Celestial Spheres and Circles

Duhem P. , To save the phenomena, translated by Doland E. and Maschler C. (Chicago and London, 1969).

Neugebauer O. , A history of ancient mathematical astronomy (Berlin. Heidelberg and New York, 1975). 1.

Gingerich O. , “Kepler's place in astronomy”, Vistas in astronomy, xviii (1975), 261–78, pp. 261–2.

Aristotle . De caelo, 269 a 32.

De caelo, 270 b 24. Cf. Plato , Cratylus, 410 B.

Aristotle , Meteorologica, 340 b 24–33.

De caelo, 289 a 20–34.

De caelo, 293 a 5.

De caelo, 290 a 28.

De caelo, 292 a 22.

Aristotle , Metaphysics, 1072 a 24.

Plato , Republic, 616 C-617 A.

The interpretation of Plato's text presents difficulties: See Dicks D. R. , Early Greek astronomy to Aristotle (London, 1970), 236, footnote 150.

Diels H. and Kranz W. , Die Fragmente der Vorsokratiker (6th ed. reprinted Dublin and Zurich, 1974), 28 A 37. Cf. commentary, vol. i, p. 242. Cicero interpreted stephane as ‘coronae similem’. Tarán L. , Parmenides (Princeton, 1965), 232–3.

Plato , Timaeus, 36 D.

Timaeus, 38 D-E.

On the problematical question whether Plato was aware of the stations and retrogressions of the other planets, see Duhem P. , Le système du monde (Paris, 1913–59), i, 110.

Timaeus, 34 A.

Timaeus, 40 B.

Gorgias, 451 C.

Aristotle , Metaphysics , 1073 b 17–33. Cf. Lassere F. , Die Fragmente des Eudoxos von Knidos (Berlin, 1966), 15 (Greek and German).

Heiberg J. L. (ed.), Commentaria in Aristotelem Graeca, vii (Berlin, 1894), 492–7. Cf. Lassere , op. cit. (ref. 21), 67–73 (fragment 124). See also Duhem , op. cit. (ref. 17), i, 111–23; Neugebauer , op. cit. (ref. 2), 624–30 and 677–85; Riddell R. C. , “Eudoxan mathematics and Eudoxan spheres”, Archive for history of exact sciences, xx(1979). 1–19.

Aristotle , Metaphysics, 1073 b 3.

Aristotle , Physics, 194a 8–13.

Aristotle , Metaphysics , 1073 b 17. On problems of interpretation of this passage. see Dicks , op. cit. (ref. 13), 178–81; Thoren V. E. , “Anaxagoras, Eudoxus and the regression of the lunar nodes”, Journal for the history of astronomy, ii (1971), 23–28.

According to Simplicius, this idea had arisen from the observation that, at the solstices, the Sun did not always rise from the same points of the horizon. Heiberg , op. cit. (ref. 22), 493. Cf. Lassere , op. cit. (ref. 21), 68.

Aristotle , Metaphysics, 1074 a 1.

Heiberg , op. cit. (ref.22), 488, line 23. ‘Regular’ here means ‘in one direction’.

ibid., 488, line 13.

ibid., 493, lines 2–3.

Duhem , op. cit. (ref. 1), 5. Cf. Duhem , op. cit. (ref. 17), i, 102–6.

Timaeus, 56 A.

ibid., 46 D-E.

Heiberg , op. cit. (ref. 22), 488. Cf. Lassere , op. cit. (ref. 21), 67 (fragment 121). The passage is translated in Krafft F. , “Physikalische Realität oder mathematische Hypothese?”, Philosophia naturalis. xiv (1973), 243–75, p. 250.

Mittelstrass J. , Die Rettung der Phänomene (Berlin, 1962); Neuzeit und Aufklärung (Berlin, 1970), 250–63.

Plato , Laws, 821 E-822 A.

Mittelstrass J. , Neuzeit und Aufklärung (ref. 35), 255.

See Wright L. , “The astronomy of Eudoxus: Geometry or physics?” Studies in history and philosophy of science, iv (1973), 165–72.

Lassere , op. cit. (ref. 21), 200.

Diels and Kranz , op. cit. (ref. 14), 59 B 21a.

Krafft F. , “Johannes Keplers Beitrag zur Himmelsmechanik”, in Krafft F. Meyer K. and Sticker B. (eds), Internationales Kepler-Symposium Weil der Stadt 1971 (Arbor scientiarum , Reihe A, Bd 1) (Hildesheim, 1973), 55–139, 64–5, footnote 22.

Maula E. , “Eudoxus encircled”, Ajatus, Yearbook of the Philosophical Society of Finland, xxxiii (1971), 201–53; Studies in Eudoxus' homocentric spheres (Commentationes humanarum litterarum , 1; Helsinki, 1974); Kasanen E. , “An algebraic view of Eudoxus' method”, and Mattila J. K. , “On some mathematical properties of Plato's ‘Great Harmonia’”, in Mattila J. K. and Siitonen A. (eds), Analysis, harmony and synthesis in ancient thought (Acta Universitatis Ouluensis, Ser. B, Humaniora 6, Historia 3; Oulu, 1977).

