Abstract
This chapter is subdivided chronologically into periods that correspond to changes in educational system, aims, and ideals: c. 500–750, c. 750–1100, c. 1100–1200, and c. 1200–1500. To the extent it is possible, Latin and lay education (in its geographical and professional diversity) are distinguished, but the scarcity of sources for lay teaching causes the Latin (first quadrivial, later broader universitarian) type to preponderate in the exposition.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A detailed investigation (which also makes clear the absence of every kind of mathematical studies) is Riché 1976.
- 2.
This group of disciplines and the collective name used about it since Boethius are the closest we can get to a unified concept of mathematics in the Medieval Latin school tradition at least until the thirteenth century; only Aristotelian philosophy brought in the notion of “more physical” mathematical disciplines (which beyond astronomy included optics and the science of weights). However, when asking about mathematics education – a modern concept – we shall need to include also mathematical activities falling outside the quadrivial framework, such as practical computation. Unfortunately, the sources are mostly mute on this account.
- 3.
The ways the problem was confronted from the beginning until the early eighth century is accounted for in Jones 1943, pp. 1–114.
- 4.
- 5.
This picture is confirmed by the Manual for My Son written by the mid-ninth-century noblewoman Dhuoda. When speaking about sacred numerology, in Augustinian style, she tacitly assumes the son to understand basic computation (and shows that she does so herself) (Riché 1975, pp. 326–334).
- 6.
Edition and German translation in Folkerts and Gericke (1993).
- 7.
A ratio was not understood as the number resulting from a division but as a relation between two numbers; it might be multiplex (of type m:1), superparticular (of type m + 1:n), multiplex superparticular (mn + 1:n), superpartient (n + p : n), multiplex superpartient (of type mn + p :n) or inverses of any of these. Depending on the numbers involved, ratios had specific names – 5 : 2 (and 10 : 4, etc.), for instance, were “duplex sesquialter”.
- 8.
See Bergmann 1985 (to be used with some care).
- 9.
Since the slave trade route to Muslim Spain passed through Lotharingia, Gerbert’s stay in Catalonia was not the only numerate cultural contact at hand. Thompson (1929) lists a number of further contact on the courtly and literate levels. Since the names given to the nine figures seem to be of mixed Magyar-Arabic-Latin-German origin (Köppen 1892, p. 45), the slave traders could be the most likely inspiration.
- 10.
Arno Borst claims in his fundamental study (1986) that the game can be traced back to c. 1030 and no further. However, Walther von Speyer’s Libellus scolasticus (ed. trans. Vossen 1962, p. 41, pp. 52ff) clearly speaks of a very similar game played around 970 (without indicating the name, which may indeed be later) on the abacus board and using its counters. Borst dismisses this testimony, asserting that Walther does not understand what he is speaking about, and Vossen because he does not know that the abacus board belonged with geometry.
The didactical use of the game was discussed by Gillian R. Evans (1976). A recent discussion of the game and its survival is Moyer (2001). A short presentation of the way the game was played is in Beaujouan (1972, pp. 644–650).
- 11.
The existence of a Geometria Gerberti decides nothing, since it may well be a compilation from the later eleventh century.
- 12.
Known by him from one of Boethius’s treatises on Aristotelian logic, even though it is amply used by the agrimensors.
- 13.
Uta Lindgren (1976, pp. 48–59) discusses some of them in detail and comes substantially to the same result.
- 14.
An analysis of this part of the treatise and its inspiration from ancient philosophical sources is in Tannery (1922, pp. 208–210). Tannery rejects Hugue’s authorship as probably a thirteenth-century reconstruction (pp. 319–321), but better editions of the texts on which his arguments are based turn the conclusion upside-down – cf. also Baron (1955).
- 15.
Abelard, Historia calamitatum (ed. Muckle 1950).
- 16.
The medical schools of Salerno and Montpellier were older but only came to be characterized as “universities” at a moment when this term had acquired new meanings.
The whole process by which the universities emerged is much too complex to be treated justly in the present context. A recent fairly detailed description is Pedersen (1998, pp. 138–188).
- 17.
trans. Thorndike 1944, pp. 27–30.
- 18.
Now available in critical edition (Busard 2005).
- 19.
Cf. the discussion in Murdoch (1968).
- 20.
More information on mathematics teaching at medieval universities (despite various imprecisions) in the first chapter of Schöner (1994).
- 21.
Rhetorica antiqua (ed. Rockinger 1863, p. 173). In any case, since notarial documents were written in Latin, merchants needed to understand the rudiments of that language.
- 22.
The “abbacus” is not, as one might believe, the calculation board; the word (mostly in this spelling) had come to designate practical mathematics – thus already in Leonardo Fibonacci’s Liber abbaci.
