“Abbreviating” the Alfonsine Tables in Cracow: The Tabulae Aureae of Petrus Gaszowiec (1448)

Chabás J. Goldstein B. R. , The Alfonsine Tables of Toledo (Dordrecht, 2003), 243–306.

North has suggested that one cannot “categorically lay down” Alfonsine positions, due to unknown or inconsistent rounding and interpolation practices by the mediaeval table makers and users. Yet most tables in the Alfonsine corpus provide more significant figures than minutes for the mean motions and significant figures to minutes for the equations, and I have found it possible in many (but not all) cases to recompute a planetary position or even an eclipse presented in a fifteenth-century manuscript, by means of tables in the Alfonsine corpus and consistent modern rounding practice, to within ±0;01° or ±0;01h. See North J. D. , “The Alfonsine Tables in England”, in Prismata, Naturwissenschaftsgeschichtliche Studien: Festschrift für Willy Hartner, ed. by Maeyama Y. Satzer W. G. (Wiesbaden, 1977), 269–301, p. 284; and idem, “Just whose were the Alfonsine Tables?”, in From Baghdad to Barcelona: Studies in the Islamic exact sciences in honour of Prof. Juan Vernet, ed. by Casulleras J. Samsó J. (Barcelona, 1996), i, 453–75, p. 455.

Birkenmajer A. , “Piotr Gaszowiec”, Études d'histoire des sciences en Pologne (Studia Copernicana, iv; Wrocław, 1972), 527–30; Markowski M. , “Kanony Piotra Gaszowca do ‘złotych tablic astronomicznych’ w rękopisach Badeńskiej Biblioteik Krajowej w Karlsruhe, Biblioteki Uniwersyteckiej w Padwie, Archiwum praskiego zamku w Pradze i Biblioteki Jagiellońskiej w Krakowie”, Biuletyn Biblioteki Jagiellońskiej, xxxvi (1986), 1986–22; idem, “Piotr Gaszowiec twórcą krakowskiej komputystyki o zasięgu międzynarodowym”, Studia mediewistyczne, xxv/1 (1988), 69–117; Nadolski B. , “Humanistyczne mowy lekarza Piotra Gaszowica”, Pamietnik literacki, xxviii (1931), 1931–72. Autograph copies of the orations are preserved in Cracow, Biblioteca Jagiellońska (BJ) 126, ff. 73v–77r. See Catalogus codicum manuscriptorum medii aevi latinorum qui in Bibliotheca Jagellonica Cracoviae asservantur (Wrocław, 1980–), i, 112–15.

BJ 557 (Liber magistri Petri Gaschowiiec, doctoris medicine, heredis de Losmerz Polonicali, comparatus X florenis, f. IIIr). See Catalogus (ref. 3), iii, 382–3.

Rosińska G. , Scientific writings and astronomical tables in Cracow: A census of manuscript sources (XIVth—XVIth centuries) (Wrocław, 1984), #1256.

Eisermann F. , Verzeichnis der typographischen Einblattdrucke des 15. Jahrhunderts im Heiligen Römischen Reich Deutscher Nation (Wiesbaden, 2004), A-163; Gomółka B. , “Zagadnienie autorstwa kalendarza astrologiczno-medycznego na rok 1474”, Biuletyn Biblioteki Jagiellońskiej, xxviii (1978), 1978–29. Gomółka offers indirect evidence for her attribution of A-163 to Gaszowiec. The almanac names two local saints, Hedwig and Stanislaus, and thus must have been prepared by someone from Cracow. This sheet, the only known incunabula almanac printed in Cracow, was for 1474, the year Gaszowiec died, which might explain the absence of almanacs for subsequent years. Gaszowiec was a member of the faculty of medicine, and the university statues in the 1470s required the chair holder in astrology to prepare an annual prognostication. No other university professor is known who might have been able to compute the almanac for 1474. It is not clear whether the university statute required a single-sheet almanac or a multi-leaved practica, and none of the latter is known to have been printed in Cracow before 1500. By 1479, the Cracow university astronomer, Johannes de Glogovia, did author a practica for Cracow, printed in Merseburg (GW M13940).

