Nicole Oresme on Motion and the Atomization of the Continuum
Main Article Content
Abstract
As Aristotle classically defined it, continuity is the property of being infinitely divisible into ever-divisible parts. How has this conception been affected by the process of mathematization of motion during the 14th century? This paper focuses on Nicole Oresme, who extensively commented on Aristotle’s Physics, but also made decisive contributions to the mathematics of motion. Oresme’s attitude about continuity seems ambivalent: on the one hand, he never really departs from Aristotle’s conception, but on the other hand, he uses it in a completely new way in his mathematics, particularly in his Questions on Euclidean geometry, a tantamount way to an atomization of motion. If the fluxus theory of natural motion involves that continuity is an essential property of real motion, defined as a res successiva, the ontological and mathematical structure of this continuity implies that continuum is in some way “composed” of an infinite number of indivisibles. In fact, Oresme’s analysis opened the path to a completely new kind of mathematical continuity.
Downloads
Download data is not yet available.
Article Details
Section
ARTICLES
References
Bibliography
Primary sources
Aristotle, Physics, translated by R. P. Hardie and R. K. Gaye, in The Complete Works of Aristotle I, edited by J. Barnes (Princeton (NJ): Princeton University Press, 1984), 315-446.
The Complete Works of Aristotle, 2 vols, edited by J. Barnes, Jonathan (Princeton (NJ): Princeton University Press, 1984).
Boethius, De institutione arithmetica libri duo, De institutione musica libri quinque. Accedit Geometria quae fertur Boetii, edited by G. Friedlein (Leipzig: Teubner, 1867).
Burton, Dan, Nicole Oresme’s “De visione stellarum (On seeing the stars)”: A Critical Edition of Oresme’s Treatise on Optics and Atmospheric Refraction (Leiden and Boston: Brill, 2007).
Oresme, Nicole, Le Livre du ciel et du monde, edited by A. D. Menut (Madison: University of Wisconsin Press, 1968).
Questiones super physicam. Books I-VII, edited by S. Caroti, J. Celeyrette, S. Kirschner and E. Mazet (Leiden: Brill, 2013).
Questiones super geometriam Euclidis, edited by H. L. L. Busard, Nicole Oresme, Questiones super geometriam Euclidis (Stuttgart: F. Steiner, 2010).
Expositio et quaestiones in Aristotelis De anima, edited by B. Patar (Louvain-la-Neuve: Éditions de l'Institut Superieur de Philosophie; Louvain and Paris: Éditions Peeters, 1995).
Tractatus de configurationibus qualitatum et motuum, edited by M. Clagett, Nicole Oresme and the Medieval Geometry of Qualities and Motions: a treatise on the uniformity and difformity of intensities known as Tractatus de configurationibus qualitatum et motuum (Madison, Milwaukee and London: The University of Wisconsin Press, 1968).
Quaestiones super de generatione et corruption, edited by S. Caroti (München: Bayerische Akademie der Wissenschaften: C. H. Beck, 1996).
Tractatus de commensurabilitate vel incommensurabilitate motuum celi, edited by E. Grant, Nicole Oresme and the kinematics of circular motion: Tractatus de commensurabilitate vel incommensurabilitate motuum celi (Madison: University of Wisconsin Press, 1971).
De communicatione ydiomatum, edited by E. Borchert, Der Einfluss des Nominalismus auf die Christologie der Spätscholastik: nach dem Traktat De communicatione idiomatum des Nicolaus Oresme (Münster: Aschendorff, 1940).
De proportionibus proportionum and Ad pauca respicientes, edited by E. Grant (Madison: University of Wisconsin Press, 1966).
Witelo, “Perspectiva, Book IV”, in Opticae Thesaurus, edited by F. Risner (Basel: Per Episcopios, 1572). For a modern but unpublished edition, see the edition and translation of Kelso, Carl, Witelonis Perspectivae liber quartus (Columbia: University of Missouri, PhD diss., 1994).
Secondary sources
Boyer, Carl B., The history of the calculus and its conceptual development (New York: Dover Publications Inc, 1959 [1939]).
Caroti, Stefano, “La position de Nicole Oresme sur la nature du mouvement (Questiones super Physicam III, 1-8): problèmes gnoséologiques, ontologiques et sémantiques”, Archives d’histoire doctrinale et littéraire du Moyen Age 61 (1994): 303-385.
