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Notes
A chair of mathematics and astronomy was founded at Cracow University in the first decade of the 15th century by a certain Stobnerus, citizen of Cracow. The document of the foundation (a copy from 1472) is still preserved in the Archives of the Jagellonian University, ms. 44, f. 45r. Professor “Stobnerianus” was responsible for mathematical and astronomical formation of students of the quadrivium, particularly for their knowledge of astronomical tables and their skills in predicting eclipses of the Sun and Moon: “... collegiatus domini Stobneri duos actus faciat. Pro uno legat in mathematica hoc ordine, videlicet Euclidem, perspectivam, arismetricam et musicam et Theoricam planetarum, demum Tabulas Alphoncii, premisso Algorismo minuciarum. [...] Pro secundo actu practicet et publicet notoria [...] eclipses, almanach et minuciones [sanguinis] pro honore Universitatis”. Cf. ms. in the Archiwum Uniwersytetu Jagiellońskiego, nr. 68, p. 18. The collection of astronomical manuscripts, still preserved in the Jagellonian Library, permits us to reconstruct the activity of Cracow mathematicians and astronomers, from as early as the end of the 14th century. It was the subject of studies exemplified by the editions (or at least partial editions) of mathematical and astronomical texts by L. A. Birkenmajer, “Krakowskie tablice syzygiów dla r. 1379 i 1380. Przyczynek do dziejów astronomii w Polsce w XIV wieku”, in Rozprawy Akademii Umiejętności, Wydział matematyczno-przyrodniczy, seria II, t.1. (Kraków, 1891), pp. 261–285. J. Dobrzycki, “New sources for the history of calendar reform”, in Proceedings No. 2. XIV International Congress of the History of Science, Tokyo & Kyoto, 1974, pp. 35–36. J. Dobrzycki, “The Tabulae resolutae”, in M. Comes, R. Puig, and J. Samsó (eds.), De astronomia Alphonsi Regis (Barcelona, 1987), pp. 71–77. R. L. Kremer and J. Dobrzycki, “Alphonsine Meridians: Tradition Versus Experience in Astronomical Practice c. 1500”, Journal for the History of Astronomy 29(2):187–199. G. Rosińska, “Une table astronomique de Laurent de Raciborz. Le commentaire qui l’accompagne”, Mediaevalia Philosophica Polonorum XIX:141–147 (1974) (there are some doubts concerning the attribution to the table “Radices ad meridianum Cracoviensem A.D. 1420 completo et valent ad tabulas sequentes pro instrumentis Campani” to Laurent of Raciborz, but there is no doubt that this table was compiled in Cracow about 1420). G. Rosińska, “Sandivogius de Czechel et l’école astronomique de Cracovie vers 1430”, Organon 9:217–229 (1973). G. Rosińska, “Instrumenty astronomiczne na Uniwersytecie krakowskim w XV wieku” (“Astronomical instruments at Cracow University in the 15th century”). Studia Copernicana, Vol. XI. Wrocaw, Ossolineum 1974 (in Polish and Latin, English summary).
G. F. Vescovini, “Bianchini Giovanni”, in Dizionario Biografico Degli Italiani, Vol. X (1968), pp. 194–196, where the references to manuscript sources and early prints are given.
L. Thorndike, “Giovanni Bianchini in Paris Manuscripts”, Scripta Mathematica, Vol. 16, 1950, pp. 5–12 and 169–180. L. Thorndike, “Giovanni Bianchini in Italian manuscripts”, Scripta Mathematica, Vol. 19, 1953, pp. 5–13. E. Poulle, La bibliothèque scientifique d’un imprimeur humaniste au XV e siècle. Catalogue des manuscrits d’Arnaud de Bruxelles à la Bibliothèque Nationale de Paris (Librairie Droz: Genève, 1963), pp. 38–44, pp. 59–60, pp. 73–74. G. Rosińska, Scientific Writings and Astronomical Tables in Cracow. A Census of Manuscript Sources. Studia Copernicana, vol. XXII, Wrocław 1984. Nrs. 218, 429, 485, 875, 1702 and II. 1–67. W. Kokott signals the adaptation of Bianchini’s planetary tables to the latitude of Leipzig (51°): “Syzygia as Pivots: An Unusual Mid-Fifteenth-Century Working Ephemeris”, Journal for the History of Astronomy 29:129.
