Reflections on the Origin of Copernicus's Cosmology

See Swerdlow Noel M. , “The derivation and first draft of Copernicus's planetary theory: A translation of the Commentariolus with commentary”, Proceedings of the American Philosophical Society , cxvii (1973), 423–512; Goldstein Bernard R. , “Copernicus and the origin of his heliocentric system”, Journal for the history of astronomy , xxxiii (2002), 219–35; and Clutton-Brock Martin , “Copernicus's path to his cosmology: An attempted reconstruction”, Journal for the history of astronomy , xxxvi (2005), 197–216.

As Clutton-Brock , op. cit. (ref. 1), 197, rightly emphasizes.

The tangled story of the manuscripts is told by Rosen Edward in the introduction to his translation of Commentariolus , in Copernicus Nicholas , Minor works (Complete works , iii; Warsaw , 1985), 76–80 and throughout his footnotes. Yet a third copy, designated by Rosen as “A”, is described by Dobrzycki Jerzy , “The Aberdeen copy of Copernicus's Commentariolus”, Journal for the history of astronomy , iv (1973), 124–7. Suffice it to say that the task of reconstruction of the Ur-text has been Herculean. Except for one place where I cite the Latin text, all references to Commentariolus are to Rosen's translation.

Rosen , op. cit. (ref. 3), 175. See also Biskup Marian , Regesta Copernicana (Studia Copernicana , viii; Wroclaw , 1978), 63, no. 91; and Hajdukiewicz Ludwik , Biblioteka Macieja z Miechowa (Wroclaw, 1960), 205 and 218. The date of the entry is 1 May 1514.

Commentariolus 84 and note 110.

This was the view maintained by Birkenmajer Antoni Ludwik : Mikotaj Kopernik (Cracow, 1900), 166. See Biskup , Regesta (ref. 4), 65–66, items 98 and 102. The observation is dated 11 March 1515; note that the second in 1516 was the result of a calculation, not an observation. Swerdlow , “Derivation” (ref. 1), 430, points out that Copernicus could not have derived the motion of the Earth's apsidal line as presented in De revolutionibus until 1522 or 1524, but that is compatible with the weaker claim that around 1515 Copernicus began to doubt his assumption in Commentariolus.

This is the Latin translation of the Syntaxis by Gerard of Cremona: Almagestum seu magnae constructionis liber (Venice, 1515). Copernicus's copy is now catalogued at Uppsala University Library as Copernicana 17. See Czartoryski Pawel , “The library of Copernicus”, Studia Copernicana , xvi (Science and history: Studies in honor of Edward Rosen; Wroclaw, 1978), 355–96, esp. p. 372, no. 17. On Copernicus's annotations in the codex, see Birkenmajer , Mikołj Kopernik (ref. 6), 242–92. See also Gingerich Owen , “Copernicus and the impact of printing”, Copernicus yesterday and today , ed. by Arthur Beer Strand K. Aa. ( Vistas in astronomy , xvii (1975)), 201–20.

See Rosen Edward , “Biography of Copernicus”, in Three Copernican treatises 3rd edn (New York, 1971), 345, where he argues for 1511–13. Swerdlow Even , despite the doubts about an earliest date expressed in 1973 ( op. cit. (ref. 1), 431), seems to have accepted 1510, following Rosen's conclusions about Copernicus's departure from Lidzbark-Warmiński (Heilsberg in German), as the plausible year of composition: Noel Swerdlow and Otto Neugebauer, Mathematical astronomy in Copernicus's De revolutionibus (Berlin, 1984), 6–7. Copernicus made the move to Frombork in the fall of 1510 at the latest. See Biskup , Regesta (ref. 4), 54, no. 65 and note 1.

See Wilson Curtis , “Rheticus, Ravetz, and the ‘necessity’ of Copernicus' innovation”, The Copernican achievement , ed. by Westman Robert S. (Berkeley and Los Angeles, 1975), 17–39, esp. p. 19. Wilson based his conclusion on “findings of Birkenmajer and Rosen”. But in his later introduction to the translation of Commentariolus 80, Rosen says that the treatise was written between the latter half of 1508 and early 1514.

