The place of Musica in medieval classifications of knowledge

Journal of Musicology

Volumn 24, Issue 1, 2007, Pages 3-71

  Dyer, Joseph ^a  

^a NONE

Author keywords

Boethius; Classifications of knowledge; Divisio scientiarum; Musica speculativa; Quadrivium

Indexed keywords

EID: 57749157439     PISSN: 02779269     EISSN: None     Source Type: Journal    
DOI: 10.1525/jm.2007.24.1.3     Document Type: Article

References (158)

1. For a recent explanation of the distinction see Wolfgang Fuhrmann, Herz und Stimme: Innerlichkeit, Affekt und Gesang im Mittelalter, Musiksoziologie 13 (Kassel: Bärenreiter, 2004), 19-47
2. Fernand Van Steenberghen, "L'organisation des études au moyen âge et ses repercussions sur le mouvement philosophique," Revue philosophique de Louvain 52 (1954): 583-92
3. Martin Grabmann, Die Geschichte der scholastischen Methode, 2 vols. (Freiburg im Breisgau: Herder, 1909; repr. Graz: Akademische Druck- und Verlagsanstalt, 1957), 2:28-54
4. e siècles," Bulletin Du Gange [Archivum Latinitatis Medii Aevi] 46-47 (1988): 39-64
5. Modern understanding of "science" inevitably involves the empirical observation of natural phenomena, the use of experimentation, and application of strict principles of demonstration to arrive at an explanation of the phenomena, but natural science was considered a philosophical discipline in the Middle Ages. The word scientia had a wide range of meaning for the Middle Ages, being the equivalent of both "knowledge" or " philosophy" in general. Boethius, like Aristotle before him, defined scientific knowledge as a "doctrine built on first principles and derived by strict demonstration," according to Charles Lohr, "The New Aristotle and 'Science' in the Paris Arts Faculty," in L'enseignement des disciplines, 263-64 [251-69]
6. Pierre Riché, Education and Culture in the Barbarian West, trans. John Contreni (Columbia: Univ. of South Carolina Press, 1976)
7. David L. Wagner, ed., The Seven Liberal Arts in the Middle Ages (Bloomington: Indiana Univ. Press, 1983)
8. Günther Glauche, Schullektüre im Mittelalter. Entstehung und Wandlungen des Lektürekanons bis 1200 nach den Quellen dargestellt, Münchener Beiträge zur Mediävistik und Renaissance-Forschung 5 (Munich: Arbeo Gesellschaft, 1970)
9. Excerpts from the Problemata are printed in Andrew Barker, Greek Musical Writings, 2 vols. (Cambridge: Cambridge Univ. Press, 1984), 2:85-97
10. Ancius Manlius Severinus Boethius, De arithmetica, ed. Henricus Oosthout and Iohannes Schilling, Corpus Christianorum: Series Latina 94A (Turnhout: Brepols, 1999), 10-14
11. On the quadrivium see John North, "The Quadrivium," in A History of the University in Europe 1: Universities in the Middle Ages, ed. Hilde de Ridder-Symoens (Cambridge: Cambridge Univ. Pres, 1992), 337-59, though North attributes to music a more important part in the university curriculum than I would be inclined to do (pp. 343-44)
12. Pearl Kibre, "The Quadrivium in the 13th-century Universities (with Special Reference to Paris)" in Arts libéraux et philosophie au moyen age, 185-87, an erudite article in which the references to music treatises could have been better informed
13. According to Hans Martin Klinkenberg, "die divisio philosophiae gehört zu den großen philosophischen Problemen;" see "Divisio philosophiae" in Scientia und Ars im Hoch- und Spätmittelalter, ed. Ingrid Craemer-Ruegenberg and Andreas Speer, Miscellanea Medievalia 22, 2 vols. (Berlin: Walter de Gruyter, 1994), 1:3-19, esp. p. 3