“Oresme on Motion (Questiones super Physicam III, 2-7)”, Vivarium 31, 1 (1993): 8-36.
“Modi rerum and materialism: a note on a quotation of a condemned articulus in some fourteenth-century Parisian De anima commentaries”, Traditio 55 (2000): 211-234.
“Configuratio, ymaginatio, atomisme et modi rerum dans quelques écrits de Nicole Oresme”, in Méthodes et statuts des sciences à la fin du Moyen Âge, edited by Ch. Grellard (Villeneuve d’Ascq: Presse universitaire du Septentrion, 2004), 127-140.
“Nicole Oresme and Modi Rerum”, Noctua 1, 1 (2014): 1-27.
Celeyrette, Jean, “Apparences et imaginations chez Nicole Oresme: Question III. 1 sur la Physique et question sur l’apparence d’une chose”, Revue d’histoire des sciences 60, 1 (2007): 83-100.
Celeyrette, Jean and Grellard, Christophe, Nicole Oresme philosophe: Philosophie de la nature et philosophie de la connaissance à Paris au XIVe siècle (Turnout: Brepols, 2014).
Cesalli, Laurent, “Ontologie ‘nominale’ et ‘adverbiale’ chez Nicole Oresme”, in Nicole Oresme philosophe: Philosophie de la nature et philosophie de la connaissance à Paris au XIVe siècle, edited by J. Celeyrette and Ch. Grellard (Turnout: Brepols, 2014), 163-183.
Clagett, Marshall, The science of mechanics in the Middle Ages (Madison: University of Wisconsin Press, 1959).
Archimedes in the Middle Ages, 5 vols. (Madison and Philadelphia: University of Wisconsin Press and American Philosophical Society, 1964-1984).
Debroise, Philippe, Mathématiques de l’intensité et Merveilles de la nature, Étude sur le Tractatus de configurationibus qualitatum et motuum (Paris: Université Paris Diderot, Doctoral thesis, 2019).
Damerow, Peter; Freudenthal, Gideon; Mclaughlin, Peter and Renn, Jürgen, Exploring the Limits of Preclassical Mechanics: A Study of Conceptual Development in Early Modern Science: Free Fall and Compounded Motion in the Work of Descartes, Galileo, and Beeckman (New York: Springer, 2004).
Di Liscia, Daniel A., “La conclusio pulchra, mirabilis et bona: una ingeniosa demostración atribuible a Nicole Oresme”, Mediaevalia. Textos e estudos 37 (2018): 139-168.
Drake, Stillman, “Impetus Theory and Quanta of Speed before and after Galileo”, Physics 16 (1974): 47-65.
“Impetus Theory Reappraised”, Journal of the History of Ideas 36 (1975): 27-46.
“Free Fall from Albert of Saxony to Honoré Fabri”, Studies in History and Philosophy of Science 5 (1975): 347-366.
“A Further Reappraisal of Impetus Theory”, Studies in History and Philosophy of Science 7 (1976): 319-336.
Duhem, Pierre, Études sur Léonard de Vinci, 3 (Paris: Hermann, 1984 [1908]).
Le Système du monde. Histoire des doctrines de Platon à Copernic, vol. 7 (Paris: Hermann, 1956).
Edwards, Charles Henry Jr., The historical development of the calculus (New York: Springer Verlag, 1979).
Grellard, Christophe, Méthodes et statuts des sciences à la fin du Moyen Âge (Villeneuve d’Ascq: Presse universitaire du Septentrion, 2004).
Grellard, Christophe and Robert, Aurélien, Atomism in Late Medieval Philosophy and Theology (Leiden and Boston: Brill, 2009).
Kirschner, Stefan, “Oresme’s Theory of Motion”, in Nicole Oresme philosophe: Philosophie de la nature et philosophie de la connaissance à Paris au XIVe siècle, Studia Artistarum 39, edited by J. Celeyrette and Ch. Grellard (Turnhout: Brepols Publishers, 2014).
“Oresme on Intension and Remission of Qualities in His Commentary on Aristotle’s ‘Physics’”, Vivarium 38, 2 (2000): 255-274.
“A Possible Trace of Oresme’s Condicio-Theory of Accidents in an Anonymous Commentary on Aristotle’s Meteorology”, Vivarium 48, 3 (2010): 349-367.
