Salta al menu principale di navigazione Salta al contenuto principale Salta al piè di pagina del sito

Focus

V. 23 N. 1 (2026)

Johann Scheubel, Wilhelm Xylander e il trattamento numerico della geometria euclidea nell’Europa della prima età moderna

DOI
https://doi.org/10.57617/gal-103
Inviata
novembre 18, 2025
Pubblicato
2026-05-22

Abstract

I commenti latini e tedeschi ai primi sei libri degli Elementi di Johann Scheubel (1494–1570), professore all'Università di Tübingen, e di Wilhelm Xylander (1532–1576), professore all'Università di Heidelberg, si distinguono nella tradizione euclidea del XVI secolo per il loro ampio uso dell'aritmetica e dell'algebra cossica (nella tradizione di Christoph Rudolff e Michael Stifel) nelle rispettive esposizioni delle proposizioni geometriche di Euclide, un’impostazione connessa soprattutto al loro retroterra protestante e al contesto pedagogico in cui operavano. Tale approccio divenne più frequente nella tradizione euclidea del XVII secolo in tutta Europa, in particolare nell'Europa settentrionale, e si sviluppò per soddisfare il programma pedagogico del crescente numero di professori di matematica e per integrare i nuovi sviluppi della disciplina. Concentrandosi principalmente sui commenti di Scheubel e Xylander, questo articolo mira a offrire un quadro dell'evoluzione dell'approccio numerico alla geometria euclidea nell'Europa dell'inizio dell'età moderna.