Maula E. , “The spider in the sphere: Eudoxus' arachne”, Philosophia (Athens), 5–6 (1975–76), 225–58. There is an account in Finnish, with illustrations of a reconstruction of the arachne and the temple, in IBM Katsaus (Helsinki) xvii (1978), 4–24.

Aristotle, De caelo, 292 a 5–20.

The general belief that Heraclides of Pontus introduced an epicycle to explain the appearances of Venus is based on a misinterpretation. Neugebauer , op. cit. (ref. 2), 694–6 and 758. Cf. Toomer G. J. , in Gillispie C. (ed.). Dictionary of scientific biography (New York, 1970–80), xv, 202–5. There is evidence that Heraclides proposed a rotation of the Earth on its axis. Proclus, Commentaire sur le Timée, translated by Festugière A. J. (Paris, 1966–68), iv, 176.

Neugebauer , op. cit. (ref. 2), 263–4 and 949–50.

Again our source is Simplicius, in his commentary on Aristotle's Physics. Diels H. (ed.), Commentaria in Aristotelem Graeca, ix (Berlin, 1882), 291–2. This passage is translated in Krafft , op. cit. (ref. 34), 254–5 and Duhem , op. cit. (ref. 1), 10–11.

Krafft , op. cit. (ref. 34), 256–9.

Sosigenes lived in the second century A.D. and should not be confused with the calendar reformer of the first century B.C.

Schramm M. , Ibn al-Haythams Weg zur Physik (Wiesbaden, 1963), 15–63.

Heiberg , op. cit. (ref. 22), 505, lines 1–9. The date of Sosigenes's book must be later than A.D. 164. Despite the failure of homocentric astronomy to ‘save the appearances’, attempts to revive the theory were made in the sixteenth century by Amici G. B. , De Motibus corporum coelestium (Venice, 1536) and Fracastoro G. , Homocentrica (Venice, 1538).

Heiberg , op. cit. (ref. 22), 509, line 30–510, line 3. Cf. Schramm , op. cit. (ref. 50), 57.

Heiberg , op. cit. (ref. 22), 510, lines 3–8. Cf. Schramm , op. cit. (ref. 50), 60. Sosigenes held the circular motions to be uniform: Heiberg , 488, lines 3–24. Cf. Krafft , op. cit. (ref. 34), 250.

Sosigenes is here referring to the diurnal motion.

Heiberg , op. cit. (ref. 22), 510, lines 15–19. Cf. Schramm , op. cit. (ref. 50), 61.

and Duhem , op. cit. (ref. 17), ii, 67.

Heiberg , op. cit. (ref. 22), 510, lines 19–24.

Duhem , op. cit. (ref. 17), ii, 67, misinterprets this passage.

Schramm , op. cit. (ref. 50), 62.

Neugebauer , op. cit. (ref. 2), 834–5.

This work consists of two books. Only the first part of Book I is extant in Greek but the whole work is extant in Arabic. The Greek text together with German translation of this and Book II of the Arabic version have been published in Heiberg J. L. , Claudii Ptolemaei opera quae exstant omnia (Leipzig, 1907), vol. ii. An English translation of the remaining part of Book I, together with a facsimile of the complete Arabic text, have been published by Goldstein B. R. , “The Arabic version of Ptolemy's Planetary hypotheses”, Transactions of the American Philosophical Society, n.s., lvii (1967), Pt 4. See also Duhem , op. cit. (ref. 17), ii, 86–99 (the concluding part of Book I was unknown to Duhem); Pedersen O. , Survey of the Almagest (Odense, 1974), 391–7 (Pedersen confuses the two Sosigenes); Neugebauer , op. cit. (ref. 2). 917–26; Palter R. , “An approach to the history of early astronomy”. Studies in history and philosophy of science, i (1970). 93–133. Izydora Dambska has suggested that the models of the Planetary hypotheses were envisaged by Ptolemy simply as a teaching aid. Dambska I. , “L'Épistemologie de Ptolemée”, in Avant, avec, après Copernic (Paris, 1975), 31–37. Yet this seems inconsistent with Ptolemy's supposition that the spheres are contiguous, because it is inconceivable that a vacuum should exist in nature. See Goldstein , op. cit. (ref. 60), 8.

Heiberg , op. cit. (ref. 60), 73.

Goldstein , op. cit. (ref. 60), 7; Ptolemy , Handbuch der Astronomie , translated by Manitius K. (reprinted Leipzig, 1963), ii, 93 (Translation of the Almagest).

Goldstein , op. cit. (ref. 60), 8. On the influence of Ptolemy's theory of planetary distances among the Arabs, see Hartner W. , “Medieval views on cosmic dimensions and Ptolemy's Kitāb al-Manshūrāt”, in Oriens-occidens (Hildesheim, 1968), 319–48.

Heiberg , op. cit. (ref. 60), 111–13.

Plato , Republic, 616 D.

Heiberg , op. cit. (ref. 60), 114.