Martinus’ Summa is preserved in Cracow, Jagellonian Library (Biblioteka Jagiellońska), ms. BJ 1927, ff. 250r–318r and in Oxford, Bodleian Library, ms. Can. misc. 499, ff. 222ra–258vb. Marcin Rex (Król) de Premislia (alias de Żurawica) see L. A. Birkenmajer, “Marcin Bylica z Olkusza oraz narzędzia astronomiczne, które zapisał Uniwersytetowi Jagiellońskiemu w roku 1493”. Kraków 1892, pp. 21–27 (Cap. III. Marcin z Żórawic, jeden z profesorów Marcina Bylicy). Z. Kuksewicz, “Marcin Król z Żurawicy. Stan Badań”, in Materiały i Studia Zakładu Historii Filozofii Starżytnej i Średniowiecznej PAN. Seria A: Materiały do historii filozofii sredniowiecznej w Polsce, Vol. I, pp. 118–140. Zathey, “Biblioteka Jagiellońska w latach 1364–1492”, in Historia Biblioteki Jagiellońskiej (Kraków, 1966), pp. 108–109 where a list of Martinus’ treatises preserved in the Jagellonian Library is given. J. Dianni, “Pierwszy znany traktat rękopiśmienny w literaturze matematycznej w Polsce. Algorismus minutiarum Martini Regis, de Premislia”. (“Algorismus minutiarum by Marcin Król”. in Polish, English summary), Kwartalnik Historii Nauki i Techniki (Quarterly Journal of the History of Science and Technology) 12:269–288 (1967, nr. 2. G. Rosińska, “Nieznany traktat astronomiczny Marcina Króla z Żurawicy” (“An unknown astronomical treatise by Martinus Król of Żurawica” (in Polish, Latin, English summary), Kwartalnik Historii Nauki i Techniki 17:227–233, nr. 2, and G. Rosińska, “Scientific Writings and Astronomical Tables in Cracow...” op. cit. nrs. 751, 767, 688, 1203, 1211, 1213, 1228, 2228, 2243. According to L. A. Birkenmajer, Martinuswas in Prague in 1445 and in Bologna in 1447–1449, where he taught astronomy in 1449: L. A. Birkenmajer, “Marcin Bylica z Olkusza oraz narzędzia astronomiczne które zapisał Uniwersytetow Jagiellońskiemu w roku 1493.” Kraków, 1892, pp. 22–27 and pp. 113–119. U. Dallari, “I Rotuli dei Lettori Legisti e Artisti dello Studio Bolognese dal 1384 al 1799”. Vol. 1, Bologna 1888, p. 26, col. 2: “Ad lecturam Astronomie: D. M. Johannes de Fondis arcium et medicine doctor, D. M. Martinus de Polonia arcium doctor”. Martinus died in Cracow in 1452, see M. Kowalczyk, “Przyczynki do biografii Henryka Czecha i Marcina Króla z Żurawicy”, in Biuletyn Biblioteki Jagiellońskiej. Vol. 22 (1971), pp. 87–91.
G. Rosińska, “A Chapter in the History of the Renaissance Mathematics: Negative Numbers and the Formulation of the Law of Signs (Ferrara, Italy ca. 1450)”. Kwartalnik Historii Nauki i Techniki 41:53–70 (1996), nr.3–4. G. Rosińska, “Decimal Positional Fractions. Their Use for the Surveying Purposes (Ferrara, 1442)”. Kwartalnik Historii Nauki i Techniki 40:17–31 (1995), nr. 4.
A. Gerl, “Trgonometrisch-astronomisches Rechnen kurz vor Copernicus. Der Briefwechsel Regiomontanus-Bianchini”. Boethius Bd. XXI, Stuttgart 1989, where the author seeks to present Bianchini as a (possible) partner of Regiomontanus in discussions of problems of the spherical astronomy. See p. 26, pp. 265–269, pp. 331–335.
