Copernicus,Amico,Fracastoro and Ṭūsī's Device: Observations on the Use and Transmission of a Model

See for example Neugebauer O. , The exact sciences in Antiquity , 2nd edn (Providence, 1957), 197, 203–4; Roberts V. , “The solar and lunar theory of Ibn ash-Shāṭir: A pre-Copernican Copernican model”, Isis , xlviii (1957), 428–32; Kennedy E. S. Roberts V. , “The planetary theory of Ibn al-Shāṭir”, Isis , 1 (1959), 227–35; Abbud F. , “The planetary theory of Ibn al-Shāṭir: Reduction of the geometric models to numerical tables”, Isis , liii (1962), 492–9; Roberts V. , “The planetary theory of Ibn al-Shāṭir: Latitudes of the planets”, Isis , lvii (1966), 208–19; Kennedy E. S. , “Late medieval planetary theory”, Isis , lvii (1966), 365–78; idem, “Planetary theory in the medieval Near East and its transmission to Europe”, in Atti del convegno internazionale sul tema: Oriente e Occidente nel Medioevo: Filosofia e scienze (9–15 aprile 1969) (Rome, 1971), 595–607; Hartner W. , “Trepidation and planetary theories: Common features in late Islamic and early Renaissance astronomy”, ibid. , 609–29; idem, “Copernicus, the man, the work and its history”, Proceedings of the American Philosophical Society , cxvii (1973), 413–22; Swerdlow N. M. , “The derivation and first draft of Copernicus's planetary theory: A translation of the Commentariolus with commentary”, ibid. , 423–512, pp. 456, 467–9, 488–9, 500, 504; Hartner W. , “Ptolemy, Azarquiel, Ibn al-Shāṭir, and Copernicus on Mercury: A study of parameters”, Archives internationales d'histoire des sciences , xxiv (1974), 5–25; idem, “The Islamic astronomical background of Nicolaus Copernicus”, Studia Copernicana , xiii (Colloquia Copernicana, iii), ed. by Gingerich O. Dobrzycki J. (Warsaw, 1975), 7–16; idem, “Ptolemaische Astronomie im Islam und zur Zeit des Regiomontanus”, in Regiomontanus-Studien , ed. by Günther Hamaan (Vienna, 1980), 109–24; Swerdlow N. M. Neugebauer O. , Mathematical astronomy in Copernicus's De revolutionibus (New York, 1984), 46–48; Saliba G. , “The astronomical tradition of Marāgha: A historical survey and prospects for future research”, Arabic sciences and philosophy , i (1991), 67–99, pp. 68–69, 74–81.

The first to show, in modern times, that the theorem demonstrated by Copernicus ( De revolutionibus , III, 4) was already known to Naṣīr al-Dīn al-ṭūsī and used by him in his planetary theory, seems to be Dreyer J. L. E. , History of the planetary systems from Thales to Kepler (Cambridge, 1906; reprinted as A history of astronomy from Thales to Kepler (New York, 1953)), 269, n. 1.

Swerdlow N. M. , “Aristotelian planetary theory in the Renaissance: Giovanni Battista Amico's homocentric spheres”, Journal for the history of astronomy , iii (1972), 36–48, p. 37.

Di Bono M. , Le sfere omocentriche di Giovan Battista Amico nell 'astronomia del Cinquecento. Con il testo del “De motibus corporum coelestium …” (Genoa, 1990), 70–71.

Ragep F. J. , “The two versions of the ṭūsī couple”, in From deferent to equant: A volume of studies in the history of science in the ancient and medieval Near East in honor of E. S. Kennedy , ed. by King D. A. Saliba G. (New York, 1987), 329–56, p. 331; Saliba G. , “The role of the Almagest commentaries in medieval Arabic astronomy: A preliminary survey of ṭūsī's redaction of Ptolemy's Almagest”, Archives intemationales d'histoire des sciences , xxxvii (1987), 3–20, pp. 11, 15. The Ḥall is the only of ṭūsī's works quoted in our paper whose dating is uncertain. Saliba's argument (op. cit.) that it must follow the taḥrīr al majistī of 1247, seems convincing.

ṭūsī claimed that he was the inventor of the mechanism and there seems no motive to doubt him, because, as Hartner, “Trepidation” (ref. 1), 631, affirms, declarations of the genre are rare in either Antiquity or the Middle Ages, when being able to attribute one's own invention to a past authority increased its value.