14. James A. Weisheipl, "The Concept of Scientific Knowledge in Greek Philosophy," in Mélanges à la mémoire de Charles De Konnick (Québec: Presses de l'Université Laval, 1968), 487-507
15. For the classical background I am indebted to Pierre Hadot, "Les divisions des parties de la philosophie dans l'Antiquité," Museum Helveticum 36 (1979): 201-23
16. see also James A. Weisheipl, "The Nature, Scope, and Classification of the Sciences," in Science in the Middle Ages, ed. David C. Lindberg (Chicago: Univ. of Chicago Press, 1978), 464-66
17. see also the analysis in Maurer, The Division and Methods of the Sciences, Medieval Sources in Translation 3, 4th rev. ed. (Toronto: Pontifical Institute of Mediaeval Studies, 1986), 27-28
18. Weisheipl, "The Nature," 465
19. William D. Ross argues that Aristotle fairly portrayed Plato's theory on this point; Aristotle, 5th ed. rev. (London: Methuen, 1949), 157-59
20. Augustine, De civitate dei 8.4
21. Sancti Aurelii Augustini episcopi opera, 5/1-2, ed. Emanuel Hoffmann, 2 vols., Corpus Scriptorum Ecclesiasticorum Latinorum 40 (Vienna: F. Tempsky, 1899), 1:359
22. De celo et mundo 2.9; 290b 12-291a 28
23. trans. John L. Stocks in The Works of Aristotle, ed. William D. Ross, The Works of Aristotle, vol. 2 (Oxford: Clarendon Press, 1930)
24. Metaphysics 6.1-2 (1025b 1-1026a 33)
25. cf. Metaphysics 1.1, 980b 25-981b 10
26. cf. Metaphysics 1064a 10 and 16-18)
27. Calvin Bower, "Boethius," New Grove Dictionary of Music and Musicians, 2nd ed. (New York: Grove's Dictionaries, 2000), 2:784-86
28. Margaret T. Gibson, ed., Boethius: His Life, Thought and Influence (Oxford: Blackwell, 1981)
29. Leo Schrade ("Music in the Philosophy of Boethius," Musical Quarterly 33 [1947]: 188-200) does not seem to discriminate between the Isagoge commentaries and De Trinitate
30. In Isagoge Porphyrii Comm. ed. prima, 1:3-4, ed. Samuel Brand, Corpus Scriptorum Ecclesiasticorum Latinorum 48 (Vienna: F. Tempsky 1906), 7-10; hereafter CSEL
31. The Isagoge has been translated by Edward W. Warren, Porphyry the Phoenician: Isagoge, Mediaeval Sources in Translation 6 (Toronto: Pontifical Institute of Mediaeval Studies, 1975)
32. See Ralph McInerny, Boethius and Aquinas (Washington: The Catholic Univ. of America Press, 1990)
33. On the sources see Enders, "Zum Philosophie-Verständnis," 450n29
34. on De Trinitate, see McInerny, Boethius and Aquinas, 121-24
35. Ancius Manlius Severinus Boethius, De Trinitate, ed. Hugh F. Stewart, Edward K. Rand, and S. Jim Tester, Boethius: The Theological Tractates, revised ed. (Cambridge: Harvard Univ. Press, 1973 [1990]), 8-12
36. see also Grabmann, Die scholastische Methode, 1:163-77
37. Thomas Aquinas, In Boethii De Trinitate, q 5, a 1, sed e contra, ed. Decker, Expositio super librum Boethii De Trinitate, 163-64
38. Maurer, The Division and Methods of the Sciences, 12
39. On the reception of Boethius previous to the 12th century, see Calvin Bower, "The Role of Boethius' De institutione musica in the Speculative Tradition of Western Musical Thought," in Boethius and the Liberal Arts: A Collection of Essays, ed. Michael Masi, Utah Studies in Literature and Linguistics 18 (Bern: Peter Lang, 1981), 157-74
40. The treatise on music has been translated into English with copious annotations by Calvin Bower, Fundamentals of Music (New Haven: Yale Univ. Press, 1989), 17 (all English translations are taken from this volume)