L. A. Birkenmajer, Mikołaj Kopernik (Kraków 1900), pp. 21–11, p. 25, p. 28, pp. 60–64, p. 162, pp. 228–229.
J. L. Jervis, “Cometary Theory in Fifteenth-Century Europe”. (Studia Copernicana, vol. XXVI). Wrocław 1985. Appendix B: “Toscanelli’s Mathematical Computations”, pp. 162–169. M. Folkerts, “Regiomontanus als Mathematiker”. Centaurus 21(3–4):214–245 (1977). E. Glowatzki and H. Göttsche, “Die Tafeln des Regiomontans: Ein Jahrhundertwerk”. Algorismus, Heft 2, Münich 1990. W. Kaunzner, “Über Regiomontanus als Mathematiker”, in Wien (ed.), Regiomontanus-Studien, (1980), pp. 125–145. W. Kaunzner, “Zum Stand der Westeuropaeischen Mathematik zur Zeit der Entdeckung Americas”, in S. S. Demidov et al. (eds.), Amphora. Festschrift fur Hans Wussing zu seinem 65. Geburtstag. (Basel: Boston, Berlin, 1992), pp. 362–374. N. M. Swerdlow, “Regiomontanus’s Concentric-Sphere Models for the Sun and Moon”, Journal for the History of Astronomy 30(Part 1): 1–23.
G. Arrighi (ed.), “Piero della Francesca, Trattato d’abaco. Dal Codice Ashburnhamiano 280 (359*–291*) della Biblioteca Medicea Laurenziana di Firenze”, a cura e con introduzione di Gino Arrighi. Pisa, Domus Galilaeana 1970, where a bibliography of principal works by Arrighi is given on pp. 13–13, notes 9–10.
R. Franci and L. Toti Rigatelli, “Introduzione all’aritmetica mercantile del Medioevo e del Rinascimento”, Urbino 1982; L. Toti Rigatelli, “Matematici fiorentini del tre-quattrocento”, in Symposia mathematica, Vol. 27, 1968, pp. 3–67.
J. Sessiano, “On an Algorithm for the Approximation of Surds From a Provencal Treatise”, in C. Hay (ed.), Mathematics from Manuscript to Print (Oxford: Clarendon Press, 1988), pp. 30–55.
W. V. Egmond, “Practical Mathematics in the Italian Renaissance. A Catalog of Italian Abacus Manuscripts and Printed Books to 1600”, Supplemento agli Annali dell’Istituto di Storia delle Scienze di Firenze; Firenze 1980, fasc. 1.
E. Giusti, “L’algebra del Trattato d’abaco di Piero della Francesa: Osservazioni e Congetture”, Bollettino di Storia delle Scienze Matematiche Vol. XI, 1991, pp. 55–83. S. A. Jaywardene, “The Influence of Practical Arithmetic on the Algebra of Rafael Bombelli”, Isis 64:510–523 (1973).
G. Rosińska, “The ‘Fifteenth-Century Roots’ of Modern Mathematics. The Unit segment. Its Function in Bianchini’s De Arithmetica, Bombelli’s L’Algebra and Descartes’ La Géometrie”, Kwartalnik Historii Nauki i Techniki. Vol. 41:59–62 (1996), nr. 3–4.
Nicomachus of Gerasa “Introduction to Arithmetic”. Translated into English M. L. D’Ooge. With Studies in Greek Arithmetic by F. E. Robbins and L. C. Karpinski. (New York, 1926), p. 190.
Ms. BJ 1840, ff. 19r, 37r, 43r–44r.
“When integers are divided by fractions they are multiplied by unity in proportion to the divisor. When fractions are divided by integers they are multiplied by unity in proportion to the number divisor...” G. Rosińska, “The ‘fifteenth century roots’ of modern mathematics...” op. cit. pp. 58–60.