Rosińska , op. cit. (ref. 5), # 74, 379, 1648, 53, 916; Markowski M. , Astronomica et astrologica Cracoviensia ante annum 1550 (Florence, 1990), 179–81; Catalogus (ref. 3), passim; Patera A. Podlaha A. , Soupis rukopisů knihovny metropolitní kapitoly Pražské (Prague, 1910–22), ii, 473–4; and Holder A. , Die Handschriften der Badischen Landesbibliothek in Karlsruhe, iii: Die Durlacher und Rastatter Handschriften, reprint edn (Wiesbaden, 1970), 124–7.

Other Gaszowiec autographs and glosses can be found in BJ 126, 258, 830 and 852.

Matriculated on 26 February 1490 at Cracow University as “Bernhardus Conradi Mahvlehel de Selmuha, dioc. Constanciensis”, according to Album studiosorum Universitatis Cracoviensis (Cracow, 1887–1904), ii, 6.

Johannes de Glogovia's commentary on the Tabulae resolutae (Pro generali, in tabulas motuum planetarum, quas resolutas dicimus …), with explicit dated Buda, 1513, is copied immediately prior to the Tabulae aureae in Padua ms. 643, ff. 41r–64r. Both texts are in the same hand that has been attributed to Antonio Gazio, a Paduan physician who had spent the first two decades of the sixteenth century in Hungarian, Polish and German areas. See Markowski M. , “Krakowskie dzieła astronomicze w zbiorach rękopiśmiennych biblioteki uniwersyteckiej w Padwie jako świadectwo ich recepcji przez naukę włoską”, Biuletyn Biblioteki Jagiellońskiej, xxix (1979), 43–53; Marangon P. , “Schede per una reinterpretazione dei rapporti culturali tra Padova e la Polonia nei secoli XIII—XVI”, in Italia, Venezia e Polonia tra medio evo e età moderna, ed. by Branca V. Graciotti S. (Florence, 1980), 177–9.

C5 bears a unique title, Circa inicium canonum necnon exemplorum Tabulas aureas explanantes, and includes the introduction (Pro intellectu igitur …) found in other copies of the longer canon. C5 then offers seventeen canons, none of which can be found in the other manuscripts here considered. Its sixth canon, f. 79v, illustrates how to advance Gaszowiec's radices from 1448 to 1488. Presumably C5's canon was composed around the latter year.

Parts of John of Saxony's canons 14, 17–23 appear verbatim in the longer canons. For a transcription of the titles of the eighteen canons in C3, see Markowski , op. cit. (ref. 3), 18–19.

Markowski , op. cit. (ref. 3), 19–20.

Pr, f. 237v: Tabule auree de medijs et viris motibus planetarum … per venerabilem virum magistrum Petrum de Cassovicz … pro utilitate volencium in praxi astronomie proficere ex tabulis illustris Alphoncii Romanorum ac Castelle regis abbreviate.

Of the six extant copies, only Pr (f. 236v) presents a short list of the mean motions for a single day that includes the argument of lunar latitude. The scribe, however, slipped a line in labelling these values; the lunar latitude is marked with the sign for Saturn. All the values match those of the Parisian Alfonsine Tables.

This title appears only in PrKa.

The twelfth-century Toledan Tables used cyclical radices of 30 lunar years, their adaptations for Marseille and Novara used 28 years, the adaptation for Toulouse used 24 years. The Castilian Alfonsine Tables, compiled around 1272, and their reworked versions by John of Lignères in Paris and English astronomers in the 1340s, as well as the Tabulae resolutae of 1428, used cycles of 20 Julian years. As North aptly put it, the makers and users of tables with cyclical radices “preferred to copy more and think less”, in contrast to users of the Parisian Alfonsine Tables who preferred the inverse. See North , “England” (ref. 2), 272; Chabás Goldstein , op. cit. (ref. 1), 152; de Mateo B. Porres , “Les tables astronomiques de Jean de Gmunden: Édition et étude comparative”, doctoral thesis, Paris, École Pratique des Hautes Études, 2003.

