The Diffusion of the Alfonsine Tables: The case of the Tabulae resolutae

José Chabás University Pompeu Fabra, Barcelona, Spain Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/168/1789147/106361402321147513.pdf by guest on 26 September 2021

The Alfonsine Tables were compiled during the second half of the 13th cen- tury in Toledo, Spain, and were largely diffused throughout Europe, mainly via Paris. They became the basic computing tool for European astronomers during several centuries. The Tabulae resolutae are a particular form of presenting the Alfonsine material which differs in many ways from that in the ªrst printed edition of the Alfonsine Tables (Venice, 1483). This paper focuses on the inºuence of the 15th century Viennese astronomer John of Gmunden on the genesis of the Tabulae resolutae, and analyses its contents and impact on European astronomy.

1. The Tabulae resolutae is a set of astronomical tables that circulated widely in Europe during the 15th century in manuscript form, and as printed books during the 16th century. They are strictly based in the Alfonsine Tables; moreover, they are a particular form of presenting the Alfonsine Tables which largely differs from that in the editio princeps (Venice, 1483). The most visible difference is that the Tabulae resolutae maintain the sys- tem of cyclical radices with intervals of 20 years that is explained in can- ons to the Castilian Alfonsine Tables, rather than the organization in days to be counted sexagesimally, as in the editio princeps. The history of the Alfonsine Tables is well known. Two centuries earlier King and León, called the Learned, gathered at his court a group of Muslim, Jewish, and Christian scholars. They translated scientiªc works from Arabic into Castilian, and wrote some original trea- tises on astronomy and related disciplines. Among them are the Alfonsine

I thank Peter Barker and Bernard R. Goldstein for their for their help in writing this paper, and Beatriz Porres for her useful comments.

Perspectives on Science 2002, vol. 10, no. 2 ©2003 by The Massachusetts Institute of Technology

168 Perspectives on Science 169

Tables, composed of canons and tables, and written in Toledo around 1272 by two of the king’s collaborators: Judah ben Moses ha-Cohen and Isaac ben Sid. The canons are extant in a unique manuscript in Castilian: Ma- drid, Biblioteca Nacional, MS 3306. On the other hand, the Castilian Alfonsine Tables have not been preserved in their original presentation, a fact that has given rise to all kinds of conjectures. These tables depended on the rich astronomical tradition developed in medieval Spain, mainly Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/168/1789147/106361402321147513.pdf by guest on 26 September 2021 from Islamic sources, and were to become the most inºuential astronomi- cal tables in Christian Europe during more than three centuries. In the early 14th century the Alfonsine Tables reached Paris, and in the 1320s some canons were written in Latin to explain their use. The tables were also modiªed from the original model, but it is impossible to know to what extent, for the original set of tables is missing. The Latin version of the Alfonsine Tables recast in Paris then reached Oxford and Italy, and spread very largely throughout Europe, becoming the main computing tool for European astronomers (North 1977). Along with their diffusion, the tables diversiªed. They were adapted to various meridians, other than that of Toledo, which had been maintained by the ªrst Parisian astronomers. Even their presentation kept evolving. However, the underlying models, as well as the parameters they are based on, were never challenged. These tables share other common basic fea- tures, but the variety of sets of tables in the Alfonsine tradition is so vast that the term “Alfonsine Tables”, with no further speciªcation, is no longer appropriate. The scholarship devoted to this particular subject has already reached a level of precision which requires a more accurate termi- nology.

2. The diffusion of the Parisian Alfonsine Tables in Central Europe is not well known. However, a crucial role was probably played by John of Gmunden (ca. 1380–1442), the ªrst professor in mathematics and astron- omy at the University of Vienna, where he lectured since 1408 (Mundy 1942–3; Vogel 1975). He was the author of several works on arithmetic and instruments (the quadrant of Jacob ben Makhir, the , the cylindrum, the albyon of Richard of Wallingford), and he also wrote com- mentaries to some astronomical treatises (the Theorica planetarum of Campanus, the canons of John of Murs, and the Sphaera of Sacrobosco). His original astronomical works include many calendars and ephemerides (starting from 1415), some of which were published after his death. He also composed some canons and tables, making continuous changes in them from 1422 to 1440. 170 The Diffusion of the Alfonsine Tables

