 2; 6 3;22 3;47 3;10 1;39
…
10 0;38 2;22 3;30 3;46 2;58 1;20
…
15 0;56 2;36 3;37 3;42 2;44 1; 1
…
20 1;15 2;49 3;42 3;37 2;30 0;42
…
25 1;32 3; 1 3;46 3;31 2;14 0;21
…
30 1;50 3;14 3;47 3;22 1;57 0; 0 add
in this table also derive from Heybech’s table, and correspond to its column iv, c4(ᾱ), as was the case in the Tabule Verificate for Salamanca (tv 7, col. 2). Note again that Zacut’s table for the lunar correction is presented as a single table, not as a column of a table, as in the Tabule Verificate.
Table 3 (f. 9b) is a double argument table where Zacut combined both solar and lunar components. The rows were computed for values of the lunar anomaly at intervals of 10°, ranging from 0s 0° to 6s 0°, whereas the columns were computed for values of the solar longitude at intervals of 15°, beginning with 3s 0° (as in Table 1). The entries in this double argument table are given in minutes of time.
We note that for lunar anomaly, the maximum correction takes place for argument 0°, and vanishes for argument 180°. This should indeed be so, for this correction is to be added to, or subtracted from, the value found in Table 1 when the Moon is at its epicyclic perigee; thus, the correction at argument 180°
table 2 An excerpt of the table for the second correction (New York, jtsa, ms 2574, f. 9a). Table for the correction of lunar anomaly, in hours and minutes.
0s 1s 2s 3s 4s 5s
add
0 0; 0h 4;59h 8;32h 9;42h 8;15h 4;43h
1 0;11 5; 8 8;37 9;42 8; 9 4;34
2 0;22 5;17 8;42 9;42 7;58 4;25
3 0;33 5;26 8;42* 9;42 7;55 4;16
4 0;43 5;35 8;51 9;42 7;53 4; 7
…
10 1;44 6;23 9;12 9;31 7;16 3;13
…
15 2;36 7; 1 9;27 9;17 6;42 2;25
…
20 3;25 7;34 9;36 9; 2 6; 4 1;39
…
25 4;13 8; 4 9;42 8;40 5;24 0;50
…
30 4;59 8;32 9;42 8;15 4;43 0; 0
subtract
* With jtsa, ms 2567, f. 65b, read: 8;46.
(i.e., lunar epicyclic perigee) is 0. The maximum effect of solar anomaly should be at 90° from the solar apogee and this is represented by the column for 6s 0°. Hence, this third correction is to improve the first correction displayed in Table 1, where the only variable considered was solar longitude. Table 3 then takes into account the effect of the change in lunar anomaly in the time interval due to the solar motion (where the heading is the solar longitude), and represents the term [1 – c3(ᾱ)] · c2(κ̄) in Eq. (2). This is certainly the case:
1 – c3(ᾱ) appears as the column for a solar longitude of 6s 0°, i.e., when the Sun is 90° ahead of apogee and the correction reaches its minimum; it is the complement in 1 of Heybech’s column iii, as in tv 7, column 4. Moreover, c2(κ̄) appears as the row for a lunar anomaly of 0s 0°, and it is column ii in Heybech’s table, as in tv 8, column 4. The product of the entries in this row and this column generates the rest of the table. For example, consider the entries in the row for 2s 0°: 0, 12, 23, 32, …, 47, …, 24, 12, 0. Each one is found
table 3 An excerpt of the table for the third correction (New York, jtsa, ms 2574, f. 9b). Table for the correction of all values for the lunar anomaly to be added to its value at the perigee of its epicycle: a double argument table [luaḥ meḥubberet; lit.: a combined table]
Solar
long. 3s 0° 3s 15° 4s 0° 4s 15° … 6s 0° … 8s 0° 8s 15° 9s 0°
Lunar
anom. subtract
0s 0° 0 15 29 41 60 31 16 0
0s 10° 0 15 29 40 59 30 15 0
0s 20° 0 14 28 40 58 29 15 0
1s 0° 0 14 28 39 56 29 15 0
1s 10° 0 13 27 37 44* 28 14 0
1s 20° 0 13 25 35 51 26 13 0
2s 0° 0 12 23 32 47 24 12 0
…
3s 0° 0 8 16 23 31 17 9 0
…
5s 20° 0 0 0 1 1 1 1 0
6s 0° 0 0 0 0 0 0 0 0
add
* With jtsa, ms 2567, f. 66a, read: 54.
by multiplying the corresponding entry for a lunar anomaly of 0s 0° by 47 (the entry corresponding to a solar longitude of 6s 0°). So, for 2s 0° of lunar anomaly and
3s 0° of solar longitude: 0; 0 · 0;47 = 0; 0 (entry: 0 min);
3s 15° of solar longitude: 0;15 · 0;47 = 0;11,45 = 0;12 (entry: 12 min);
4s 0° of solar longitude: 0;29 · 0;47 = 0;22,43 = 0;23 (entry: 23 min);
4s 15° of solar longitude: 0;41 · 0;47 = 0;32, 7 = 0;32 (entry: 32 min);
…
6s 0° of solar longitude: 0;60 · 0;47 = 0;47 (entry: 47 min);
…
8s 0° of solar longitude: 0;31 · 0;47 = 0;24,17 = 0;24 (entry: 24 min);
8s 15° of solar longitude: 0;16 · 0;47 = 0;12,32 = 0;13 (entry: 12 min);
9s 0° of solar longitude: 0; 0 · 0;47 = 0; 0 (entry: 0 min).
