Albert of Brudzewo's Little Commentary on George Peurbach's ‘Theoricae Novae Planetarum’

On Brudzewo see Brożek Z. P. , “Wojciech of Brudzewo”, in Markowski M. , The Cracow circle of Nicholas Copernicus, Copernicana Cracoviensa III (Cracow, 1973), 61–75; Goddu André , Copernicus and the Aristotelian tradition: Education, reading and philosophy in Copernicus's path to heliocentrism (Leiden, 2010), 36–7, 162–6.

de Brudzewo Albertus , Commentariolum super theoricas novas planetarum Georgii Purbachii in studio generali Cracoviensi per Mag. Albertum de Brudzewo diligenter corrugatum A.D. MCCCCLXXXII. Edited on the basis of the 1495 Milan edition and the codex, by L. A. Birkenmajer (Cracow, 1900). Birkenmajer also lists five manuscript copies, ibid., p. xlv. Birkenmajer's edition is now conveniently available through Google Books. Subsequent citations are to this edition. For clarity, I will refer to this book as the Little commentary in the text of the present paper, to help the reader distinguish Brudzewo's Commentariolum from Copernicus's similarly named Commentariolus.

Goddu , Copernicus and the Aristotelian tradition (ref. 1), 156, 159 ff.

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; Rosen Edward , “Copernicus' spheres and epicycles”, Archives internationales d'histoire des sciences, xxv (1975), 82–92; Swerdlow Noel M. , “Pseudodoxia Copernicana”, Archives internationales d'histoire des sciences, xxvi (1976), 108–58; Rosen Edward , “Reply to Swerdlow N. “, Archives internationales d'histoire des sciences, xxvi (1976), 301–4; Rosen Edward , Copernicus, Complete works (3 vols, London, 1973), iii, 123, n. 326.

See below ref. 55 and subsequent text. Other writers who note these passages and reach similar conclusions include Jardine Nicholas , “The significance of the Copernican orbs”, Journal for the history of astronomy , xiii (1982) 168–94, esp. 171ff, Clutton-Brock Martin , “Copernicus's path to his cosmology: An attempted reconstruction”, Journal for the history of astronomy, xxxvi (2005), 197–216, p. 211, who cites Rosen's reading of Brudzewo , and Lerner Michel , Le monde des sphères, 2nd edn (2 vols, Paris, 2008), i, 319, n. 83 and ii, 4, although he cautions (i, 130) that the subject has not been systematically studied.

Goddu , Copernicus and the Aristotelian tradition (ref. 1), 148, 157–8, 158 n. 56, 162–7, 370–80, esp. 376 n. 36.

Goddu , Copernicus and the Aristotelian tradition (ref. 1), 158.

Goddu , Copernicus and the Aristotelian tradition (ref. 1), 254–5, 381–4, prefers a version of the route to heliocentrism proposed by Goldstein Bernard R. , “Copernicus and the origin of his heliocentric system”, Journal for the history of astronomy, xxxiii (2002), 219–35.

Glogow's views are clearly presented by Jardine , “Significance of the Copernican orbs” (ref. 5), esp. 193, n. 95.

Barker Peter , “The reality of Peurbach's orbs”, in Boner Patrick J. , (ed.), Change and continuity in early modern cosmology (New York, 2011), 7–32.

Thorndike Lynn , The Sphere of Sacrobosco and its commentators (Chicago, 1949), 42–5; Lattis James M. , Between Copernicus and Galileo: Christoph Clavius and the collapse of Ptolemaic cosmology (Chicago, 1994), 38–45; Lindberg David C. , The beginnings of Western science: The European scientific tradition in philosophical, religious, and institutional context, prehistory to A.D. 1450, 2nd edn (Chicago, 2008), 265–70.

Sarnowsky Jürgen , “The defence of the Ptolemaic system in late mediaeval commentaries on Joannes de Sacrobsoco's De sphaera”, in Bucciantini Massimo Camerota Michele Roux Sophie (eds), Mechanics and cosmology in the medieval and early modern period (Florence, 2007), 29–44.