These canons are being edited by Beatriz Porres (2003), and have al- ready been the object of a ªrst survey. It turns out that John of Gmunden’s canons to his tables depend heavily on those written for the Alfonsine Ta- bles by the Parisian astronomers of the early 14th century: John of Sax- ony’s (1327), on the one hand, but especially the canons by John of Lignères written in 1322 and beginning with “Cuiuslibet arcus propositi” and “Priores astrologi”. Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/168/1789147/106361402321147513.pdf by guest on 26 September 2021 The tables appended to his canons form a voluminous and coherent set. John of Gmunden uses “physical” signs of 60°, as in the version of his ta- bles in Vienna, Nationalbibliothek, MS 5151, but some time later switches to “zodiacal” signs of 30°, as in another manuscript, also in Vi- enna, MS 5268. In Gmunden’s tables, the mean motions of the luminaries and the ªve are displayed for groups of 20 years, which is the standard period that was used in the original Castilian Alfonsine Tables, a fact explicitly mentioned in the Castilian canons. We note that this was not the period used by John of Lignères (25 years). Then follow the equations for access and recess, the 8th sphere, the and the , and the ªve planets. The tables for the solar declination and the lunar latitude use parameters well attested in medieval Spain: 23;33,30° for the obliquity of the and 5;0,0° for the maximum latitude of the Moon. The tables for the planetary latitudes and for planetary visibility and retrogradation are identical to those found among John of Lignères’s ta- bles, but also identical to those in such pre-Alfonsine sets as the Toledan Tables. It should be noted that John of Gmunden knew directly, with no intermediary sources, the Toledan Tables, for in his will, in which he leaves his books and instruments to the faculty of arts of the University of Vienna, one of the items is a “white book in parchment containing tabulas toletanas” (Mundy 1942–3, p. 198). Among the trigonometrical tables compiled by John of Gmunden there is one entitled “Tabula differentie ascensionum”. This table is not at all a common feature in the standard literature, but there is a very similar one in another pre-Alfonsine material, the twelfth-century of Azarquiel (Millás 1943–50, p. 225). The same table is found in the Ta- bulae resolutae for Salamanca, compiled by Nicholaus Polonius ca. 1460 (Chabás 1998, p. 171). Most of the other trigonometrical tables of John of Gmunden are computed for a latitude of 47;46° which corresponds to Vi- enna. The same applies to the only table for parallax in Gmunden’s tables. These are some of the tables related to John of Gmunden’s canons, but they are by no means the only ones he computed or copied in his working papers. There are many others, but two of them are of special interest. Perspectives on Science 171

First, there is a double-entry table to determine the time between mean and true syzygies, a table depending on John of Murs (Porres and Chabás 2001). Then there are also double-entry tables for the planetary latitudes that complement the tables “bipertiales” and “quadripertiales” of Toledan origin mentioned above. It is also noteworthy that the heading of these double-entry tables for the planets indicate that they were compiled in

Oxford, thus pointing to the tables of Oxford for 1348. Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/168/1789147/106361402321147513.pdf by guest on 26 September 2021 To sum up, it is clear after examination of his tables that John of Gmunden had at hand a vast quantity of astronomical material, including the best material one could possibly have at the beginning of the 15th century. It is, so to speak, a “Christian” material, which at that time means “Alfonsine”, and no direct inºuence of Arabic or Jewish astronomy is found in his tabular work. It is also clear that John of Gmunden did not make any substantial innovation in his tables; rather, all his tabular work follows the previous tradition, the Alfonsine tradition. Many of Gmunden’s tables are not among those traditionally attributed to John of Lignères, but it is indeed difªcult to be more precise on that point because no edition of these tables has been done yet. On the other hand, some of Gmunden’s tables seem related to a tradition more directly connected to the pre-Alfonsine work. This is the case, very particularly, with the tables for the colors of eclipses and for the unequal motion of the planets, both of them described in the Castilian canons to the Alfonsine Tables, but very rarely found in other sets of tables of the Alfonsine corpus (Goldstein, Chabás, and Mancha 1994).