- 23.
In Fiesole outside Florence, in the relatively benign years 1621–1626, 20 % died with the first year of life; later in the century, this rate doubled, with peaks above 50 % (Cipolla 1993, p. 221).
- 24.
Recent discussions of the social history of this institution are Ulivi (2002a) and (dealing particularly with Florence) Ulivi (2002b).
Contrary to what is often claimed (also repeatedly by Ulivi), the abbacus school does not descend from Fibonacci’s Liber abbaci – cf. Høyrup (2005). There is some (mostly indirect) evidence that the Italian tradition (as already Fibonacci) was inspired from what went on in the Iberian region, but we have no information of how teaching was organized there before the fifteenth century.
- 25.
Shelby (1970) reaches similar conclusions concerning late medieval English masons.
References
Arrighi, Gino. 1967. Un “programma” di didattica di matematica nella prima metà del Quattrocento (dal Codice 2186 della Biblioteca Riccardiana di Firenze). Atti e memorie dell’Accademia Petrarca di Lettere, Arti e Scienze di Arezzo, Nuova Serie 38: 117–128.
Baron, Roger. 1955. Hugues de Saint-Victor, auteur d’une Practica geometriae. Mediaeval Studies 17: 107–116.
Baron, Roger. 1956. Hugonis de Sancto Victore Practica geometriae. Osiris 12: 176–224.
Beaujouan, Guy. 1972. L’enseignement du “quadrivium”. In La scuola nell’Occidente latino dell’alto medioeva, 639–657, pl. i–iv. Spoleto: Centro italiano di studio sull’alto medioevo.
Bergmann, Werner. 1985. Innovationen im Quadrivium des 10. und 11. Jahrhunderts. Studien zur Einführung von Astrolab und Abakus im lateinischen Mittelalter. Stuttgart: Franz Steiner.
Borst, Arno. 1986. Das mittelalterliche Zahlenkampfspiel. Heidelberg: Carl Winter.
Brewer, J.S. (ed.). 1859. Fr. Rogeri Bacon Opera quaedam hactenus inedita. Vol. I containing I.- Opus tertium. II.- Opus minus. III.- Compendium philosophiae. London: Longman, Green, Longman, and Roberts.
Brown, Giles. 1994. Introduction: The Carolingian renaissance. In Carolingian culture: Emulation and innovation, ed. Rosamond McKitterick, 1–51. Cambridge: Cambridge University Press.
Bubnov, Nicolaus (ed.). 1899. Gerberti postea Silvestri II papae Opera mathematica, 972–1003. Berlin: Friedländer.
Burnett, Charles. 1987. Adelard, Ergaphalau and the science of the stars. In Adelard of Bath. An English scientist and arabist of the twelfth century, ed. Charles Burnett, 133–145. London: The Warburg Institute.
Burnett, Charles. 1996. Algorismi vel helcep decentior est diligentia: The arithmetic of Adelard of Bath and his circle. In Mathematische Probleme im Mittelalter. Der lateinische und arabische Sprachbereich, ed. Menso Folkerts, 221–331. Wiesbaden: Harrassowitz.
Burnett, Charles. 2010. Numerals and arithmetic in the Middle Ages, Variorum collected studies. Farnham, Surrey, Burlington: Ashgate.
Busard, H.L.L. 2005. Campanus of Novara and Euclid’s Elements. 2 vols. Stuttgart: Franz Steiner.
Busard, H.L.L., and Menso Folkerts (eds.). 1992. Robert of Chester’s (?) redaction of Euclid’s Elements, the so-called Adelard II version, vol. 2. Boston: Birkhäuser.
Cambridge economic history of Europe. 1965–1987. 2nd ed, 7 vols. Cambridge: Cambridge University Press.
Cipolla, Carlo M. 1993. Before the Industrial Revolution; European society and economy, 1000–1700. London: Routledge.
Denifle, H., and É. Chatelain (eds.). 1889. Chartularium Universitatis Parisiensis, 4 vols., 1889–1897. Paris: Frères Delalain.
Evans, Gillian Rosemary. 1976. The rithmomachia: A medieval mathematical teaching aid. Janus 63: 257–271.
Folkerts, Menso (ed.). 1970. “Boethius” Geometrie II. Ein mathematisches Lehrbuch des Mittelalters. Wiesbaden: Franz Steiner.
Folkerts, Menso, and Helmuth Gericke (eds., trans.). 1993. Die Alkuin zugeschriebenen Propositiones ad acuendos iuvenes (Aufgaben zur Schärfung des Geistes der Jugend). In Science in Western and Eastern civilization in Carolingian times, ed. P. L. Butzer and D. Lohrman, 283–362. Basel: Birkhäuser.
