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Tables for the radii of the Sun, the Moon, and the shadow from John of Gmunden to Longomontanus

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Abstract

A table in five columns for the radii of the Sun, the Moon, and the shadow is included in sets of astronomical tables from the fifteenth to the early seventeenth century, specifically in those by John of Gmunden (d. 1442), Peurbach (d. 1461), the second edition of the Alfonsine Tables (1492), Copernicus (d. 1543), Brahe (d. 1601), and Longomontanus (d. 1647). The arrangement is the same and the entries did not change much, despite many innovations in astronomical theories in this time period. In other words, there is continuity in presentation and, from the point of view of the user of these tables, changes in the theory played no role. In general, the methods for computing the entries are not described and have to be reconstructed. In this paper, we focus on the users of these tables rather than on their compilers, but we refer to modern reconstructions where appropriate. A key issue is the treatment of the size of the Moon during a solar eclipse which was not properly understood by Tycho Brahe. Kepler’s solution and that of his predecessor, Levi ben Gerson (d. 1344), are discussed.

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Notes

  1. Nallino 1903–1907, 2: 95–101.

  2. For an analysis of previous tables for these radii up to and including that of John of Genoa, see Chabás and Goldstein (2023).

  3. Almagest V.14; Toomer (1984, p. 253): “It was determined that the moon is at its greatest distance when it subtends the same angle at the eye as the sun”. Note that at greatest distance (apogee) the apparent lunar radius is smallest. This implies that an annular solar eclipse is impossible for, elsewhere, the Moon is closer to the Earth and hence its angular size is greater than at apogee.

  4. A summary of Ptolemy’s procedures can be found in Neugebauer 1975, pp. 118–141.

  5. See Van Helden (1985, pp. 15–20), which includes many valuable comments on Ptolemy’s procedure.

  6. Note that for small angles tan x ≈ sin x ≈ x.

  7. For al-Battānī’s table of the solar and lunar velocities, see Nallino 1903–1907, 2: 88.

  8. For translations and commentaries on this text, see Malpangotto (2021) and Aiton (1987).

  9. On the first two editions of the Parisian Alfonsine Tables, see Chabás (2019, pp. 237–276).

  10. For Copernicus’s lunar theory, see Swerdlow and Neugebauer (1984, pp. 193–286).

  11. Copernicus owned a copy of the second edition of PAT: Czartoryski (1978 p. 366).

  12. See Swerdlow (1973, p. 426) n.5, and Sect. 4, above.

  13. See Swerdlow and Neugebauer (1984, p. 255). The justification for this change from 2 3/5 = 13/5 is given on p. 347: Ptolemy took the apparent radii of the Sun and Moon to be equal at greatest lunar distance, 64;10 terrestrial radii, whereas Copernicus took them to be equal at lunar distance 62 terrestrial radii, and so the radius of the shadow should be greater: 64;10/62 ≈ 31/30, and (31/30) · (13/5) = 403/150. This change meant that for Copernicus an annular solar eclipse is possible.

  14. See Swerdlow (2009, p. 44, n. 20). Swerdlow refers to Thoren (1990, pp. 495–496) who claimed, incorrectly, that the apparent reduction in the size of the Moon during a solar eclipse is an effect of the solar corona. However, Swerdlow does not refer to Kepler’s correct explanation of Brahe’s dilemma, and he does not cite the lengthy discussion of this point in Straker (1981).

  15. Kepler referred to a different argument in Levi ben Gerson’s Astronomy, included in Commandino (1558), fols. 60v–61r, concerning the determination of the center of vision in the eye by means of a cross staff: see Donahue (2000, pp. 229–230). Cf. Roche (1981, p. 8), and Goldstein (1985, pp. 51–54, 143–146). On Kepler’s limited knowledge of Levi’s scientific work, see Straker (1970, pp. 218–220). The pinhole camera was addressed earlier in the Arabic treatise, On the Shape of Eclipses by Ibn al-Haytham (d. 1040), but it was not translated into either Hebrew or Latin and so it was unavailable to Levi and Kepler: see Raynaud (2016, p. 5), n. 3, et passim.

  16. Published in Kepler (1604), Ad Vitellionem Paralipomena quibus Astronomiae Pars Optica traditur, and translated in Donahue (2000).

  17. The length of a canna (French: canne) varied from place to place; e.g., the canne of Toulouse was equal to 8 spans, corresponding to about 1.8 m: Charbonnier (1994, p. 90). In the Hebrew version of ch. 56 of Levi’s Astronomy, the distance between the two panels is not specified, but in the Hebrew version of ch. 9 (quoted below), the length of the staff is 16 spans or more (where the length of a span varied between 20 and 25 cm).

  18. For a brief biography of Longomontanus, see Christianson (2000, pp. 313–319).

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Authors and Affiliations

  1. University of Pittsburgh, Pittsburgh, PA, USA

    Bernard R. Goldstein

  2. Universitat Pompeu Fabra, Barcelona, Spain

    José Chabás

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Goldstein, B.R., Chabás, J. Tables for the radii of the Sun, the Moon, and the shadow from John of Gmunden to Longomontanus. Arch. Hist. Exact Sci. 78, 67–86 (2024). https://doi.org/10.1007/s00407-023-00318-w

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