3. We turn now to the Tabulae resolutae. According to Zinner (1968, p. 259), they were ªrst computed for 1428 and for Prague by Peter Rein or Teyn in Zittau, but Dobrzycki (1987) considers that this set of tables was based on an earlier set compiled for the meridian of Wroclaw. The author of these earlier tables, with 1424 as epoch, has been identiªed by Dobrzycki as Petrus Cruciferus, from Silesia. The tables reached Cracow and by the middle of the century they were used as textbook at the University. A pro- fessor at this University, Marcin Król of Zurawica (d. 1459), composed a set of Tabulae resolutae, with canons and tables, in 1445 as epoch and for the meridian of Cracow (Birkenmajer 1972, pp. 5–7). In 1449, a disciple of Marcin Król, Andrzej Grzymala of Poznan (d. 1466), wrote some “Canones tabularum resolutarum,” the preface of which contains the signature of the author appearing in the initials of the ªrst words: “Girum recesendo zodiaci inter magnalia astronomicae longe alto ingenio Alfoncium, regem Castelle,....”These canons are dated 172 The Diffusion of the Alfonsine Tables

1448, according to Birkenmajer, and are extant in various manuscripts (Birkenmajer 1972, pp. 515–527; Rosin´ska 1984, pp. 160–161). The canons and tables of Grzymala traveled to Spain, and were adapted to the meridian of Salamanca in about 1460 by a Polish astronomer, Nicholaus Polonius, the ªrst incumbent of the chair of astronomy at the University of Salamanca (Chabás 1998).

A few years later, some other canons beginning with “Nemo integre Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/168/1789147/106361402321147513.pdf by guest on 26 September 2021 sapit . . .” were written in Poland for the Tabulae resolutae, but the name of the author remains unknown. Still later, Albertus de Brudzewo expanded these canons and added some examples for year 1482, and Johannes de Glogovia (d. 1507) then reworked the text and included various examples for the years between 1487 and 1490 (Dobrzycki 1987). During this ªrst period, the Tabulae resolutae circulated widely in manuscript form, and its canons, as well as the tables, evolved consider- ably. Because of these constant changes, it is quite difªcult to establish what exactly the Tabulae resolutae were, as is the case with any other set of astronomical tables with a signiªcant degree of diffusion. However, the printed editions helped to ªx the identity of such tables. The Tabulae resolutae were ªrst printed in Frankfurt in 1511 by Ambrosius Lacher under the title “Tabulae resolutae de motibus planetarum aliorumque super celestium mobilium”. Johannes Schöner (1477–1547) edited a new version of the these tables in 1536 for the me- ridian of Nürnberg. The book contains an enlarged set of tables with re- spect to those found in manuscript. The title of Schöner’s book is “Tabulae astronomicae quas vulgo, quia omni difªcultate et obscuritate carent, Resolutas vocant”. The Tabulae resolutae were again printed in Nürnberg six years later, in 1542, under the title “Tabulae resolutae de supputandis siderum motibus ....”Onthis occasion the tables were computed for the meridian of Heidelberg by Johannes Virdung who had been a student of both Albertus de Brudzewo and Johannes de Glogovia in Cracow (Birkenmajer 1933, p. 365). Schöner’s version was reprinted several times, ªrst in Nürnberg in 1551 within his Opera Mathematica, and again in 1561 (Grassi 1977, p. 54). The last edition of Schöner’s Tabulae resolutae was printed in Wittenberg in 1588. Note that the printed versions of the Tabulae resolutae, which are, as it has already been said, a particular form of presentation of the Alfonsine Tables, competed for almost 40 years with the compiled by , ªrst published in 1551 in Tübingen and based on Copernicus’s theories. It is also noteworthy that Copernicus himself copied some planetary latitude tables related to those in the Tabulae resolutae Perspectives on Science 173

(Swerdlow and Neugebauer 1984, p. 4). They are found in a bound vol- ume owned by Copernicus and containing the Alfonsine Tables (in the ver- sion printed in 1492 in Venice) and ’s Tabulae directionum (Augsburg, 1490). As we shall see, the extant manuscripts of the Tabulae resolutae do not contain any planetary latitude tables, so Copernicus must have taken them from one of the earliest printed editions just mentioned. Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/168/1789147/106361402321147513.pdf by guest on 26 September 2021

4. The Tabulae resolutae have been preserved in many manuscripts, mostly ex- tant at the Jagiellonian Library in Cracow and the National Library of Munich. One of the oldest, dated ca. 1448, is Cracow, MS BJ 1864 (Rosiñska 1984, p. 161; Markowski 1990, pp. 276–283). It contains a Theorica planetarum beginning by “Circulus ecentricus . . .” and the canons to the Tabulae resolutae written by Andrzej Grzymala of Poznan and begin- ning with “Girum recesendo zodiaci....”Then follow two sets of tables. The ªrst one is the Tabulae resolutae. They use signs of 30°, as opposed to the signs of 60° in the printed edi- tions of the Alfonsine Tables of 1483 and 1492. The tables can be ar- ranged in various subgroups: A. The ªrst 12 tables (ff. 36r–45r) give the radices and mean mo- tions for the apogees and the ªxed stars; access and recess of the 8th sphere; Sun, Venus, and Mercury; Moon; argument of the Moon; ar- gument of latitude of the Moon; lunar node; Mars; Jupiter; Saturn; argument of Venus; and argument of Mercury. All entries are arranged according to a system of cyclical radices ad annos collectos at 20-year intervals, for a period covering, in this case, 1428–1808. Different sub-tables display the mean motions for these 12 variables for 20 consecutive years (anni expansi), for the months of the year (whether in a leap year or not), for the days of a month and for the hours of a day. Note that 20 years is the time in- terval used in the Castilian canons to the Alfonsine Tables for col- lected years.

