Regiomontanus as a Physical Astronomer: Samplings from The Defence of Theon against George of Trebizond

The flattering comments “Iohannis de Monteregio, Germanorum decoris, etatis nostre astonomorum principis, Ephemerides” appear in Venetian editions of the 1490s and also in the title that a fifteenth-century scribe gave to Kraków, Bibliotheka Jagellionica, BJ 2729 (BB XXIII 5); see Zinner Ernst , Leben und Wirken des Joh. Müller von Königsberg genannt Regiomontanus, 2nd edn (Osnabrück, 1968), 303; and Wisłocki Władyslaw (ed.), Catalogus codicum manuscriptorum bibliothecae universitatis jagellonicae cracoviensis (2 vols, Cracow, 1877–81), ii, 643.

Swerdlow N. M. , “Regiomontanus's concentric-sphere models for the Sun and the Moon”, Journal for the history of astronomy, xxx (1999), 1–23, p. 1.

Duhem Pierre , To save the phenomena: An essay on the idea of physical theory from Plato to Galileo, transl. by Dolan E. Maschler C. (Chicago, 1969), 46.

Cohen H. Floris , “Les raisons de la transformation: La spécificité européenne”, in Blay Michel Nicolaïdis Efthymios (eds), L'Europe des sciences: Constitution d'un espace scientifique (Paris, 2001), 51–94, p. 76.

Uiblein Paul , “Die Wiener Universität, ihre Magister und Studenten zur Zeit Regiomontans”, in Hamann Günther (ed.), Regiomontanus-Studien (Österreichische Akademie der Wissenschaften, phil.-hist. Kl., ccclxiv; Vienna , 1980), 409–42; Zinner , Leben (ref. 1), 46–76.

Regiomontanus Johannes , Disputationes contra deliramenta cremonensia (Nuremberg, c. 1475), 3v–4r; facsimile in Schmeidler Felix (ed.), Joannis Regiomontani opera collectanea (Osnabrück, 1972), 518–19; Shank Michael H. , “Regiomontanus and homocentric astronomy”, Journal for the history of astronomy, xxix (1998), 1998–66, p. 161.

“Iam vero venio ad rationes contra illam antiquam fantasiam <ecentricorum> que, si esset vera secundum approbatam sententiam ponentium eos, sequitur lunam in plenilunio non plus in sesquialtero esse remotiorem quam in quadraturis, id est quando est dimidiata lumine(?), quod est falsum; quia sic in quadraturis deberet apparere notabilioris dyametri quam in plenilunio”, Vienna, ÖNB cod. 5203 108v; “ibidem invenitur Venerem posse elongari ultra minimam eius longitudinem a terra dicto modo respectu lune bene per 437 semidiametros terre quia maximam distantiam Veneris a terra 604 semidyametrorum terre, igitur necessario Venere existente in maxima sui elongatione ipsa multo fortius in 10plo minor apparebit nobis quam in minima eius distantia a nobis; et illud argumentum eque bene procedit si pontaur dyameter Veneris tertia pars dyametri lune ex eadem deductione”, Vienna, ÖNB cod. 5203, 110r.

Curtze Maximilian , “Der Briefwechsel Regiomontan's mit Giovanni Bianchini, Jacob von Speier und Christian Roder” in Curtze M. (ed.), Urkunden zur Geschichte der Mathematik im Mittelalter und der Renaissance (1902; repr. New York, 1968), 187–336; Swerdlow Noel M. , “Regiomontanus on the critical problems of astronomy”, in Levere Trevor Shea William (eds), Nature, experiment and the sciences (Boston Studies in the Philosophy of Science, cxx; Boston, 1990), 165–95, pp. 172–4.

Swerdlow Noel M. , “Science and humanism in the Renaissance: Regiomontanus's Oration on the dignity and utility of the mathematical sciences”, in Horwich Paul (ed.), World changes: Thomas Kuhn and the nature of science (Cambridge, MA, 1993), 131–68, p. 150. Jardine also noticed in the Paduan Oration the resonance between Regiomontanus's association of astronomy with “knowledge of the form of the world” and Kepler's physicalist view: Jardine Nicholas , The birth of the history and philosophy of science: Kepler's A Defence of Tycho against Ursus with essays on its provenance and significance (Cambridge, 1984), 265.