41. Helpful also is the Latin-French edition of Christian Meyer, Boèce: Traité de la Musique (Turnhout: Brepols, 2004), 44-47. Most of chap. 9 of the first book of De institutione musica is taken up with the "deception of the senses" as it relates to the definition and study of music
42. Boethius' ambivalence is discussed in Eva Hirtler, Die Musica als scientia mathematica von der Spätantike bis zum Barock, Europäische Hochschulschriften, series 36, vol. 137 (Frankfurt: Peter Lang, 1995), 25-45
43. De institutione musica 1.2; ed. Friedlein, 187-88; trans. Bower, 9-10
44. Musicae partes sunt tres: armonica - rithmica - metrica. Armonica est scientia musica quae decernit in sonis acutum et gravem. Rithmica est quae requirit incursionem verborum, utrum bene sonus an male cohaereat. Metrica est quae mensuram diversorum metrorum probabili ratione cognoscit; Roger A. B. Mynors, ed., Cassiodori Senatoris Institutiones (Oxford: Clarendon Press, 1937), 142-50 (2.5)
45. trans. Leslie Webber Jones, An Introduction to Divine and Human Readings (New York: Columbia Univ. Press, 1946; repr. 1966), 190-91
46. Cf. Aurelian of Réôme, Musica Disciplina 4, ed. Lawrence Gushee, Corpus Scriptorum de Musica 21 (n.p.: American Institute of Musicology 1975), 66-68 (hereafter CSM)
47. For this reference I am indebted to Frank Ll. Harrison, "Music at Oxford before 1500," in The History of the University of Oxford 2: Late Medieval Oxford, ed. Jeremy I. Catto and Ralph Evans (Oxford: Clarendon Press, 1992), 347-68
48. see also Gordon Leff, "The Trivium and the Three Philosophies," in A History of the University in Europe 1: Universities in the Middle Ages, 308
49. Olga Weijers, Le maniement du savoir, 192. She notes the frequency with which the introductions to commentaries on Porphyry's Isagoge contained such classifications
50. e siècle: Le Tractatus quidam de philosophia et partibus eius," Archives d'histoire doctrinale et littéraire du moyen age 49 (1982): 155-93
51. e siècle: Textes critiques et études historiques (Montréal: Institut d'Études Médiévales, 1988). These will be discussed below
52. Grabmann, Die scholastische Methode, 2:42-58
53. e siècles," in L'enseignement philosophique 40/4 (1990): 5-27
54. idem, "Le classificazioni delle scienze e l'insegnamento universitario nell XII secolo," in Le Universitá dell'Europa: Le Scuole e i Maestri. Il Medioevo, eds. Gian Paolo Brizzi and Jacques Vergier (Milan: Silvana, 1994), 19-43
55. Speculum doctrinale 1.14-18
56. which refers to a dissertation by Serge Lusignan, "Le Speculum doctrinale, livre III: Etude de la logique dans le Miroir des sciences de Vincent de Beauvais" (Ph.D diss., Université de Montréal, 1971, 117-39)
57. On the presence of philosophical concepts in theory treatises see Joseph Dyer, "Chant Theory and Philosophy in the Late 13th Century," in Cantus Planus: Papers Read at the Fourth Meeting - Pécs, Hungary, 3-8 September 1990, ed. László Dobszay (Budapest: Hungarian Academy of Sciences-Institute for Musicology 1992), 99-118
58. Hermann Abert, Die Musikanschauung des Mittelalters und ihre Grundlagen (Halle an der Saale: M. Niemeyer, 1905; repr. Tutzing: Schneider, 1964) takes up in detail only the church Fathers and the music theorists
59. as does Gerhard Pietzsch, Die Klassifikation der Musik von Boetius bis Ugolino von Orvieto, Studies zur Geschichte der Musiktheorie im Mittelalter 1 (Halle an der Saale: M. Niemeyer, 1929)