B. Mean syzygies (ff. 45v–49r). There is a table for mean conjunc- tions and another for mean oppositions. Each of them lists four variables: time of mean syzygy; mean motion of the Moon; mean argument of the Moon; mean argument of latitude of the Moon. In both tables the variables are calculated for every other 20 years for a long period (in this case, 1429–1809, i.e., they are shifted one year with respect to the mean motion tables). For each of these variables, 174 The Diffusion of the Alfonsine Tables

other sub-tables display the values for 20 consecutive years (anni expansi) and for each month in a year. This is the standard format for tables for mean syzygies in the Alfonsine corpus.

C. Positions and motions of the apogees (ff. 49v–51v): Sun and Ve- nus; Saturn; Jupiter; Mars; and Mercury. The positions are also tab-

ulated at intervals of 20 (collected) years, (in this case, from 1428 Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/168/1789147/106361402321147513.pdf by guest on 26 September 2021 to 1808, and the motions at anni expansi from 1428 to 1788). Other manuscripts containing the Tabulae resolutae display sub- tables for the motions of the apogees for months of the year (whether in a leap year or not) and for the days of a month. This is a presentation differing from that in John of Gmunden (see, for instance, Vienna, MS 5268, f. 11v), whose table contains more information. The data in the Tabulae resolutae could have been taken directly from Gmunden’s but for Jupiter. The entries for this show a difference of about 0;45° between those in a Tabulae resolutae and in Gmunden’s.

D. Equations (ff. 52r–71r) for the ªrst point in Aries, the Sun, the Moon and the ªve planets. The equation for the motion of the ªrst point in Aries (otherwise called equation of access and recess) is given for each degree of each sign and reaches a maximum of 9;0,0° at 90°. (In the different copies of the Tabulae resolutae consulted the entries for 61° and 62° are faulty [7;52,49° and 7;57,19°] and dif- fer from those in the corresponding table of the editio princeps of the Alfonsine Tables [7;51,49° and 7;56,19°].) The equation for the Sun is given for each degree of each sign and reaches a maximum of 2;10,0° at 92°–94 as is the case in the standard table of the Alfonsine corpus. For the lunar equations, the table is identical to that found in the Alfonsine corpus and lists the equation of the center (max ϭ 13;9° at 114Њ–115°), the “minuta proportionalia”, the “diversitas diametri” (max ϭ 2;40° at 103°–109°), and the equation of the ar- gument (min ϭϪ4;56,0° at 95°) for each degree of each sign. The line-by-line differences are also tabulated in all four cases. The tables for the equations of the ªve planets give the equation of the center, the “minuta proportionalia”, the “longitudo longior”, the equation of the argument, the “longitudo propior” and the “statio prima”. The entries for the ªrst station vary in the following ranges: Saturn, 3;22,44°–3;25,30°; Jupiter: 4;4,5°–4;7,11°; Mars: 5;7,28°–5;19,15°; Venus: 5;15,51°–5;18,21°; and Mercury: 4;27,17°–4;24,42°. Again, there are no novelties in this table, ex- Perspectives on Science 175 cept for the fact that the column for the ªrst station is not tabu- lated in the version of the Alfonsine Tables printed in Venice in 1483.

E. Then follows an extensive set of interpolation tables (ff. 71v– 83v) entitled “Tabula minutorum proporcionalium pro diebus intermediis in almanach faciendis”, where the entries are the suc- Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/168/1789147/106361402321147513.pdf by guest on 26 September 2021 cessive divisions by 2, 3, etc. of a given number between 0;1° and 9;0°.