Some scholars noticed the tension, but did not discuss it; see Shank , “Homocentric astronomy” (ref. 6), 159, 164 n9.

Shank Michael H. , “The ‘Notes on al-Biṭrūjī’ attributed to Regiomontanus: Second thoughts”, Journal for the history of astronomy, xxiii (1992), 15–30.

Shank , “Homocentric astronomy” (ref. 6), 158; Swerdlow , “Concentric-sphere models” (ref. 2), 2, 6–9.

Swerdlow , “Concentric-sphere models” (ref. 2), esp. pp. 2–5.

Bernard R. Goldstein has recently emphasized the importance of distance considerations in Copernicus; see Goldstein , “Copernicus and the origins of his heliocentrism”, Journal for the history of astronomy, xxxiii (2002), 219–35, passim.

Shank Michael H. , “Regiomontanus on Ptolemy, physical orbs, and astronomical fictionalism: Goldsteinian themes in Regiomontanus's ‘Defence of Theon against George of Trebizond’”, Perspectives on science, x (2003), 179–207, pp. 195, 200; and see below.

Swerdlow N. 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.

Shank, “Regiomontanus on Ptolemy” (ref. 15), 188–203. For additional background, see Zinner , Leben (ref. 1), 155–6, 262–4; and Monfasani John , Collectanea trapezuntia: Texts, documents, and bibliographies of George of Trebizond (Binghamton, NY, 1984), 672–4.

Regiomontanus Johannes , Defensio Theonis contra Georgium Trapezuntium, St Petersburg, Archive of the Russian Academy of Sciences, ms. IV-1-935. Defensio xii refers to the comet of “last year” (211v), an ambiguous note since Regiomontanus saw comets in 1468 and 1472. A few folios later, however, Regiomontanus alludes to the recent election of Pope Sixtus (219r). Although Regiomontanus calls him “tertius”, these references are consistent only with a date after c. 1473. Regiomontanus wrote some fifty folios after these references, concluding with comments pertinent to the end of Almagest xiii. Since he had included the Defensio on his advertisement (c. 1474) of the books that he planned to print, Books 12 and 13 clearly belong to the last years of his life.

“innocentia Procli summi philosophi” (297r); “illi philosopho summo Theoni” (38v).

“Omnes autem philosophi medium motum sive equalem investigant propter verum cognoscendum. Tu vero si medium motum cognoscere velis, verum prius didicisse cogeris”, Regiomontanus, Defensio (ref. 18), 31r; “Dices igitur epiciclum lune equaliter moveri respectu centri ecentrici sui, pariformiterque in quinque retrogradis, quod falsum est et omnibus philosophis contrarium” (34v). The one place I have found so far in which Regiomontanus does suggest a separation between the roles of philosopher and astronomer is a marginalium, also in Book 12 (215r: “qualiter astronomi equalitatem motus considerant et qualiter philosophi”). It faces, however, a passage in which Regiomontanus criticizes the eccentric and epicycle of the Moon and warns George of Trebizond that he will never be a worthy astronomer if he cares only about accommodating his calculations to the appearances: “Neque eccentricus neque epicyclus lunaris officium suum exhibet cuius presertim gratia assumptus est ab astronomis. Nam si hoc solum cures quo lineamento aut quovis alio adminiculo calculum tuum apparentibus concines, astronomi dignitatem haudquaquam consequeris” (215r).

“Since these matters are treated daily in our schools, I may keep the point brief, for we have come not to teach the rudiments of this art (although the expositor could not aspire even to these), but to protect the integrity of those who philosophize honourably and to restrain the slanderous tongue of this story-teller” (“Verum cum hec in scholis nostris quotidie agitentur, breviorem libuit facere sermonem: Non enim docturi venimus huius artis inchoamenta quamvis ne ad illa quidem aspirare expositor potuerit, sed ut innocentiam ingenue philosophantium protegamus ac linguam huius nugatoris maleditam coerceamus”), Regiomontanus , Defensio (ref. 18), 66r—v.

