The Pre-Telescopic Treatment of the Phases and Apparent Size of Venus

Price D. J. , ‘Precision instruments: To 1500’, in A history of technology , ed. by Singer C. Holmyard E. J. Hall A. R. (7 vols, New York and London, 1954–78), iii, 582–619 (espec. p. 583).

See Smart W. M. , Text-book on spherical astronomy , 5th edn (Cambridge, 1962), 414.

The brightness of Venus is measured in negative magnitudes that range from −3.9 to −4.7 (where −4.7 is of course at the brighter end of the scale); this is a small variation in brightness, and is difficult to detect with the naked eye: cf. Price D.J. , ‘Contra-Copernicus: A critical re-estimation of the mathematical planetary theory of Ptolemy, Copernicus, and Kepler’, in Critical problems in the history of science , ed. by Clagett M. (Madison, 1959), 197–221 (espec. pp. 212–14). The phases vary over half the synodic period of Venus, i.e., from superior conjunction to inferior conjunction, and this is about 292 days. I am informed by Dr Leroy E. Doggett of the U.S. Naval Observatory that the formula in actual use for the Astronomical almanac is the following: V = −4.4 + 5 log (d*r) + 0.09 (i/100) + 2.39 (i/100)² — 0.65 (i/100)³ where V is the magnitude of Venus, d is the geocentric distance of Venus in astronomical units, r is the heliocentric distance of Venus, and i is the phase angle in degrees. Dr Doggett adds that ‘The constants were determined from observations of phases in the range 0.9° — 170.7°. Outside this range (i.e., close to conjunction), Earth-based photometric observations are not possible’ (private communication, 11 July 1995). Cf. Danjon A. , Astronomie générate , 2nd edn (Paris, 1959), 377–9. In 1994 ( The astronomical almanac for the year 1994 (Washington and London, 1993), A4, A8, E60–62; with additional information provided by Dr Doggett), at the limit of visibility near superior conjunction (elongation near 0°; phase angle 1.2°), the magnitude of Venus was −3.9; and at the limit of visibility near inferior conjunction (elongation 7°; phase angle 170.7°), the magnitude was −4.0. At greatest elongation (about 47°; phase angle 90°) its magnitude was −4.3; at maximum brightness (elongation about 39°; phase angle 120°) its magnitude was −4.7. Note that Venus is not visible to the naked eye at superior conjunction, but its magnitude stays the same for a long time around superior conjunction; and it is also not visible for somewhere between 2 and 19 days near inferior conjunction, i.e., for 1 to 10 days before and after it (depending on the season at which it takes place and the geographical latitude): See Huber P. , Astronomical dating of Babylon I and Ur III (Malibu, 1982), 11. For telescopic views of Venus at inferior conjunction, maximum brightness, and greatest elongation, see Byrd D. , ‘Eye on the sky’, Astronomy , xvi, no. 7 (July 1988), 54–56. For the elongation corresponding to maximum brightness, see Herschel J. , Outlines of astronomy (London, 1849), 279; Rudaux L. de Vaucouleurs G. , Larousse encyclopedia of astronomy (New York, 1959), 187; Flammarion G. C. Danjon A. , The Flammarion book of astronomy (New York, 1964), 279. The time interval from greatest elongation to maximum brightness is about 35 days and the time interval from maximum brightness to inferior conjunction is also about 35 days (these intervals may vary by a few days): cf. Müller G. , ‘Helligkeitsbestimmungen der grossen Planeten und einiger Asteroiden’, Publicationen des Astrophysikalischen Observatoriums zu Potsdam , viii (1893), 193–371 (espec. p. 324); Danloux-Dumesnils M. , Eléments d'astronomie fondamentale (Paris, 1985), 232–3.

Cf. Goldstein B. R. Sawyer F. W. III , ‘Remarks on Ptolemy's equant model in Islamic astronomy’, in Prismala: Festschrift für Willy Hartner , ed. by Maeyama Y. Salzer W. G. (Wiesbaden, 1977), 165–81.

Though this introduction was written by a Lutheran theologian, Andreas Osiander, for a while it was thought to be by Copernicus. Kepler was the first to call attention to the true identity of its author. See Jardine N. , The birth of history and philosophy of science (Cambridge, 1984), 150–4; Wrightsman B. , ‘Andreas Osiander's contribution to the Copernican achievement’, in The Copernican achievement , ed. by Westman R. S. (Berkeley and Los Angeles, 1975), 213–43.