60. see also Frank Hentschel, Musik - und die Geschichte der Philosophie und Naturwissenschaften im Mittelalter: Fragen zur Wechselwirkung von 'Musica' und 'Philosophia' im Mittelalter, Studien und Texte zur Geistesgeschichte des Mittelalters 62 (Leiden: E. J. Brill, 1998)
61. Jacques Maritain, Philosophy of Nature, 16
62. [Jules] Alexandre Clerval, Les écoles de Chartres au Moyen Âge: du Ve au XVIe siècle (Paris: A. Picard et fils, 1895; repr. Frankfurt am Main: Minerva Verlag, 1965)
63. Raymond Klibansky, "The School of Chartres," in Twelfth-Century Europe and the Foundations of Modern Society, ed. Marshall Clagett, Gaines Post, and Robert Reynolds (Madison: Univ. of Wisconsin Press, 1961), 3-14
64. Richard W. Southern, Medieval Humanism and Other Studies (Oxford: Oxford University Press, 1970), 61-85
65. Jean Châtillon, "Les écoles de Chartres et de Saint-Victor," in La scuola nell'Occidente latino dell'alto medioevo, Settimane di studio sull'alto medioevo 19, 2 vols. (Spoleto: Centro Italiano di Studi sull'Alto Medioevo, 1972), 2:798-801
66. His character is eloquently portrayed in Richard W. Southern, The Making of the Middle Ages (New Haven: Yale Univ. Press, 1953), 199-206
67. Winthrop Wetherbee, "The School of Chartres," in A Companion to Philosophy in the Middle Ages, ed. Jorge J. E. Garcia and Timothy B. Noone (Oxford: Blackwell, 2003), 37-39; Wetherbee contests the position of Richard Southern, who had denied that the scholars of Chartres were harbingers of new learning, but only "the last of the Carolingian grammarians."
68. PL 172:247. Steven P. Marrone renarks that "a Platonic worldview did shape most approaches to nature in western Europe in the 12th century" in "Medieval Philosophy in Context," in The Cambridge Companion to Medieval Philosophy, ed. Arthur S. McGrade (Cambridge: Cambridge Univ. Press, 2003), 26
69. see also Schueller, The Idea of Music, 351-53
70. Edouard Jeanneau, ed., Guillaume de Conches: Glosae super Platonem (Paris: J. Vrin, 1965)
71. Carmelo Ottaviano, Un brano inedito della Philosophia di Guglielmo di Conches (Naples: A. Morano, 1935), 26ff
72. cf. Cassiodorus, Institutiones 2.5.5 (musicae partes sunt tres: armonica - rithmica - metrica), ed. Mynors, 144. Cassiodorus' division of musica instrumentalis was percussionalia - tensibilia - inflatilia, a division not unlike that of Boethius
73. See also the discussion in Weisheipl, "The Nature," 474
74. and idem, "Classification of the Sciences in Medieval Thought," 65-66 ("a successful combination of the Boethian and Stoic classifications of science")
75. Weijers, Le maniement, 189-90
76. Schueller, The Idea of Music, 348-51 and Appendix C (outline), 439-41
77. Dahan, "Tractatus," 173
78. Pietzsch, Die Klassifikation, 75-77
79. The definition of music draws on a frequent etymology: "musica ab aqua vocabulum sumpsit, eo quod nulla euphonia, id est bona sonoritas, sine humore fieri possit." On this etymology see Noel Swerdlow, "Musica dicitur a Moys, quod est aqua," Journal of the American Musicological Society 20 (1967): 3-9
80. De institutione musica 1.2; ed. Friedlein, 187-89; trans. Bower, 9-10 (with references to Boethius' sources). There is a summary of Hugh's (Boethius') classification of music in the Liber Excerptionum (PL 177:198 [Appendix ad Hugonis Opera Dogmatica])
81. The ternary subdivisions were repeated in later sources; see F. Alberto Gallo, "Die Musik in der Einteilung der Wissenschaften bei Egidius Romanus und Johannes Dacus," International Musicological Society: Report of the 11th Congress, Copenhagen 1972, ed. Henrik Glahn, Søren Sørensen, and Peter Ryom, 2 vols. (Copenhagen: Wilhelm Hansen, 1974), 1:388-90