For the text, see Copernicus Nicolaus , Die humanistischen, ökonomischen und medizinischen schriften , ed. by Stefan Kirschner Andreas Kühne , in Gesamtausgabe , v: Opera minora (Berlin, 1999), 5–8, esp. p. 6, lines 31–36: “Huic vir doctus adest, Enee vt fidus achates, / Hoc opus ex greco in verba latina trahens, / Qui celerém lune cursum alternosque meatus / fratris cum profugis tractat et astra globis, / Mirandum omnipotentis opus, rerumque latentes / Causas scit miris querere principijs.” Some, for example Birkenmajer and Jerzy Dobrzycki, interpret Corvinus's remarks as a reference to the heliocentric theory in Commentariolus. See Birkenmajer , op. cit. (ref. 6), 168; Dobrzycki Jerzy , “Notes on Copernicus's early heliocentrism”, Journal for the history of astronomy , xxxii (2001), 223–5, thinks that Copernicus formulated the theory before 1508, but does not commit himself to a date of composition.

See Rosen , “Biography” (ref. 3), 338–9. Aside from Rosen's claim that “brother” refers to the Sun, Corvinus also implies that the stars move. In the German translation, the editors agree with the interpretation of “brother” as “Sun”, p. 9: “… und die wechselnden Läufe des ‘Bruders’ (d. h. der Sonne) sowie die Sterne mit den Planeten behandelt….”.

See Clutton-Brock , op. cit. (ref. 1), 203. To my knowledge, Clutton-Brock is the first in print to suggest a connection between Corvinus's comments and the Capellan arrangement.

See Czartoryski , “Library” (ref. 7), 366, no. 2.

The information on its publication shows that it appeared in November, and allowing for time to reach Cracow, 1493 is the earliest that Copernicus could have purchased it.

The full title with description of the last text reads (fol. 154r): “Tabella Sinus recti, per gradus et singula minuta divisa. Ad tabulas directionum magistri Johannis de regiomonte necessarias cum quibus exemplis, partes eiusdem tabelle multum concordant.”.

Copernicus wrote tables on folios 113v, 136r, and began one note on fol. 169v in addition to the sixteen folios (recto and verso) of the “Uppsala Notebook”.

See Birkenmajer , op. cit. (ref. 6), 57–58 and 192–7, for a description of the codex, and for his dating. Rosińska , “Identyfikacja ‘szkolnych tablic astronomicznych’ kopernikana” [“Identification of Copernicus's school (latitude) tables”], Kwartalnik historii nauki i techniki , xxix (1984), 637–44; “Kwestia ‘krakowskich autografów’ kopernika w kodeksie Copernicana 4 biblioteki uniwersyteckiej w Uppsali” [“The question of Copernicus's Cracow autograph in codex Copernicana 4 at Uppsala University Library”], Kwartalnik historii nauki i techniki , xlvi (2001), 71–94. In these articles she reports that Copernicus relied on Johannes Bianchini for corrections in his latitude tables, and also dates them to Copernicus's Cracow period.

This is the folio that Birken majer calls “D” and Swerdlow ( op. cit. (ref. 1)) calls “U”. Swerdlow's Plate 1, p. 428 has a photograph of U, and on p. 429 he provides a transcription.

As we have indicated, following Rosen, the manuscripts of Commentariolus are faulty, but there is sufficient agreement between the numbers in the treatise and the bottom half of U to use the numbers in U to correct the faulty numbers in Commentariolus. See Rosen's translation, 84–90 and especially all of the footnotes related to the numbers. In De revolutionibus, Copernicus cites numbers almost identical to the numbers in the upper part of U precisely to emphasize his agreement with Ptolemy's conclusions. Compare De revolutionibus , v. 9 ( Rosen translation, 254, line 13), where Saturn's semidiameter is 1090^p. For Jupiter (v. 14, p. 262, 1.27) the semidiameter is 1916; for Mars (v. 19, p. 269, line 24) it is 6583 ; for Venus (v. 21, p. 271, 1. 44) it is 7193; and for Mercury (v. 27, p. 282, 1. 35) it is 3763, all normed to 10,000.