Pedersen Compare Olaf , “The Theory of the Planets”, in Grant Edward (ed.), A source book in medieval science (Cambridge, MA, 1974), 451–65, with Benjamin Francis S. Toomer Gerald J. , Campanus of Novara and medieval planetary theory: Theorica planetarum (Madison, WI, 1971). See also Pedersen Olaf , “The origins of the Theorica planetarum”, Journal for the history of astronomy, xii (1981), 113–23, and Pedersen Olaf , “The decline and fall of the Theorica planetarum: Renaissance astronomy and the art of printing”, Studia Copernicana, xvi (1978), 157–85.

In the present paper I will retain the contemporary term theorica (pl. theoricae) as far as possible, for a variety of uses. A theorica (small ‘t’) is a model offering a basis for calculating planetary positions against the fixed stars, expressed as angles from a fixed line of reference. The theoricae for individual planets may be quite separate; the appearance of particular features in one should lead to no expectation that a similar feature will appear in adjacent planets, or as a universal feature of the theoricae of all planets. Second, a Theorica (capital ‘T’) is a book presenting theoricae. The most important instances are the anonymous Theorica planetarum often attributed to Gerard of Cremona, and Georg Peurbach's Theoricae novae planetarum, discussed below. On the term ‘theorica’ and the contrasting term ‘practica’, see now Westman Robert S. , The Copernican question: Prognostication, scepticism and celestial order (Berkeley, 2011), 40–3.

For an extended discussion of the relationship between astrology and astronomy in the early modern period, see Westman , Copernican question (ref. 14.).

Peurbach Georg , Theoricae novae planetarum de Georgii Purbachii (Nuremberg, 1474). The book appears to have been printed at some point between the years 1472 and 1474; Zinner E. , Regiomontanus: His life and work, translated by Brown E. (Amsterdam, 1990), 22. For an English translation see Aiton Eric J. , “Peurbach's Theoricae novae planetarum: A translation with commentary”, Osiris, iii (1987), 5–43. A first edition at the Library of the University of Vienna's Institute for Astronomy is available at (www.univie.ac.at/hwastro/books/theoColMed.pdf). On the concept of a partial orb, see below, text to ref. 25 and ref. 43, and Figure 1. In general see Barker , “Reality of Peurbach's orbs” (ref. 10).

Grant Edward , Planets, stars and orbs (Cambridge, 1996), 278ff; Lerner, Monde des sphères (ref. 5), i, 115.

de Virduno Bernardus , Tractatus super totam astrologiam , ed. by Hartmann Polycarp ( Werl , Westphalia, 1961). For an argument — Admitted by the author to be less than conclusive — That dates the Tractatus before 1284, see: Poulle Emmanuel , “Bernard de Verdun et le turquet”, Isis, lv (1964), 200–8.

As Lerner perhaps implies when noting Duhem's claim that Bernard had many supporters in Paris at the end of the thirteenth century and into the fourteenth century. Lerner , Monde des sphères (ref. 5), i, 118, and 311, n. 44.

Kren Claudia , “Homocentric astronomy in the Latin West: The De reprobatione ecentricorum et epiciclorum of Henry of Hesse”, Isis , lix (1968), 269–81. Grant Edward , “Eccentrics and epicycles in medieval cosmology”, in Mathematics and its applications to science and natural philosophy in the Middle Ages, ed. by Grant Edward Murdoch John E. (Cambridge, 1987), 129–214.

Grant , Planets, stars and orbs (ref. 17), 281–3.

Pedersen Olaf Larsen B. Dalsgaard , A fifteenth century planetary theory: Nova theorica planetarum magistri Johannis Lauratii … de Fundis ( Aarhus , 1961). A transcription of MS Utrecht 3 H 15 fol. 56 r–63 r.