Riferimenti bibliografici

  1. Manuscript sources
  2. Biblioteca Apostolica Vaticana, Pal. Lat. 1350, 1–320. https://doi.org/10.11588/diglit.12343
  3. Printed sources
  4. Alingham, William. Geometry epitomiz’d: being a compendious collection of the most useful propositions in the first, third, fourth, fifth and sixth books of Euclid. Together with their uses, in several practical parts of the mathematicks. Also, Euclid’s second book and doctrine of proportion algebraically demonstrated. With some of the most useful problems required in practise. London: J. Moxon and B. Beardwell, 1684.
  5. Axworthy, Angela. Le Mathématicien renaissant et son savoir. Le Statut des mathématiques selon Oronce Fine. Paris: Classiques Garnier, 2016.
  6. Axworthy, Angela. “The hybridization of practical and theoretical geometry in the sixteenth-century Euclidean tradition.” Journal of Interdisciplinary History of Ideas 11/22 (2022), 4:1–104. https://doi.org/10.13135/2280-8574/7333
  7. Axworthy, Angela. “Renaissance approaches to the terminology of mathematics.” Le Français Préclassique 26 (2024): 61–68.
  8. Axworthy, Angela. “Mathematics in the Accademia del Cimento: from a language of Nature to a language of Reason.” Physis 59/2 (2024): 463–499. 10.1400/298756
  9. Baur, Ludwig. Dominicus Gundisalvi. De divisione philosophiae. Münster: Aschendorff, 1903.
  10. Betsch, Gerard. “Praxis geometrica und Kartographie and der Universität Tübingen um 16. und frühen 17. Jahrhundert.” In Zum 400. Geburtstag von Wilhelm Schickard: zweites Tübinger Schickard-Symposium, edited by Friedrich Seck, 185–226. Thorbecke: Sigmaringen, 1995.
  11. Betsch, Gerard. “Südwestdeutsche Mathematici aus dem Kreis um Michael Mästlin.” In Der „mathematicus“: zur Entwicklung und Bedeutung einer neuen Berufsgruppe in der Zeit Gerhard Mercators, edited by Irmgard Hantsche, 121–150. Bochum: Brockmeyer, 1996.
  12. Beutel, Tobias. Geometrischer Lust-Garten. Darinnen die edele und höchstnützliche schöne Kunst Geometria, Aus den Euclide gepflantzet. Leipzig: Christian Michael, 1660.
  13. Billingsley, Henry. The Elements of Geometrie of the most auncient Philosopher Euclide of Megara Faithfully (now first) translated into the Englishe toung, by H. Billingsley, Citizen of London. Whereunto are annexed certaine scholies, Annotations, and Inventions, of the best Mathematiciens, both of time past, and in this our age. London: John Daye, 1570.
  14. Bos, Henk. Redefining Geometrical Exactness Descartes’ Transformation of the Early Modern Concept of Construction. New York: Springer, 2001.
  15. Brunn, Lucas. Euclidis Elementa Practica, Oder Außzug aller Problematum und Handarbeiten auß den 15. Büchern Euclidis. Nürnberg: Simon Halbmayer, 1625.
  16. Cajori, Florian. A History of Mathematical Notations, vol. 1. Chicago: The Open Court, 1928.
  17. Camerarius, Joachim. Εὐκλείδου στοιχείων βίβλια ἓξ. Euclidis elementorum geometricorum libri sex conversi in latinum sermonem à Ioach. Camerario. Leipzig: Valentinus Papa, 1549.
  18. Campanus de Novara. Preclarissimum opus elementorum Euclidis megarensis una cum commentis Campani perspicacissimi in artem geometriam. Venezia: Erhard Ratdolt, 1482.
  19. Cataldi, Pietro Antonio. I primi sei libri de gl’Elementi d’Euclide ridotti alla Prattica, dove si mostrano le Inventioni delle Regole Geometriche, & Algebratiche necessarie, & di continuo uso. Bologna: Sebastiano Bonomi, 1620 [First ed. 1613].
  20. Clavius, Christoph. Euclidis elementorum libri XV, accessit XVI de solidorum regularium comparatione, omnes perspicuis demonstrationibus, accuratisque scholijs illustrati. In Opera omnia, vol. 1. Mainz: Anton Hierat, 1611–1612 [First ed. 1574].
  21. Commandino, Federico. Euclidis Elementorum libri XV. Unà cum scholijs antiquis. Pesaro: Camillo Francischino, 1572.