Copernicus N. , De revolutionibus orbium coelestium (Nuremberg, 1543), fol. i-v. Copernicus (De revolutionibus, i, 10, fol. 8r) gives the ratio of greatest to least geocentric distance for Venus as 6 to 1. Osiander's implied ratio of 4 to 1 for Venus's greatest to least distance is peculiar: According to Ptolemy's Planetary hypotheses, this ratio is 104 to 16 although, with the parameters in the Almagest, it is 104;25 to 15;35 which is closer to 7 to 1 than to 6 to 1; see also ref. 28, below. Osiander, of course, assumed that the apparent diameter of a planet varies inversely with its distance from the observer, and that this followed from a knowledge of optics.

Carmody F.J. , Al-Biṭrūjī: De motibus celorum (Berkeley and Los Angeles, 1952); Goldstein B. R. , Al-Bịrūjī: On the principles of astronomy (2 vols, New Haven, 1971).

Goldstein , op. cit. (ref. 7), i, 125.

Goldstein B. R. , The Arabic version of Ptolemy's Planetary Hypotheses (Philadelphia, 1967), 7: ‘The occultation of a large body [the Sun] may not be perceptible on account of the remainder of the solar body which would still be exposed, for when the Moon eclipses part of the Sun, equal to, or even greater than, the diameter of one of the planets, the eclipse is not perceptible. Moreover, such events could only take place at long intervals …’.

Goldstein B. R. , ‘Some medieval reports of Venus and Mercury transits’, Centaurus , xiv (1969), 49–59; reprinted in Goldstein B. R. , Theory and observation in ancient and medieval astronomy (London, 1985).

Arafat W. Winter H. J. J. , ‘The light of the stars: A short discourse by Ibn al-Haytham’, The British journal for the history of science , v (1971), 282–8 (espec. p. 287).

Al-Bīrūnī , The book of instruction in the art of astrology , ed. and transl. by Wright R. R. (London, 1934), para. 156.

Copernicus , op. cit. (ref. 6), fol. 8r; for prior discussions in Latin, see Ariew R. , ‘The phases of Venus before 1610’, Studies in the history and philosophy of science , xviii (1987), 91–92.

See Bowen A. C. Goldstein B. R. , Homocentrics and history: Aristotle's testimony about Eudoxus and Callippus , Chap. 1: ‘The question of evidence’ (in preparation).

Cf. Bowen A. C. Goldstein B. R. , ‘Geminus and the concept of mean motion in Greco-Latin astronomy’, in Archive for history of exact sciences (in press).

de Virduno Bernardus , Tractatus super totam astrologiam , ed. by Hartmann P. (Werl, 1961), 68: ‘Similiter in Mercurio et maxime in Venere hoc apparet. Cum enim incipiunt apparere procedendo, id est a Sole secundum successionem signorum, recedendo apparent minoris quantitatis quam cum apparere incipiunt retrocedendo, id est a Sole contra successionem signorum protendendo. Ergo hoc necesse est contingere eo, quod aliquando sunt remotiora aliquando propinquiora obtutibus nobis.’ The translation is based on a slight emendation, reading obtutis nobis and construing obtutus (obtueo) analogously to exercitus (exerceo) in classical usage, since obtutus, at least in classical times, was apparently limited to acts of observation and not extended to the agents. Cf. Duhem P. , Le système du monde (10 vols, Paris, 1954), iii, 445. Duhem misinterprets the passage; according to him the claim would be that the apparent diameter of Venus is greater at superior conjunction than at inferior conjunction. See also Dreyer J. L. E. , ‘Mediaeval astronomy’, in Toward modern science , ed. by Palter R. M. (2 vols, New York, 1961), i, 235–56 (espec. p. 251).

The same argument is made in a text by an unknown author, previously thought to be Regiomontanus. This text was probably written in Vienna in the early fifteenth century, and the author was heavily indebted to Guido de Marchia (early fourteenth century), a Franciscan as was Bernard of Verdun. See Shank M. , ‘The ‘Notes on al-Bịrūjī’ attributed to Regiomontanus: Second thoughts’, Journal for the history of astronomy , xxiii (1992), 15–30.

Goldstein , op. cit. (ref. 9), 8.