A particularly important example is Ciruellus Petrus , Uberrimum sphere mundi come[n]tu[m] intersertis etia[m] questionibus d[omi]ni Petri de Aliaco (Paris, 1498). For a discussion see Barker, “Reality of Peurbach's orbs” (ref. 10), 15–16.

Glogoviensis Ioannes , Introductorium compendiosum in tractatum sphere materialis magistri Ioannis de Sacrobusto, quem abbreviavit et Almagesti sapientis Ptolemei Claudii philosophi alexandrini et Pheludio progeniti per magistrum Joannem Glogoviensem feliciter recollectum. As in the case of Brudzewo's book, which is my main focus, Glogow's work was not printed until after Copernicus left Cracow. Astronomy students would have had ready access to manuscript copies. Thorndike L. , History of magic and experimental science (8 vols, New York, 1923–58), iv, 450, n. 46 gives the first printed edition as: Cracow Joh. Haller , 28 April 1506. Zwiercan notes a second Cracow edition in 1513: Zwiercan M. , “Jan of Glogow”, in Markowski , The Cracow circle (ref. 1), 95–110, p. 106. I have consulted the edition published by Ioannes Knoblauch: Strasbourg, 1518; subsequently referred to as John of Glogow, Introductorium. See also Jardine , “Significance of the Copernican orbs” (ref. 5), esp. 193, n. 95.

Glogow , Introductorium (ref. 24), k ii R.

Glogow , Introductorium (ref. 24), k ii R—V: Tertius vero est eccentricus … et ille defert corpus solis: Et ad motu[um] eius movetur corpus solis q[uod] est ei infixum.

Glogow , Introductorium (ref. 24), k iiii R: Quilibet trium superiorum tres orbes habet reales a se divisos secundum imaginationem trium orbium solis.

Glogow , Introductorium (ref. 24), k iiii R: … ergo in celo est realis orbis in theoricis que manuducunt nos in cognitionem celestium….

Glogow , Introductorium (ref. 24), k iiii R: … theoriste orbes istas planetarum qui sunt reales orbes spissitudinem in earum substantia habentes vocant circulos: Cum tunc secundum veritatem non sunt circuli. (Reading “theoriste” here as a reference to those who construct theoricae, and in context, those earlier theoricae that used only circles rather than orbs.).

Glogow , Introductorium (ref. 24), k iii V: Equans est circulus imaginarius cuius imaginatio ab astronomis sic est inventa q[uem] a eadem planete non equaliter moventur super super [sic] centro mundi, nec semper moventur equaliter super centro deferentium suorum….

Glogow , Introductorium (ref. 24), k iiii R: Notandum pro intellectu textus q[uod] caput et cauda draconis non est stella nec pars celi realis. Cf. Evans James , History and practice of ancient astronomy (Oxford, 1998), 316.

Sabra A. I. , “The Andalusian revolt against Ptolemaic astronomy: Averroes and al-Biṭrūjī”, in Mendelsohn E. (ed.), Transformation and tradition in the sciences (Cambridge, 1984), 133–53.

Averroes's objections to Ptolemaic astronomy appear in three main sources: His commentary on Aristotle's Metaphysics, his commentary on Aristotle's On the heavens, and a separate work that is also an exposition of the Metaphysics, usually known as the Epitome. All three were widely available during the Middle Ages and Renaissance. Modern editions are: Genequand Charles , Ibn Rushd's Metaphysics: A translation with introduction of Ibn Rushd's Commentary on Aristotle's Metaphysics, Book Lam (Leiden, 1986); Carmody Francis J. , Averrois Cordubensis commentum magnum super libro De celo et mundo Aristotelis, ed. by Endres Gerhard (Leuven, 2003); and Arnzen Rüdiger , Averroes on Aristotle's ‘Metaphysics’: An annotated translation of the so-called ‘Epitome’ (Berlin, 2010).