I must point out one error in Clutton-Brock's reconstruction. Line 6 of U reads as follows. In the lefthand margin appears the number 376, and to the right of that number: Mercurii ecce<ntricitas> 225<6> Epi<cyclus> a cu<m> b. [10] 6 …/ 100 Clutton-Brock reads the letters between ‘a’ and ‘b’ as an ‘m’. In fact the letters are ‘cu’ with a horizontal stroke over them, clearly an abbreviation for the word “cum”. For a description of Copernicus's handwriting, see Goddu André , “Copernicus's annotations: Revisions of Czartoryski's ‘Copernicana'”, Scriptorium , lviii (2004), 202–26, esp. pp. 210–11 for the formation of the letter c, and pp. 213–14 for his letter u.

The upper part of U lacks an entry for Venus, but it can be supplied by the calculation from the lower part where Venus does appear. The number for Mercury is 2256, based on a scale of 6,000, as Swerdlow, op. cit. (ref. 1), 505, explains. When adusted to 10,000, the number is 3760. In the margin, Copernicus wrote “376”, clearly based on a scale of 1,000.

Actually, Copernicus says “38 fere”, that is, nearly 38; the precise number is 37;58;35, as a simple computation confirms: 25 × 10,000 / 6583 = 37.976 = 37;58;35.

Wilson Curtis , “Rheticus, Ravetz” (ref. 9), 18–25, has effectively refuted Ravetz's J. E. account in Astronomy and astrology in the achievement of Nicolaus Copernicus (Warsaw, 1965); hence, I do not consider it here. See also Gingerich Owen , “‘Crisis’ versus aesthetic in the Copernican revolution”, Vistas in astronomy , xvii (1975), 85–93, esp. p. 90, note 19.

See Epytoma Joannis de monte regio In Almagestum Ptolomei , facsimile reprint of Venice, 1496 edition (Hain 13806), in Regiomontanus Joannes , Opera collectanea, Milliara , x. 2, ed. by Felix Schmeidler (Osnabrück, 1972).

See Swerdlow , “Derivation” (ref. 1), 426–9 and 471–8.

Goldstein , op. cit. (ref. 1), 220.

Goldstein, op. cit. (ref. 1), 221–2.

Goldstein, op. cit. (ref. 1), 220. On p. 221, he uses the expression “initial commitment” to a heliocentric system. I do not ignore Clutton-Brock's account, but clarifying the order of the steps in his reconstruction is complicated. In fairness to him, I should emphasize that while he follows Swerdlow's analysis of U, he has recognized the need for a number of intermediate steps and consideration of external conditions.

Of course, there are disagreements about dating the earliest parts of De revolutionibus. The Preface was composed in 1542, but Book I is another matter. On the basis of Copernicus's paper watermarks, in part, Rosen thought 1515 likely, but most others think that the mid-1520s are likelier. Watermarks can provide a terminus post quem, but determining when someone stops using the paper is conjectural. Copernicus may have begun some composition before 1520, but his administrative and military obligations between 1515 and 1522 were very disruptive. In 1519 he wrote his short treatise on monetary reform. A cease-fire was negotiated with the Teutonic Knights in 1521, and the Treaty of Cracow (1525) settled the war. In the period 1522–24 Copernicus appears to have returned to work in a more intensive way on astronomical matters, leading to the writing of the Letter to Werner. He probably began the writing of Book I in the period of the mid- to late 1520s.

See Wilson , “Rheticus, Ravetz” (ref. 9), 38–39, for Copernicus's acceptance of the principle as a necessary axiom of the astronomical art.