Ibid ., 120. Cf. 131. Here Ptolemy speaks of the planets each possessing a force to produce “in its own place and about its own centre a motion of uniform rotation”. Plato had envisaged axial rotation. Timaeus, 39 D-E. Cf. Proclus, op. cit. (ref. 45), iv, 164.

Heiberg , op. cit. (ref. 60), 141.

ibid., 142. Since nothing hinders it, the aether moves with the primitive motion; that is, the diurnal motion of the outermost sphere.

ibid., 117.

Also influential were the writings attributed to Dionysius the Areopagite, which contain a thinly disguised version of Proclus's doctrines. On the Dionysian problem see Tigerstedt E. N. , The decline and fall of the Neoplatonic interpretation of Plato (Commentationes humanarum litterarum, lii; Helsinki, 1974), 21–22.

Proclus , Hypotyposis astronomicarum positionum, translated by Manitius K. (Leipzig, 1909), 220–1.

Neugebauer , op. cit. (ref. 2), 918.

Proclus , op. cit. (ref. 45), iv, 85.

Proclus , op. cit. (ref. 72), 222–3.

ibid., 236–9.

Proclus , op. cit. (ref. 45), iv, 125. Cf. iv, 79, 164 and iii, 308.

ibid., iv, 79 and 125.

ibid., iv, 164.

Ibid ., iv, 189. Cf. 125.

On the Arabic translations of the Almagest and the Latin translations based on these, see Kunitzsch P. , Der Almagest. Die Syntaxis Mathematica des Claudius Ptolemäus in arabisch-lateinischer Überlieferung (Wiesbaden, 1974), 15–112. See also Kunitzsch P. , “New light on al-Battāni's Zij”, Centaurus, xviii (1974), 270–4. For a general survey, see Rybka E. , “Mouvement des planètes dans l'astronomie des peuples de l'Islam”, in Convegno internazionale. Oriente e occidente nel medioevo: Filosofia e scienze (Rome, 1971), 579–94.

Steinschneider M. , “Die arabischen Übersetzungen aus dem Griechischen”, Zeitschrift der deutschen morganländischen Gesellschaft, 1 (1896), 161–219, and 337–417, p. 211. Cf. Schramm , op. cit. (ref. 50), 16.

Carmody F.J. , The astronomical works of Thabit b. Qurra (Berkeley, 1960), 19.

Maimonides M. , Le guide des égarés , translated by Munk S. (Paris, 1856–66; reprinted Osnabrück, 1964), ii, 189–90 and iii, 100; Magnus Albertus , Opera omnia (Archendorff, 1971), v, pt 1, 28–30.

See Hartner W. in Dictionary of scientific biography (ref. 45), i, 511.

For editions and translations, see Sabra A. I. in Dictionary of scientific biography (ref. 45), iv, 541–5, p. 544. On al-Fargāni, see also Suter H. , Die Mathematiker und Astronomen der Araber und ihre Werke (Leipzig, 1900; reprinted New York and London, 1972), 18–19 and Mieli A. , La science arabe (Leiden, 1938), 82–88.

al-Battani , Opus astronomicum, ed. and trans, by Nallino C. A. (Milan, 1899–1903; reprinted Frankfurt-am-Main, 1969), Pt 1, 120–124.

The reconciliation through Alhazen of the traditional astronomy and optics with Aristotle's physics is the central theme of Schramm's book (ref. 50). For other accounts of Alhazen's astronomy, see Hartner W. , “The Mercury horoscope of Marcantonio Michiel of Venice”, in Oriens-occidens (ref. 63), 440–495, pp. 480–3 and Duhem , op. cit. (ref. 17), ii, 119–29.

W. Hartner remarks that falak corresponds to sphaira, sphaera or orbis and da'ira to kuklos or circulus, but authors in any of the three languages seldom aim at perfect consistency. See his article on falak in the Encyclopedia of Islam, reprinted in Oriens-occidens (ref. 63), 265–7, p. 266.

Kohl K. , “Über den Aufbau der Welt nach Ibn al Haitam” (translation of the third part of Alhazen's work), Sitzungsberichte der physikalisch-medizinischen Societät in Erlangen , liv-lv (1922–23), 140–79, pp. 150–1. Cf. the Latin translation in Vallicrosa J. M. Millás , Las traducciones orientales en los manuscritos de la Biblioteca Catedral de Toledo (Madrid, 1942), 285–312, p. 287.

Kohl , op. cit. (ref. 90), 157.

Kohl interprets this as “die ähnlich gelagerte Sphäre”, meaning that its centre and axis coincide with those of the ecliptic; ibid ., 153–4. Cf. Hartner's description “assimilated; i.e. concentric or parecliptic”, Hartner , op. cit. (ref. 88), 481.

Kohl , op. cit. (ref. 90), 148.

Wiedemann E. , “Ibn al Haitam, ein arabischer Gelehrter”, in Festschrift für J. Rosenthal (Leipzig, 1906), 149–78, p. 164.