82. Didascalicon 2.12; ed. Buttimer, 33
83. see also Daniel F. Blackwell, "The artes liberates as Remedies," Theologische Zeitschrift 45 (1989): 117 [115-24]
84. cf. Cassiodorus, Institutiones 2.20; ed. Mynors, 130.4-8
85. see also in the same collection Verena Epp, "Ars und scientia in der Geschichtsschreibung des 12. Jahrhunderts," 2:829-45
86. John of Afflighem, quoting Guido, is just as insulting ("to whom should I better compare the singer than to a drunken man"); De Musica cum tonario, ed. Joseph Smits van Waesberghe, CSM 1 (Rome: American Institute of Musicology, 1950), 52
87. cf. also the anonymous Victorine Excerptionum allegoricarum libri XXIV, 1.21 (PL, 177:201)
88. On the artes mechanice see Guy H. Allard and Serge Lusignan, eds., Les arts méchaniques au moyen-age, Cahiers d'Études Médiévales 7 (Montréal: Bellarmin, 1982)
89. and Mariken Teeuwen, The Vocabulary of Intellectual Life in the Middle Ages, Études sur le Vocabulaire Intellectuel du Moyen Âge 10 (Turnhout: Brepols, 2003), 367-69
90. and by Bernhard Bischoff, "Eine verschollene Einteilung der Wissenschaften," Archive d'histoire doctrinale et littéraire du moyen age 25 (1958): 5-20
91. e siècle: Le Tractatus quidam de philosophia et partibus eius," 166-68
92. e siècle"; the Tractatus is also discussed by Grabmann, Geschichte der scholastischen Methode, 2:45-48
93. Tractatus, ed. Dahan, 182-87
94. Ed. Dahan, 186. There is also a sidelong glance toward the nine Muses
95. The modern edition of Raoul's commentary is In Anticlaudianum Alani commentum, ed. Jan Sulowski, Źródła do Dziejów Nauki i Techniki 13 (Warsaw: Zakład Narodowy im. Ossolińskich, 1972); for the capitula concerning the classification system and music see pages 39-41 and 189-97, respectively
96. Ed. Robert Bossuat, Alain de Lille, Anticlaudianus, Texte critique avec une introduction et des tables (Paris: Vrin, 1955)
97. trans. James J. Sheridan, Anticlaudianus, or the Good and Perfect Man (Toronto: Pontifical Institute of Mediaeval Studies, 1973)
98. Table 4 is adapted from from Grabmann, Geschichte der scholastischen Methode, 2:52
99. Edited in Baur, De divisione (n1 above). As is clear from Table 5, the Boethian divisions were not neglected; see Alexander Fidora, "Die Rezeption der boethianischen Wissenschaftseinteilung bei Dominicus Gundissalinus," in "Scientia" und "Disciplina, " 209-22
100. Domingo Gundisalvo, De Scientiis, ed. Manuel Alonso Alonso (Madrid: Imp. y Editorial Maestre, 1954)
101. e siècle, Philosophes Médiévaux 9 (Louvain: Publications Universitaires, 1966), 122
102. Baur, De divisione, 3-19
103. Anton-Hermann Chroust, "The Definitions of Philosophy in the De divisione philosophiae of Dominicus Gundissalinus," The New Scholasticism, 25 (1951): 253-81. There are six definitions
104. De scientiis, ed. Alonso Alonso, 105-7
105. Don M. Randel, "Al-Fārābī and the Role of Arabic Music Theory in the Latin Middle Ages," Journal of the American Musicological Society 29 (1976): 173-88
106. Jerome of Moravia [Moray (Scotland)?] quoted this passage in his Tractatus de Musica, ed. Simon Cserba, Freiburger Studien zur Musikwissenschaft 2 (Regensburg: F. Pustet, 1935), 22-23 ("Divisio musicae secundum Alphorabium") from the Speculum doctrinale of the encyclopedist Vincent of Beauvais
107. On Jerome's geographical origin, see Michel Huglo, "La Musica du Fr. Prêcheur Jérome de Moray," in Max Lütolf zum 60, Geburtstag, ed. Bernhard Hangartner and Urs Fischer (Basel: Fischer, 1994), 113-16
