Regiomontanus's Concentric-Sphere Models for the Sun and Moon

The Tabulae primi mobilis and the sine tables have been carefully analysed by Glowatzki E. Göttsche H. , Die Tafeln des Regiomontanus: Ein Jahrhundertwerk . Algorismus. Heft 2 (Munich, 1990). In the former, they find 1072 errors of ±1‘, 48 of ±2’, 4 of ±3‘, 2 of ±4’, 1 of ±5’. The sine table for R = 10,000,000 contains 1820 errors of ±1, 1 of +2, and 12 of −2 in 5400 entries. Is there anyone who could do that today?.

The best guide to the Tabulae primi mobilis, although they are nowhere mentioned, is the edition of Johann Werner's De meteoroscopiis libri sex by Björnbo A. and, following his death, J. Würschmidt in Abhandlungen zur Geschichte der mathematischen Wissenschaften , xxiv/2 (1913). The reason is that Werner, using his instrument, the meteoroscope, which he also calls a saphea, a universal astrolabe containing stereographic projection of circles of longitude and latitude, provides solutions for all of the problems in the Tabulae primi mobilis, and a good many more besides. It is a very interesting work, very well edited, and should be better known. A description and illustration of such an instrument may be found in North John , Horoscopes and history (Warburg Institute Surveys and Texts 13; London, 1986), 67–69. Obviously, the precision of using the Tabulae primi mobilis far exceeds that of any instrument that could be made.

Nuremberg Cent V app. 56c, f. 39v:

Aequationem octauae spherae secundum Alfonsi fundamenta numerare.

Aequationem octauae spherae secundum imaginationem Tebith computare.

Aequationem solis colligere.

Aequationem argumenti lunae dinumerare.

Sed quo ruit calamus ille molestus atque audax? Forsitan omnia astronomorum quaesita ad hanc unicam tabulam appellet? Dico ego bonam partem huiusmodi quaesitorum per hanc repertum iri tabulam, si prius concentricam astronomiam totam fundauerimus. Quid illud? Diuersitates motuum planetarum per concentricos saluare pulcrum erit. Iam soli et lunae uiam dedimus, de reliquis autem quaedam initialia iacta sunt, quibus completis aequationes omnium planetarum per hanc tabulam numerare licebet. De hac re nihil amplius impraesentiarum, ne legendi scripta mea maius patiamini fastidium quam ego scribendo uoluptatem habeam. Si quid harum rerum audentius aequo dixisse uideor, posteris litteris rationes meas luculentius (sic puto) accipietis. Mallem tamen uoce quam calamo hisce de rebus disserere, facilius enim multo et expeditius foret. Dum id fieri nequit, littere uoci officia sumant, [margin: Quae, quicquid afferent, uestro iudicio limandum erit].

Curtze's M. text in Abhandlungen zur Geschichte der mathematischen Wissenschaften , xii (1902), 218, is, as usual, faulty. While mine may not be perfect, it is at least better and intelligible. For this passage, I have not checked the editions of von Murr (1786) and Magrini S. , Atti e memorie della deputazione ferrarese di storia patria , xxii/3 (1917), which are always preferable to Curtze's. Regiomontanus follows this with the, possibly deleted, remark that he intends to go to Milan on some business, and it appears that he did, for the following spring he reports in his oration on the mathematical sciences delivered in Padua that he saw Giovanni de Dondi's astrarium kept safely by the Duke of Milan in his Castle of Pavia.

Zinner E. , “Neue Regiomontan-Forschungen und ihre Ergebnisse”, Sudhoff's Archiv , xxxvii (1953), 107–8. Zinner again mentions the letter in Leben und Wirken des Joh. Müller von Königsberg genannt Regiomontanus 2nd edn (Osnabrück, 1968), 151, and the remarks in the letter to Bianchini, 101. There is also a reference by Grossing H. , “Regiomontanus und Italien”, Regiomontanus-Studien , ed. by Hamann G. , Sitzungsberichte der österreichische Akademie der Wissenschaften , Phil.-hist. Kl., ccclxiv (1980), 223–4, p. 234, and discussions by Gerl A. , Trignometrisch-astronomisches Rechnen kurz vor Copernicus (Boethius 21; Stuttgart, 1989), 210–13, and “The most recent results of research on Regiomontanus”, in Zinner E. , Regiomontanus: His life and work , transl. by Brown E. (Amsterdam, 1990), 335–8. I would politely suggest that not one of these accounts is based upon a study of the text or can withstand scrutiny.

In the heading on f. 18r, the work is even addressed to Ilkusch: “Johannes Germanus ad Martinum Ilkusch Cracoviensis in theoricas veteres a Gerardo aiunt Cremonensis editas.”.

Carmody F. J. , “Regiomontanus' Notes on al-Biṭrūjī's astronomy”, Isis , xlii (1951), 121–30, which Carmody wrote in connection with his edition of Michael Scot's Latin translation, Al-Bitrūjī, De motibus celorum (Berkeley, 1952), and “The planetary theory of Ibn Rushd”, Osiris , x (1952), 556–86.

Shank M. H. , “The ‘Notes on al-Biṭrūjī’ attributed to Regiomontanus: Second thoughts”, Journal for the history of astronomy , xxiii (1992), 15–30. The eclipses are those of 17 June 1433, which seems to have been total, not annular, and 31 January 1310, which was annular and was dated by Marcho to 1309 since the year in Paris began at Easter. Shank identifies Marcho G. with the Francisan Guy de la Marche. See also Shank M. H. , “Regiomontanus and homocentric astronomy”, Journal for the history of astronomy , xxix (1998), 157–66. I wish to thank Michael Shank and Richard Kremer, who are engaged in a far more extensive study of Regiomontanus's interest in concentric-sphere models, for encouraging me to retrieve my transcription and translation and write this paper.

Zinner , Regiomontanus (ref. 4), 61; Shank , op. cit. (ref. 7), 17.

Goldstein B. R. , Al-Biṭrūjī: On the principles of astronomy (2 vols, New Haven, 1971), 7–12.

Shank , op. cit. (ref. 7), 25–26.

Swerdlow N. M. , “Regiomontanus on the critical problems of astronomy”, Nature, experiment, and the sciences , ed. by Levere T. H. Shea W. R. (Dordrecht, 1990), 165–95, pp. 173–4. Let me correct a mistranslation in this paper that has bothered me for years. On p. 172, paragraph 3, lines 2–3, for “familiarly known” substitute “obstinate” (inveterati, i.e. “chronic”). Then on p. 183, paragraph 2, delete the first sentence. Regiomontanus's point is that Jupiter and Saturn do not present the long-standing problems of Mars.

If one asks why Regiomontanus did not simply put the mean sun at K’ and the true Sun at L’, with ZCK’ in advance of λ_A by &αbar;, the answer must be that ZK’ and ZL’ are not physical or structural quadrants, as KS̄ and LS, but just a way of measuring the equation δ in the ecliptic; and such an arrangement would not be a correction or modification of Biṭrūjī's model, in which the quadrant from the epicycle to the Sun, Moon or planet is essential.

This is the solution in De triangulis omnimodis IV, 28, for two sides and an included angle, adapted to an exterior angle; V, 2 is equivalent to the law of cosines, although using versed sines, but is not in a form suitable to this problem and one does not know whether Regiomontanus even knew it when he wrote this work.