Averroes , De caelo 2.6 comm. 35 (288a 14); cf. Carmody Francis J. , “The planetary theory of Ibn Rushd”, Osiris, x (1952), 556–86, p. 572.

Evans , History and practice of ancient astronomy (ref. 31), 305–12.

Averroes , Metaphysica 12.8 comm. 44–5, cf. Carmody , “Planetary theory of Ibn Rushd” (ref. 34), 567.

Genequand , Ibn Rushd's Metaphysics (ref. 33), 176.

As indicated above, text to refs 17–22. On the reception of Averroes's criticisms see Carmody, “Planetary theory of Ibn Rushd” (ref. 34); Kren , “Homocentric astronomy in the Latin West” (ref. 20); Grant , “Eccentrics and epicycles in medieval cosmology” (ref. 20). A recent summary appears in Lindberg, Beginnings of Western science (ref. 11), 254–62.

Averroes , De caelo 2.6 comm. 35 (288a 14); cf. Carmody , “Planetary theory of Ibn Rushd” (ref. 34), 570.

Glogow's Questiones de motu discusses Averroes's criticism of eccentrics and epicycles at length according to Zwiercan, “Jan of Glogow” (ref. 24), 107–8.

Glogow , Introductorium (ref. 24), k iiii V: Un[de] et magister Thadeus de parma (ut scribit magister Joannes Danckonis in theoricis suis) tenet q[uam] luna habet spheralem motum pr[a]eter motum epiciculi ad salvandam apparentiam macul[a]e lun[a]e.

Brudzewo , Commentariolum (ref. 2), 5–6: [S]icut arguit Commentator, quod non sit ponenda nona sphaera super octavam, quia si poneretur, frustra erit: Nam non influeret aliquid istis inferioribus. Orbibus enim omnen influentiam, quam habet, a stella habet, seu ratione stellae vel stellarum existentium in ipso. Cum ergo in nona sphaera nulla sit stella, per consequens non influit, aliquid istis inferioribus, ergo erit frustra. Item in contrarium videtur, quod sint multo plures quam octo aut novem…. Item pro ipso Sole assignantur tres orbes et in reliquis planetis plures quam tres, ut patet per Theoricas. In determinatione istius dubitationis etsi plures possent aduci philosophorum et astronomorum ac docotorum katholicorum varietates et determinationes secundum quas tractant de numero sphaerarum coelestium mobilium, sed cum non sit praesentis intentionis tantam varietatem pertractare, veritatemque examinare probabiliorem, cuius sit adducuere dumtaxat ea quae sunt pro faciliora intelligentia conformationeque eorum probabili, quae dicuntur in his Theoricis novis Georgii Peurbachii.

Brudzewo does not identify the doctori katholici, but an obvious candidate is Henry of Langenstein, who discussed the order of the heavens in his Lecturae super Genesim, a work he began at Vienna in 1385. See Steneck Nicholas H. , Science and creation in the Middle Ages: Henry of Langenstein (d. 1397) on Genesis (Notre Dame, IN, 1976).

Brudzewo , Commentariolum (ref. 2), 6–7: Sphaera sive orbis dicitur unus tripliciter. Uno modo, quia est pars coeli sphaerica, non separata a toto, nec suppositaliter in se existens; illo modo stella dicitur una sphaera et sic essent tot sphaera seu orbes quot ipsae stellae. Secundo, sphaera vel orbis dicitur unus quilibet ille, qui est suppositaliter existens, sive sit concentricus mundo sive non; et illo modo accipitur orbis cum dicitur Sol habet tres orbes. … Tertio modo accipitur orbis pro orbe concentrico mundo, vel pro aggregato ex omnibus orbibus, qui requiruntur et sufficiunt ad salvandum motum unius planetae tam secundum longitudinem, quam secundum latitudinem.