“superscr. Tu vero At novus hic astronomie instaurator ac veterum explosarumque opinionum resuscitator ‘frustra’ autumat ‘poni nonam spheram a multis; ipsam enim octavam spheram esse quam sicut totum non vi nec raptu sed natura ceteras veluti partes ad occasum sequi’”, Regiomontanus, Defensio (ref. 18), 144v (apart from its infinitive construction, the quotation matches the wording in the Trebizond commentary in Vienna, ÖNB cod. 3106, 154v).

“Tu vero affirmas octavam spheram esse sicut totum ceterasque tanquam partes naturaliter eam sequi ad occasum. Nemine profecto adeo insanum crediderim ut spheram lunarem esse partem octave perperam enunciet, tot mediis tantisque existentibus planetarum spheris. At dicat quispiam totum celeste corpus ad occasum ferri secum ducens ceteras stellarum spheras naturaliter, quomodo igitur motus earum interea ad ortum naturali quavis [crossed out: Lege] serie potest evenire? Quippe cum nemo unquam philosophus eidem corpori simplici duos presertim contrarios motus naturales concesserit”, Regiomontanus, Defensio (ref. 18), 144v.

Grant Edward , Planets, stars, and orbs: The medieval cosmos, 1200–1687 (Cambridge, 1994), 497, 711 (Question 184).

Here Regiomontanus's revision while he was composing 144v (ref. 23) is informative: He first wrote lege, which he crossed out immediately in favour of serie; that is, he retained the idea of necessary connection, but made it more material and mechanical.

Shank, “Notes” (ref. 11), 24, 26; and idem, “Rings in a fluid heaven: The equatorium-driven physical astronomy of Guido de Marchia (fl. 1309)”, Centaurus, xlv (2003), 175–203, pp. 182–5. He may also have been familiar with Theon of Smyrna, available in Bessarion's library (Shank, “Regiomontanus on Ptolemy” (ref. 15), 185, n. 4). When discussing the arrangement of the planets, Theon of Smyrna speculates that each planet has its own hollow sphere that carries within its thickness a solid (epicyclic) sphere that in turn carries the planet (Part 3, chap. 32). It is here that he also mentions the possibility that the Sun, Mercury, and Venus may all be carried by a single hollow sphere: Dupuis J. , Théon de Smyrne, philosophe platonicien: Exposition des connaissances mathématiques utiles pour la lecture de Platon traduites pour la première fois du grec en français (Paris, 1892), 301–3.

“Nescio si forte epicurea somnia narrare velis, ut communi quadam vertigine putes totum celi orbem ad occasum ferri, stellis interea ac planetis nihilominus more animantium multifariam hac atque illac cursitantibus.” “Nescio si forte insaniendo totum celi orbem ferri ad occidentem opineris, stellis autem nihilominus atque planetis animantium more multifariam hoc atque illac quasi pro libito cursitantibus.” Regiomontanus, Defensio (ref. 18), 145v, 147r. Regiomontanus returns to the issue briefly in Book 13 (259v, 264r).

Siorvanes Lucas , Proclus: Neo-Platonic natural philosophy and sciences (New Haven, 1996), 296–7, who points to Proclus, In Timaeum, 3. 144.4, and notes that this view goes back to Aristotle and Plato, who ascribe reason to the celestial bodies — Not indeterminacy. See also Proclus , Commentaire sur le Timée, transl. by Festugière A. J. (5 vols, Paris, 1966–68), iii, 285 (240,29–241,4).

“Equidem quamvis motum stellarum ad occasum vi aut raptu quodam fieri minime dixerim, siquidem violentia eternitati repugnans nihil habet loci in celestibus, eum tamen motum neque proprie naturalem affirmarim, cum ad ortum ferri eas natura propria iubeat. Is itaque occasualis motus preter naturam, ut iuniores autumant, haberi potest quasi medius inter naturalem et violentum, aut potius naturalis simpliciter, non quidem a natura propria sed communi inditus. Frivola igitur est unica ratiuncula tua quam haud satis expresse sed transitu quodam insinuas vi et raptu motum talem fieri non posse, quod ne discipulus quidem nuper astronomicis initiatus rudimentis diceret. Ceterum si motus ille ad occasum naturalis est, cur tam multifariam stelle ab universali motu deficiunt? quando eodem nisu aut eadem naturali obedientia mobili primo obsequuntur”, Regiomontanus, Defensio (ref. 18), 144v–145r.