Gmunden prepared at least five different versions of his tables, three of which remain extant. The most complete copy is found in Vienna, ÖNB 5268, copied in the 1430s. See ff. 3r–6r, 7v–11r of this manuscript for the version with mean motions for intervals of 1, 2, …, 20, 40, 60, …, 100, 200, …, 2000 years, with radices for 1436 complete. Regiomontanus's autograph of these tables, Nuremberg, Stadtbibliothek Cent. VI, 18, ff. 3v–15r, copied around 1452 with radices for 1440 complete, lists mean motions up to only 1000 years. No copies of the three versions of Gmundens tables have yet been identified in the BJ. However, a copy of what Porres has called the “posthumous version” of those tables is preserved in BJ 609, ff. 1r–74v (tables), 94r–127r (canons), with similar intervals of mean motions but extending to 8000 years, copied in Prague in 1446 and bound there. It is not clear when this codex came to Cracow. Fifteenth-century Viennese astronomical materials can also be found in BJ 613 and BJ 1840. See Porres , op. cit. (ref. 17), 38–42, 65–67, 78–80; de Mateo B. Porres , “Die astronomischen Tafeln des Johannes von Gmunden: Seine Lehre und Forschung an und außerhalb der Universität Wien”, in Johannes von Gmunden (ca. 1384–1442): Astronom und Mathematiker, ed. by Simek R. Chlench K. (Vienna, 2006), 105–25; Neske I. , Die Handschriften der Stadtbibliothek Nürnberg, v: Die lateinischen mittelalterlichen Handschriften, Varia, 13.–15. und 16–18. Jahrhundert (Wiesbaden, 1997), 172–4; Catalogus (ref. 3), passim; Rosińska, op. cit (ref. 5), # 1808, 2102.

Kremer R. L. Dobrzycki J. , “Alfonsine meridians: Tradition versus experience in astronomical practice c. 1500”, Journal for the history of astronomy, xxix (1998), 187–99.

To place the meridian at 1;26,25h east, the radix for mean Jupiter should read 7s 15;02,25,59°. Perhaps Gaszowiec or an early scribe confused “53” for “59” in copying that radix?.

Pr: 11s 29;57,59,30.

Pa: 6s 15;26,35,20; Pr: 6s 15;29,35,19.

Pa: 4s 29;54,23,40.

This table is missing in C1 and Pr, the earliest and latest copies of the tables.

If I slightly clean the tabular values for 1° to 29° of mean elongation (by reading 37 for 27s at 5°, 34 for 24s at 21°, and dropping three other values that appear to be more erratically miscopied), the Parisian Alfonsine rate of mean elongation (mean synodic month of 29;31,50,07,37d) matches to the nearest 0;0,1h in 23 of the remaining 26 tabular values.

The editio princeps, Alphonso X, Tabulae astronomicae (Venice, 1483), sig. c8r, lists the aux of the Sun and Venus to only three significant digits. Gaszowiec assumed the value for that aux at Incarnation to be 1s 11;25,23,4°.

For example, the Tabulae resolutae include tables for the planetary auxes at twenty- and one-year intervals, from 1428 through 1808, tables that clearly show the slow decrease in the annual motion of the aux over the years. For an early copy, see BJ 1864, ff. 49v–51v.

BJ 550, f. 6v. Copied on paper dated by watermarks to the 1390s and bound around 1400 in Prague, the manuscript reached Cracow by 1440. In addition to the Parisian Alfonsine Tables (ff. 1r–24r), it contains John of Lignères's tables for eclipses, excerpts from John of Saxony's canons for the Alfonsine Tables, and the Theorica planetarum and notes thereon, and is a typical compendium of astronomical texts for the university student. See Catalogus (ref. 3), ii, 327–32. The copy of the Alfonsine Tables is nearly complete, containing all the tables included in E. Poulle, Les tables alphonsines avec les canons de Jean de Saxe (Paris, 1984) except for Poulle's # 10.

BJ 459, 546, 547, 548, 549, 551, 552, 553, 563, 564, 570, 578, 602, 604, 612, 618. See also Rosińska , op. cit. (ref. 5), 463–75.

Three of the twelve values in BJ 550 differ from my computed value by ±1 sexagesimal sixth. The other nine values match exactly.

Ka changed to 0;11,28,13,12 by a later hand.