Brudzewo , Commentariolum (ref. 2), 7: Item notandum pro aliquibus suppositionibus, quas Aritosteles in philosophia probat esse veras. Prima. Coelum est corpus simplex (primo coeli). — Secunda. Cuiuslibet corporis simplicis non est nisi unus motus simplex secundum naturam propriam (primo coeli). — Tertia. Motus conveniens alicui corpori praeter naturam propriam necessario convenit alteri secundum naturam propriam (primo et secundo coeli). Quarta. Unus orbis non movetur pluribus motibus ab eadem intelligentia, nec idem orbis movetur a pluribus intelligentiis sibi aeque primo appropriatis (istud satis ostendit Aristotelis XXII^ma Metaphysicae). — Quinta potest addi: Quod sphaera inferior non influit motuum suum superiori, sed potius e converso superior inferiori.

Grant , Planets, stars and orbs (ref. 17), 514–68.

Brudzewo , Commentariolum (ref. 2), 9: Ergo a superioribus duo sibi convenient; tertius vero qui est latitudinis, sibi erit proprius, qui est motus accessionis et recessionis capitis Arietis et Librae ab Alberto seu trepidationis dicitur ab Alphonso et eum sequentibus. Sic ergo octavae sphaereae unus motus de his tribus erit proprius. Similiter nonae sphaerae unus de reliquis duabus erit proprius, scilicet ab occidente in oriens; alter vero sibi conveniet praeter naturam propriam, scilicet diurnus. His ergo solus finaliter ipsi primo mobili est attribuendus; ergo sphaerae mobiles erunt decem.

Brudzewo , Commentariolum (ref. 2), 18: Philosophus enim naturalis considerat motum coelestem totius sphaerae et totius coeli prout est unus in omnibus, scilicet diurnum ipsum comparando in tarditate et velocitate iuxta extensionem magnitudinum, in quibus est, et sic dicit: Lunam tardius moveri quam Saturnum eo, quod orbis Lunae minor sit quam Saturni, utroque in eodem tempore faciente revolutionem, scilicet diurnam. Astronomus autem, non tantum totius coeli motum considerat et totius sphaerae, sed etiam cuiuslibet coeli et orbis, tam totalem quam partialem motum seorsum tractat, et hoc quoad revolutionem secundum motum proprium, ex qua arguit tarditatem velocitatemque uniuscuiusque. Ideo dicit Lunam velocissime moveri, quia citius proprio motu circumgyrat suum circulum quam aliquis planetarum, quamquam Luna vadens 13 gradibus, minus (philosophice) movetur quam Sol uno gradu, quia unus gradus sphaerae Solis valet fere duobus de viginti Lunae, sicut demonstrat Ptolomeus tertio Almagesti.

Brudzewo , Commentariolum (ref. 2), 19: Ibi enim Sol non in circulo, qui est figura plana unica superficie contenta, sed in orbe, qui est corpus solidum et sphaericum, in rei veritate movetur.

Brudzewo , Commentariolum (ref. 2), 22: Theorica Solis principali divisone dividitur in tres partes. In prima parte ponit Magister [Peurbachius] divisionem totius sphaerae solaris in orbes reales partiales…. In secunda determinat de motibus illorum orbium … in quibus polis et axibus…. In tertia aptando illos orbes ad circulos imaginarios, definit circulum eccentricum et ipsius consequentia declarat, … quibus utuntur in tabulando Solis motum.

Brudzewo , Commentariolum (ref. 2), 86: “Quantum est in se, ad motum orbium non est opus aequante. Nihil enim aequans facit ad motuum orbis realis, cum sit circulus imaginarius, sed quantum ad opus astronomorum….” The equant is first referred to as an imaginary circle on p. 80: “… et propter hoc ordinaverunt aequantes circulos imaginatos…”.

Brudzewo , Commentariolum (ref. 2), 23: Tot orbes habet Sol, quot requiruntur et sufficiunt ad salvandum motum Solis in zodiaco diversum.