108. Calvin Bower, "Natural and Artificial Music: The Origins and Development of an Aesthetic Concept," Musica Disciplina 25 (1971): 17-33
109. De divisione philosophiae, p. 39, ll. 10-14
110. Robert Kilwardby, De ortu scientiarum, ed. Albert G. Judy, Auctores Brittanici Medii Aevi 4 (London: The British Academy, 1976). The extracts available at TML:KILDOS must be read in conjunction with the complete text
111. See also D. E. Sharp, "The De ortu scientiarum of Robert Kilwardby (d. 1279)," The New Scholasticism, 8 (1934): 1-30
112. For other schematic reductions of Kilwardby's classification see Weisheipl, "The Nature," 480 [chart] and Weijers, Le maniement, 194
113. A listing of each chapter and brief summary of contents may be found in Baur, De divisione, 369-80
114. Marie-Dominique Chenu, "Arts méchaniques et oeuvres serviles," Revue des sciences philosophiques et théologiques 29 (1940): 313-15
115. Franco Alessio, "La filosofia e le 'artes mechanicae' nel secolo XII," Studi Medievali, ser. terza, 6/1 (1965): 71-161
116. Kilwardby's treatment of music receives due attention in Eva Hirtler, Die Musica als scientia mathematica, 67-86
117. De ortu 18 (§ 127), ed. Judy, 51
118. Cf. Boethius, De institutione arithmetica 1.21-24; ed. Friedlein, 45-51
119. Cf. Aristotle, Posterior Analytics 1.13 (78b 32-39)
120. Operantes enim licet considèrent in eo quod agunt, tamen non perscrutantur de causa propter seipsam sed respectu eius quod agunt; De ortu 40 (§ 394), ed. Judy 138
121. cf. Aristotle: "the end of theoretical knowledge is truth, while that of practical knowledge is action;" Metaphysics 2.1 (993b 20), trans. Ross, The Works of Aristotle, vol. 8
122. The citations from Eustratios are supplied by Judy, De ortu, 143-44
123. Cf. the commentary of Eustratios: "sunt autem operativae artes quae post operationem per artem factam non relinquunt opus manens;" De ortu 44 (§ 412), ed. Judy, p. 144, n. 3
124. Ed. Pesce, Guido d'Arezzo's Regule rithmice, 330-31
125. See also Max Haas, "Musik zwischen Mathematik und Physik: Zur Bedeutung der Notation in der 'Notitia artis musicae' des Johannes de Muris," in Festschrift Arno Volk, ed. Carl Dahlhaus and Hans Oesch (Cologne: Gerig, 1974), 31-46
126. Beati Alberti Magni Ratisbonensis Episcopi Ordinis Praedicatorum Opera Omnia, ed. Auguste Borgnet, 38 vols. (Paris: Louis Vivès, 1890-95), 2:66
127. e siècle," in Philosophy and Learning: Universities in the Middle Ages, ed. Maarten J. F. E. Hoenen, Jakob Hans Josef Schneider, and Georg Wieland (Leiden: E.J. Brill, 1995), 137-99
128. Olga Weijers, Le maniement du savoir, 30-32 ("la littérature d'introduction")
129. Lafleur, Quatre Introductions, 177-253
130. On the introductory functions of the "accessus" genre, see Edwin A. Quain, "The Medieval accessus ad auctores," Traditio 3 (1945): 215-64
131. Teeuwen, The Vocabulary of Intellectual Life, 215-17
132. Accessus, 183 (11. 45-56)
133. see also Weijers, Le maniement du savoir, 58
134. trans. Barker, Greek Musical Writings, 2:58-61
135. Christian Meyer, "Un abrégé universitaire des deux premiers livres du De institutione musica de Boèce," Archives d'histoire doctrinale et littéraire du moyen age 65 (1998): 91-121, esp. 109
136. Cf. Aristotle, Metaphysics 6.1 (1026a 18-19)
137. Prout recipit aliquam differentiam alterius generis supra ipsum, sicut sonorum, et de hac est musica; Lafleur, Quatre Introductions, 265 (11. 172-73)