I thus reject Barker's Peter Clutton-Brock's explanation as possible but superfluous. Copernicus's reticence on numerous topics is evidence of his caution on some matters and of his reluctance to take positions on matters not essential to his project. See Barker, “Copernicus, the orbs, and the equant”, Synthèse , lxxxiii (1990), 317–23. On the issue of spheres and models, Copernicus sometimes indicates that he is uncertain which of several alternative models is true. Yes, Copernicus was a realist about cosmological hypotheses and the large spheres that move the planets, but he was also pragmatic about geometrical hypotheses and mechanisms. See De revolutionibus , iii. 25; v. 4 and 16, and Swerdlow Neugebauer , op. cit. (ref. 8), 159–61. Copernicus devoted his efforts to trying to explain the observations to the best of his ability. Commentators have been driven by the internal logic of their own arguments to attribute global or universal conclusions to Copernicus on matters about which he was undecided or uncertain. In some cases his reticence is a reaction to lack of consensus among natural philosophers. As Clutton-Brock rightly emphasizes, there were competing views about the reality of epicycles. Copernicus's reticence suggests that the issue was not crucial to his project.

It seems that Copernicus did not regard Maragha-like devices as a violation of the axiom, perhaps because he believed that his circular models moved uniformly around their proper centres. Notice too how Copernicus minimizes the departure from a perfect circle in the general model for the superior planets in De revolutionibus , v. 4, as “imperceptibly noncircular”.

See Gingerich , “Crisis” (ref. 23), 89–91, for emphasis on the “fixed symmetry” of the parts in Copernicus's system and on complex sociocultural structures. Birkenmajer , op. cit. (ref. 6), 187–90, also concluded that the problem with the equant led Copernicus gradually to other doubts, the first of which in U and in Commentariolus was the attempt, with exception of the Earth, to eliminate eccentrics. But nowhere does Birkenmajer so much as suggest that Copernicus puzzled over geo-heliocentric and heliostatic heliocentric conversions.

These are expressed in Commentariolus 81, and De revolutionibus , i. 4.

See Regiomontanus , Epytoma , v. 22, pp. 144–5 of the facsimile; Copernicus , Commentariolus 85, and De revolutionibus , iv. 2.

Copernicus emphasizes this inconsistency in De revolutionibus , i. 10, and he implies it in Commentariolus 82, where he says that consideration of the lengths of the spheres' radii is sufficient to grasp the advantages in his arrangement. On Goldstein's discussion of the principle of planetary order, we should add that the interpretation of Aristotle's De caelo , ii. 10 (that the farthest from the Earth are the slowest and the closest are the fastest) was commonplace at Cracow in the 1490s. I cite, for example, Versoris Johannes , Quaestiones de caelo et mundo (Cologne, 1488), Biblioteca Jagiellonica, Inc. 596, fol. 21rb: “Secunda pars patet quia ut dictum est celum stellatum motu proprio solum facit unam circulationem in triginta sex millibus annorum; Saturnus in triginta annis; Jupiter in duodecim annis; Mars in duobus annis; Sol, Venus, et Mercurius quasi uniformiter in uno anno; et luna in una mense. Causam vero utriusque partis conclusionis assignat Aristoteles, quia quanto aliquis orbis est propinquior supremo tanto supremus magis prevalet in ipso, et ideo velocius movet ipsum motu diurno.” In discussing the speed of the planetary spheres, Versor read De caelo as ordering the spheres relative to the centre, that is, the Earth, according to their periods around the Earth. Following Aristotle, however, he referred to distance from the sphere of the stars for the dynamic explanation. See Markowski Mieczysłw , “Kształtowanie się krakowskiej szkoły astronomicznej” [“Formation of the Cracow School of Astronomy”], Historia astronomii w Polsce [History of astronomy in Poland ], i, ed. by Eugeniusz Rybka (Wrocłw, 1975), 70, for the influence of Versor's text of Aristotle and his commentary. In short, the principle seems to have been a commonplace by the late fifteenth century. Copernicus very likely took the principle for granted, and undoubtedly noticed that it did not work for Mercury and Venus. In fact, Versor himself noted it as he went on to discuss the problem of ordering Mercury and Venus above or below the Sun. Finally, we may mention Copernicus's friendship with Giese Tiedemann . Giese wrote marginal comments on the works of Aristotle in editions that contained the commentaries by Versoris Johannes . Copernicus Giese may have met as early as 1501 and very likely were in contact between 1503 and 1510 in Lidzbark. See Pociecha Włdysłw , “Giese Tiedeman Bartłomiej”, Polski słownik biograficzny (Cracow, 1959), vii, 454–6.