This usage of “preternatural” in the science of motion is obviously very different from that of natural history, with its monsters and other occurrences outside the ordinary course of nature. See Daston Lorraine Park Katherine , Wonders and the order of nature (Cambridge, MA, 1998), 121ff; and Daston , “Preternatural philosophy”, in Daston (ed.), Biographies of scientific objects (Chicago, 2000), 15–41. To the contrary, the daily motion is the fundamental motion of the universe, the archetype of the unceasing, the predictable, and the expected.

Regiomontanus's expression recentiores suggests that he did not know that this type of analysis goes back at least to Simplicius, On Aristotle's On the Heavens, transl. by Hankinson R. J. (Ithaca, 2002), I, 1–4, pp. 39–40.

For Sacrobosco's use of the expressions rapere and raptus firmamenti, see Thorndike Lynn , The Sphere of Sacrobosco and its commentators (Chicago, 1949), 79, 101–2. Regiomontanus also used raptus when discussing comets in his De cometae magnitudine, longitudineque ac de loco eius vero problemata XVI (Nuremberg, 1531), f. B2r, reproduced in Jervis Jane , Cometary theory in fifteenth-century Europe (Dordrecht, 1985), 181. I discuss raptus in a forthcoming paper, “Rough edges: Raptus and the medieval disturbances of the celestial—terrestrial boundary”, first presented at the European Science Foundation's Mechanics and Cosmology workshop at the Max Planck Institut für Wissenschaftsgeschichte, Berlin, November 2006.

Aristotle, De caelo, in Aristotelis opera cum Averrois commentariis, v (Venice, 1562), 10vH, 11rF (De caelo I), 161vM (De caelo II, 6: “…cum violenta sit et preter naturam”, referring to the possible motion of the Earth).

Oresme Nicole , Le livre du ciel et du monde, ed. by Menut Albert Denomy Alexander (Madison, 1968), 76 (my translation, as Menut did not understand that hors nature was Oresme's direct translation of praeter naturam, which cannot be rendered as either “unnatural” or “violent” — As Menut does — Without undermining Oresme's point: p. 77).

See Aschbach Joseph , Geschichte der Wiener Universität im ersten Jahrhunderte ihres Bestehens (3 vols, Vienna, 1865), i, 429–30. Although the last digit of the year is in doubt, this question belongs to a collection of quodlibetal questions, now in Melk 958 (formerly 6/A8; paginated, not foliated), that belonged to another Viennese master, Urban of Melk, who evidently supervised the quodlibetal disputations of 1429.

Angerer gives a definition of passum because he wishes it to include the material aspects of the heavens, which other definitions exclude: “by passum one must understand the subject of inherence or denomination of motion, whether that be matter, or form, or the composite whole, or some other thing. Hence [when] something is moved locally through some local motion, it is a passum in such a local motion, whether it be substance or accident. By passum, one should not understand only the form or the composite of matter and form, as some, however, want to maintain; and I prove this [proposition] thus. I suppose that that heaven is not composed of matter and form, but is a simple substance, and I take the heaven in the sense of ‘sphere’, excluding the intelligence” (“… ad primum dicendum est quod per passum debet intelligi subiectum inhesionis aut denominationis motus, sive illud sit materia sive forma, vel totum compositum vel quid aliud; unde quicquid movetur localiter per aliquem motum localem, hoc est passum in tali motu locali, sive sit substantia sive accidens. Nec per passum debet solum intelligi forma vel compositum ex materia et forma, sicut tamen aliqui volunt dicere, et hoc probo sic; et suppono quod celum non sit compositum ex materia et forma, sed est una simplex substantia et capio celum pro orbe seclusa intelligentia”), Melk, Stiftsbibliothek 958 (6/A8), 415.