138. For revealing insights into what Boethius might have said about musica mundana and humana, see David S. Chamberlain, "Philosophy of Music in the Consolatio of Boethius," Speculum 45 (1970): 80-97
139. Arnoul's Divisio scientiarum is edited in Lafleur, Quatre Introductions, 295-347
140. Arnoul here spells out what was only implied in the second division of Philosophica disciplina
141. trans. Ross, The Works of Aristotle, vol. 8
142. Hentschel, Sinnlichkeit und Vernunft, 288-92
143. Physica (Translatio vetus), eds. Fernand Bossier and Jozef Brams, Aristoteles Latinus 7/1 (Leiden: E.J. Brill, 1990), 51
144. "Similar evidence is supplied by the more physical of the branches of mathematics, such as optics, harmonics, and astronomy;" Physics 2.2 (194a 6)
145. trans. Robert P. Hardie and Russel K. Gaye, The Works of Aristotle, ed. William D. Ross, vol. 2 (Oxford: Clarendon Press, 1930)
146. For a discussion of Raoul's reconciliation of geometry and optics see Frank Hentschel with Martin Pickavé, "Quaestiones mathematicales: Eine Textgattung der Pariser Artistenfacultät im frühen 14. Jahrhundert," in Nach der Verurteilung von 1277: Philosophie an der Universität von Paris im letzten Viertel des 13. Jahrhunderts, eds. Jan A. Aertsen, Kent Emery, Jr., and Andreas Speer, Miscellanea Medievalia 28 (Berlin: Walter de Gruyter, 2001), 632-34
147. Physica (Translatio vetus); ed. Fernand Bossier and Jozef Brams, 51
148. Regrettably, the current English translation in the widely disseminated Loeb Classical Library edition misinterprets the passage; The Physics, ed. Philip H. Wicksteed and Francis M. Cornford (Cambridge: Harvard Univ. Press, 1980), 121
149. Hentschel, Sinnlichkeit und Vernunft, 290 (25-26) and 292 (45)
150. Post. Analytics 1.13 (78b 35-79a 15), trans. Barnes, 21
151. Jean Gagné, "Du quadrivium aux scientiae mediae," in Arts libéraux et philosophie au moyen age, 978 [975-86]; Gagné also notes parallels in Arabie philosophy
152. See also Siegfried Neumann, Gegenstand und Methode der theoretischen Wissenschaften nach Thomas von Aquin aufgrund der Expositio super librum Boethii, Beiträge zur Geschichte des Philosophie und Theologie des Mittelalters, Texte und Untersuchungen 41/2 (Münster in Westfalen: Aschendorff, 1965), 106-13
153. Eva Hirtler, "Die musica im Ubergang von der scientia mathematica zur scientia media" in Musik - und die Geschichte der Philosophie und Naturwissenschaften im Mittelalter, 19-37, esp. 33-37
154. Hentschel, Sinnlichkeit und Vernunft, 132-37 (with many references to related texts)
155. This topic is not taken up in Josef Hermann Burbach, Studien zur Musikanschauung des Thomas von Aquin, Kölner Beiträge zur Musikforschung 34 (Regensburg: Bosse, 1966)
156. Et ideo nihil prohibet, si in quantum cum naturali communicant, materiam sensibilem respiciunt; in quantum enim cum mathematica communicant, abstractae sunt, In Boethii De Trinitate, q. 5 a. 3 ad 6; ed. Decker, 189, trans. Armand Maurer, The Division and Methods of the Sciences [=introduction and translation of qq. 5 and 6 of In Boethii de Trinitate], 4th rev. ed., Medieval Sources in Translation 3 (Toronto: Pontifical Institute of Medieval Studies, 1986), 45
157. Olaf Pedersen, "Du quadrivium à la physique: Quelques apercu̧s de l'évolution scientifique au Moyen Âge," in Artes Liberales: Von der antiken Bildung zur Wissenschaft des Mittehlters, 107-23
158. Hans Martin Klinkenberg, "Divisio philosophiae," in Scientia und Ars im Hoch- und Spätmittelalter, 1:3-19, esp. p. 7