Astronomy in Salamanca in the Mid-Fifteenth Century: The Tabulae Resolutae

The teaching of astronomy at the University of Salamanca is mostly known through the inclusion of Copernicus among the authors in the curriculum of its chair of astrology: According to the University statutes of 1561, Copernicus's treatise could be chosen, “al voto de los oyentes” (“upon the vote of the audience”), among other works by Ptolemy, Regiomontanus, and Geber. During most of the sixteenth century the University of Salamanca was prominent in the teaching of science, and was decidedly open to innovation. See Brotóns Navarro Víctor , “The reception of Copernicus in sixteenth-century Spain”, Isis , lxxxvi (1995), 52–78; Brotóns Navarro V. , “Contribución a la historia del copernicanismo en España”, Cuadernos Hispanoamericanos , cclxxxiii (1975), 3–24; and Vernet Juan , “Copernicus in Spain”, Colloquia Copernicana , i ( Studia Copernicana , v (1972)), 271–91.

Burgos Cantera F. , “El judío salmantino Abraham Zacut”, Revista de la Academia de Ciencias de Madrid , xxvii (1931), 63–398; and Burgos Cantera F. , Abraham Zacut (Madrid, 1935).

Burgos Cantera , opera cit. (ref. 2); Chabás J. Goldstein B. R. , Abraham Zacut and Iberian astronomy in the late 15th century (in preparation).

Dreyer J. L. E. , “On the original form of the Alfonsine Tables”, Monthly notices of the Royal Astronomical Society , liii (1920), 243–62, called attention to that manuscript and its relationship to the tables drawn by the astronomers of Alfonso X. See also Beaujouan G. , “La science en Espagne aux XIVe et XVe siècles”, Conférence du Palais de la Découverte , D 116 (Paris, 1967), 5–45; and Beaujouan G. , “L'astronomie dans la péninsule ibérique à la fin du moyen âge”, Agrupamento de estudos de cartografia antiga , xxiv (Coimbra, 1969), 3–22, reprinted in Beaujouan G. , Science médiévale d'Espagne et d'alentour (Aldershot, 1992), I and X.

For Jacob ben David Bonjorn, see Chabás J. , “The astronomical tables of Jacob ben David Bonjorn”, Archive for history of exact sciences , xlii (1991), 279–314; Chabás J. , L'astronomia de Jacob ben David Bonjorn (Barcelona, 1992). For the tables of Ibn al-Kammād, see Chabás J. Goldstein B. R. , “Andalusian astronomy: al Zij al-Muqtabis of Ibn al-Kammād”, Archive for history of exact sciences , xlvii (1994), 1–41. For John of Lignères, see Poulle E. , “John of Lignères”, Dictionary of scientific biography , vii, 122–8; and Saby M.-M. , “Les canons de Jean de Lignères sur les tables astronomiques de 1321”, unpublished thesis, École Nationale des Chartes, Paris (1987) — a summary with the same title appeared in École Nationale des Chartes: Position des thèses , 1987, 183–90.

For the transcription of the text, see Porres B. Chabás J. (in preparation).

For the development of the Tabulae resolutae in Poland, see Dobrzycki J. , “The Tabulae resolutae”, in Comes M. Puig R. Samsó J. (eds), De astronomia Alphonsi Regis (Barcelona, 1987), 71–77.

The period covered by these tables is in general 1428–1808; it is indeed so in the manuscripts at the Jagiellonian Library (Cracow) containing the Tabulae resolutae we have consulted: MSS 1864, and 1865. For other such manuscripts see the Catalogus codicum manuscriptorum medii aevii Latinorum qui in Bibliotheca Jagellonica Cracoviae asservantur (6 vols, Cracow, 1980–); Rosińska G. , Scientific writings and astronomical tables in Cracow: A census of manuscript sources (XIVth-XVIth centuries) ( Studia Copernicana , xxii (1984)), 509–21; and Markowski M. , Astronomica et astrologica Cracoviensia ante annum 1550 (Florence, 1990), 276–82.

Rico y Sinobas M. , Libros del saber de astronomía del Rey Alfonso X de Castilla (5 vols, Madrid, 1863–67), iv, 133–4. Other manuscripts with tables in the Alfonsine tradition meeting these requirements were described by Dreyer , op. cit. (ref. 4), 249ff., and North J. D. , “The Alfonsine Tables in England”, Prismata (Wiesbaden, 1977), 269–301 (esp. pp. 273ff).