“Secundo noto quod duplex est motus quantum ad propositum sufficit, scilicet naturalis et preter naturam. Motus naturalis est qui fit in a principio activo extrinseco vel intrinseco conf<er>ente vim passo. Motus autem preter naturam est duplex, quidam preter naturam, id est contra naturam et dicitur violentus, et est motus qui fit a principio activo extrinseco passo non conferente vim, sed potius renitente vel resistente. Alius est motus preter naturam, id est nec secundum inclinationem naturalem, nec contra inclinationem naturalem. Et describitur sic: Est motus qui fit a principio activo extrinseco, passo non conferente <vim> nec renitenti. Exemplum primi, videlicet de motu naturali: Sicut motus gravis deorsum vel levis sursum. Exemplum secundi: Sicut motus gravis sursum non pro repletione vacui, aut levis deorsum. Exemplum tertii: Sicut motus circularis ignis in spera sua quo movetur in sequendo primum mobile ab oriente in occidens per meridiem; similiter motus supreme regionis aeris”, Melk, Stiftsbibliothek 958 (6/A8), 415.

See ref. 33 above. Michael Scot's translation from the Arabic, which the Giunta edition juxtaposes to Moerbeke's (the latter in roman font, the former in italics) uses accidentalis or non naturalis where Moerbeke uses preter naturam.

The same distinction also appears in Giordano Bruno's De magia: “Duplex est rerum motus: Naturalis et praeternaturalis; naturalis qui est a principio intrinseco, praeternaturalis qui a principio extrinseco; item naturalis qui est conveniens naturae, consistentiae vel generationi, praeternaturalis qui non. Et hic est duplex: Violentus, qui est contra naturam; et ordinatus seu coordinabilis, qui non repugnat naturae”, Bruno Giordano , “De magia”, in Tocco F. Vitelli H. (eds), Jordani Bruni Nolani opera latine conscripta (3 vols, Florence, 1891), iii, 417.

The distinction between natural, violent, and preternatural motions appears in Muzio Vitelleschi's discussions of Aristotle's Physics at the Collegio Romano in the sixteenth century, in Galileo's De motu antiquiora, and elsewhere; see Wallace William A. , Galileo and his sources (Princeton, 1984), 162–3, 209, 242–3.

In the Latin translation of al-BiṬrūjī, the term is incurtatio; see Carmody Francis J. (ed.), Al-Biṭrūjī's De motibus celorum: A critical edition of the Latin translation of Michael Scot (Berkeley, 1952), 168. Regiomontanus, who often strove for philological hypercorrectness, may have snubbed incurtatio as a neologism.

“Et si motum probas relictitium marg.: Vide Simplicium qui sequitur Eudoxum; vide Alpetragium opinionem secutus Eudoxi vetustissimi aut Alpetragii recentioris; quo pacto servabis quantitatem relictionis tam multiplicem? Nempe aiunt quo amplius distat sphera quevis a supremo celo, eo magis deficere a motu universali. Sphera igitur fixarum immunis erit ab hac relictione, quandoquidem immediate, ut illi putant, sub primo mobili statuitur. Aut si a centris fixarum sumi oportet distantiam, quomodo illud constare potest ut relictio solaris dupla ferme sit ad relictionem martialem, cum tamen remotio solis a primo celo non sit dupla ad remotionem Martis, sed vix unam de 189 partibus ei superaddens”, Regiomontanus, Defensio (ref. 18), 145r.

As Goldstein has shown, Copernicus's rethinking of such a relationship played an important role in his reorganization of the cosmos; see Goldstein , “Copernicus” (ref. 14), 220–6.

See the criticisms of al-Biṭrūjī in Bernard of Verdun and in Guido de Marchia, with parts of which Regiomontanus was acquainted; see Shank , “Notes” (ref. 11), 25. For Marchia's argument in relation to the Sun, Venus, Mercury, and the Moon, see Vienna , ÖNB cod. 5266, 237rb, cited in Shank, “Rings” (ref. 26), 186. Regiomontanus's argument here replicates that of Proclus in the Hypotyposes; see Halma L'abbé , Hypothèses et époques des planètes de C. Ptolémée et Hypotyposes de Proclus Diadochus (Paris, 1820), 68–69.