Kohl , op. cit. (ref. 90), 142–3. The authenticity of the appendix is confirmed by its agreement with Alhazen's On the light of the Moon. See the commentary by Schramm, op. cit. (ref. 50), 67–69 and 130–41. Cf. Sabra A. I. , “The physical and mathematical in Ibn al-Haytham's theory of light and vision”, Commemoration volume of Birūni International Congress in Tehran (Tehran, 1976), 439–78. (The author kindly sent me a corrected post-print).

See Sabra A. I. in Dictionary of scientific biography (ref. 45), vi, 189–210, p. 198 and p. 207.

Sachau E. C. , Alberuni's India (London, 1888; reprinted Delhi, 1964). ii. 69.

Hartner , op. cit. (ref. 63), 347 and op. cit. (ref. 79), 267. The Arabic text is printed in Goldstein, op. cit. (ref. 60).

There is a German translation of the astronomical part in Ethé H. , Zakarija ben Muhammed Mahmūd el-Kazwini's Kosmographie (Leipzig, 1868), 31–181.

There is a German translation; Rudloff G. and Hochheim A. , “Die Astronomie des…al-Ǵagmini”, Zeitschrift der deutschen morgenländischen Gesellschaft, xlvii (1893), 213–75.

ibid., 235.

Aristotelis opera cum Averrois commentariis (Venice, 1562–74; reprinted Frankfurt-am-Main, 1962), vol. viii, f. 329v, col. 1 (Comment on Metaphysics, 1073 b 17).

Cf. Aristotle, De caelo, 286 a 15.

Averroes , op. cit. (ref. 102), vol. v, f. 118v, col. 1 (Comment on De caelo, 288 a 13).

Ibid ., f. 116r, col. 1 (Comment on De caelo, 287 b 15). Cf. Carmody F. J. , “The planetary theory of Ibn Rushd”, Osiris, x(1952), 556–86, pp. 583–4.

Averroes , op. cit. (ref. 102), vol. viii, f. 329v, col. 2. Quotations from Averroes in French translation may be found in Duhem, op. cit. (ref. 17), ii, 133–9, and in English translation in Carmody, op. cit. (ref. 105).

This is hearsay quoted by Maimonides. See al-Bitrūji , On the principles of astronomy, translated by Goldstein B. R. (New Haven and London, 1971), i, 4.

Ibid ., Cf. Carmody , op. cit. (ref. 105), 558.

Alpetragius objects to Ptolemy that, since the eccentric and epicycle lie in the interior of a single (concentric) sphere, the motions imply that this sphere would have to be fluid. al-Bitrūji, op. cit. (ref. 107), i, 60.

A similar device had also been used by al-Zarqallu to explain the imaginary trepidation of the equinoxes. Goldstein B. R. , “On the theory of trepidation”, Centaurus, x (1964), 232–47. On al-Zarqallu, see Vallicrosa J. M. Millás , Estudios sobre Azarquiel (Madrid and Grenada, 1943–50).

al-Bitrūji , op. cit. (ref. 107), 8–9.

Duhem , op. cit. (ref. 17), ii, 156.

Ch. xi of this work has been translated into French by de Vaux Carra , in Tannery P. , Recherches sur l'histoire de l'astronomie ancienne (Paris, 1893), 337–61. See also Hartner W. , “Nasir al-Din al-Tūsi's lunar theory”, Physis, xi (1969), 287–304 (which includes a critical analysis of Carra de Vaux's interpretations); Livingston J. W. , “Nasir al-Din al-Tūsi's al-Tadhkirah”, Centaurus, xvii (1973), 260–75; Saliba G. , “The first non-Ptolemaic astronomy at the Maraghah school”, Isis, lxx (1979), 571–6.

Kennedy E. S. and Roberts V. , “The planetary theory of Ibn al-Shātir”, Isis, 1 (1959), 227–35; Kennedy E. S. , “Late medieval planetary theory”, Isis, lvii (1966), 365–78; Abbud F. , “The planetary theory of Ibn al-Shātir”, Isis, liii (1962), 492–9; Rosinka G. , “al-Tūsi and al-Shātir in Cracow?”, Isis, lxv (1974), 239–43. Copernicus (and indeed al-Tūsi) may have obtained the so called Tūsi-couple from a similar device of Proclus. Veselovsky I. N. , “Copernicus and Nasir al-Din al-Tūsi”, Journal for the history of astronomy, iv (1973), 128–30. Cf. Proclus , Les commentaires sur le premier livre des Elémens d'Euclide , transl. by ver Eecke P. (Bruges, 1948), 96.

See Hartner W. , “Trepidation and planetary theories”, in Convegno internazionale (ref. 81), 609–32, esp. 610–14.

Tannery , op. cit. (ref. 113), 355–7. In fact, al-Tūsi refers explicitly only to Alhazen's attempt to account for the motions in latitude.

Carra de Vaux describes this scheme as “l'idée bien étrange” of Alhazen; ibid., 357. al-Tūsi also used the couple, projected on to a spherical surface enclosing the epicyclic sphere (in the fashion of the polar circles of Alpetragius) to explain the oscillation of the plane of the epicycle needed by Ptolemy in his latitude theory; ibid ., 357–9. On Ptolemy's theory, see Pedersen , op. cit. (ref. 60), 359–61.