Consider, for example, the entry for the position of the solar apogee for 1348: 2s 29;41,51,9° (f. 39r). It can be recomputed as follows. The table for the motion of the apogees and fixed stars gives the value 0s 9;54,13,14° for 1348 (f. 33r). This is the progress in longitude since the Incarnation and accounts for precession. Now, in the editio princeps of the Alfonsine Tables the radix for the Incarnation is given as 2s 11;25,23°. Furthermore, the entry for 1348 for access and recess is 2s 8;32,6,35° (f. 33v), which implies an equation of access and recess (trepidation) of 8;22,15°. Adding this last value and that of precession to the radix for the Incarnation yields 2s 29;41,51°, in full agreement with the entry in the table.

For the Toledan Tables, see Toomer G. J. , “A survey of the Toledan Tables”, Osiris , xv (1968), 5–174 (esp. pp. 78–79).

Toomer , op. cit. (ref. 11), 60–68.

For the zij of Ibn al-Kammād, see Chabás Goldstein , op. cit. (ref. 5), 33; for the zij of al-Khwārizmī, see Suter H. , Die astronomischen Tafeln des Muhammad ibn Mūsā al-Khwārizmī (Copenhagen, 1914), 138–67. There are, however, some discrepancies in the entries: 3s 24;44° for the 1st station of Saturn at apogee (instead of 3s 22;44° as on f. 48v); 5s 7;18° for the 1st station of Mars at apogee (instead of 5s 7;28° as on f. 53v); 6s 34;9° for the 2nd station of Venus at apogee (instead of 6s 14;9°); and the entries for the 1st station of Mercury at apogee and perigee have been interchanged.

For the zij of al-Battānī, see Nallino C. A. , Al-Battānī sive Albatenii Opus astronomicum (3 vols, Milan, 1899–1907), ii, 61–64; for the Toledan Tables, see Toomer , op. cit. (ref. 11), 34–35.

Neugebauer O. , “Thābit ben Qurra On the solar year and On the motion of the eighth sphere”, Proceedings of the American Philosophical Society , xvi (1962), 264–99 (esp. p. 276, sentence 87); Morelon R. , Th¯bit ibn Qurra: Oeuvres d'astronomie (Paris, 1987), p. lii.

See Nallino , op. cit. (ref. 14), i, 72. For this particular value of precession implicit in On the solar year and explicit in the Expositio of John of Murs, see Goldstein B. R. , “Historical perspectives on Copernicus's account of precession”, Journal for the history of astronomy , xxv (1994), 189–97 (esp. pp. 190–2). I am grateful to Professor Goldstein for this calculation.

Chabás J. Goldstein B. R. , “Some astronomical tables of Abraham Zacut preserved in Segovia” (in review).

Toomer , op. cit. (ref. 11): 106 (I), 111 (II, III), 103 (IV), 98–99 (Toledo), 104 (V), 108 (VI), and 109 (VII).

Neugebauer O. Schmidt O. , “Hindu astronomy at Newminster in 1428”, Annals of science , viii (1952), 221–8, report one such table in MS Ashmole 191.II (f. 137r), where the entries were derived by the equivalent of the expression 150/12 x tan δ. M. Lesley, “Bīrūnī on rising times and daylight lengths”, Centaurus , v (1957), 121–41 (esp. pp. 125–7), adds three other such tables, found in the zij of Ibn Yūnis, the Almanac of Azarquiel, and the zij of al-Baghdādī. See also Toomer , op. cit. (ref. 11), 33.

Millás J. , Estudios sobre Azarquiel (Madrid and Granada, 1943–50), 225.

A more accurate table of this kind, but based on ε = 23;35°, is found in the zij of al-Marrākushī (13th century); see Sédillot J.-J. Sédillot L.-A. , Traité des instruments astronomiques des Arabes (Paris, 1834; reprinted Frankfurt a. M., 1984), 209.

Dobrzycki , op. cit. (ref. 7), 75.

On Grzymala, see Birkenmajer A. , “Andrezj Grzymala de Poznan, astronome et médecin du XVe siècle”, Studia Copernicana , iv (1972), 515–27.

It has been proposed, without conclusive evidence, to identify Nicolás Polonio with Nicholaus of Tuchow, an astronomer educated at the University of Paris, and author of an almanac of the planets for the year 1455 extant in Paris, Bibliothèque de France, MS Lat. 7427, but also with Nicole de Poulaine, astronomer at the Court of the Duke of Bourgogne towards 1466–68; see Beaujouan G. , op. cit. (ref. 4), 27 and 13, respectively; and Boudet J.-P. , Lire dans le ciel: La bibliothèque de Simon de Phares, astrologue du XVe siècle (Brussels, 1994), 100–1.

MS Salamanca, AUSA 1, f. 2r.

For a recent survey on Nebrija, see Codoñer C. González J. A. (eds), Antonio de Nebrija: Edad Media y Renacimiento (Salamanca, 1994).