Thus, Zacut replaces two columns in two different tables in the Tabule Ver-ificate by one double argument table where the multiplication is already done, thus facilitating the task of the computer; all that is left is linear interpolation in both the horizontal and vertical directions in the table. The use of a dou-ble argument tadou-ble is consistent with Zacut’s preference for this kind of tadou-ble, which appears already in his Ḥibbur, and this distinguishes him from the tra-dition, always within the Alfonsine corpus, represented by Heybech and the Tabule Verificate. On the other hand, the entries in these three tables are not identical with those in the Tabule Verificate, but both sets are internally consis-tent and differ very slightly.
Chapter 3 of the canons for these tables (jtsa, ms 2574, ff. 12b–13a) concerns the time interval from mean to true syzygy, with instructions on how to find this interval from the three tables, but nothing is said about the origin of this table or the way its entries were computed. Zacut adds a worked example (f. 12b) for finding true conjunction for Tishri 5274am [= 30 Aug. 1513]: mean conjunction took place 3(d) 18;6,30h after noon.12 According to the text, the Sun’s position was then 5s 16;29° and the lunar anomaly was 8s 12;47°. The first correction with solar longitude 5s 16;29° as argument is 3;38,30h to be subtracted. The result is 14;28h [= 18;6,30h – 3;38,30h]. With argument 8s 12° for the lunar anomaly, the second correction is stated to be about 9;10h [ms: 10, written as a word] to be subtracted. The result is then given in the text as 5;22h, although it should be 5;18h [= 14;28h – 9;10h]. The third correction, with the two arguments, is 0;22h to be subtracted. Thus, the final result, as given in the text, is 5;0h. This result can be checked using the tables themselves. In Table 1 the entry for 5s 16° is 3;38h;
in Table 2 the entry for 8s 12° is 9;8h and for 8s 13° it is 9;11h. So 9;10h, the value given in the text for the second correction, agrees with computation using the table. In Table 3, with 5s 15° (rounded from 5s 16° for the solar longitude) and 8s 10° (rounded from 8s 12° for the lunar anomaly) as arguments, the entry is 0;22h. Hence, the total correction is –13;10,30h (= –3;38,30h – 9;10h – 0;22h),13 and the result should be: 18;6,30h – 13;10,30h = 4;56h (text: 5;0h).
It is most likely that for this refinement of a technique in the Alfonsine cor-pus Zacut depended on tables in Latin he had seen in Salamanca many years before he arrived in Jerusalem. Zacut then transmitted his new method for finding Δt in Hebrew, thus contributing to the enlargement of the Alfonsine
12 3(d) means weekday 3, i.e., Tuesday. And indeed 30 Aug. 1513 (jdn 2273923) was a Tuesday.
13 The absolute value of this amount for the total correction is close to its maximum: see Richard L. Kremer, “Wenzel Faber’s tables for finding true syzygy”, Centaurus, xlv (2003), 305–329, p. 314 (table 2).
corpus, a body of astronomical material with no theoretical changes or modi-fications in the models or the parameters that depended, directly or indirectly, on the Alfonsine Tables compiled in Toledo in the 1270s and diffused from Paris throughout Europe and beyond in the 1320s.
Epilogue
There is one known instance of a later text that depended on Zacut’s tables of 1513 for finding Δt: a Geniza fragment in Hebrew, ms A 697-1, at the John Rylands University Library, Manchester, England.14 In this brief fragment of an anony-mous calendrical text for 5557am (= 1796–1797),15 the goal is to compute the times of true opposition (full-moon) for each month in the year 5557am as a function of true solar longitude and mean lunar anomaly at mean opposition.
On the first line Zacut is credited for the method of determining true opposi-tions, and the text includes values computed from his first two correction tables (but there is no evidence of his third table). For example, the solar longitude in this text for opposition in Tishri 5557 is 6s 23;47° whose correction is given as 3;33h, and the lunar anomaly at that time is 3s 20;26° whose correction is given as 9;2h. These are exactly the values for these arguments in Zacut’s Tables 1 and 2, where the arc minutes of the arguments have been ignored.
14 We are grateful to Y. Tzvi Langermann for bringing this manuscript to our attention.
15 The date given in the text is not easy to read but 5557am is confirmed by recomputing the astronomical data with Zacut’s tables for 1513.