McGinnis, John, “A Medieval Arabic Analysis of Motion at an Instant: The Avicennan Sources to the forma fluens/fluxus formae Debate”, The British Journal for the History of Science 39, 2 (2006): 189-205.
Maier, Anneliese, “Die scholastische Wesensbestimmung der Bewegung als forma fluens oder fluxus formae und ihre Beziehung zu Albertus Magnus”, Angelicum 21 (1944): 97-111.
“La doctrine de Nicolas d’Oresme sur les ‘configurationes intensionum’”, Revue des sciences philosophiques et théologiques 32, 1/2 (1948): 52-67. Also in Maier, Anneliese, Ausgehendes Mittelalter I: Gesammelte Aufsätze zur Geistesgeschichte des 14. Jahrhunderts (Rome: Edizioni di Storia e Letteratura, 1964), 335-352.
Ausgehendes Mittelalter I: Gesammelte Aufsätze zur Geistesgeschichte des 14. Jahrhunderts (Rome: Edizioni di Storia e Letteratura, 1964).
Mazet, Edmond, “La théorie des séries de Nicole Oresme dans sa perspective aristotélicienne. ‘Questions 1 et 2 sur la Géométrie d’Euclide’”, Revue d’histoire des mathématiques 9, 1 (2003): 33-80.
“Richard Swineshead et Nicole Oresme: deux styles mathématiques”, in Nicole Oresme philosophe: philosophie de la nature et philosophie de la connaissance à Paris au XIVe siècle, edited by J. Celeyrette and Ch. Grellard (Turnhout: Brepols, 2014), 105‑137.
Molland, Georges A., “The Atomisation of Motion: A Facet of the Scientific Revolution”, Studies in History and Philosophy of Science 13, 1 (1982): 31-54.
Murdoch, John E., “Review of Nicole Oresme’s ‘Quaestiones super geometriam Euclidis’ edited by H.L.L. Busard”, Scripta Mathematica 27 (1964): 67-91.
“Beyond Aristotle: Indivisibles and Infinite Divisibility in the Later Middle Ages”, in Atomism in Late Medieval Philosophy and Theology, edited by Ch. Grellard and A. Robert (Leiden and Boston: Brill), 15-38.
Murdoch, John E. and Sylla, Edith D., “The Science of Motion”, in Science in the Middle Ages, edited by D. C. Lindberg (Chicago and London: University of Chicago Press, 1978).
Pabst, Bernhard, Atomentheorien des lateinischen Mittelalters (Darmstadt: Wissenschaftliche Buchgesellschaft, 1994).
Pasnau, Robert, Metaphysical Themes: 1274-1671 (Oxford: Clarendon Press, 2011).
Pyle, Andrew, Atomism and its Critics. Problem Areas Associated with the Development of the Atomic Theory of Matter from Democritus to Newton (Bristol: Toemmes Press, 1995).
Quillet, Jeannine, Autour de Nicole Oresme, Actes du colloque Oresme organisé à l’Université de Paris XII (Paris: Vrin, 1990).
Rommevaux, Sabine, Les nouvelles théories des rapports mathématiques du XIVe au XVIe siècle (Turnhout: Brepols, 2014).
Sattler, Barbara M., The Concept of Motion in Ancient Greek Thought, Foundations in Logic, Method and Mathematics (Cambridge: Cambridge University Press, 2020).
Souffrin, Pierre, “La quantification du mouvement chez les scolastiques. La vitesse instantanée chez Nicole Oresme”, in Autour de Nicole Oresme, Actes du colloque Oresme organisé à l’Université de Paris XII, edited by J. Quillet (Paris: Vrin, 1990), 63 83.
Souffrin, Pierre and Weiss, Jean-Pierre, “Le Traité des configurations des qualités et des mouvements. Remarques sur quelques problèmes d’interprétation et de traduction”, in Nicolas Oresme. Tradition et innovation chez un intellectuel du XIVe siècle, edited by P. Souffrin and A.-P. Segonds (Paris: Les Belles Lettres, 1988).
Souffrin, Pierre and Segonds, Alain-Philippe, Nicolas Oresme. Tradition et innovation chez un intellectuel du XIVe siècle (Paris: Les Belles Lettres, 1988).