On avoiding splitting and vacua in the case of the Sun, Commentariolum (ref. 2), 25–6. On the Moon, p. 47: Primo enim habet tres orbes. — Si in Luna ponatur ecentricus, necessario ponendi sunt alii duo orbes circumpositi ecentrico ex causa circa Solem assignata, ne scilicet sequeretur scissio sphaerarum et commixtio vacui, et sic ecentricus orbis est una ratio sive causa ponendi orbes Augem Lunae deferentes.

Brudzewo , Commentariolum (ref. 2), 28 (following n. 6): Huius oppositum Commentator [Averroes] putebat destruens eccentricos, verum in hoc sentiens tanquam philosophus, cuius non est nisi motum totius sphaerae considerare, non autem partialis orbis, quod Astronomiae proprium est. Cf. p. 25, text to notes 5–7: … sphaeras secundum se totas eccentricas salvare, quod Commentator [Averroes] destruit, verum dicens tanquam philosophus, cuius non est nisi motum totius sphaerae considerare. See also Lerner , Monde des sphères (ref. 5), i, 361, n. 4.

Brudzewo , Commentariolum (ref. 2), 51: … Sol in uno atque eodem loco aliquando minus, aliquando magis eclipsari videtur. Quae non fierent nisi Luna haberet specialem orbem, ratione cuius suus motus iam fieret tardus, iam velox, iam mediocris; ratione cuius etiam iam accederet ad terram, iam ab ea elongaretur. Et hoc testatur Commentator IIdo Coeli commento tricesimo quarto inquiens: “Nihil invenitur in libris mathematicorum ad probandum eccentricos et epicyclos, nisi id quod apparet in eclipsi Lunae”; habet igitur Luna epicyclum.

Brudzewo , Commentariolum (ref. 2), 13, text to n. 4: Addit Albertus: “Est autem attendendum quod non puto unquam fuisse depraehensos ab aliqui mortalium omnes motus coelorum”; p. 26 para. 2: Qui quidem eccentrici an veraciter existant in sphaeris planetarum, nemo mortalium novit, nisi fateamur illos (ut nonulli aiunt), similiter et epicyclos, revelatione spirituum propalatos, si non extunc sola imaginatione mathematicorum effictos, sicut testatur Albeon….

Rosen quotes these two passages from Brudzewo against the claims that astronomers at the time of Copernicus's education adopted solid spheres to control planetary motions, and specifically that Brudzewo held eccentrics and epicycles to be “existentially improbable” ( Rosen , Copernicus, Complete works (ref. 4), iii, 123, n. 326), but gives no account of the immediate context for the remarks he is quoting, or of the wider context. See also ref. 5 above.

On similar passages in other writers, see Barker Peter Goldstein Bernard R. “Realism and instrumentalism in sixteenth century astronomy: A reappraisal”, Perspectives on science, vi (1998), 232–58, pp. 247–9.

Brudzewo , Commentariolum (ref. 2), 13 following text to n. 4: [E]t ideo etiam de substantiis mobilibus et praecipue de numero earum incertum est. Sed id, quod rationabilius dici poterit, ut videtur, iam a nobis dictum est. Quia hoc est certum, sphaeras esse causas esse et vitae, et differentias sphaerarum causus esse differentiarum, quae sunt in esse et vita. Et ideo videtur talis numerus esse earum, qualis nunc dictus est.

Swerdlow Noel M. Neugebauer Otto , Mathematical astronomy in Copernicus' De revolutionibus (New York, 1984), i, 45–7; Saliba George , Islamic science and the making of the European Renaissance (Cambridge, MA, 2001), esp. chap. 6.

She also claims that the device, as it applies to the Moon problem, does not seems to be original with Brudzewo, but can be traced back at least half a century to the work of Sandivogius of Czechel (fl. 1430) who in turn credits a double epicycle model to the author of the Theorica planetarum, and to Ptolemy himself. Rosińska Grażyna , “Naṣīr al-Dīn al-ṭūsī and Ibn al-Shāṭir in Cracow?”, Isis, lxv (1974), 239–43, pp. 241–3.