  22. Corry, Leo. “Geometry and arithmetic in the medieval traditions of Euclid’s Elements: A view from Book II.” Archive for History of Exact Sciences 67 (2013): 637–705. https://doi.org/10.1007/s00407-013-0121-5
  23. Dasypodius, Conrad. Εὐκλείδου τῶν πέντε καὶ δέκα Στοιχειῶν, ἐκ τῶν τοῦ Θέωνος συνουσιῶν τὸ πρῶτον. Euclidis quindecim elementorum geometriae primum: ex Theonis commentariis Graecè, & Latine. Cui accesserunt Scholia, in quibus quæ ad percipienda geometriae elementa spectant, breviter & dilucide explicantur. Strasbourg: Christianus Mylius, 1564.
  24. Dasypodius, Conrad. Euclidis quindecim elementorum Geometriae secundum: ex Theonis commentarijs Graecè, & Latinè. Item, Barlaam monachi Arithmetica demonstratio eorum, quae in secundo libro elementorum sunt in lineis & figuris planis demonstrata. Strasbourg: Christianus Mylius, 1564.
  25. Day, Mary S., Scheubel as an Algebraist, Being a Study of Algebra in the Middle of the Sixteenth Century, Together with a Translation of and a Commentary upon an Unpublished Manuscript of Scheubel’s Now in the Library of Columbia University, Columbia University. New York: AMS Press, 1972.
  26. De Risi, Vincenzo. “Euclid upturned: Borelli on the foundations of geometry.” Physis 57/2 (2022): 1–23. https://www.doi.org/10.1400/290985
  27. Digges, Leonard and Thomas. A geometrical practise, named Pantometria divided into three bookes, longimetra, planimetra, and stereometria, containing rules manifolde for mensuration of all lines, superficies and solides, with sundry straunge conclusions both by instrument and without, and also by perspective glasses, to set forth the true description or exact plat of an whole region. London: Henry Bynneman, 1571.
  28. Dou, Jan Pieterszoon. De ses eerste Boucken Euclidis, Van de beginselen ende fondamenten der Geometri. Waer by gevoecht sijn eenige nutticheden, uyt de selve Boecken ghetrocken; Mitsgaders de Specien in Geometrische figuren, als ‘tmaecken, veranderen, ‘t samenvougen, aftrecken, vermenichvuldigen, ende deelen. Leiden: Orlers, 1607.
  29. Drüll, Dagmar. Heidelberger Gelehrtenlexikon 1386–1651. Berlin; Heidelberg: Springer, 2002. https://doi.org/10.1007/978-3-642-56189-4
  30. Dybvad, Christoffer. In Geometriam Euclidis prioribus sex Elementorum libris comprehensam Demonstratio Numeralis. Arnhem; Leiden: Christopher Guyot, 1603.
  31. Fine, Oronce. Protomathesis opus varium, ac scitu non minus utile quàm iucundum, nunc primùm in lucem foeliciter emissum. Paris: Gerard Morrhe and Jean Pierre, 1532.
  32. Fine, Oronce. In sex priores libros geometricorum elementorum Euclidis Megarensis demonstrationes. Paris: Simon de Colines, 1536.
  33. Frischlin, Nicodemus, and Wilhelm Xylander. Carmen de Astronomico Horologio Argentoratensi. Strasbourg: Nicolas Wyriot, 1575.
  34. Gall, Dorothee. “Xylander, Guilielmus.” Brill’s New Pauly Supplements I Online, vol. 6. https://doi.org/10.1163/2214-8647_bnps6_COM_00765
  35. Gestrinius, Martin. In geometriam Euclidis demonstrationum libri sex. Uppsala: Aeschillus Matthiae, 1637.
  36. Giusti, Enrico. Euclides Reformatus. La teoria delle proporzioni nella scuola galileiana. Torino: Bollati Boringhieri, 1993.
  37. Grendler, Paul. “The Universities of the Renaissance and Reformation.” Renaissance Quarterly 57/1 (2004): 1–42. https://doi.org/10.2307/1262373
  38. Harrison Jr., Richard L. “Melanchthon’s role in the reformation of the University of Tübingen.” Church History 47/3 (1978): 270–278.
  39. Hartmann, Alfred. “Birk, Sixt (Xystus Betul[e]ius).” Neue Deutsche Biographie 2 (1955): 256.
  40. Heath, Thomas L. The Thirteen Books of Euclid’s Elements. New York: Dover, 1956.
  41. Heeffer, Albrecht. “The Genesis of the Algebra Textbook: From Pacioli to Euler.” Almagest 3/1 (2012): 26–61. https://doi.org/10.1484/J.ALMA.5.100794
  42. Henrion, Denis (or Didier). Les quinze livres des Elements geometriques d’Euclide. Traduicts en François par D. Henrion Professeur és Mathematiques, imprimez, reveus & corrigez du vivant de l’Autheur: avec des Commentaires beaucoup plus amples & faciles. Paris: Isaac Dedin, 1632.