For a translation of the text see de Vaux Carra B. , “Les sphères célestes selon Nasir-Eddin Attusi”, Appendix VI in Tannery P. , Recherches sur l'histoire de l'astronomie ancienne (Paris, 1893), 337–61, p. 348. See also Hartner W. , “Naṣīr al-Dīn al-ṭūsī's lunar theory”, Physis , xi (1969), 287–304, p. 288; idem, “Trepidation” (ref. 1), 614–15; Ragep, op. cit. (ref. 5), 332.

See for example Neugebauer , op. cit. (ref. 1), 203; Koyré A. , La révolution astronomique: Copernic, Kepler, Borelli (Paris, 1961), 104, n. 7 (but referring to Copernicus); and Kennedy, “Late medieval planetary theory” (ref. 1), 369.

Ragep , op. cit. (ref. 5), 332–3.

Kennedy , “Late medieval planetary theory” (ref. 1), 369. As an aside, one can note that Koyré, op. cit. (ref. 8), and Kennedy , “Planetary theory” (ref. 1), 601, affirm erroneously that the velocities were the same.

This is also generally the reason why this version of the device and that used by Copernicus in De revolutionibus are considered the same.

The demonstration is reported also by Quṭb al-Dīn al-Shīrāzī in the Tuḥfat al-shāhiyyah and in the Nihāyat al-idrāk fī dirāyat al-aflāk and by Ibn al-Shāṭir in the Nihāyat al-su'ūl fī taḥṣīl al-uṣūl. Cf. Livingston J. W. , “Naṣīr al-Dīn al-ṭūsī's al-Tadhkirah: A category of Islamic astronomical literature”, Centaurus , xvii (1973), 260–75, pp. 271–3.

Hartner , “Trepidation” (ref. 1), 614.

Hartner , op. cit. (ref. 7), 289; idem, “Trepidation” (ref. 1), 614–15; idem, “Copernicus” (ref. 1), 416–17. As an aside, one can note that Benedetti (Io. Baptistae Benedicti … Diversarum speculationum mathematicarum, etphysicarum liber … (Turin, 1585), Cap. XXXV, “Motum rectum curvo posse comparari etiam disentiente Aristotele”, 194–5) bases his refusal of the Aristotelian distinction on other arguments. He does not display knowledge of this theorem, even having read Copernicus.

Ragep , op. cit. (ref. 5), 333–4.

Ibid. , 329, 331.

Ragep F. J. , “Cosmography in the ‘Tadhkira’ of Naṣīr al-Dīn al-ṭūsī” (Ph. D. dissertation, Harvard University, 1982); idem, op. cit. (ref. 5); idem, Naṣīr al-Dīn al-ṭūsī's Memoir on astronomy (al-Tadhkira fi 'ilm al-hay'a) (New York, 1993); Saliba , op. cit. (ref. 1).

Ragep , op. cit. (ref. 5), 330.

Saliba , op. cit. (ref. 1), 84–86.

Saliba G. , “Theory and observation in Islamic astronomy: The work of Ibn al-Shāṭir of Damascus”, Journal for the history of astronomy , xviii (1987), 35–43; idem, op. cit. (ref. 1), 82–83.

Ragep , op. cit. (ref. 5), 332.

Saliba , op. cit. (ref. 5), 13–15.

Ragep , op. cit. (ref. 5), 342.

Saliba G. Kennedy E. S. , “The spherical case of the ṭūsī couple”, Arabic sciences and philosophy , i (1991), 285–91, p. 285.

If the expression “curvilinear version” of Ragep does not immediately communicate the idea that one is dealing with a great circle arc on a sphere, the formula “spherical version” of Saliba and Kennedy seems to imply that the other is a plane version; whereas both are spherical, even if the proceeding one has parallel axes and the motion functions as if it were in the plane.

Saliba Kennedy , op. cit. (ref. 24) display the spheres from the internal to the external, but this is very odd, because in all the models of this type one always specifies that the motions are transmitted from the external to the internal — which is obvious, given that all of them originate from the prime mover.

Ragep , op. cit. (ref. 5), 348.

Saliba Kennedy , op. cit. (ref. 24), 291.

In the taḥīr, there is no mention of solid spheres, cf. Saliba, op. cit. (ref. 5), 14.