Goddu , Copernicus and the Aristotelian tradition (ref. 1), esp. 154–9 and 261–9. For a detailed rebuttal see Barker Peter Vesel Matjaž , “Goddu's Copernicus”, Aestimatio, ix (2012), in press.

Gabbey Alan , “Innovation and continuity in the history of astronomy: The case of the rotating Moon”, in Barker P. Ariew R. (eds), Revolution and continuity (Washington, DC, 1991), 114–20; Lerner , Monde des sphères (ref. 5), i, 114, text to n. 17 ff.

Goddu , Copernicus and the Aristotelian tradition (ref. 1), 156–7: “In a commentary on the Theorica planetarum, Sandivogius of Czechel (fl. 1430) described a lunar model to explain why, if the Moon moves on an epicycle, we see the spots on the Moon always oriented the same way. His solution placed the Moon on a second epicycle that moves at the same rate as the first epicycle but in the opposite direction. Now it is obvious that this is not the problem that either Maragha astronomers or Copernicus solved. But the point is the geometrical device itself, not the problem it proposes to solve…. What has been established, however, is that Albert of Brudzewo repeated the solution in his Commentariolum, a text on which Copernicus very likely received instruction. The point is that Copernicus could have learned of the device in a lecture on Brudzewo's Commentariolum. The application of the solution to the problem with the Ptolemaic lunar model and the ‘replacement’ of the equant would have been Copernicus's contribution”.

Brudzewo , Commentariolum (ref. 2), 67–8: Propter ergo salvare istum motum apparentem in Luna, quidam imaginantur epicyclum talem in Luna, quod habet alium intra se inclusum, qui movet epicyclum deferentem Lunam motu declinationis et reflexionis, quod non videtur esse inconveniens.

I would like to say a special word of thanks to Noel Swerdlow for advice on both the translation and the meaning of this passage and the passage in the next note, while absolving him of responsibility for the opinions I have expressed about them.

Brudzewo , Commentariolum (ref. 2), 68, n. 1 (= folio 51 R, left margin): Lunam quidam imaginantur habere duos epiciclos, unum maiorem, alterum minorem, in quo est eius corpus situatum, et ita epiciclus superior tantum, motu declinationis et reflexionis movetur. Et pro tanto illa macula, quae in Luna aspicitur semper una et eadem apparet propter istum epicyclum: Quod non esset, si talis epicyclus non esset.

Cf. Rosińska , “Naṣīr al-Dīn al-Ṭūsī and Ibn al-Shāṭir in Cracow?” (ref. 59), 241, also n. 8a and n. 9. I see no connection between the passages considered here, and the annotation quoted by Rosińska in her n. 10.

Brudzewo , Commentariolum (ref. 2), 68, fig. 9. Compare Nicholas Copernicus, De revolutionibus orbium coelestium (Nuremberg, 1543), Bk IV.

The effect of the prosneusis is that, as the centre of the lunar epicycle moves from the apogee to the perigee of the eccentric, the true apogee of the epicycle leads the mean apogee, and as the centre of the epicycle moves from perigee to apogee of the eccentric, the mean apogee of the epicycle leads the true apogee. I thank Noel Swerdlow for valuable discussion of the arrangement of the epicycles in Brudzewo, and the concept of prosneusis. On the latter, see: Ptolemy , Almagest V:5; Toomer G. J. , Ptolemy's Almagest (Princeton, 1998), 227; Pedersen Olaf , A survey of the Almagest (Odense, 1974), 192 ff.

Glogow , Introductorium (ref. 24), K iiii V: Luna habet spheralem motum pr[a]eter motum epiciculi ad salvandam apparentiam macul[a]e lun[a]e. Lerner, Monde des sphères (ref. 5), i, 114 and n. 21, pp. 307–8.