The dating of Bianchini’s Arithmetica to ca. 1440 results from the fundamental character of this work with the respect to other Bianchini’s mathematical and astronomical writings (in some of which the Arithmetica is quoted). Furthermore, Bianchini accquired his training in practical mathematics before 1427 (the year of Bianchini’s coming to Ferrara). As for the dating of Bianchini’s Compositio instrumenti, essential for the dating of the introduction of the arithmetic of decimal fractions into European mathematics, I refer to the arguments in favor of the year 1442 given in G. Rosińska, “Decimal positional fractions...” op. cit. pp. 28–29, note 5.
G. Rosińska, “Kwadratura kola i ‘liczba π’ w nauczaniu matematyki na Uniwersytecie krakowskim w pierwszej polowie XV wieku. Recepcja Archimedesa De mensura circuli poprzez Tomasza Bradwardina Geometria speculativa”. (“The quadrature of circle and ‘number π’ in the teaching of mathematics at Cracow university in the first half of the 15th century. Reception of Archimedes’ De mensura circuli by means of Thomas Bradwardinus’ Geometria speculativa”). In Polish and Latin. English summary. Kwartalnik Historii Nauki i Techniki. Vol. 42, nr. 2, 2000, pp. 49–62.
Treatise Circulus obliquus qui signifer nuncupatur dividitur in sex signa... is preserved in the Jagellonian mss.: BJ 459, ff. 3ra–4vb; BJ 563, ff. 240ra–242rb; BJ 602, ff. 131r–133r. All mss. come from the 20s of the 15th century.
BJ 1929, ff. 151r–183v.
Algorismus minutiarum: mss. BJ 1844, ff. 275r–295v; BJ 1859, ff. 41r–56r; BJ 1927, ff. 189r–211v. J. Dianni, “Pierwszy znany traktat rękopiśmienny...” op. cit. pp. 269–289. Geometria practica: mss. BJ 1865, ff. 39r–51v; BJ 1968, ff. 1r–10v. L. A. Birkenmajer [ed.], “Marcina Króla z Przemyśla Geometria praktyczna” (Warszawa, 1895).
All quotations of Martinus’ Summa come from the ms. BJ 1927.
From the point of view of paleography both readings, veritas and varietas, are admissible; veritas, however, seems to be justified by the context. I consulted also the ms. Cm 499 f. 241vb.
A similar text from the Flores Almagesti, ms. BJ 558, f. 17r was published by L. A. Birkenmajer, “Flores Almagesti. Ein angeblich verlorengegangener Traktat Giovanni Bianchini’s, Mathematikers und Astronomen von Ferrara aus dem XV. Jahrhundert” (Extrait du Bulletin de l’Académie des Sciences de Cracovie). Cracovie 1911. Birkenmajer was the first to consider Bianchini’s authorship of the sine table (R = 60.10), on the base of the 15th century note in the margin of the table in the ms. BJ 600 p. 268.
An historical perspective on the foundations of the “pragmatist” concept of real number, present in astronomy since Babylonian times, is given in N. Bourbaki, “Éléments d’histoire des mathématiques”. Nouvelle édition revue, corrigée et augmentée. (Paris: Hermann 1974), pp. 184–195, pp. 184–191.
Fragments of the Tables of tangent and cosecant was published by G. Rosińska, “Tables trigonométriques de Giovanni Bianchini”, Historia Mathematica. Vol. 8, 1981, pp. 49–50.
Ms. BJ 556, f. 5ra–7va: Addiciones Canonum primi mobilis ordinate per dominum Iohannem Blanchinum.
In fact, Stevin states in the title of his work (the Franch version, Leyde 1585) as follows: “La Disme. Enseignant facilement expedier par nombres entiers sans rompuz, tous comptes se rencontrans aux affaires des Hommes”. See E. J. Dijksterhuis’ comment on this subject in E. J. Dijksterhuis, “Simon Stevin. Science in Netherlands around 1600”, in R. Hooykas and M. G. J. Minnaert (eds.), The Hague (Martin Nijhoff, 1970), p. 19.