F. Daily motions of the Sun and the Moon (ff. 83v–84r): In a single table we ªnd the entries corresponding to the Sun and the Moon. The entries, and the extremal values, in this table are frequently found in the Alfonsine corpus: 0;2,23°/h Ϫ0;2,34°/h (Sun), and 0;29,37°/h Ϫ0;36,53°/h (Moon), but the lunar velocity table is completely different from the corresponding table in the editio princeps, and its maximum is also found in a table ascribed to John of Lignères (Goldstein 1992, p. 11).

G. Equation of time (ff. 84v–85r). The entries are given in degrees and minutes of arc for each degree of every zodiacal sign. The extremal values are:

Min ϭ 0; 0° (Aqu 18°–25°) max ϭ 5;21° (Tau 25°–27°) min ϭ 2;49° (Leo 5°) Max ϭ 7;57° (Sco 8°–9°)

These values do not agree with those in the 1483 edition of the Alfonsine Tables. It has been shown that the table in the editio princeps implies a longitude of the apogee of 82;45°, which corre- sponds to al-Battânî’s time, and indeed the same table is found in the of al-Battânî and the Toledan Tables (Chabás 1998, pp. 169–170). On the other hand, the entries in the Tabulae resolutae correspond to a longitude of the apogee of 87;23°, a value which agrees with the time of King Alfonso X, and the same table is found in the tables attributed to John of Lignères, and John of Gmunden, among others.

H. The table for the planetary stations and retrogradations (f. 85v) gives only four values for each planet: the positions of the ªrst and the second stationary points at apogee and perigee. It is not found 176 The Diffusion of the Alfonsine Tables

in the editio princeps of the Alfonsine Tables, but it is among John of Gmunden’s tables. Its origin goes far back in time, for it is already found in the zij of the pre-Alfonsine astronomer Ibn al-Kammâd (12th century), and it derives ultimately from al-Khwârizmî’s zij.

I. Then follows a sexagesimal multiplication table (ff. 86r–92r) of

60 ϫ 60 entries. Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/168/1789147/106361402321147513.pdf by guest on 26 September 2021

J. Rising times (ff. 92v–97r). The ªrst table is the standard one for normed right ascension (based on an obliquity of 23;35°, also found in al-Battânî’s zij, the Toledan Tables, and the editio princeps of the Alfonsine Tables). The two other tables give the rising times (ascensiones) and the length of the day (partes horarum) for the middle of the seventh climate (latitude ϭ 50°) and its end (latitude ϭ 51°).

The tables in MS BJ 1864 that follow are not part of the Tabulae resolutae. They are mainly tables for the mean motions and the equations of the planets presented in the same way as the tables compiled by John of Gmunden. The material in this set of Tabulae resolutae just described is almost the same as that found in more than 20 other manuscripts containing these ta- bles. We note, however, that in this set some important tables are missing: there are no latitude tables for the Moon and the planets, no parallax ta- bles, no eclipse tables, no star table, and no table for geographical coordi- nates, among others. The printed editions of the Tabulae resolutae, which span the years from 1511 to 1588, differ from the manuscripts in the sense that they incorporate a whole set of tables which could compete with any other version of the Alfonsine Tables, such as those of the editions of 1483 and 1492, and later on to compete with the Prutenic Tables, based on Co- pernicus’s theories, and ªrst printed in 1551. The reason for the lack of such tables in the manuscripts is simple: the ªrst Tabulae resolutae to be compiled seem to have been basically intended to compute calendars, al- manacs, or ephemerides. This is what the heading of one of the tables seems to imply, for it speciªes its use: “in almanach faciendis”. For this purpose, tables of mean motions and equations of the planets, on the one hand, and tables for syzygies, on the other, are about all that is needed. The Tabulae resolutae were also easier to use, and this is another reason for their success. Anyone computing the positions of the luminaries and the planets, in order to cast , write astrological judgments or prognostications, or elaborate calendars, would ªnd that the Tabulae resolutae gave many more intermediate data needed in the calculation pro- Perspectives on Science 177 cess than the standard versions of the Alfonsine Tables, such as those printed at Venice in 1483 and 1492. Since the tables of John of Gmunden were compiled at more or less the same time as the ªrst version of the Tabulae resolutae, one may ask whether the Tabulae resolutae were based on his tables. Although it seems implausi- ble that Gmunden gave the Tabulae resolutae its characteristic format, it is in turn clear that throughout their continuous elaboration, the Tabulae Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/168/1789147/106361402321147513.pdf by guest on 26 September 2021 resolutae greatly beneªted from the vast amount of Alfonsine material dif- fused throughout Central Europe, largely due to the efforts of John of Gmunden.

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