Gibson, Strickland (ed.). 1931. Statuta antiqua Universitatis Oxoniensis. Oxford: The Clarendon Press.
Goldthwaite, Richard A. 1972. Schools and teachers of commercial arithmetic in Renaissance Florence. Journal of European Economic History 1: 418–433.
Goldthwaite, Richard A. 2009. The economy of Renaissance Florence. Baltimore: Johns Hopkins University Press.
Grabmann, Martin. 1934. Eine für Examinazwecke abgefaßte Quaestionensammlung der Pariser Artistenfakultät aus der ersten Hälfte des XIII. Jahrhunderts. Revue néoscolastique de philosophie 36: 211–229.
Grabmann, Martin. 1941. I divieti ecclesiastici di Aristotele sotto Innocenzo III e Gregorio IX. Roma: Saler.
Haskins, Charles Homer. 1927. The renaissance of the twelfth century. Cambridge, MA/London: Harvard University Press.
Høyrup, Jens. 1988. Jordanus de Nemore, 13th century mathematical innovator: An essay on intellectual context, achievement, and failure. Archive for History of Exact Sciences 38: 307–363.
Høyrup, Jens. 2005. Leonardo Fibonacci and abbaco culture: A proposal to invert the roles. Revue d’Histoire des Mathématiques 11: 23–56.
Høyrup, Jens. 2009. The rare traces of constructional procedures in ‘practical geometries’. In Creating shapes in civil and naval architecture, ed. Horst Nowacki and Wolfgang Lefèvre, 367–377. Leiden/Boston: Brill.
Jones, Charles W. (ed.). 1943. Bedae Opera de temporibus. Cambridge, MA: The Mediaeval Academy of America.
Keyser, R., P.A. Munch, and C.R. Unger (ed., trans.). 1848. Speculum regale – Konungs-skuggsjá – Konge-speilet. Christiania: Carl C. Werner.
Kink, Rudolf. 1854. Geschichte der kaiserlichen Universität zu Wien. I. Geschichtliche Darstellung der Entstehung und Entwicklung der Universität bis zur Neuzeit. II. Statutenbuch der Universität. Wien: Carl Gerold.
Klebs, Arnold C. 1938. Incunabula scientifica et medica. Osiris 4: 1–359.
Köppen, Friedrich Th. 1892. Notiz über die Zahlwörter im Abacus des Boethius. Bulletin de l’Académie des Sciences de St. Petersbourg 35: 31–48.
Lemay, Richard. 1976. The teaching of astronomy in medieval universities, principally at Paris in the fourteenth century. Manuscripta 20: 197–217.
Lindgren, Uta. 1976. Gerbert von Aurillac und das Quadrivium. Untersuchungen zur Bildung im Zeitalter der Ottonen. Wiesbaden: Steiner.
Moyer, Ann E. 2001. The philosophers’ game. Rithmomachia in medieval and Renaissance Europe, with an edition of Ralph Lever and William Fulke, The most noble, auncient, and learned playe (1563). Ann Arbor: University of Michigan Press.
Muckle, J.T., C.S.B. (eds.). 1950. Abelard’s letter of consolation to a friend (Historia Calamitatum). Mediaeval Studies 12: 163–213.
Murdoch, John E. 1968. The medieval Euclid: Salient aspects of the translations of the Elements by Adelard of Bath and Campanus of Novara. Revue de Synthèse 89: 67–94.
Pacioli, Luca. 1494. Summa de Arithmetica Geometria Proportioni et Proportionalita. Venezia: Paganino de Paganini.
Paetow, Louis John (ed., trans.). 1914. The battle of the seven arts. A French poem by Henri d’Andeli, trouvère of the thirteenth century. (Memoirs of the University of California, 4:1. History, 1:1). Berkeley: University of California Press.
Pedersen, Olaf. 1998. The first universities: Studium generale and the origins of university education in Europe. Cambridge: Cambridge University Press.
Pirenne, Henri. 1929. L’instruction des marchands au Moyen Âge. Annales d’Histoire économique et sociale 1: 13–28.
Rashdall, Hastings. 1936. The universities of Europe in the Middle Ages. Edited by F.M. Powicke and A.B. Emden. Oxford: The Clarendon Press.
Riché, Pierre (ed.). 1975. Dhuoda, Manuel pour mon fils. Trans. Bernard de Vregille and Claude Mondésert. Paris: Éditions du Cerf.
Riché, Pierre. 1976. Education and culture in the Barbarian West, sixth through eighth centuries. Columbia: University of South Carolina Press.
