The Abacus And The Cross Part 11

35 "Little Red Riding Hood": "Little Red Riding Hood": Riche, Riche, Les grandeurs de l'an mil Les grandeurs de l'an mil, 199-200.

36 the cla.s.sics: the cla.s.sics: Riche, Riche, Gerbert d'Aurillac Gerbert d'Aurillac, 42.

36 "quench your thirst": "quench your thirst": Gerbert, 205; translated by Darlington, "Gerbert the Teacher," 465. "The features of my friend remain fixed in my heart" is from Gerbert, 83, 93; "Equally in leisure and in work" is from Gerbert, 92; "Although really still a learner" is from Gerbert, 45, all in Lattin's translation. Gerbert, 205; translated by Darlington, "Gerbert the Teacher," 465. "The features of my friend remain fixed in my heart" is from Gerbert, 83, 93; "Equally in leisure and in work" is from Gerbert, 92; "Although really still a learner" is from Gerbert, 45, all in Lattin's translation.

37 Gerbert's writing: Gerbert's writing: Riche, Riche, Gerbert d'Aurillac Gerbert d'Aurillac, 250; Lattin, introduction to The Letters of Gerbert The Letters of Gerbert, 27-28. Gerbert's "Ciceronian eloquence" is remarked on by Richer of Saint-Remy, vol. 2, 51.

37 kinds of logic: kinds of logic: Fulbert of Chartres, Fulbert of Chartres, The Letters and Poems of Fulbert of Chartres The Letters and Poems of Fulbert of Chartres, translated by Frederick Behrends, xxv-xxvi. Contreni gives the "G.o.d is not anywhere" lesson, "The Pursuit of Knowledge in Carolingian Europe," in Sullivan, 123.

38 "father and light": "father and light": Teta E. Moehs, "Gerbert of Aurillac as Link Between Cla.s.sicism and Medieval Scholarship," in M. Tosi, ed., Teta E. Moehs, "Gerbert of Aurillac as Link Between Cla.s.sicism and Medieval Scholarship," in M. Tosi, ed., Gerberto Gerberto, 342. See also Pascale Bourgain, "L'Hommage de Gerbert a Boece," in Olivier Guyotjeannin and Emmanuel Poulle, Autour de Gerbert d'Aurillac Autour de Gerbert d'Aurillac, 296-299. "For these cares ... ," Gerbert, 91.

38 "so far as life": "so far as life": Boethius, Boethius, The Consolation of Philosophy The Consolation of Philosophy, translated by E. Watts, 12.

39 "practice disputation": "practice disputation": Richer of Saint-Remy, translated by Frederick Behrends in his introduction to Richer of Saint-Remy, translated by Frederick Behrends in his introduction to The Letters and Poems of Fulbert of Chartres The Letters and Poems of Fulbert of Chartres, x.x.x. C. Stephen Jaeger cites Alcuin in The Envy of Angels The Envy of Angels, 32; for "n.o.bility of soul," see 56, 93, 102, 113.

39 "all things human and divine": "all things human and divine": Richer of Saint-Remy, vol. 2, 77. His definition makes it clear that "philosophy" had a much larger meaning for Gerbert than it does today. Richer of Saint-Remy, vol. 2, 77. His definition makes it clear that "philosophy" had a much larger meaning for Gerbert than it does today.

40 "made his escape": "made his escape": William of Malmesbury, William of Malmesbury, Gesta Regum Anglorum Gesta Regum Anglorum, translated by R. A. B. Mynors, R. M. Thomson, and M. Winterbottom, vol. 1, 279 (hereafter cited as "William of Malmesbury").

Chapter III: The Ornament of the World.

41 "The abbot inquired": "The abbot inquired": Richer of Saint-Remy, translated by Darlington, "Gerbert the Teacher," 460. For Gerbert's letters on Spain, see Gerbert, 115, 117. Borrell's son is quoted by Pierre Riche, Richer of Saint-Remy, translated by Darlington, "Gerbert the Teacher," 460. For Gerbert's letters on Spain, see Gerbert, 115, 117. Borrell's son is quoted by Pierre Riche, Gerbert d'Aurillac Gerbert d'Aurillac, 234.

42 Borrell's Spain: Borrell's Spain: Paul H. Freedman, Paul H. Freedman, The Diocese of Vic The Diocese of Vic, 116-118. On the Visigoths' sophisticated code of law, see Richard Fletcher, The Quest for El Cid The Quest for El Cid, 13-14.

43 Song of Roland: Maria Rosa Menocal, The Ornament of the World The Ornament of the World, 56-58.

46 "searched with great zeal": "searched with great zeal": J. L. Berggren, J. L. Berggren, Episodes in the Mathematics of Medieval Islam Episodes in the Mathematics of Medieval Islam, 4; he also discusses al-Khwarizmi's work, 6-9. On star tables, see David A. King, "Islamic Astronomy," in Christopher Walker, ed., Astronomy Before the Telescope Astronomy Before the Telescope, 143-174.