  43. Homann, Frederick A. Hugh of St Victor. Practical geometry = Practica geometriae. Milwaukee: Marquette University Press, 1991.
  44. Howard, Thomas A. Protestant Theology and the Making of the Modern German University. Oxford: Oxford University Press, 2006. https://doi.org/10.1093/0199266859.001.0001
  45. Høyrup, Jens. The World of the Abbaco. Abbacus Mathematics Analyzed and Situated Historically Between Fibonacci and Stifel. Cham: Birkhäuser, 2024. https://doi.org/10.1007/978-3-031-25164-1
  46. Hughes, Barnabas. “Johann Scheubel’s revision of Jordanus de Nemore’s De numeris datis: An analysis of an unpublished manuscript.” Isis 63/2 (1972): 221–234. https://doi.org/10.1086/350886
  47. Hughes, Barnabas. “The private library of Johann Scheubel, sixteenth-century mathematician.” Viator 3 (1972): 417–432. https://doi.org/10.1484/J.VIATOR.2.301677
  48. Karpinski, Louis. Robert of Chester’s Latin translation of the Algebra of al-Khowarizmi. New York: Macmillan, 1915.
  49. Köpf, Ulrich. Die Universität Tübingen und ihre Theologen. Tübingen: Mohr Siebeck, 2020.
  50. Knobloch, Eberhard. “Géométrie pratique, géométrie savante.” Albertiana 8 (2005): 27–56.
  51. Lee, Eunsoo. “Let the Diagram Speak: Compass Arcs and Visual Auxiliaries in Printed Diagrams of Euclid’s Elements.” Endeavour 42/2–3 (2018): 78–98.
  52. Loget, François. “De l’algèbre comme art à l’algèbre pour l’enseignement: Les manuels de Pierre de La Ramée, Bernard Salignac et Lazare Schoner.” Revue de Synthèse 132/4 (2011): 495–527. https://doi.org/10.1007/s11873-011-0170-3
  53. L’Huillier, Hervé. “Practical Geometry in the Middle Ages and the Renaissance.” In Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, edited by Ivor Grattan-Guinness, 185–191. London: Routledge, 1994.
  54. Le Tenneur, Jacques. Traité des quantitez incomensurables: ou sont decidees plusieurs belles Questions des Nombres Rationaus & Irrationaus. Les erreurs de Stevin refutées. Et le Dizième Livre d’Euclide illustré de nouvelles demonstrations plus faciles & plus succinctes que les ordinaires, & reduit à 62. propositions. Paris: Jean Dedin, 1640.
  55. Malet, Antoni. “Renaissance notions of number and magnitude.” Historia mathematica 33 (2006): 63–81.
  56. Malet, Antoni. “Euclid’s swan song: Euclid’s Elements in early modern Europe.” In Greek Science in the Long Run: Essays on the Greek Scientific Tradition (4th c. BCE-17th c. CE), edited by Paula Olmos, 205–234. Newcastle upon Tyne: Cambridge Scholars Publishing, 2012.
  57. Masotti, Arnoldo. “Tartaglia (also Tartalea or Tartaia) Niccolò.” Dictionary of Scientific Biography, vol. 13, edited by Charles Coulston Gillispie, 258–262. New York: Charles Scribner’s Sons, 1976.
  58. Meißner, Heinrich. Des Gantzen, In 15 Büchern bestehenden, Teutschen Euclidis. Hamburg: Wieringen, 1690.
  59. Menghini, Marta. “From practical geometry to the laboratory method: The search for an alternative to Euclid in the history of teaching geometry.” In Selected Regular Lectures from the 12th International Congress on Mathematical Education, edited by Sung Je Cho, 561–587. Cham: Springer, 2015. https://doi.org/10.1007/978-3-319-17187-6_32
  60. Morel, Thomas. “Bringing Euclid into the mines: Classical sources and vernacular knowledge in the development of subterranean geometry.” In Translating Early Modern Science, edited by Sietske Fransen et al., 154–181. Leiden: Brill, 2017. https://doi.org/10.1163/9789004349261_008
  61. Naets, Jürgen. “How to Define a Number? A General Epistemological Account of Simon Stevin’s Art of Defining.” Topoi 29 (2010): 77–86. https://doi.org/10.1007/s11245-009-9068-1
  62. Neal, Katherine. From Discrete to Continuous. The Broadening of Number Concepts in Early Modern England. Dordrecht: Springer Science & Business Media, 2002.