Copernicus probably read Ficino's translation of Timaeus (Florence, 1484) in the episcopal library at Lidzbark-Warmiński. See Czartoryski , “The library” (ref. 7), 382. As Rosen , transl. of Commentariolus, 126, n. 327, indicates, Copernicus knew the treatise by Martianus Capella before he wrote Commentariolus.

Copernicus emphasizes these questions in De revolutionibus , i. 10. He does not refer to the variations in distance of Venus from the Earth, which in the Ptolemaic model results from the size of the epicycle, not from observation. Ironically, Andreas Osiander in the “Letter to the Reader” does emphasize variation in distance as a consequence of Ptolemy's model for the purpose of rejecting models as either true or probable. I owe emphasis on the details about Venus to Owen Gingerich.

Rosen transl., 82.

Rosen transl., 11, the concluding paragraph of chap. 4.

I leave aside his commitment to the Earth's axial rotation. His arguments in De revolutionibus , i. 5–9 suggest that he adopted axial rotation first, but the order of that argument has a clearly rhetorical motive, namely to coax readers into considering other motions of the Earth.

De revolutionibus , i. 10, Opera omnia , ii (Warsaw and Cracow, 1975), 21, *II. 10–12: “Inuenimus igitur sub hac ordinatione admirandam mundi symmetriam ac certum harmoniae nexum motus et magnitudinis orbium, qualis alio modo reperiri non potest.”.

De revolutionibus, Opera omnia , ii, for example, pp. 278 and 290, and consult the index for each planet under “eccentrotes”.

I cite the edition by Leopold Prowe, amending the incorrect numbers that go back to Tycho Brahe's manuscript and that Prowe or Curtze misreported. See Rosen's transl., 107, nn. 166–8. See Prowe Leopold , Nicolaus Coppernicus , ii (Berlin, 1884), 194–6.

In “Derivation” (ref. 1), 465–6, Swerdlow translates the relevant passages from Commentariolus thus: “For where the semidiameter of the great sphere is given 25 parts, the semidiameter of the sphere of Mars will receive 38 parts, … I call the semidiameter the distance from the center of the sphere to the center of the first epicycle…. The sizes of the epicycles are as follows: In the case of Saturn the semidiameter of the first is 19;41 parts where the semidiameter of the great sphere was assumed to be 25 parts, and the second epicycle has a semidiameter of 6;34 parts.” Rosen refused to accept the suggestion that Copernicus related the semidiameter of an orb to the semidiameter of an epicycle via Epitome , xii. 1–2. See his comments in his translation of Commentariolus , n. 169.

See Birkenmajer , op. cit. (ref. 6), chap. 7, Appendix 2, pp. 202–10. Compare Swerdlow, “Derivation” (ref. 1), 471–80. Copernicus shifted from variable semidiameters in the upper part to a constant semidiameter in the lower part. Clutton-Brock also notes that Copernicus's models of the inferior planets in De revolutionibus retain “moving eccentrics”. In fact, it remained one of his general alternatives. See De revolutionibus , iii. 20–25; v. 4, 23, and 25–30. The transition from variable semidiameters in U, upper part to a constant semidiameter in U, lower part could be explained as a move that Copernicus regarded as a simplification.