Thijssen, Johannes M.M.H., “The Debate Over the Nature of Motion: John Buridan, Nicole Oresme and Albert of Saxony. With an Edition of John Buridan’s Quaestiones Super Libros Physicorum, Secundum Ultimam Lecturam, Book III, Q. 7”, Evidence and Interpretation in Studies on Early Science and Medicine, edited by E. Sylla and W. R. Newman, 14, 1-3 (2009): 186-210.
Wilson, Curtis, William heytesbury. Medieval Logic and the Rise of Mathematical Physics (Madison: University of Wisconsin Press, 1960).
Youschkevitch, Adolf P., Geschichte der Mathematik im Mittelalter (Leipzig: Teubner, 1963).
Primary sources
Aristotle, Physics, translated by R. P. Hardie and R. K. Gaye, in The Complete Works of Aristotle I, edited by J. Barnes (Princeton (NJ): Princeton University Press, 1984), 315-446.
The Complete Works of Aristotle, 2 vols, edited by J. Barnes, Jonathan (Princeton (NJ): Princeton University Press, 1984).
Boethius, De institutione arithmetica libri duo, De institutione musica libri quinque. Accedit Geometria quae fertur Boetii, edited by G. Friedlein (Leipzig: Teubner, 1867).
Burton, Dan, Nicole Oresme’s “De visione stellarum (On seeing the stars)”: A Critical Edition of Oresme’s Treatise on Optics and Atmospheric Refraction (Leiden and Boston: Brill, 2007).
Oresme, Nicole, Le Livre du ciel et du monde, edited by A. D. Menut (Madison: University of Wisconsin Press, 1968).
Questiones super physicam. Books I-VII, edited by S. Caroti, J. Celeyrette, S. Kirschner and E. Mazet (Leiden: Brill, 2013).
Questiones super geometriam Euclidis, edited by H. L. L. Busard, Nicole Oresme, Questiones super geometriam Euclidis (Stuttgart: F. Steiner, 2010).
Expositio et quaestiones in Aristotelis De anima, edited by B. Patar (Louvain-la-Neuve: Éditions de l'Institut Superieur de Philosophie; Louvain and Paris: Éditions Peeters, 1995).
Tractatus de configurationibus qualitatum et motuum, edited by M. Clagett, Nicole Oresme and the Medieval Geometry of Qualities and Motions: a treatise on the uniformity and difformity of intensities known as Tractatus de configurationibus qualitatum et motuum (Madison, Milwaukee and London: The University of Wisconsin Press, 1968).
Quaestiones super de generatione et corruption, edited by S. Caroti (München: Bayerische Akademie der Wissenschaften: C. H. Beck, 1996).
Tractatus de commensurabilitate vel incommensurabilitate motuum celi, edited by E. Grant, Nicole Oresme and the kinematics of circular motion: Tractatus de commensurabilitate vel incommensurabilitate motuum celi (Madison: University of Wisconsin Press, 1971).
De communicatione ydiomatum, edited by E. Borchert, Der Einfluss des Nominalismus auf die Christologie der Spätscholastik: nach dem Traktat De communicatione idiomatum des Nicolaus Oresme (Münster: Aschendorff, 1940).
De proportionibus proportionum and Ad pauca respicientes, edited by E. Grant (Madison: University of Wisconsin Press, 1966).
Witelo, “Perspectiva, Book IV”, in Opticae Thesaurus, edited by F. Risner (Basel: Per Episcopios, 1572). For a modern but unpublished edition, see the edition and translation of Kelso, Carl, Witelonis Perspectivae liber quartus (Columbia: University of Missouri, PhD diss., 1994).
Secondary sources
Boyer, Carl B., The history of the calculus and its conceptual development (New York: Dover Publications Inc, 1959 [1939]).
Caroti, Stefano, “La position de Nicole Oresme sur la nature du mouvement (Questiones super Physicam III, 1-8): problèmes gnoséologiques, ontologiques et sémantiques”, Archives d’histoire doctrinale et littéraire du Moyen Age 61 (1994): 303-385.
“Oresme on Motion (Questiones super Physicam III, 2-7)”, Vivarium 31, 1 (1993): 8-36.
“Modi rerum and materialism: a note on a quotation of a condemned articulus in some fourteenth-century Parisian De anima commentaries”, Traditio 55 (2000): 211-234.
“Configuratio, ymaginatio, atomisme et modi rerum dans quelques écrits de Nicole Oresme”, in Méthodes et statuts des sciences à la fin du Moyen Âge, edited by Ch. Grellard (Villeneuve d’Ascq: Presse universitaire du Septentrion, 2004), 127-140.