  63. Omodeo, Pietro D. “The German and European network of the professors of mathematics at Helmstedt in the sixteenth century.” In The Circulation of Science and Technology: Proceedings of the 4th International Conference of the ESHS, edited by Antoni Roca-Rosell, 294–301. Barcelona: Societat Catalana d’Història de la Ciència i de la Tècnica, 2012.
  64. Overfield, James H. Humanism and Scholasticism in Late Medieval Germany. Princeton: Princeton University Press, 1984. https://doi.org/10.2307/j.ctvckq7sd
  65. Pejlare, Johanna, and Staffan Rodhe. “On the relations between geometry and algebra in Gestrinius’ edition of Euclid’s Elements.” In History and Pedagogy of Mathematics. Proceedings of the 2016 ICME Satellite Meeting, edited by Luis Radford et al., 513–523. Montpellier: IREM de Montpellier, 2016. https://publications.lib.chalmers.se/records/fulltext/242063/local_242063.pdf
  66. Peletier, Jacques. L’Algèbre. Lyon: Jean de Tournes, 1554.
  67. Peletier, Jacques. In Euclidis elementa geometrica demonstrationum libri sex. Lyon: Jean de Tournes and Guillaume Gazeau, 1557.
  68. Ramus, Petrus. Algebra. Paris: André Wechel, 1560.
  69. Ramus, Petrus. Arithmeticae libri duo; Geometriae septem et viginti. Paris: André Wechel, 1569.
  70. Reich, Ulrich. Schriftenreihe des Stadtarchivs Kirchheim unter Teck. 500 Jahre Johann Scheubel. Kirchheim unter Teck: Gottlieb & Osswald, 1994.
  71. Reich, Ulrich. “Johann Scheubel (1494–1570): Geometer, Algebraiker und Kartograph.” In Der „mathematicus“: zur Entwicklung und Bedeutung einer neuen Berufsgruppe in der Zeit Gerhard Mercators, edited by Irmgard Hantsche, 151–182. Bochum: Brockmeyer, 1996.
  72. Reich, Ulrich. Johann Scheubel und die älteste Landkarte von Württemberg 1559. Karlsruhe: Hochschule für Technik, 2000.
  73. Reich, Ulrich. “Scheubel, Johann.” Neue Deutsche Biographie 22 (2005): 709–710. https://www.deutsche-biographie.de/pnd119337711.html#ndbcontent
  74. Rommevaux, Sabine. “Aperçu sur la notion de dénomination d’un rapport numérique au Moyen Âge et à la Renaissance.” Methodos: Savoirs et textes 1 (2001): 223–243.
  75. Roth, Rudolph (von). Urkunden zur Geschichte der Universität Tübingen aus den Jahren 1476 bis 1550. Tübingen: H. Laupp, 1877.
  76. Scheubel, Johann. De numeris et diversis rationibus seu regulis computationum opusculum. Leipzig: Michael Blum, 1545.
  77. Scheubel, Johann. Compendium arithmeticae artis. Basel: Johannes Oporinus, 1549.
  78. Scheubel, Johann. Euclidis Megarensis, Philosophi & Mathematici excellentissimi, Sex libri priores, de Geometricis principiis, Graeci & Latini, unà cum demonstrationibus propositionum, absque literarum notis, veris ac proprijs, & alijs quibusdam, usum earum concernentibus, non citra maximum huius artis studiosorum emolumentum adiectis. Algebrae porro regulae, propter numerorum exempla, passim propositionibus adiecta, his libris praemissae sunt, eaedemque demonstratae. Basel: Hervagius, 1550.
  79. Scheubel, Johann. Algebrae compendiosa facilisque descriptio. Paris: Guillaume Cavellat, 1551.
  80. Scheubel, Johann. Iacobi Fabri Stapulensis in Arithmetica Boëthi epitome, unà cum difficiliorum locorum explicationibus & figuris (quibus antea carebat) nunc per Ioannem Scheubelium adornatis & adiectis. Accessit Christierni Morssiani Arithmetica practica. Basel: Henri Estienne, 1553.
  81. Scheubel, Johann. Das sibend, acht und neünt Bůch, des hochberuͤmbten Mathematici Euclidis Megarensis, in welchen der operationen unnd regulen aller gemainer rechnung, ursach grund und fundament, angezaigt wirt, zů gefallen allen den, so die kunst der Rechnung liebhaben, durch Magistrum Johann Scheybl, der loͤblichen universitet zů Tübingen des Euclidis und Arithmetic Ordinarien, auß dem latein ins teütsch gebracht, unnd mit gemainen exempeln also illustrirt unnd an tag geben, das sy ein yeder gemainer Rechner leichtlich verstehn, unnd jene nutz machen kan. Augsburg: Valentin Ottmar, 1555.