Ka changed to 0;02,18,55,04 by a later hand.

Ka changed to 0;00,34,42,10 by a later hand, who adds in the margin: “NB. Blanchini in anno 55 dat in motu augium communium”.

Read 36 for 35 seconds.

Pa erroneously lists values for 1h and the days in the rows labelled months, i.e., duplicates six rows and thus does not provide values for the months.

Ka changed to 0;00,24,10,52 (the correct value) by a later hand.

Ka changed to 0;00,16,09,01 by a later hand; perhaps a third hand added in margin the original value of 0;00,18,08,09.

Pr: 1 day; the correct value is 16 days.

Ka changed to 0;00,01,24,15 by a later hand, perhaps an attempt to render a value for the stated argument of 15 days, which would be 0;00,01,25,00.

See Poulle E. Gingerich O. , “Les positions des planètes au Moyen Âge: Application du calcul électronique aux tables alphonsines”, Académie des Inscriptions & Belles-lettres, Comptes rendus des séances de l'anneé 1967, 531–48, pp. 541–42, 544.

Although Pa lacks this table, it does contain the table of planetary stations (f. 73r) that often accompanies the SMM table.

As Toomer first showed, the Toledan tables incorporate into their mean motions a fixed rate of precession of about 0;0,0,5,55°/day, a value in agreement with that of Ptolemy's Almagest. Earlier Islamic tables frequently had used a fixed value of precession of 0;0,0,8,58°/day. See Toomer G. J. , “A survey of the Toledan Tables”, Osiris, xv (1968), 5–174, pp. 44–45.

Dobrzycki J. , “The ‘Tabulae Resolutae’”, in De astronomia Alphonsi regis, ed. by Comes M. Puig R. Samsó J. (Barcelona, 1987), 71–77; de Mateo B. Porres , “Sirení středovêkych astronomickych tabulek ve strední evrope v 15. století”, in Astronomie ve středově ké vzdělanosti, ed. by Hadravová A. Hadrava P. (Prague, 2003), 39–51.

Pa, ff. 41r–64v, however, does bear a copy of Johannes de Glogovia's canons to the Tabulae resolutae (Rosińska, op. cit. (ref. 5), #1651). The SMM tables do not appear in later printed editions of the Tabulae resolutae. See Lacher A. , Tabulae resolutae de motibus planetarum aliorumque super celestium mobilium (Frankfurt, 1511); Schöner J. , Tabulae astronomicae … resolutas vocant (Nuremberg, 1536); and Virdung J. , Tabulae resolutae de supputandis siderum motibus (Nuremberg, 1542).

For dates and provenances of these codices, see Catalogus (ref. 3); Rosińska , op. cit. (ref. 5); Markowski , op. cit. (ref. 7); Holder , op. cit. (ref. 7); Truhlár J. , Catalogus codicum manu scriptorum latinorum qui in C.R. bibliotheca publica atque universitatis Pragensis asservantur (2 vols, Prague, 1905–6), ii, 42–43. PNk = Prague, Národní knihovna; ÖNB = Vienna, Österreichische Nationalbibliothek. Three of the copies, BJ 611(7), BJ 600(88) and BJ 605(25), present expanded versions of the SMM, at intervals from 1 to 31 days (the latter is incomplete, with entries only for argumentum solis, although the folio is lined to provide columns for all eight tables of the set). BJ 1852 lists values only for the first days of the months from January to December.

Toruń, Biblioteka Uniwersytecka, rps. 74, contains the Tabulae resolutae on ff. 251r–68r (Rosińska, op. cit. (ref. 5), IV, 1–7, 9, 12, 15–22) and a copy of the Parisian Alfonsine Tables on ff. 33r–55v, but not SMM. Brodský Pavel , Illuminované rukopisy českého původu v polských sbírkách (Prague, 2004), 105, places the origin of this manuscript in Prague, around 1425. However, its paper (Piccard Ochsenkopf, vi, 257) has been localized to Wrocław, 1419–28, and it includes several glossed radices for 1420 (ff. 85r, 105r) which might indicate an origin not later than that year.