Goldstein B. R. Swerdlow N. , ‘Planetary distances and sizes in an anonymous Arabic treatise preserved in Bodleian Ms. March 621’, Centaurus , xv (1970–71), 135–70 (espec. p. 150); reprinted in Goldstein , op. cit. (ref. 10). A similar remark appears in a letter from Regiomontanus (d. 1476) to Bianchini (d. after 1469) concerning outstanding problems in astronomy. Regiomontanus noted that ‘the surface of Venus ought to appear to our sight sometimes as 1, but sometimes as 45, which has never become known to anyone observing. Further, its apparent diameter will sometimes be 0;12,30°, that is, two-fifths the apparent diameter of the Moon, which certainly has never been perceived in the heavens.’ Although Regiomontanus recognized these problems, he offered no solution to them. See Swerdlow N. M. , ‘Regiomontanus on the critical problems of astronomy’, in Nature, experiment, and the sciences , ed. by Levere T. H. Shea W. R. (Dordrecht and Boston, 1990), 165–95, espec. p. 173. On 45 to 1 as the ratio for the apparent area of Venus's disk at least and greatest distances, see ref. 28, below.

Goldstein B. R. , ‘Levi ben Gerson's contributions to astronomy’, in Studies on Gersonides: A fourteenth-century Jewish philosopher-scientist , ed. by Freudenthal G. (Leiden, 1992), 3–19 (espec. p. 9).

Goldstein B. R. , The astronomy of Levi ben Gerson (1288–1344) (New York and Berlin, 1985), 105.

Goldstein , op. cit. (ref. 21), 69. Levi described the use of this instrument for determining the apparent size of the Moon in his Astronomy, chap. 75 (MS Paris, heb. 724, 146a:24–146b:1, and MS Paris, heb. 725, 113b: 14–24). For a list of 45 dated planetary observations by Levi ben Gerson, see Goldstein B. R. , ‘A new set of fourteenth century planetary observations’, Proceedings of the American Philosophical Society , cxxxii (1988), 371–99.

Cf. Steavenson W. H. , ‘Shadows cast by Venus’, Journal for the British Astronomical Association , lxvi (1956), 264–5; Herschel , op. cit. (ref. 3), 272.

Most of the observations of Venus reported in the Almagest were taken when Venus was at its maximum elongation: See Almagest x; Toomer G. J. , Ptolemy's Almagest (New York, Berlin, 1984), index, s. v. Venus, observations of.

Levi ben Gerson's Astronomy, chap. 56 (MS Paris heb. 724, 104b:4–8, 105a:13, and MS Paris heb. 725, 79b:13–15, 80a:13–14). The Latin version of this chapter, with English translation and commentary, was published as an Appendix to Mancha J. L. , ‘Astronomical use of pinhole images in William of Saint-Cloud's Almanach planetarum (1292)’, Archive for history of exact sciences , xliii (1992), 275–98. Levi later revised his value for the solar eccentricity on the basis of an examination of eclipse observations: See Goldstein B. R. , ‘Medieval observations of solar and lunar eclipses’, Archives internationales d'histoire des sciences , xxix (1979), 101–56 (espec. p. 104); reprinted in Goldstein , op. cit. (ref. 10).

See Levi ben Gerson's Astronomy, chap. 44 (MS Paris heb. 724, 84a:20–22, and MS Paris heb. 725, 62a:16–18). In chap. 15 ( Goldstein , op. cit. (ref. 21), 98), Levi argued that ‘if you find that the sizes of the [Sun] are equal [at apogee and perigee], you can conclude that the solar sphere is not eccentric to the centre of the world as Ptolemy assumed; but if you find that the sizes are different, you can conclude that its centre is eccentric to the centre of the world’.

Drake S. O'Malley C. D. , The controversy on the comets of 1618 (Philadelphia, 1960), 184.

Drake S. , Galileo: Dialogue concerning the two chief systems of the world (Berkeley and Los Angeles, 1962), 334; cf. Ariew , op. cit. (ref. 13). In contrast to Osiander, for whom the ratio of the area of Venus's disk at least and greatest distances was more than 16 to 1, Galileo has a little less than 40 to 1 which implies a ratio of greatest to least distance between 6 to 1 and 7 to 1.

Cf. Van Helden A. , Measuring the universe (Chicago, 1985), 71.

Van Helden , op. cit. (ref. 29), 68.