Rosińska , “Naṣīr al-Dīn al-Ṭūṣī and Ibn al-Shāṭir in Cracow?” (ref. 59), 240, text to n. 8; Goddu , Copernicus and the Aristotelian tradition (ref. 1), 165.

Hartner Willy , “Copernicus, the man, the work and its history”, Proceedings of the American Philosophical Society , cxvii (1973), 413–22; Saliba , Islamic science (ref. 58), 199–201; Saliba , “Revisiting the astronomical contacts between the world of Islam and Renaissance Europe: The Byzantine connection”, in Magdalino Paul Mavroudi Maria (eds), The occult sciences in Byzantium (Geneva, 2006), 361–73.

On Copernicus's broader indebtedness to Islamic science see Swerdlow Neugebauer , Mathematical astronomy (ref. 58), i, 41–8; Saliba , Islamic science (ref. 58), 193–232. On the Moon model see Roberts Victor , “The solar and lunar theory of Ibn ash-Shāṭir: A pre-Copernican Copernican model”, Isis, xlviii (1957), 428–32; Swerdlow Neugebauer , Mathematical astronomy (ref. 58), i, 193; Pedersen Olaf , Early physics and astronomy, rev. edn (Cambridge, 1993), 272–4, p. 273: “A comparison [of Copernicus's lunar model] with the lunar theory of Ibn ash-Shāṭir shows that not only is the general structure of the two models the same, but the geometrical parameters are also identical”, although, Pedersen continues, the similarity in parameter values may indicate that both authors used minimum and maximum values from Ptolemy's Almagest as the basis for their calculations.

Goddu , Copernicus and the Aristotelian tradition (ref. 1), e.g. 387ff.

Westman Robert S. , “The astronomer's role in the sixteenth century: A preliminary survey”, History of science , xviii (1980), 105–47; Jardine , “Significance of the Copernican orbs” (ref. 5); Granada Miguel A. , “The defense of the movement of the Earth in Rothmann, Maestlin and Kepler: From heavenly geometry to celestial physics”, in Bucciantini M. (eds), Mechanics and cosmology in the medieval and early modern period (Florence, 2007).

Barker Peter , “Copernicus and the critics of Ptolemy”, Journal for the history of astronomy , xxx (1999), 343–58; Westman , Copernican question (ref. 14), chap. 3, esp. pp. 99a–100b.

As Dobrzycki and Kremer state in “Peurbach and Marāgha astronomy?”, Journal for the history of astronomy , xxvii (1996), 187–237: “We know of no extant text by Peurbach or Regiomontanus in which the Ptolemaic models are criticized explicitly on the grounds that they violate uniform, circular motion” (p. 211).

On the Commentariolus see Swerdlow , “Derivation and first draft” (ref. 4). On Copernicus's silence about orbs in De revolutionibus, see Barker Peter , “The Hypotyposes orbium coelestium (Strasbourg, 1568)”, in Granada M. A. Mehl E. (eds), Nouveau ciel nouvelle terre — La révolution Copernicienne dans l'Allemagne de la Réforme (1530–1630) (Paris, 2009), 85–108, esp. pp. 88–94. For theorica representations of Copernican mathematical models, see Magini Giovanni Antonio , Novae coelestium orbium theoricae, congruentes cum observationibus N. Copernici (Venice, 1589).

See especially the works by Rosen, cited in ref. 4 above. More recently, see Goddu , Copernicus and the Aristotelian tradition (ref. 1), 257, n. 137, who rejects Swerdlow's proposal on related grounds. For additional difficulties with Goddu's account of theorica orbs see Barker Vesel , “Goddu's Copernicus” (ref. 60).

For an examination and evaluation of recent proposals for Copernicus's route to heliocentrism by Swerdlow , “Derivation and first draft” (ref. 4), Goldstein , “Copernicus and the origin of his heliocentric system” (ref. 8) and Westman, Copernican question (ref. 14), see Barker Peter , “Why was Copernicus a Copernican?”, Metascience, forthcoming.