Rosińska , op. cit. (ref. 5), 509–21, defines the Tabulae resolutae as comprised of twenty-two tables plus another five that “usually follow the Tabulae resolutae and the Tabulae Alphonsi“in the manuscripts. The Tabula minutorum proporcionalium belongs to the latter set (VI, 27). Rosińska does not associate the SMM with the Tabulae resolutae, even though it appears in 18 of the 23 manuscripts she identifies as bearing copies of the latter. See Table 3 and Chabás J. , “The diffusion of the Alfonsine Tables: The case of the Tabulae resolutae”, Perspectives on science, x (2002), 168–78, p. 175.

Rosińska , op. cit. (ref. 5), 521. These intervals were undoubtedly selected to match maximal planetary speeds. The maximal shift in the true longitude of Venus over 7 days is about 8;45°; the maximal shift for Jupiter over 11 days is about 2;40°. Mercury can shift in 7 days by more than 13°, so its true longitude would have to be computed for shorter intervals.

See BJ 1865, f. 64r—v; BJ 1920, ff. 99v–100r; Rosińska , op. cit. (ref. 5), # 160, 161.

Alfonsine computation of true longitudes for Mars and Venus, the planets with the greatest daily variation in true longitude, indicates that the almanac-making procedures of the Tabulae resolutae produce deviations between the true and averaged longitudes of those planets that can reach ±0;10° and ±0;15°, respectively, over seven-day intervals.

For wry comments on the relative labours of “copying” versus “thinking”, see North , “England” (ref. 2), 272.

Markowski, op. cit. (ref. 3), 12.

Following Pa, f. 65r.

C5C6: 56.

C5PrKaC6 and many mss in Table 3: 29.

Pr: 25mins, 40secs.

C5PrC6: 14.

Corrected from 37 in PNk X.B.3 and C5PrKaC6; C1: 49.

C5PrKaC6: 0.

This row is not included in PNk X.B.3 and most manuscripts of Table 3; it appears only in C5PrKaC6.

Corrected from 36 in C5PrKaC6.

C5PrC6: 1.

Ka: 39; C1 and ÖNB 5245, BJ 2650, BJ 611(7), BJ 2252, BJ 1858, BJ 600(88): 36. This set of seven copies consistently deviate from the Prague manuscript and probably were independently computed for a precessional motion of 0;0,0,5,42°/d.

C1: 56,47.

C1: 55,55.

C1: 25,20; C5PrKaC6: 29,27.

C1: 48,38.

C1: 47,42.

C1: 17,26.

C1: 51.

C1: 30.

C5PrC6: 7.

C5PrC6: 3;38,25,7,30°.

C5PrC6: 10 (and erroneously list day as 18).

C5PrC6: 26;33,42,9,49°.

C5PrC6: 28.

C1 and ÖNB 5245, BJ 2650, BJ 611(7), BJ 2252, BJ 1858, BJ 600(88), BJ 605(96): 45. As with the Centrum Saturni (see above), these manuscripts have used a precessional motion of 0;0,0,5,42°/d for days 7, 8 and 28; obvious scribal errors have corrupted their remaining entries for the Centrum Jovis.

C5PrKaC6: 36.

C1: 58,31.

C1: 8,16.

C5PrC6: 0;38,53,30,12°.

C1: 18 (later hand corrects to 17), 1.

C1: 33,3; C5PrC6: 49.

C1: 42; C5PrC6: 17.

C1: 34,0,19.

C5PrKaC6: 26.

C5: 6.

C5PrKaC6: 44.

C5PrC6: 16.

Not included in C1.

C5PrKaC6: 11;3.

Not included C1 and most of the manuscripts of Table 3.

Pr: 7.

C1: 27.

Corrected from 16 in PNk X.B.3 and C5PrKaC6.

Corrected from 26 in PNk X.B.3 and C5PrKaC6.

C5C6: 48.

C1Ka: 51.

C5PrC6: 48.

Pr: 50.

C1: 6.

C5PrKa: 26; C6: 29.

Corrected from 0 in PNk X.10.B; C1C5PrKaC6: 9.

C1: 2,1.

PNk X.B.3 and C1: 41.

C5PrC6: 41.

C5PrC6: 48.

C5PrKaC6: 42.

C5PrC6: 27.

Pr: 50.