Riché, Pierre. 1979. Les Écoles de l’Occident chrétien de la fin du V e siècle au milieu du XI e siècle. Paris: Aubier Montaigne.
Robertson Jr., D.W. (ed., trans.). 1958. Augustine. On Christian Doctrine. Indianapolis: Bobbs-Merrill.
Rockinger, Ludwig (ed.). 1863. Briefsteller und Formelbücher des elften bis vierzehnten Jahrhunderts. 2 vols. München: Georg Franz,
Schöner, Christoph. 1994. Mathematik und Astronomie an der Universität Ingolstadt im 15. und 16. Jahrhundert. Berlin: Duncker & Humblott.
Shelby, Lon R. 1970. The education of medieval English master masons. Mediaeval Studies 32: 1–26.
Shelby, Lon R. (ed.). 1977. Gothic design techniques. The fifteenth-century design booklets of Mathes Roriczer and Hanns Schmuttermayer. Carbondale: Southern Illinois University Press.
Siraisi, Nancy G. 1973. Arts and sciences at Padua. The studium of Padua before 1350. Toronto: Pontifical Institute of Mediaeval Studies.
Stahl, William Harris. 1971. Martianus Capella and the seven Liberal Arts. I. The quadrivium of Martinaus Capella. New York: Columbia University Press.
Tannery, Paul. 1922. Mémoires scientifiques, tôme V: Sciences exactes au Moyen Age, 1887–1921. Toulouse/Paris: Édouard Privat/Gauthier-Villars.
Taylor, Jerome. 1961. The Didascalicon of Hugh of St. Victor. A medieval guide to the arts. New York/London: Columbia University Press.
Thompson, James Westfall. 1929. The introduction of Arabic science into Lorraine in the tenth century. Isis 12: 184–193.
Thorndike, Lynn. 1944. University records and life in the Middle Ages. New York: Columbia University Press.
Travaini, Lucia. 2003. Monete, mercanti e matematica. Le monete medievali nei trattati di aritmetica e nei libri di mercatura. Roma: Jouvence.
Tummers, Paul M.J.E. 1980. The commentary of Albert on Euclid’s Elements of geometry. In Albertus Magnus and the sciences: Commemorative essays 1980, ed. James A. Weisheipl, O.P., 479–499. Toronto: The Pontifical Institute of Medieval Studies.
Ulivi, Elisabetta. 2002a. Scuole e maestri d’abaco in Italia tra Medioevo e Rinascimento. In Un ponte sul Mediterraneo: Leonardo Pisano, la scienza araba e la rinascita della matematica in Occidente, ed. Enrico Giusti, 121–159. Firenze: Edizioni Polistampa.
Ulivi, Elisabetta. 2002b. Benedetto da Firenze (1429–1479), un maestro d’abbaco del XV secolo. Con documenti inediti e con un’Appendice su abacisti e scuole d’abaco a Firenze nei secoli XIII–XVI. Bollettino di Storia delle Scienze Matematiche 22: 3–243.
Ullman, Berthold L. 1964. Geometry in the mediaeval quadrivium. In Studi di Bibliografia e di Storia in Onore di Tammaro de Marinis, IV, ed. G. Mardersteig, 263–285. Verona: Biblioteca Apostolica Vaticana.
van de Vyver, A. 1936. Les plus anciennes Traductions latines médiévales (Xe-XIe siècles) de Traités d’Astronomie et d’Astrologie. Osiris 11: 658–691.
Victor, Stephen K. 1979. Practical geometry in the Middle Ages. Artis cuiuslibet consummatio and the Pratike de geometrie. Philadelphia: The American Philosophical Society.
Villani, Giovanni. 1823. Cronica. 8 vols. Firenze: il Magheri.
Vossen, Peter (ed., trans.). 1962. Der Libellus Scolasticus des Walther von Speyer: Ein Schulbericht aus dem Jahre 984. Berlin: de Gruyter.
Wason, Robert P. 2002. Musica practica: Music theory as pedagogy. In Cambridge history of Western music theory, ed. Thomas Christensen, 46–77. Cambridge: Cambridge University Press.
Weisheipl, James A., O.P. 1964. Curriculum of the Arts Faculty at Oxford in the early fourteenth century. Mediaeval Studies 26: 143–165.
Winkelmann, Eduard. 1886. Urkundenbuch der Universität Heidelberg. Erster Band: Urkunden. Heidelberg: Carl Winter.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Høyrup, J. (2014). Mathematics Education in the European Middle Ages. In: Karp, A., Schubring, G. (eds) Handbook on the History of Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9155-2_6
Download citation
DOI: https://doi.org/10.1007/978-1-4614-9155-2_6
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-9154-5
Online ISBN: 978-1-4614-9155-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)