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The end of an error: Bianchini, Regiomontanus, and the tabulation of stellar coordinates

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Abstract

Giovanni Bianchini’s fifteenth-century Tabulae primi mobilis is a collection of 50 pages of canons and 100 pages of tables of spherical astronomy and mathematical astrology, beginning with a treatment of the conversion of stellar coordinates from ecliptic to equatorial. His new method corrects a long-standing error made by a number of his antecedents, and with his tables the computations are much more efficient than in Ptolemy’s Almagest. The completely novel structure of Bianchini’s tables, here and in his Tabulae magistrales, was taken over by Regiomontanus in the latter’s Tabulae directionum. One of the tables Regiomontanus imported from Bianchini contains the first appearance of the tangent function in Latin Europe, which both used as an auxiliary quantity for the calculation of stellar coordinates.

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Notes

  1. On the Toledan Tables, see Toomer (1968) and especially Pedersen (2002).

  2. There is a wealth of literature on the Alfonsine Tables and related works; begin with Chabás and Goldstein (2003).

  3. For a survey of astronomical tables in the late Middle Ages, see Chabás and Goldstein (2012). This work, and many of their research papers, explore the notion of “user-friendliness” in the tables of this time period; see especially Chabás and Goldstein (2013). The ALFA project (see the acknowledgments), within whose auspices this paper was written, aims to refine and complement this perspective on Alfonsine astronomy.

  4. The set of tables dealing with the motions of the planets is the subject of Chabás and Goldstein (2009).

  5. Chabás (2016); see also Rosińska (1981a).

  6. A collection of letters between Bianchini and Regiomontanus on astronomical topics also discusses this problem, among others. This correspondence was first published in Curtze (1902) and was studied thoroughly in Gerl 1989. A translation of part of the most important letter (Regiomontanus’s last) appears in Swerdlow (1990, 170–174).

  7. See Pedersen (2011, 99–101).

  8. Almagest I.14–16; see Toomer (1984, 69–74).

  9. The method may be found in Almagest VIII.5 (Toomer 1984, 410–413), where Toomer makes the same error that Bianchini is about to correct (note 201, p. 411); the table of oblique ascensions is Almagest II.8 (Toomer 1984, 100–103). See Pedersen 2011, 97–99.

  10. See Nallino (1899–1907: the Latin translation, vol. 1, 31–32; commentary by Nallino, vol. 1, 192–193; and Arabic text, vol. 3, 46–48.

  11. See King (1972, 293–295).

  12. See Kennedy (1956).

  13. See North (1976): Latin edition and translation, vol. 1, 126–134 (especially pp. 130–131) and associated commentary, vol. 2, 71. See also the edition and translation, vol. 1, 158–159 and associated commentary, vol. 2, 79–80; and the edition/translation, vol. 1, 438–439 and associated commentary, vol. 2, 306.

  14. See Saby (1987), vol. 1, 57.

  15. Porres de Mateo (2003), vol. 2, 464.

  16. On the chapter on arithmetic in the Flores Almagesti, see Rosińska (1995) (law of signs for negative numbers). On the chapter on algebra, see Rosińska (1997–98). On Bianchini’s concept of number in the chapters on arithmetic and algebra, see Rosińska (1998). On the Flores Almagesti as an inspiration for university teaching especially in Cracow, see Rosińska (2006).

  17. The planetary tables were studied in detail in Chabás and Goldstein (2009).

  18. A table of contents of our treatise in this manuscript is given in Boffito (1908, 457–460).

  19. On the Tabulae magistrales, see Rosińska (1981a) and Chabás (2016).

  20. Here and elsewhere, for tables and passages from the Tabulae primi mobilis we shall refer to folio numbers in the manuscript Cracow BJ 556. This table is on f. 59r.

  21. Cracow BJ 556, f. 12v.

  22. Cracow BJ 556, f. 13r.

  23. The table is given to seconds, but in all but one manuscript the seconds place is 0. In Cracow BJ 556, the seconds place is 30 in five entries.

  24. Here and elsewhere, for tables and passages from the Flores Almagesti we shall refer to folio numbers in the manuscript Vatican, BAV, Vat. Lat. 2228. This passage is on f. 44v. Bianchini adheres to the unusual practice of referring to himself by name when he believes he has contributed something original.

  25. Vatican, BAV, Vat. Lat. 2228, ff. 48v–49r.

  26. On Bianchini’s application of Menelaus’s Theorem in general, see Zepeda (2013, 323–329).

  27. Cracow, BJ 556, f. 13r.

  28. Vatican, BAV, Vat. Lat. 2228, ff. 46r–47r. See also Gerl (1988, 236–240).

  29. Vatican, BAV, Vat. Lat. 2228, f. 47r.

  30. Cracow, BJ 556, ff. 12v–13r; the table itself is on f. 59v.

  31. Vatican, BAV, Vat. Lat. 2228, f. 48r–v.

  32. Cracow, BJ 556, f. 13v.

  33. Vatican, BAV, Vat. Lat. 2228, f. 49r–v.

  34. This table is found under this title only in the manuscript Cracow BJ 556, ff. 55r–56v. However, the two tables from the Tabulae magistrales from which it is assembled are present in every manuscript we have consulted that includes tables.

  35. See Chabás (2016) for an analysis of the Tabulae magistrales.

  36. Curtze (1902, 238). Regiomontanus’s solution, which differs from the one used by Bianchini, is on pp. 258–259. In Flores Almagesti MS Cracow BJ 558 ff. 22v, 23r in the section where Bianchini describes his trisection and its use in constructing a sine table, there are marginal notes by Regiomontanus indicating that he believes that Bianchini’s method is better than Ptolemy’s, and that the trisection is due to the Banū Mūsā.

  37. See a Latin edition in Curtze (1902, 239–241), an account of the correspondence in Byrne (2007, 176–198), and another in Zinner (1990, 60–69). Swerdlow (1990) is an analysis especially of Regiomontanus’s final letter to Bianchini.

  38. This was noticed first in Rosińska (1981b), especially 571–573 and 577; see also Rosińska (1997–98, 140, 142). See Swerdlow (forthcoming) for a detailed discussion of the astrological aspects of the Tabulae directionum.

  39. Swerdlow (1990, 168, 193).

  40. This table is also found along with Regiomontanus’s tangent table in Paris BNF 10265, f. 222v, a manuscript otherwise filled with Bianchini’s works. (A marginal note stating Regiomontanus’s name is found about 20 folios earlier in the same manuscript.)

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Acknowledgements

This project was undertaken under the auspices of the ALFA Project (supported by Grant #723085 of the European Research Council) and with financial support from the Paris Observatory. I am extremely grateful to the principal investigator of the ALFA project, Matthieu Husson, for his support and friendship throughout the project, as well as to Richard Kremer, one of the two members of the advisory board. However, I am thankful most of all to the other member of the advisory board, José Chabás, who suggested that I turn my attention to Bianchini’s Tabulae primi mobilis and offered encouragement throughout my work on it. It is to him that this paper is dedicated. Darcy Otto was an invaluable colleague as we read through the text together. Noel Swerdlow’s critiques and editorial guidance improved the paper immeasurably. Finally, my thanks to the entire ALFA team (a special nod to A. J. Misra), who provided valuable comments, insights, and collegial conversations.

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    Glen Van Brummelen

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Communicated by Noel Swerdlow.

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Van Brummelen, G. The end of an error: Bianchini, Regiomontanus, and the tabulation of stellar coordinates. Arch. Hist. Exact Sci. 72, 547–563 (2018). https://doi.org/10.1007/s00407-018-0214-2

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