  82. Schöll, Fritz. “Xylander, Wilhelm.” Allgemeine Deutsche Biographie 44 (1898): 582–593. https://www.deutsche-biographie.de/pnd124331025.html#adbcontent
  83. Schöne, Hermann. Heronis alexandrini opera quae supersunt omnia, vol. 3. Leipzig: Teubner, 1899.
  84. Shelby, Lon R. “The geometrical knowledge of mediaeval master masons.” Speculum 47/3 (1972): 395–421. https://doi.org/10.2307/2856152
  85. Staigmüller, Hermann. “Johannes Scheubel, ein deutscher Algebraiker des XVI. Jahrhunderts.” Abhandlungen zur Geschichte der Mathematik 9 (1899): 429–469.
  86. Stevin, Simon. L’Arithmetique de Simon Stevin de Bruges: Contenant les computations des nombres Arithmetiques ou vulgaires: Aussi l’Algebre, avec les equations de cinc quantitez. Ensemble les quatre premiers livres d’Algebre de Diophante d’Alexandrie, maintenant premierement traduicts en François. Encore un livre particulier de la Pratique d’Arithmetique, contenant entre autres, Les Tables d’Interest, La Disme; Et un traicté des Incommensurables grandeurs: Avec l’Explication du Dixiesme livre d’Euclide. Leiden: Christophe Plantin, 1585.
  87. Sthen, Johannes. Arithmetices Euclideae Liber primus, Aliâs in ordine reliquorum Septimus: Qui citra praecedentium Sex librorum geometricorum opem eruditè persequitur, cum reliquis duobus sequentibus, vera principia ac solidiora fundamenta Logistices, id est, ut vocant, Arithmetices Practicae. Wittenberg, 1564.
  88. Tartaglia, Niccolò. Euclide megarense philosopho, solo introduttore delle scientie mathematice, diligentemente reassettato, et alla integrita ridotto per il degno Professore di tal Scientie Nicolo Tartalea, Brisciano, Secondo le due Tradottioni, e per commune commodo & utilita di latino in volgar tradotto, con una ampla espositione dello istesso tradottore di novo aggionta. Talmente chiara, che ogni mediocre ingegno, senza la notitia, over suffragio di alcun’altra scientia con facilita, sera capace à poterlo intendere. Venezia: Ruffinelli, 1543.
  89. Van Schooten, Frans. De propositien van de XV boucken der elementen Euclidis demonstratis. Leiden: Govert Basson, 1617.
  90. Vitrac, Bernard. Euclide. Les Éléments. Livres V-VI: Proportions et similitude. Livres VII-IX: Arithmétique. Paris: Presses universitaires de France, 1994.
  91. Williams, Kim et alii. The Mathematical Works of Leon Battista Alberti. Basel: Springer, 2010.
  92. Xylander, Wilhelm. Pselli, doctissimi viri, perspicuus Liber de quatuor mathematicis scientiis. Basel: Johannes Oporinus, 1556.
  93. Xylander, Wilhelm (Holtzmann). Die Sechs Erste Bücher Euclidis, Vom anfang oder grund der Geometri, in welchen der rechte grund, nitt allain der Geometri (versteh alles kunstlichen, gwisen, und vortailigen gebrauchs des Zirckels, Linials oder Richtscheittes und andrer werckzeüge, so zu allerlai abmessen dienstlich) sonder auch der fürnemsten stuck und vortail der Rechenkunst, furgeschriben und dargethon ist, auß Griechischer sprach in die Teütsch gebracht, aigentlich erklärt, auch mit verstentlichen Exempeln, gründlichen Figurn geziert, dermassen vormals in Teütscher sprach nie gesehen worden. Basel: Oporinus, 1562.
  94. Xylander, Wilhelm. Diophanti Alexandrini Rerum arithmeticarum libri sex. Basel: Eusebius Episcopius, 1575.
  95. Xylander, Wilhelm. Opuscula Mathematica. Aphorismi Cosmographici; De minutiis; De Surdorum Numerorum natura & tractactione; De usu Globi & Planisphærii tractatus. Heidelberg: Jacob Müller, 1577.
  96. Zamberti, Bartolomeo. Euclidis megarensis philosophi platonici mathematicarum disciplinarum Ianitoris, habent in hoc volumine quicunque ad mathematicam substantiam aspirant, elementorum libros XIII cum expositione Theonis insignis mathematici. Venezia: Johannes Tacuinus, 1505.

Articoli simili

1 2 3 4 > >> 

Puoi anche Iniziare una ricerca avanzata di similarità per questo articolo.