See Duhem Pierre , Le système du monde 10 vols, (Paris, 1913–59), iv, 153; Hugonnard-Roche H. , L'oeuvre astronomique de Thémon Juif, maître parisien du XIVe siècle (Geneva and Paris, 1973), 114; Avi-Yonah Reuven , “Ptolemy vs. al-Biṭrūjī: A study of scientific decision-making in the Middle Ages”, Archives internationales d'histoire des sciences, xxxv (1985), 1985–47, 132–3.

Toomer G. J. , Ptolemy's Almagest (Berlin, 1984), 419–20.

Theon of Smyrna, who endorses the traditional ordering of the planets, cites a poem by Alexander of Aetolia that places “the Sun in the middle of the planets”, and also comments on the disagreements among the “mathematicians”, who place the Sun immediately after the Moon, but disagree about the trans-solar order of Mercury and Venus: Dupuis , Théon de Smyrne (ref. 26), 229, 233, 303; Ricklin Thomas , “Le coeur, soleil du corps: Une redécouverte symbolique du XIIe siècle”, in Il cuore / The heart (Micrologus, xi; Florence , 2003), 123–43, esp. pp. 126–8.

“In primis itaque necessarium esse affirmat ut membrum mundi quod cordi proportionatur medium ipsius mundi locum obtineat. Non advertit homo ille latentem in hac re discrepantiam. Detur enim solem habere similitudinem cordis idcircoque proportionatum, id est medium in mundo locum obtinere. Iam videndum est si sol medius planetarum existens similem in regione celesti situm habeat. Constat autem si totam celi profunditatem plinensi <mg ptolemaici> fuerimus et eam proceritati, verbi gratia humani corporis adaptaverimus, solem ipsum situ suo pedis locum potius quam cordis occupare. Crassitudo enim pedis humani iuxta talos a vigesima parte totius proceritatis humane haud milium discrepat. Et remotio solis ab elementari regione unde mundus superior initium sumit vigesima ferme pars est totius celestis altitudinis. <mg vide per calculum> Qui ergo solem cordi assimilat ideoque ei locum quem nunc in celo habet iuste tributum esse affirmat, aut cor humanum in loco tali vel pedis ponendum aut solem in sphera Iovis statuendum esse insinuat ut medium in altitudine celi locum possideat marg. vide exactius locum hunc, quorum certe neutrum natura permittit. At si quis dicat solem possidere medium locum non quidem spatiali et continua sed discreta quadam equidistantia quod tres sub se et totidem supra se habeat planetas, cur ergo excludet tantam stellarum multitudinem que in octava sphera consistunt quasi non sint membra celestis corporis? Que tamen maximam vim in mundum istum inferiorem habent si Ptolemeo ipsi credimus. <mg terrenis corporibus Ptolemeo ipso decernente insinuerunt(?)> Quare si medius inter stellas soli tribuendus est locus sub Saturno utique esse non poterit ne plures supra quam infra habeat”, Regiomontanus, Defensio (ref. 18), 154v–155r.

Segonds and Siorvanes both emphasize that Proclus presents different physical, astronomical, and metaphysical orderings of the planets; see Siorvanes , Proclus (ref. 28), 307–8; and also Segonds Alain , “Proclus: Astronomie et philosophie”, in Pépin Jean Saffrey H. D. (eds), Proclus: Lecteur et interprète des anciens (Paris, 1987), 319–34, pp. 326–9.

Segonds , “Proclus” (ref. 49), 327–8.

“Verum si quis rationes illas repudiare velit, quoniam suppositione eccentricorum atque epicyclorum innituntur, per eos enim predicte remotiones invente sunt, hoc saltem reclamare non poterit remotionem solis a supremo celo vix esse duodecimam partem intercapedinis duorum luminarium, quod certe naturali inscitie penitus contrarium intelligitur”, Regiomontanus , Defensio (ref. 18), 145v.

“Si planetis altioribus unicam posueris diversitatem, epicyclus in concentrico aut ecentricus sine epicyclo eidem sufficiens erit occasio”; “In Venere idem et Mercurio videri necesse est”, Regiomontanus, Epytoma Almagesti (Venice, 1496), f. N4r—v; facsimile in Schmeidler Felix (ed.), Joannis Regiomontani opera collectanea (Osnabrück, 1972), 243–4. See also Swerdlow , “Derivation” (ref. 16), 472–4.