Carmody F. J. , al-Bitrūji: De motibus celorum (Berkeley and Los Angeles, 1952). On Michael Scot, see Haskins C. H. , Studies in the history of mediaeval science (New York, 1960), 272–98. On the influence of the Arabs, see Petri W. , “Tradition und Fortschritt in der Astronomie des Mittelalters”. in Convegno internazionale (ref. 81), 633–44; Cimino M. , “L'astronomia araba e le sua diffusione”, ibid., 647–74.

“Compotus Roberti Grossecapitis”, in Bacon R. , Opera hactenus inedita , ed. by Steele R. (Oxford, 1909–40), fasc. vi, 282. Grosseteste reasoned that the stars and their spheres could not be of the same nature. “De generatione stellarum”, in Baur L. , Die Philosophischen Werke des Robert Grosseteste (Beiträge zur Geschichte der Philosophie des Mittelalters , ix, Münster, 1912), 32–36, p. 32. On the complementary roles of reason and experience in Grosseteste's methodology, see Crombie A. C. , Robert Grosseteste and the origins of experimental science (Oxford, 1953), esp. 87–90.

Bacon , op. cit. (ref. 119), fasc. iv, 437–8.

See Aristotle , Physics, 261 b 29–262 a 1.

Bacon , op. cit. (ref. 119), fasc. iv, 419.

Ibid ., 429. This objection had also been stated by Grosseteste. “De motu supercaelestium”, in Baur , op. cit. (ref. 119), 99.

Bacon , op. cit. (ref. 119), 430.

Ibid ., 434–5. Cf. 437–9.

Bate H. , Speculum divinorum et quorundum naturalium , Pt 22, ch. 16: “Expositio Sosigenes in praemissis iuxta Callippum”. This chapter exists only in manuscript. The first fasc. only of a critical edition of the Speculum by G. Wallerand appeared in Louvain in 1931. The first two volumes of a new critical edition by van de Vyner E. have been published in the series Philosophes médiévaux (Louvain and Paris), iv (1960) and x (1967). Moerbeke's translation of Simplicius's commentary on De caelo was first printed in Venice in 1540.

Aquinas St Thomas , Opera omnia (Rome, 1882 -), iii, 186.

ibid., 187.

Aquinas had in mind chiefly Aristotle's counteracting spheres.

ibid., 188.

Ibid ., 11. Cf. Litt T. , Les corps célestes dans l'univers de Saint Thomas d'Aquin (Philosophes médiévaux, vii, Louvain and Paris, 1963), 352.

Aquinas , op. cit. (ref. 127), iii, 188. Cf. Litt , op. cit. (ref. 131), 354, footnote 6.

Campanus of Novara, Theorica planetarum, trans, by Benjamin F. S. and Toomer G. J. (Madison, Milwaukee and London, 1971), 180–5. From section iv, line 302, it is clear that the epicycles and deferents were regarded as circles.

Ibid ., 34. Another possible source is De sphaera of Robert Grosseteste, in Baur, op. cit. (ref. 119), 10–32.

For an English translation by Pedersen O. , see Grant E. (ed.), A source book in medieval science (Cambridge, Mass., 1974), 451–65.

Thorndike L. , The Sphere of Sacrobosco and its commentators (Chicago, 1948), 77. The definition of Euclid also implies that a sphere has one surface.

Ibid ., 145. He also considers whether the celestial spheres are continuous or contiguous. Cf. Aristotle, Metaphysics, 1069 a 3.

Thorndike , op. cit. (ref. 136), 77–78. By accident the sphere is divided into the right sphere and the oblique sphere. See Neugebauer , op. cit. (ref. 2), 31.

Thorndike , op. cit. (ref. 136), 79.

Aristotle , Meteorologica, 339 a 33–339 b 16.

Thorndike , op. cit. (ref. 136), 78.

de Virduno Bernardus , Tractatus super totam astrologiam , ed. by Hartmann Polykarp (Franziskanische Forschungen, xv; Werl, 1961). There is a translation of an extract on the necessity of epicycles and eccentrics and their embodiment in solid spheres, in Grant E. , op. cit. (ref. 135), 520–4.

Buridan J. , In Metaphysicen Aristotelis quaestiones (Paris, 1518: Reprinted Frankfurt-am-Main, 1964), f. 73r, col. 2. An extract (ff. 73r-74r) is translated in Grant E. , op. cit. (ref. 135), 524–9. See also Duhem , op. cit. (ref. 17), iv, 124–42.

See Duhem , op. cit. (ref. 17), iv, 161–2.

Le livre du ciel et du monde, ed. by Menut A. D. and Denomy A. J. (Madison, Milwaukee and London, 1968), 581.

Questiones Marsilii super quattuor libros sententiarum (1501, reprinted Frankfurt-am-Main, 1966), f. 243r, col. 1. Cf. Duhem , op. cit. (ref. 17), iv, 167.

Wilpert P. (ed.), Nikolaus von Kues Werke (Berlin, 1967; reprint of the Strasbourg edition of 1488), i, 5.