See Swerdlow , “Derivation” (ref. 1), 471–80. But Birkenmajer came very close to the solution, for he recognized that the upper part of U replaces an epicycle on a concentric with an eccentric that is equal to the radius of the concentric's epicycle. For Copernicus, however, the “eccentric” or “semidiameter” is the distance of the Earth from the mean Sun. See Birkenmajer , op. cit. (ref. 6), 203–4, esp. p. 203: “The heliocentric system in De revolutionibus is eccentro-epicyclic, and cannot be directly compared with either Commentariolus or U. Nevertheless, without any numerical changes in De revolutionibus, we can replace an eccentric with an equivalent concentric with epicycle, the radius of which equals the eccentric, without violating in any way the kinematic foundations of the system. In that case, all three mechanisms would be qualitatively equivalent, allowing for a direct comparison of the quantitative data in De revolutionibus with the data in Commentariolus and U.” Birkenmajer did not see the relevance of Epitome , xii. 1–2 to his suggestion. Incidentally, the working translation, although selective, of Birkenmajer's study that Owen Gingerich and Jerzy Dobrzycki supervised is very helpful. It is available on microfilm: “Nicholas Copernicus: Studies on the works of Copernicus and biographical materials”, Ann Arbor, Michigan, University Microfilms, 1976. I have attempted a less literal translation that preserves the sense of the original.

Swerdlow , “Derivation” (ref. 1). 459 and 505, refers to column numbers (c) from the Alfonsine Tables. In his bibliography Swerdlow cites a 1518 edition of the Alfonsine Tables and a 1587 edition of Regiomontanus's Tabula, not the editions, of course, that Copernicus used. Whatever his source, Swerdlow's reconstruction is remarkable.

Birkenmajer , op. cit. (ref. 6), 64, surmised that Copernicus realized that he could not fit the note on fol. 269v. In fact, however, he did fit it onto one folio, but perhaps he thought at first that he might need more than one folio. At that point both fol. 15r and 15v were empty. In any case, he resumed writing on fol. 15v, that is, U. Dating U is still a matter for conjecture. Fol. 15r records a date of 1532. Fol. 16r is troublesome for other reasons. Swerdlow , “Derivation” (ref. 1), 426, following Prowe, Coppernicus (ref. 44), ii, 206–44, evidently assumed that fol. 16r was blank. In fact, there is a table on that folio as well. Birkenmajer , op. cit. (ref. 6), 190, described the table, but drew no conclusions about its purpose. There are doubts about the authenticity of the handwriting, and even if they are Copernican, the writing of the number 4 (distinctively gothic in style) suggests a date earlier than 1500. Birkenmajer , op. cit. (ref. 6), 169, referring to earlier Copernican tables in the codex, noted differences between Copernicus's earlier style of writing numbers as opposed to his later style. Fol. 16v can be dated 1500. For all of these reasons, dating U is difficult. The style of the numbers shows that it is probably post-1499, but I cannot be more precise at this time.

Regiomontanus's propositions interpret Ptolemy's Almagest , xii. [1] and [2], transl. by Toomer Gerald (Princeton, 1984), 554–5. Copernicus did make corrections and additions to the tables in Copernicana 4, which may account for some problems in his calculations. From his copies of the Tables of Sines and the Alfonsine Tables in Copernicana 4, Copernicus used the following numbers. For Venus, he had a value of 7190 (a number rounded off to 7200). 7190 × 6 = 43140. On f. 262r of the Table of Sines, 43140 corresponds approximately to 45”58′, and on f. 73r of the Alfonsine Tables, we find the equation of argument given as 45$59′, a very close match. For Mercury, Copernicus gives us 2256<0>, coresponding approximately to 22$5′ (f. 257r), corresponding closely to 22$2′, which Copernicus underlined in his copy (f. 76v). For Mars we must once again make a calculation, for Copernicus provides a number normed to 10,000, namely 6583. 6583 × 6 = 39498, corresponding to 41$10′ (f. 260v), exactly the number for the mean position of the epicycle in the equation of argument on f. 81r. On f. 80r, Copernicus wrote “minus parte 38” (misread by Birkenmajer as “32”) at the top, but this correction seems to have affected only the numbers in the second column, the equation of centre on f. 80v. For Jupiter, Copernicus calculated 1917. 1917 × 6 = 11502, corresponding approximately to 11$3′ (f. 255v), exactly the equation of argument for Jupiter (f. 84v). Finally, for Saturn Copernicus calculated 1083. 1083 × 6 = 6498, very close to 6497, the sine value for 6$13′ (f. 255v), corresponding exactly to the equation of argument for Saturn (f. 88v).