Swerdlow , “Derivation” (ref. 16), 471–8. The significance of Epitome xii, 1–2 for Copernicus is independent of Swerdlow's speculation (carefully labelled as such) about Copernicus's rationale for choosing the heliocentric over the “Tychonic” model; see Goldstein , “Copernicus” (ref. 14), 221–2; and Goddu André , “Reflections on the origins of Copernicus's cosmology”, Journal for the history of astronomy, xxxvii (2006), 37–53, pp. 39–40.

Ragep F. Jamil , “^cAlī Qushjī and Regiomontanus: Eccentric transformations and Copernican revolutions”, Journal for the history of astronomy, xxxvi (2005), 359–71, pp. 359–60.

In Shank , “Regiomontanus on Ptolemy” (ref. 15), 185 n. 4, I speculated that Theon of Smyrna's discussions may have stimulated Regiomontanus's interest.

Rigo Antonio , “Bessarione, Giovanni Regiomontano e i loro studi su Tolomeo a Venezia e Roma (1462–1464)”, Studi veneziani, n.s., xxi (1991), 83–95.

“In hoc autem duodecimo eosdem reminiscitur inequalitatum modos tamquam stationi atque regressui probe accommodabiles; non enim ostendere pergit superscr. voluit Ptolemeus in presentiarum quopacto omnimoda motuum diversitas per utranque viam eque servetur, quamvis hoc putet expositor; nanque id iampridem de sole et luna singulatim docuit; sed ut proportiones linearum parilium inter visum et circumferentiam tam eccentrici quam epicycli interceptarum utrobique sint eedem; illud enim negocio regressuum utroque modo salvandorum conducibile est, tam in tribus altioribus quam in duobus humilioribus quamvis in Venere et Mercurio id ipsum procedere expositor abiurat testem citans autorem non intellectum: ‘Nam Veneris’, inquit, ‘atque Mercurii motus epicyclum necessario flagitat, aliorum si epicyclum subtraxeris servatur quidem sed eccentricitatis proportio adeo magna fit ut incredibilis videatur’”, Regiomontanus, Defensio (ref. 18), 208v.

Swerdlow , “Derivation” (ref. 16), 473–4.

See Ptolemy: “If the anomaly related to the Sun is represented by the eccentric hypothesis (which is a viable hypothesis only for the three [outer] planets which can reach any elongation from the Sun), in which the centre of the eccentre moves [uniformly] about the centre of the ecliptic with the speed of the [mean] Sun towards the rear [i.e. in the order] of the signs, while the planet moves on the eccentre in advance [i.e., in the reverse order] of the signs with a speed [uniform] with respect to the centre of the eccentre and equal to the [mean] motion in anomaly, and if a line is drawn in the eccentre through the centre of the ecliptic (i.e., the observer) in such a way that the ratio of half the whole line to the smaller of the two segments of the line formed by [the position of] the observer is equal to the ratio of the speed of the eccentre to the speed of the planet, then when the planet arrives at the point in which the above line cuts the arc of the eccentre near the perigee, it will produce the appearance of station.” In commenting on Ptolemy's claim, Toomer's puzzlement is noteworthy: “This type of eccentric model is in fact applicable to the inner planets as well, provided that, for the speed of the eccentre, one uses, not the speed of the mean Sun, but the sum of the speeds of the mean Sun and the planet's anomaly (which sum is the same as the modern heliocentric mean motion). I do not understand why Ptolemy does not recognise this”, Toomer , Almagest (ref. 22), 555–6.

E.g., Regiomontanus , Defensio (ref. 18), 186v. Although I have not checked Regiomontanus's Latin manuscript of the Almagest, the Venice, 1515, edition of the Gerard of Cremona translation is unambiguous: “Et si fuerit diversitas que est propter solem necessaria etiam propter radicem in qua agitur secundum orbem egredientis centri; et illud non est possibile ut sit nisi in stellis tribus tantum que elongantur a sole longitudine tota” (131v). While he was finishing the Epitome, he would have had access to the same Bessarion codex that George of Trebizond had borrowed for his translation and commentary, presumably the one Bessarion had judged the best of the six Greek Almagests in his library: Monfasani, Collectanea trapezuntia (ref. 17), 749.