Saliba , op. cit. (ref. 1), 81.

Saliba Kennedy , op. cit. (ref. 24), 286.

Regarding the succession proposed, it must be remembered that, as mentioned earlier (ref. 5), the dating of the Ḥall is still uncertain. Therefore the reconstruction we propose of the evolution of ṭūsī's thought is at least partly hypothetical. Copernicus's sequence may not be opposed to ṭūsī's, should the Ḥall be earlier than the taḥrīr al-majisṭī, but it still remains different to it.

Prowe L. , Nicolaus Coppernicus (Berlin, 1883–84; reprinted Osnabrück, 1967), 197–8. “It is still possible that such a motion be composed from two spheres. Since these are concentric, [the higher] one carries around the inclined poles of the other, and the lower revolves the poles of the sphere carrying the epicycles in the direction opposite to [the motion of] the higher sphere with twice the velocity. And the poles [of the sphere carrying the epicycles] are inclined from the poles of the next higher sphere by the same amount that the poles of this sphere are inclined from the poles of the highest sphere” (Swerdlow, op. cit. (ref. 1), 483).

It diverges because of the lack of use of the third sphere that we will speak of later.

Prowe , op. cit. (ref. 33), 201. “This is necessarily brought about by two small nested spheres with axes parallel to the axis of the sphere provided: That the center of the larger epicycle or of the whole of this (sic) be distant as far from the center of the small sphere directly connecting [with it] as the center of this small sphere is from the center of the outer small sphere …, and that the motion of the outer small sphere complete two revolutions in a tropical (sic!) year while the inner small sphere, which has a motion in the opposite direction with twice the number of returns, turn around four times in the same period” (Swerdlow, op. cit. (ref. 1), 503).

Swerdlow , op. cit. (ref. 1), 504–5. In Di Bono, op. cit. (ref. 4), 40, n. 61, we claimed that Copernicus used only the version with equal radii, either the plane or spherical version as the case may have been.

Nicolai Copernici Torinensis De revolutionibus orbium coelestium libri VI (Nuremburg, 1543), III, 4, f. 67r.

One is dealing with a simple harmonic motion.

Hartner , “Trepidation” (ref. 1), 619.

Whether he is able to carry out his intentions is another question. On this argument see for example Moesgaard K. P. , “The 1717 Egyptian years and the Copernican theory of precession”, Centaurus , xiii (1968), 120–38; Hartner , “Trepidation” (ref. 1), 619–22; Swerdlow Neugebauer , op. cit. (ref. 1), 129–48.

In the Commentariolus the precession was regular and so it did not require this particular mechanism.

Swerdlow Neugebauer , op. cit. (ref. 1), 47. Since the Commentariolus is far more concerned with describing the physical models, while De revolutionibus with the mathematics that follows from those models, me fact that De revolutionibus is silent on this point does not imply, according to Swerdlow, that the underlying physical models have been changed or replaced, unless there are notable differences. However, regarding the reciprocation device and the models that make use of it, we believe that some differences can be found, though direct references in this sense are lacking.

We agree with Moesgaard , op. cit. (ref. 40), 129–30, on this point.

Also for Mercury we do not see the motive for returning to the spherical version with parallel axes present in the Commentariolus, one that moreover does not appear elsewhere, when Copernicus could have easily used the plane version of De revolutionibus, that in this case did not even need to be approximated. For a deep analysis of Copernicus's models of Mercury see Swerdlow N. M. , “Copernicus's four models of Mercury”, Studia Copernicana , xiii (ref. 1), 141–55.

On Amico see Swerdlow , op. cit. (ref. 3); Di Bono , op. cit. (ref. 4); idem, “Il modello omocentrico di Giovan Battista Amico”, Rinascimento , 2nd ser., xxxii (1992), 275–89; Peruzzi E. , “Un contemporaneo di Telesio: Il cosentino Giovan Battista Amico e la teoria delle sfere omocentriche”, in Bernardino Telesio e la cultura napoletana , ed. by Sirri R. Torrini M. (Naples, 1992), 241–56.

Ioannis Baptistae Amici Cosentini opusculum de motibus corporum coelestium iuxta principia peripatetica, sine eccentricis et epicyclis, denuo aeditum odditis nonnullis quae rem locupletiorem facilioremque reddunt (Venice, 1537), ff. 11r–14r; now also in Di Bono, op. cit. (ref. 4), 84–85, 159–69.