The principle may have been inspired by Plotinus. Duhem P. , Études sur Léonard de Vinci (reprinted Paris, 1955), ii, 128. The idea of a coincidence of opposites is to be found in Heraclitus but there is no mention there of the infinite. Diels and Kranz, op. cit. (ref. 14), 22 B 60 and 22 B 103.

Wilpert , op. cit. (ref. 147), i, 6.

Ibid ., 39. Cf. Koyré A. , From the closed world to the infinite universe (Baltimore, 1957), 5–27.

On a manuscript of Cusanus relating to the celestial motions, see Duhem , op. cit. (ref. 17), x, 313–19.

P. H. Michel has remarked that, in naming all four elements on the surface of the Sun, Cusanus is seen to have observed sunspots. Le soleil à la Renaissance (Bruxellesand Paris, 1965), 401.

Wilpert , op. cit. (ref. 147), ii, 583.

On the continuing influence of Aristotle, see Grant E. , “Aristotelianism and the longevity of the medieval world view”, History of science, xvi (1978), 93–106.

The letter is printed in Klibansky R. , The continuity of the Platonic tradition during the Middle Ages (London, 1939), 45–47.

De sole & lumine libri duo, ch. 4. Marsili Ficini Florentini opuscula (Venice, 1503), sig. A iv r.

Ficino translated both Plato and the Hermetic writings.

Ficino Marsilio , Théologie platonicienne , trans, by Marcel R. (Paris, 1964–70), i, 151–2. Cf. Wolfson H. A. , “The problem of the souls of the spheres from the Byzantine commentaries on Aristotle through the Arabs and St. Thomas to Kepler”, Dumbarton Oaks papers, no. 16 (Washington, 1962), 65–93.

Ficino , op. cit. (ref. 158), i, 160.

Plato , Timaeus, 63 D.

Cf. Horský Z. , “Le cosmologie de Marsile Ficin”, Acta historiae rerum naturalium necnon technicarum , Special Issue no. 2 (Prague, 1966), 57–68, p. 63; “Le rôle du Platonisme dans l'origine de la cosmologie moderne”, Organon, iv (1967), 47–54.

Ficino , op. cit. (ref. 158), i, 162. Cf. 166.

Rosen E. , Three Copernican treatises (New York, 1971), 143.

Pontano G. G. , Opera omnia (Venice, 1519), iii, f. 145v.

Ibid ., f. 146r. Cf. Duhem , op. cit. (ref. 1), 54–56.

Rice E. F. , “Humanist Aristotelianism in France. Jacques Lefèvre d'Étaples and his circle”, in Levi A. H. T. (ed.), Humanism in France at the end of the Middle Ages and in the early Renaissance (Manchester, 1970), 132–3.

Cassirer E. , Individuum und Kosmos in der Philosophie der Renaissance (Berlin, 1927; reprinted Darmstadt, 1969), 93. Lefèvre's edition of Cusanus was the third, not the first as Cassirer states.

Rice , op. cit. (ref. 166), 143.

Stapulensis Iacobus Faber , Introductorium astronomicum, theorias corporum coelestium duobus libris complectens (Paris, 1517), f. lv (first published in 1503).

Ibid.

Duhem , op. cit. (ref. 1), 56–59.

Schanz P. , Der Cardinal Nicolaus von Cusa als Mathematiker (1872, reprinted Wiesbaden, 1967), 2–3.

Peurbach's Theoricae novae planetarum was first completed in about 1472 by his student Regiomontanus, who also completed and published an Epitome of the Almagest, started by Peurbach. Both are reprinted in facsimile in Regiomontanus, Opera collectanea , ed. by Schmeidler F. (Osnabrück, 1972). Albertus de Brudzewo wrote a commentary on Peurbach which was used in lecture courses at Cracow in the time Copernicus was there. See Rybka E. , op. cit. (ref. 81), 579.

Fine O. , La theorique des cielz, mouvemens et termes practiques des sept planetes (Paris, 1528); Toscannella O. , Le nuove teoriche de i pianeti di Georgio Peurbachio (Venice, 1566).

Regiomontanus , op. cit. (ref. 173), 755–6.

ibid., 758.

ibid., 757.

ibid., 99 and 192.

Stapulensis Iacobus Faber , op. cit. (ref. 169). Clichtove's definitions are given on ff. 3r-3v and Lefèvre's commentary on ff. 4r-5v.

Maestlin M. , Epitome astronomiae (Tübingen, 1597), 12–15.

Petri Nonii Salaciensis opera (Basel, 1592), 106.

ibid., 197.

Münster S. , Rudimenta mathematica (Basel, 1551), 60.

Kennedy and Roberts , op. cit. (ref. 114). See also Rosińska G. , “L'École astronomique de Cracovie et la révolution Copernicienne”, Avant, avec, après Copernic (ref. 60), 89–92.

Hartner W. , “Copernicus, the man and his work”, Proceedings of the American Philosophical Society, cxvii (1973), 421–2.

Jarzebowski L. , Biblioteka Mikolaja Kopernika (Torun, 1971).