Earlier, in Almagest ix, 5, Ptolemy had already stated that the eccentric hypothesis would not work for Mercury; Toomer , Almagest (ref. 22), 442.

Shank , “Regiomontanus on Ptolemy” (ref. 15), 195–6, 200–1.

I discussed this passage briefly in Shank , “Regiomontanus on Ptolemy” (ref. 15), 200–2. The Latin I give below corrects an uncertain reading with a definitive one, thanks to the high-resolution digital images made possible by the National Science Foundation and produced by Dr Aleksandr Eliseyev (Russian Academy of Sciences, St Petersburg): “Ad astronomiam attinere ut non modo calculum apparentibus accommodet sed et figuras corporum celestium una cum lege motuum edoceat; alias enim fictitiam tradere artem. Equalitatem motuum celestium operepretium tutandam esse”, Regiomontanus, Defensio (ref. 18), 210v. I have modified my translation accordingly, and otherwise improved it slightly.

“Deinde autem hoc quoque una confessum esto, stellas non sua sponte veluti more animalium proficisci sed ad motus orbium suorum quibus imposite sunt circumferri; placeatque in hoc peripateticis atque aliis plurimis egregie philosophantibus fidem habere non sine efficacibus argumentis ac validissimis rationibus comparatam. Nam qui celestia corpora more pecudum in aruis oberrantium hac atque illac cursitare opinatur preterquam quod ab exemplari vilissimo excelse atque nobilissime rei qualitatem aucupari tentat; hanc celo iniuriam facit quod ipsum quasi intranquillum fluidum et perpetue cessioni atque incisioni obnoxium intelligit, quam absurdam instabilitatem ne cogitare quidem fas est in nobilissimis illis corporibus. Quis enim nisi mentis inops hoc genus lationis incertum asserere ausit, cum stellas fixas tam magnitudine quam viribus impares equali graduum numero quamvis motu ad orientem tardissimo ferri animadvertat? Quod profecto evenire non posset nisi unde et cognomentum traxere in uno et eodem orbe configerentur ad motumque eius qualemcumque circumducerentur”, Regiomontanus , Defensio (ref. 18), 210v–211r.

Regiomontanus, who often insists on the correct terminology, calls them “stelle crinite” — Hairy stars — Using the classical expression that appears in Suetonius and Pliny, among others ( Suetonius , The lives of the twelve Caesars, i, 88; vi, 36; and x, 23; Pliny , Natural history, ii, 32).

“quod profecto evidens est exploramentum omnes hasce sublimiores stellas uno et eodem orbe consertas esse ad motumque eius una circumferri. Preterea hanc sententiam stelle crinite motibus suis corroborant. Quippe que motum universalem imitate regionem aeris supremam motu suo non findunt sed ductu eius orbis in quo collucescunt circumferuntur. Non enim ulla in eis est vis animalis unde tam nobilis tamque rapidus fieri motus posset; materia earum/ fumea ut philosophi perhibent existente. Nam quod nonnulli philosophastri lucentem illam massam ductu planete generatoris ferri autumant, nuge sunt. Vise etenim sepe sunt huiuscemodi secundaria sidera multum a zodiaco septentrionem versus remote ut que neque motu sed ne genitura quidem a planetis omnino dependere censeantur superscr. agnoscuntur > quod ea potissimum que superiori anno sub visa maiori cernebatur insinuant cuius profecto diversitas aspectus efficere non potuit ut quamvis sub tractu planetarum nata sub sidere tamen aquilonio illucesceret”, Regiomontanus , Defensio (ref. 18), 211r–211v.

“When the matter begins to gather in the lower region independently the comet appears by itself. But when the exhalation is constituted by one of the fixed stars or the planets, owing to their motion, one of them becomes a comet”, Aristotle, Meteorologica, transl. by Webster E. W. (Oxford, 1923), i 7, 344a 34–37.