On this question cf. Di Bono, op. cit. (ref. 4), 82–84; idem, op. cit. (ref. 45), 280–2.

Note 16 in Di Bono , op. cit. (ref. 4), 86 was incorrect. The correct figure and direction of rotation are furnished by Swerdlow, op. cit. (ref. 3), 41.

Amico , op. cit. (ref. 46), ff. 14r–15r; now also in Di Bono , op. cit. (ref. 4), 86, 169–71.

Ragep , op. cit. (ref. 5), 334–5, 347–8.

Amico , op. cit. (ref. 46), ff. 26r–28r; now also in Di Bono , op. cit. (ref. 4), 113–15, 196–9. Peurbach G. , Novae theoricae planetarum Georgii Peurbachii astronomi celeberrimi … a Petro Apiano … adomnem veritatem redactae, & eruditis figuris illustratae (Venice, 1534), ff. 34r–39r (i.e. 40r).

Di Bono , op. cit. (ref. 4), 70–71.

Hieronymi Fracastorii Homocentrica (Venice, 1538), f. 61v.

Swerdlow , op. cit. (ref. 1), 469.

Swerdlow Neugebauer , op. cit. (ref. 1), 47. Saliba , op. cit. (ref. 1), 78, underlines the importance of this debt, that not only consists in the use of ṭūsī's device and 'Urḍī's lemma — a developed form of Apollonius's theorem that permits the transformation of eccentric models into epicyclic ones — but in the fact that he uses them in the same points of the models.

Hartner , “Trepidation” (ref. 1), 610.

This fact, however, does not prevent the work from remaining unknown to Shams al-Dīn al-Khafrī, who in the sixteenth century pursued his criticism of the Ptolemaic astronomy in the tradition of the Marāgha school and produced alternative planetary models using ṭūsī's device. Cf. Saliba G. , “A sixteenth-century Arabic critique of Ptolemaic astronomy: The work of Shams al-Dīn al-Khafrī”, Journal for the history of astronomy , xxv (1994), 15–38, esp. p. 34.

Roberts , “The solar and lunar theory” (ref. 1), 432.

Neugebauer O. , A history of ancient mathematical astronomy (New York, 1975), 1035, 1456; Swerdlow Neugebauer , op. cit. (ref. 1), 47–48.

Swerdlow Neugebauer , op. cit. (ref. 1), 48. If this were the case, however, one cannot understand why these models did not appear in the works of other authors.

Hartner , “Trepidation” (ref. 1), 616–19; idem, “Copernicus” (ref. 1), 421.

It is true that we now know of the existence of a plane version with equal radii, also in ṭūsī, but if we want to argue about some given fact and not simply about hypotheses, we can, for the moment, leave it, since Hartner's proof is based on Tadhkira's figure.

Agazzi E. , (ed.), Storia delle scienze (Rome, 1984), i, 9; Di Bono , op. cit. (ref. 45). 280–2.

Veselovsky I. N. , “Copernicus and Naṣimacr;r al-Dīn al-ṭūsī”, Journal for the history of astronomy , iv (1973), 128–30.

Procli Diadochi Lycii philosophi Platonici ac mathematici probatissimi in primum Euclidis Elementorum librum commentariorum … libri IIII a Francisco Barocio … recens editi (Padua, 1560), 61; Proclus , Les commentaires sur le premier livre des Elements d'Euclide , transl. and ed. by Ver Eecke P. (Bruges, 1948), 95–96 and n. 4.

Copernicus , op. cit. (ref. 37), V, 25, f. 164v. Neugebauer O. , “On the planetary theory of Copernicus”, Vistas in astronomy , x (1968), 89–103 (now reprinted in Neugebauer O. , Astronomy and history: Selected essays (New York, 1983), 491–505), 99, in an article on Copernicus of a purely mathematical nature, cites indifferently the form imagined by Proclus or by ṭūsī for this motion, and it seems that he was the first to do so.

Nicolai Copernici De revolutionibus codicis propria auctoris manu scripti imago phototypa (Warsaw and Cracow, 1973), 75r.

On this point see Swerdlow , op. cit. (ref. 44), 146, n. 5 and 155, n. 8. On Copernicus's library see Jarzebowski L. , Biblioteka Mikolaja Kopernika (Torun, 1971), 74.