Swerdlow N. M. , “The derivation and first draft of Copernicus' planetary theory. A translation of the Commentariolus with commentary”, Proceedings of the American Philosophical Society, cxvii (1973), 423–512, p. 425. A translation of the passage in Regiomontanus is given on pp. 472–5. Cf. Regiomontanus, op. cit. (ref. 173), 243–4. The translation of the Commentariolus by Hugonnard-Roche H. Rosen E. and Verdet J. P. , Introductions à l'astronomie de Copernic (Paris, 1975), is recommended for reliability.

Facsimile and translation in Swerdlow, op. cit. (ref. 187), 428–9. According to Swerdlow, the analysis of the second anomaly would lead to the Tychonic and Copernican systems as alternatives.

See the diagram showing the intersections of the orbits of Mars and the Sun in Tycho's system in Dreyer J. L. E. , A history of astronomy from Thales to Kepler (1953), 364.

Swerdlow , op. cit. (ref. 187), 466.

Ibid ., 465. Cf. Prowe L. , Nicolaus Coppernicus (Berlin, 1883–84), ii, 195.

Rosen E. , “Copernicus' spheres and epicycles”, Archives internationales d'histoire des sciences, xxv (1975), 82–92; “Copernicus' axioms”, Centaurus, xx (1976), 44–49; Swerdlow N. M. , “Pseudodoxia Copernicana”, Archives internationales d'histoire des sciences, xxvi (1976), 108–58. The controversy is reviewed by E.J. Aiton in Zentralblatt für Mathematik, ccclvi (1978), 8–9. A major obstacle to the understanding of the controversy is a lack of clarity which leaves the reader uncertain as to just what positions are being defended or attacked. For example, Swerdlow's diagram and explanation show that his solid sphere has one surface. This is how Rosen interprets it. But in his reply, Swerdlow explains that his solid spheres are what Münster calls ‘orbes’, which is what he thinks Rosen means by hollow spheres.

See, for example, Horský Z. , “Mathématique et physique dans l'astronomie de Copernic”, Avant, avec, après Copernic (re!. 60), 119–23.

Rosen , “Copernicus' spheres and epicycles” (ref. 192), 85.

Copernicus, On the revolutions , translated by Rosen E. (Warsaw and London, 1978), 20. Cf. De revolutionibus (facsimile reprint, New York and London, 1965), ff. 8v-9r. Rheticus relates this point in the Narratio prima, using the same terminology. Johannes Kepler, Mysterium cosmographicum (Tübingen, 1596), appendix, 116.

Copernicus, On the revolutions (ref. 195), 333–4.

Prowe , op. cit. (ref. 191), 184–5.

ibid., 186.

Copernicus, De revolutionibus (ref. 195), f. iii v.

ibid., i. 4r.

Kepler , op. cit. (ref. 195), 99 and 116.

ibid., 120.

Krafft , op. cit. (ref. 34), 264–72. Cf. op. cit. (ref. 41), 73–78. See also Krafft F. , “Copernicus retroversus I”, Colloquia Copernicana (Warsaw, 1975), 113–23; “Progressus retrogradus”, in Diemer A. (ed.), Die Struktur wissenschaftlicher Revolutionen und die Geschichte der Wissenschaften (Meisenheim am Glan, 1977), 20–48.

On the extension of Osiander's interpretation to physical hypotheses, see Wrightsman B. , “Andreas Osiander's contribution”, in Westman R. S. (ed.), The Copernican achievement (Berkeley, Los Angeles and London, 1975), 213–43, p. 241.

Copernicus , On the revolutions of the heavenly spheres , trans, by Duncan A. M. (London and New York, 1976), 46. Cf. Copernicus, De revolutionibus (ref. 195), f. 7r.

Krafft , “Copernicus retroversus I” (ref. 203), 122.

Copernicus , De revolutionibus (ref. 195), f. iiii r.

Heiberg , op. cit. (ref. 22), 488, lines 3–24. Cf. Krafft , op. cit. (ref. 34), 250; Rosen , op. cit. (ref. 163), 58–59. Copernicus wondered whether there could be found a more reasonable arrangement of circles from which every apparent inequality would be derived and in which everything would move uniformly about its own centre, as the rule of absolute motion required.

Krafft , “Copernicus retroversus I” (ref. 203), 122.

Westman R. S. , “The Melanchthon circle, Rheticus and the Wittenberg interpretation of the Copernican theory”, Isis, lxvi (1975), 165–93, esp. pp. 166–7. Cf. Westman , op. cit. (ref. 204), 285–345.

Letter to Georg and Hulderich Fugger, 1552. Bretschneider C. G. and Bindseil H. E. (eds), Corpus Reformatorum (Halle, 1834–60), vii, 951. For a survey of Melanchthon's contributions to science, see Bernhardt , Philipp Melanchthon als Mathematiker und Physiker (Wittenberg, 1865; reprinted Wiesbaden, 1973).

Bretschneider , op. cit. (ref. 211), xiii, 216–17. Blumenberg H. , Die Genesis der kopernikanischen Welt (Frankfurt-am-Main, 1975), 384, points to differences between the first (1549) and second (1550) editions, chiefly to a change of emphasis in which the criticism of the Copernican theory acquires a didactic aim.