G. Bianchini, “Compositio instrumenti. (Cod. Lat. α. T. 6. 19) della Biblioteca Estense di Modena”. Acura di Paolo Garuti con introduzione di Gino Arrighi, in Rendiconti di Classe Lettere e Scienze Morali e Storiche Vol. 125(1), 1991, pp. 95–127. (Istituto Lombardo Accademia di Scienze e Lettere, Milano 1992).
Ms. Modena B.E. Lat. 145, f.6. Ed. P. Garuti, op. cit. p. 116. See G. Rosińska, “Decimal...” op. cit. p. 24.
Ms. Modena B.E. Lat. 145, f.6. Ed. P. Garuti, op. cit. p. 116. See G. Rosińska, Decimal...” op. cit. p. 24.
L. A. Birkenmajer, “Stromata Copernicana”. Kraków 1924, pp. 103–126.
Ms. BJ 1840, f. 58v.
Ms. BJ 1840, f. 1r: Algorismus novus de integris; f. 8r: Algorismus de minutiis vulgaribus; f. 10r: Algorismus de minutiis phisicalibus; 10v: De regula proportionum sive aliter Regula Mercatorum dicta; f.11v: Tercia pars est de proporcionibus. Unde quinque sunt genera proportionum; f.14r: Opus Algorismi iucundissimi; f. 19r: Algorismus novus de integis mgri Georgii Pawrbachii [!] Wienensis; f.37r: Algorismus novus de integris compendiose studioseque more Italorum compilatus; f.40v: Canon multiplicationis; Glogoviensis’ Comentary on Sacrobosco’s Omnia que a primeva...; f. 58r: Inquit Ptholomeus in sapienciis Almagesti [...] Astronomie studium super Arismetrice et Geometrie demonstracionibus certissimis est fundatum... with Glogoviensis’ Commentary on ff. 65r–69v; f. 80r: Ex quo artes mathematice propter difficultatem Arismetrice...; f. 92v: Enigmata; f. 94r: Tabula algoristica de numeris quadratis, cubicis et de eorum radicibus.
I have argued elswhere that the “fourth trigonometric table”, linked with the name of Copernicus because ascribed to him or to Rheticus—namely the large table of sines/cosines, appended to Copernicus’ “De lateribus et angulis triangulorum” [...] Vittembergae 1542—is essentially Regiomontanus’ Table, calculated by him on the R = 107 and one minute intervals. It was first published in 1541 (Norimbergae, apud Iohannem Petreium). A 15th century manuscript copy of this Table is still preserved in ms. BJ 606, f. 171r–180r, where its title runs as follows: “Tabula sinuum nova Bude confecta [?] per magistrum Johannem de Regio monte 1468”. See G. Rosińska, “Nie przypisujmy Rhetykowi dziela Regiomontana...”. Kwartalnik Historii Nauki i Techniki. Vol. 28:615–619, 1983.
E. Rosen: “Copernican scholars have not yet discovered what model, if any, Copernicus followed in transforming Ptolemy’s sexagesimal Table of Chords into an early form of the Modern Table of Natural Sines”, in Nicholas Copernicus Complete Works. Vol. I. “On the Revolutions...” Ed. J. Dobrzycki, Translation and Commentary by E. Rosen. Warsaw-Cracow 1978, p. 363.
N. Copernicus. Complete Works. Vol. IV. “The Manuscripts of Nicholas Copernicus’ Minor Works. Facsimiles”. Warsaw-Cracow 1992, plate XXX. 62. See also P. Czartoryski, “The Library of Copernicus”, in Science and History. Studia in Honor of Edward Rosen. (Studia Copernicana Vol. XVI) (Wrocław, 1978), p. 366.
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Rosińska, G. (2006). “Mathematics for Astronomy” at Universities in Copernicus’ Time: Modern Atitudes Toward Ancient Problems. In: Feingold, M., Navarro-Brotons, V. (eds) Universities and Science in the Early Modern Period. Archimedes, vol 12. Springer, Dordrecht . https://doi.org/10.1007/1-4020-3975-1_2
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