47 Al-Khwarizmi's science: Al-Khwarizmi's science: Michael C. Weber argues that it reached Spain during al-Khwarizmi's lifetime; see Thomas Glick, Steven J. Livesey, and Faith Wallis, eds., Michael C. Weber argues that it reached Spain during al-Khwarizmi's lifetime; see Thomas Glick, Steven J. Livesey, and Faith Wallis, eds., Medieval Science, Technology, and Medicine: An Encyclopedia Medieval Science, Technology, and Medicine: An Encyclopedia, 333. Glick says the star tables, at least, had arrived "in the mid-ninth century," in From Muslim Fortress to Christian Castle From Muslim Fortress to Christian Castle, 47. According to Georges Hourani, "Indian numerals and the Arabic zero as explained by Khwarizmi were known in Andalusia before 859"; see "The Early Growth of the Secular Sciences in Andalusia," 148.

48 magnificent land: magnificent land: Muhammad Ibn Hauqal, Muhammad Ibn Hauqal, Configuration de la terre (Kitab surat al-Ard) Configuration de la terre (Kitab surat al-Ard), edited and translated (into French) by J. H. Kramers and G. Wiet. For Hasdai ibn Shaprut's letter, see Hasdai ibn Shaprut, "The Epistle of R. Chisdai, son of Isaac ... " in Jewish Travellers Jewish Travellers, translated by Elkan Nathan Adler, 22-32. For the calendar, see the anonymous work Le Calendrier de Cordoue Le Calendrier de Cordoue, edited by R. Dozy and translated (into French) by Ch. Pellat.

49 paper-making: paper-making: The name "the papermaker" is cited by Glick, The name "the papermaker" is cited by Glick, Islamic and Christian Spain Islamic and Christian Spain, 279. Menocal locates a tenth-century paper mill near Valencia, 34, and describes "astonishing" Cordoba, 32. Hourani discusses the 400,000 (or 40,000) books in the caliph's library, 148.

50 "ornament of the world": "ornament of the world": Menocal translates from Hrosvit of Gandersheim's poem "Pelagius," 12. For the complete poem, see Hrotsvit of Gandersheim, Menocal translates from Hrosvit of Gandersheim's poem "Pelagius," 12. For the complete poem, see Hrotsvit of Gandersheim, Hrotsvit of Gandersheim: A Florilegium of Her Works Hrotsvit of Gandersheim: A Florilegium of Her Works, translated by Katharina Wilson, 29-40. Stephen C. McCluskey notes Racemundo's contributions to the Calendar of Cordoba in Astronomies and Cultures in Early Medieval Europe Astronomies and Cultures in Early Medieval Europe, 166-170. Menocal discusses the dhimma dhimma, 72ff.

51 "group of doctors": "group of doctors": Hourani translates Ibn Juljul's anecdote, 155-156. Hourani translates Ibn Juljul's anecdote, 155-156.

52 "the numbers and the figures": "the numbers and the figures": Menso Folkerts, "The Names and Forms of the Numerals on the Abacus in the Gerbert Tradition." David Juste brings up the number of ma.n.u.scripts of the astrolabe treatise to argue that Gerbert did not write it; see Menso Folkerts, "The Names and Forms of the Numerals on the Abacus in the Gerbert Tradition." David Juste brings up the number of ma.n.u.scripts of the astrolabe treatise to argue that Gerbert did not write it; see Les Alchandreana primitifs Les Alchandreana primitifs, 219-294.

52 "surpa.s.sed Ptolemy": "surpa.s.sed Ptolemy": William of Malmesbury, vol. 1, 279. Richer of Saint-Remy is cited by Darlington in "Gerbert the Teacher," 460. Ademar de Chabannes is quoted by Marco Zuccato, who develops the idea of "Gotmar's circle" in "Gerbert of Aurillac and a Tenth-Century Jewish Channel for the Transmission of Arabic Science to the West," 750-752. William of Malmesbury, vol. 1, 279. Richer of Saint-Remy is cited by Darlington in "Gerbert the Teacher," 460. Ademar de Chabannes is quoted by Marco Zuccato, who develops the idea of "Gotmar's circle" in "Gerbert of Aurillac and a Tenth-Century Jewish Channel for the Transmission of Arabic Science to the West," 750-752.

56 Abbot Garin: Abbot Garin: Andre Bonnery, Andre Bonnery, L'abbaye Saint-Michel de Cuixa L'abbaye Saint-Michel de Cuixa; he quotes Miro, 9. Also Pierre Riche, Les grandeurs de l'an mil Les grandeurs de l'an mil, 116-117. For his letters, see Gerbert, 63, 70, 73, 91. Zuccato identifies Hasdai as "Joseph the Wise" in "A Tenth-Century Jewish Channel," 754.