See Swerdlow , “Derivation” (ref. 1), 472. Although he adopts the main steps in Swerdlow's reconstruction, Clutton-Brock has tried to fill in the gaps in Swerdlow's account, for he recognizes that additional intermediate steps are necessary. Still, what neither Swerdlow nor his supporters have explained satisfactorily is why Copernicus should have seen in Epitome , xii. 1–2 possibilities that completely eluded Regiomontanus, who continued to defend geocentrism. Because Copernicus had already arrived at his theory, he was able to interpret those propositions in a way that evidently never occurred to Regiomontanus. Swerdlow's appeal to a Kepler-like objection as an explanation (“Derivation” (ref. 1), 472, note 6) of Regiomontanus's refusal to believe the heliocentric conversion is unpersuasive. As a planet moves on an epicycle, it is either moving in a circle around an empty point or around a point on the surface of a sphere, not around a body. It was partly Kepler's rejection of the existence of spheres that led him to seek real bodies as partial causes of motion.

Birkenmajer , op. cit. (ref. 6), 110, has shown that the Swedes took about two thousand books from Varmian libraries. We do not know what happened to Copernicus's copy of Epitome. The copy of the same edition at Uppsala University Library, Inc. 34: 33, is not of Varmian provenance, and is absent any annotations, although other works bound with it in the codex are annotated.

It may not be superfluous to point out that the single-epicycle, eccentric or eccentreccentric models on which Copernicus finally settled in De revolutionibus reduced the sizes of epicycles to roughly the size of the small epicycle in the double-epicycle model. Perhaps that was another reason for him to prefer the later solution, although he always recognized the kinematic equivalence between all of the solutions. It seems that Copernicus saw no compelling reason to consider one of these as true to the exclusion of the others. He would go no further than to say about alternative arrangements, believing that he had exhausted the alternatives, that one of them must be true. See De revolutionibus , iii. 20.

As Evans James , The history and practice of ancient astronomy (Oxford, 1998), 425–7, emphasizes with respect to the question, why 1543? Copernicus understood Ptolemy more deeply than any of his precedecessors. Again, see Gingerich, “Crisis” (ref. 23), 90–91, where he places great emphasis on complex sociocultural structures.

I agree with Clutton-Brock's turning to such issues in the concluding section of his article. In addition to intellectual contexts, Gingerich, “Crisis” (ref. 23), 91, reminds us of several other major changes. Columbus's voyage in 1492 while Copernicus was a student at Cracow and other explorations required revision of Ptolemy's geography and vision of the Earth. The Reformation challenged the authority of the Church, and served as a motive for political ambitions and reforms.

As Eisenstein Elizabeth , The printing press as an agent of change (2 vols, Cambridge, 1979), ii, 603, and Gingerich, “Copernicus and the impact of printing” (ref. 7) emphasize.

See Czartoryski , “Library” (ref 7), and compare with Goddu, “Copernicus's annotations” (ref. 20).

See di Bono Mario , “Copernicus, Amico, Fracastoro and Tusi's device: Observations on the use and transmission of a model”, Journal for the history of astronomy , xxvi (1995), 133–54, for a critical evaluation of the usual scenario.

Ptolemy rounds the ratios in the Almagest off to 7: 1 for Mars, and 104: 16 for Venus, as compared with 88: 34 for Mercury, 37: 23 for Jupiter, and 7: 5 for Saturn. The same gaps are reflected, of course, in the cosmological distance scale in Planetary hypotheses, the results of which were available in the theorica literature known to Copernicus and in the Epitome.