Bretschneider , op. cit. (ref. 211), xiii, 195. Cf. 222–3.

Bernhardt , op. cit. (ref. 211), 72.

Bretschneider , op. cit. (ref. 211), xiii, 244.

Ibid. Cf. Melanchthon's remark “Our eyes see eight spheres”, ibid., xiii, 224.

ibid., xiii, 228.

ibid., xiii, 223.

ibid., xiii, 395.

ibid., xviii, 12.

Reinhold E. , Theoricae novae planetarum Georgii Purbachii (Paris, 1553), ff. lv.–2r.

Cf. the marginal note in his copy of De revolutionibus: “The axiom of astronomy; celestial motion is circular and uniform or made up of circular and uniform parts”. Gingerich O. , “From Copernicus to Kepler: Heliocentrism as model and as reality”, Proceedings of the American Philosophical Society, cxvii (1973), 513–22, p. 515. See also Birkenmajer A. , “Le commentaire inédit d'Erasmus Reinhold sur le De revolutionibus de N. Copernic”, in Etudes d'histoire des sciences en Pologne (Warsaw, 1972), 761–62.

Reinhold E. , Theoricae novae planetarum Georgii Purbachii (Wittenberg, 1553), f. 27v. These statements are added in the Wittenberg edition of 1553 following the commentary on ff. 6v.-7r of the Paris edition of the same year. The Wittenberg edition contains extensive new commentaries on the solar theory. Cf. Westman R. S. , “The astronomer's role in the sixteenth century: A preliminary survey”, History of science, xviii (1980), 105–47.

Peucer C. , Elementa doctrinae de circulis coelestibus (Wittenberg, 1551), sig. C 7v. In the preface he gave a list of astronomers ending with Copernicus.

ibid., sig. C 8v.

ibid., sig. T2v.

Bicard Ariel , Quaestiones novae in libellum de Sphaera Ioannis de Sacro Bosco (Paris, 1569), f. 5r.

ibid., f. 66v.

Pseudo-Aristotle, De mundo, 391 b 9–12.

Bicard , op. cit. (ref. 227), f. 10v.

ibid., ff.7r–7v.

Euclides optica et catoptrica…eadem Latine reddita per I. Penam (Paris, 1557), preface, aa iii r.

Frisius Gemma , De radio astronomico et geometrico liber (Antwerp, 1545), f. 29v. Cf. Le ray astronomique, in Apian P. , Cosmographie, corrigée et augmentée par G. Frison avec…autres traités (Antwerp, 1581). On Gemma Frisius, see Waterbolk E. H. , “The ‘reception’ of Copernicus' teachings by Gemma Frisius”, LIAS: Sources and documents relating to the early modern history of ideas, i (1974), 225–42.

Kepler J. , Gesammelte Werke (Munich, 1937 -), iv, 335.

Pena , op. cit. (ref. 232), bb ii v.

ibid., aa iv r. Kepler is rather unfair to Pena when he states that he “timidly dismisses the motion of the Earth which Copernicus proved”. Pena asserted correctly that optics left the question undecided.

Dreyer J. L. E. , Tychonis Brahi Dani opera omnia (Copenhagen, 1913–29), i, 27.

Ibid ., iv, 1–378. Tycho also wrote a German treatise on the comet; ibid., 379–96. There is an English translation in Christianson J. R. , “Tycho Brahe's German treatise on the comet of 1577”, Isis, lxx (1979), 110–40. Cf. Hellman C. D. , The comet of 1577: Its place in the history of astronomy (New York, 1971).

Dreyer , op. cit. (ref. 237), iii, 111. In a letter to Rothman (14January 1595), Tycho refers to Pena's view that air extends to the heavens; ibid., vi, 320. Christianson suggests that Tycho was influenced by Paracelsus. For later opinions on the spheres, see Donahue W. H. , “The solid planetary spheres”, in Westman (ed.), op. cit. (ref. 204), 244–75.

Ramus P. , Scholae in liberales artes (Basel, 1578), 1033.

Burmeister K. H. , Georg Joachim Rhetikus (Wiesbaden, 1967–68), iii, 173–81; Dreyer , op. cit. (ref. 237), vi, 89.

P. Rami Scholarum mathematicarum (Basel, 1569), 49–50 and 66.

Russell J. L. , “Kepler and scientific method”, Vistas in astronomy, xviii (1975), 733–45, p. 745.

Aiton E. J. , “Johannes Kepler and the astronomy without hypotheses”, Japanese studies in the history of science, xiv (1975), 49–71; “Johannes Kepler in the light of recent research”, History of science, xiv (1976), 77–100. See also Jardine N. , “The forging of modern realism: Clavius and Kepler against the sceptics”, Studies in history and philosophy of science, x (1979), 141–73.

Even Maestlin objected to Kepler's reasoning involving physical (efficient) causes. Aiton E. J. , “Johannes Kepler and the Mysterium cosmographicum”, Sudhoffs Archiv, lxi (1977), 173–94, p. 180.