“Nicole Oresme and Modi Rerum”, Noctua 1, 1 (2014): 1-27.
Celeyrette, Jean, “Apparences et imaginations chez Nicole Oresme: Question III. 1 sur la Physique et question sur l’apparence d’une chose”, Revue d’histoire des sciences 60, 1 (2007): 83-100.
Celeyrette, Jean and Grellard, Christophe, Nicole Oresme philosophe: Philosophie de la nature et philosophie de la connaissance à Paris au XIVe siècle (Turnout: Brepols, 2014).
Cesalli, Laurent, “Ontologie ‘nominale’ et ‘adverbiale’ chez Nicole Oresme”, in Nicole Oresme philosophe: Philosophie de la nature et philosophie de la connaissance à Paris au XIVe siècle, edited by J. Celeyrette and Ch. Grellard (Turnout: Brepols, 2014), 163-183.
Clagett, Marshall, The science of mechanics in the Middle Ages (Madison: University of Wisconsin Press, 1959).
Archimedes in the Middle Ages, 5 vols. (Madison and Philadelphia: University of Wisconsin Press and American Philosophical Society, 1964-1984).
Debroise, Philippe, Mathématiques de l’intensité et Merveilles de la nature, Étude sur le Tractatus de configurationibus qualitatum et motuum (Paris: Université Paris Diderot, Doctoral thesis, 2019).
Damerow, Peter; Freudenthal, Gideon; Mclaughlin, Peter and Renn, Jürgen, Exploring the Limits of Preclassical Mechanics: A Study of Conceptual Development in Early Modern Science: Free Fall and Compounded Motion in the Work of Descartes, Galileo, and Beeckman (New York: Springer, 2004).
Di Liscia, Daniel A., “La conclusio pulchra, mirabilis et bona: una ingeniosa demostración atribuible a Nicole Oresme”, Mediaevalia. Textos e estudos 37 (2018): 139-168.
Drake, Stillman, “Impetus Theory and Quanta of Speed before and after Galileo”, Physics 16 (1974): 47-65.
“Impetus Theory Reappraised”, Journal of the History of Ideas 36 (1975): 27-46.
“Free Fall from Albert of Saxony to Honoré Fabri”, Studies in History and Philosophy of Science 5 (1975): 347-366.
“A Further Reappraisal of Impetus Theory”, Studies in History and Philosophy of Science 7 (1976): 319-336.
Duhem, Pierre, Études sur Léonard de Vinci, 3 (Paris: Hermann, 1984 [1908]).
Le Système du monde. Histoire des doctrines de Platon à Copernic, vol. 7 (Paris: Hermann, 1956).
Edwards, Charles Henry Jr., The historical development of the calculus (New York: Springer Verlag, 1979).
Grellard, Christophe, Méthodes et statuts des sciences à la fin du Moyen Âge (Villeneuve d’Ascq: Presse universitaire du Septentrion, 2004).
Grellard, Christophe and Robert, Aurélien, Atomism in Late Medieval Philosophy and Theology (Leiden and Boston: Brill, 2009).
Kirschner, Stefan, “Oresme’s Theory of Motion”, in Nicole Oresme philosophe: Philosophie de la nature et philosophie de la connaissance à Paris au XIVe siècle, Studia Artistarum 39, edited by J. Celeyrette and Ch. Grellard (Turnhout: Brepols Publishers, 2014).
“Oresme on Intension and Remission of Qualities in His Commentary on Aristotle’s ‘Physics’”, Vivarium 38, 2 (2000): 255-274.
“A Possible Trace of Oresme’s Condicio-Theory of Accidents in an Anonymous Commentary on Aristotle’s Meteorology”, Vivarium 48, 3 (2010): 349-367.
McGinnis, John, “A Medieval Arabic Analysis of Motion at an Instant: The Avicennan Sources to the forma fluens/fluxus formae Debate”, The British Journal for the History of Science 39, 2 (2006): 189-205.
Maier, Anneliese, “Die scholastische Wesensbestimmung der Bewegung als forma fluens oder fluxus formae und ihre Beziehung zu Albertus Magnus”, Angelicum 21 (1944): 97-111.