Marciana Library (Venice), Cod. Gr.Z.306 (coll. 1026).

Swerdlow takes no account of the possibility of a derivation of ṭūsī's device from Proclus, and thinks that Rheticus himself, after reading De revolutionibus and writing the Narratio prima, may have informed Copernicus of the passage, hence the addition in the manuscript.

Cf. Veselovsky , op. cit. (ref. 64); Neugebauer , op. cit. (refs 59 and 66).

Copernicus, among other things, would have had to follow the lessons of Alessandro Achillini, one of the supporters of the homocentric model, even though not one of the most significant. Cf. Copernico N. , Opere, ed. by Barone F. (Turin, 1979), 42, n. 85, and 75; Di Bono , op. cit. (ref. 4), 62–65.

Swerdlow N. M. , rev. of Le sfere omocentriche di Giovan Battista Amico nell'astronomia del Cinquecento, by Di Mario Bono , Journal for the history of astronomy , xxiii (1992), 211–15, p. 213; Fracastoro , op. cit. (ref. 53), ff. 60v–61v.

Starting from Birkenmajer's L. A. studies, Mikolaj Kopernik (Cracow, 1900), Stromata Copernicana (Cracow, 1924), and also from Commentariolum super Theoricas novas planetarum Georgii Purbachii in Studio generali Cracoviensi per Mag. Albertum de Brudzewo diligenter corrogatum, ed. by Birkenmajer L. A. (Cracow, 1900), Rosinska G. , “Naṣīr al-Dīn al-ṭūsī and Ibn al-Shāṭir in Cracow?”, Isis , lxv (1974), 239–43, puts forward the hypothesis of a Cracovian influence on Copernicus instead of an Arab one, since “Adalbertus of Brudzewo knew the composition of circular movements into a linear one and while discussing the theory of the moon he mentions three times a geometrical construction including an additional epicycle” (p. 240). It does not seem to us however that Albert of Brudzewo's assertions on linear motion have to do with Copernicus's demonstration, nor his second lunar epicycle with Copernicus's. Albert's added epicycle served, in fact, to explain why one always sees the same face of the Moon, whereas in the models of Marāgha and of Copernicus it served for a completely different purpose. For the arguments that supporters of the homocentric systems used, based on this phenomenon, against the existence of the epicycles see for example Amico, op. cit. (ref. 46), f. 8v, now also in Di Bono , op. cit. (ref. 4), 80, 147–9; Fracastoro , op. cit. (ref. 53), f. 69 [66]v; Lattis J. M. , “Homocentrics, eccentrics and Clavius's refutation of Fracastoro”, Physis , xxviii (1991), 699–725, pp. 720–2.

Swerdlow , op. cit. (ref. 1), 504.

Unless, as we mentioned above, he realized the defects of the version with oblique axes, something which however is not valid for the version with parallel axes.

The high number of coincidences between Copernican and Arab models is usually considered to be an argument and is taken as evidence of the derivation of Copernicus's models from Arab sources. But if this is the case, it becomes very difficult to explain how such a quantity of models and information, which Copernicus would derive from Arab sources, has left no trace — apart from ṭūsī's device — in the works of the other western astronomers of the time.

As Neugebauer, op. cit. (ref. 66), 90, writes: “The basic identity of the Copernican methods with the Islamic ones needs no special emphasis in each individual case. The mathematical logic of these methods is such that the purely historical problem of the contact or transmission, as opposed to independent discovery, becomes a rather minor one”.

We know that Amico knew Latin, Greek and Hebrew.

Ragep , op. cit. (ref. 5), 348.

Di Bono , op. cit. (ref. 4), 43.

Amico , op. cit. (ref. 46), f. 8r; now also in Di Bono , op. cit. (ref. 4), 80, 147.

Peruzzi E. , “Note e ricerche sugli “Homocentrica” di Girolamo Fracastoro”, Rinascimento , 2nd ser., xxv (1985), 247–68, pp. 252–3.

Ibid. , 253.

Fracastoro , op. cit. (ref. 53), III, 25, f. 61v.

From this point of view, one can consider a less original form with regard to the spherical version of ṭūsī with parallel axes and radii in the ratio of 1:2, which surely cannot be a re-elaboration taken from Eudoxus.