“La doctrine de Nicolas d’Oresme sur les ‘configurationes intensionum’”, Revue des sciences philosophiques et théologiques 32, 1/2 (1948): 52-67. Also in Maier, Anneliese, Ausgehendes Mittelalter I: Gesammelte Aufsätze zur Geistesgeschichte des 14. Jahrhunderts (Rome: Edizioni di Storia e Letteratura, 1964), 335-352.
Ausgehendes Mittelalter I: Gesammelte Aufsätze zur Geistesgeschichte des 14. Jahrhunderts (Rome: Edizioni di Storia e Letteratura, 1964).
Mazet, Edmond, “La théorie des séries de Nicole Oresme dans sa perspective aristotélicienne. ‘Questions 1 et 2 sur la Géométrie d’Euclide’”, Revue d’histoire des mathématiques 9, 1 (2003): 33-80.
“Richard Swineshead et Nicole Oresme: deux styles mathématiques”, in Nicole Oresme philosophe: philosophie de la nature et philosophie de la connaissance à Paris au XIVe siècle, edited by J. Celeyrette and Ch. Grellard (Turnhout: Brepols, 2014), 105‑137.
Molland, Georges A., “The Atomisation of Motion: A Facet of the Scientific Revolution”, Studies in History and Philosophy of Science 13, 1 (1982): 31-54.
Murdoch, John E., “Review of Nicole Oresme’s ‘Quaestiones super geometriam Euclidis’ edited by H.L.L. Busard”, Scripta Mathematica 27 (1964): 67-91.
“Beyond Aristotle: Indivisibles and Infinite Divisibility in the Later Middle Ages”, in Atomism in Late Medieval Philosophy and Theology, edited by Ch. Grellard and A. Robert (Leiden and Boston: Brill), 15-38.
Murdoch, John E. and Sylla, Edith D., “The Science of Motion”, in Science in the Middle Ages, edited by D. C. Lindberg (Chicago and London: University of Chicago Press, 1978).
Pabst, Bernhard, Atomentheorien des lateinischen Mittelalters (Darmstadt: Wissenschaftliche Buchgesellschaft, 1994).
Pasnau, Robert, Metaphysical Themes: 1274-1671 (Oxford: Clarendon Press, 2011).
Pyle, Andrew, Atomism and its Critics. Problem Areas Associated with the Development of the Atomic Theory of Matter from Democritus to Newton (Bristol: Toemmes Press, 1995).
Quillet, Jeannine, Autour de Nicole Oresme, Actes du colloque Oresme organisé à l’Université de Paris XII (Paris: Vrin, 1990).
Rommevaux, Sabine, Les nouvelles théories des rapports mathématiques du XIVe au XVIe siècle (Turnhout: Brepols, 2014).
Sattler, Barbara M., The Concept of Motion in Ancient Greek Thought, Foundations in Logic, Method and Mathematics (Cambridge: Cambridge University Press, 2020).
Souffrin, Pierre, “La quantification du mouvement chez les scolastiques. La vitesse instantanée chez Nicole Oresme”, in Autour de Nicole Oresme, Actes du colloque Oresme organisé à l’Université de Paris XII, edited by J. Quillet (Paris: Vrin, 1990), 63 83.
Souffrin, Pierre and Weiss, Jean-Pierre, “Le Traité des configurations des qualités et des mouvements. Remarques sur quelques problèmes d’interprétation et de traduction”, in Nicolas Oresme. Tradition et innovation chez un intellectuel du XIVe siècle, edited by P. Souffrin and A.-P. Segonds (Paris: Les Belles Lettres, 1988).
Souffrin, Pierre and Segonds, Alain-Philippe, Nicolas Oresme. Tradition et innovation chez un intellectuel du XIVe siècle (Paris: Les Belles Lettres, 1988).
Thijssen, Johannes M.M.H., “The Debate Over the Nature of Motion: John Buridan, Nicole Oresme and Albert of Saxony. With an Edition of John Buridan’s Quaestiones Super Libros Physicorum, Secundum Ultimam Lecturam, Book III, Q. 7”, Evidence and Interpretation in Studies on Early Science and Medicine, edited by E. Sylla and W. R. Newman, 14, 1-3 (2009): 186-210.
Wilson, Curtis, William heytesbury. Medieval Logic and the Rise of Mathematical Physics (Madison: University of Wisconsin Press, 1960).
Youschkevitch, Adolf P., Geschichte der Mathematik im Mittelalter (Leipzig: Teubner, 1963).