57 school at Ripoll: school at Ripoll: Joseph Pijoan, "Oliba de Ripoll (971-1046)," and Freedman, 25-26. On the ma.n.u.script Ripoll 225, see Paul Kunitzsch, "Al-Khwarizmi as a Source for the Joseph Pijoan, "Oliba de Ripoll (971-1046)," and Freedman, 25-26. On the ma.n.u.script Ripoll 225, see Paul Kunitzsch, "Al-Khwarizmi as a Source for the Sententie astrolabii Sententie astrolabii," in D. A. King and G. Saliba, eds., From Deferent to Equant From Deferent to Equant, 227. The libraries in Ripoll and Vic are compared by Michel Zimmerman, "La Catalogne de Gerbert," in Nicole Charbonnel and Jean-Eric Iung, eds., Gerbert l'Europeen Gerbert l'Europeen, 84.

58 Lobet: Lobet: Gerbert, 69. Lattin identifies Lobet in "Lupitus Barchinonensis." Gerbert, 69. Lattin identifies Lobet in "Lupitus Barchinonensis."

58 Vigila: Vigila: See Juste, See Juste, Les Alchandreana primitifs Les Alchandreana primitifs, 246; Marcel Des...o...b..s, "Un astrolabe carolingien et l'origine de nos chiffres arabes," 9; and Georges Ifrah, The Universal History of Numbers The Universal History of Numbers, 362.

58 Miro Bonfill: Miro Bonfill: See Juste, See Juste, Les Alchandreana primitifs Les Alchandreana primitifs, 242-247. That "the wisest scholar" could have been Maslama of Madrid is my speculation, not Juste's. For the letter, see Gerbert, 70. On the Elne stone, see Flavio Nuvolone's works, particularly "Elna e l'iscrizione attribuita a Gerberto d'Aurillac: Gerberto si recorda del vescovo e delle martiri?"

Chapter IV: The Schoolmaster of Reims.

63 mission to Rome: mission to Rome: John of Salerno, "The Life of Saint Odo of Cluny," translated by Gerard Sitwell, 60-61; Richer of Saint-Remy, vol. 2, 229. For Gerbert's meeting with the pope, see Marco Zuccato's translation in "Gerbert of Aurillac and a Tenth-Century Jewish Channel for the Transmission of Arabic Science to the West," 748. John of Salerno, "The Life of Saint Odo of Cluny," translated by Gerard Sitwell, 60-61; Richer of Saint-Remy, vol. 2, 229. For Gerbert's meeting with the pope, see Marco Zuccato's translation in "Gerbert of Aurillac and a Tenth-Century Jewish Channel for the Transmission of Arabic Science to the West," 748.

65 "Socratic disputations": "Socratic disputations": Gerbert, 82. Gerbert, 82.

65 courtiers: courtiers: Stephen C. Jaeger, Stephen C. Jaeger, The Origins of Courtliness The Origins of Courtliness, 31, 45.

65 "directed toward him": "directed toward him": Richer of Saint-Remy, translated by Darlington, "Gerbert the Teacher," 463. "Divinity wished to illuminate Gaul" is the translation of Jason Glenn in Richer of Saint-Remy, translated by Darlington, "Gerbert the Teacher," 463. "Divinity wished to illuminate Gaul" is the translation of Jason Glenn in Politics and History in the Tenth Century Politics and History in the Tenth Century, 63.

66 Adalbero: Adalbero: See Richer of Saint-Remy, vol. 2, 29-49. For the letters, see Gerbert, 188, 200, 221. Glenn, 25-48, describes Adalbero's reforms and his rebuilding of the Reims cathedral; he also translates some of Richer of Saint-Remy's descriptions of the monk's clothing, 74. Pierre Riche gives Adalbero's biography in See Richer of Saint-Remy, vol. 2, 29-49. For the letters, see Gerbert, 188, 200, 221. Glenn, 25-48, describes Adalbero's reforms and his rebuilding of the Reims cathedral; he also translates some of Richer of Saint-Remy's descriptions of the monk's clothing, 74. Pierre Riche gives Adalbero's biography in Gerbert d'Aurillac Gerbert d'Aurillac, 36; he discusses "prince-bishops" in Les grandeurs de l'an mil Les grandeurs de l'an mil, 137, 141-142. Geoffrey Koziol calls Adalbero an "imperial mole" and explains the crowns over the altars in Begging Pardon and Favor Begging Pardon and Favor, 113-116. In The Diocese of Vic The Diocese of Vic, Freedman describes a cathedral as a "club" and explains the difference between monks and canons, 44.

68 proto-university: proto-university: Lists of Gerbert's students are found in Glenn, 64; Anna Marie Flusche, Lists of Gerbert's students are found in Glenn, 64; Anna Marie Flusche, The Life and Legend of Gerbert of Aurillac The Life and Legend of Gerbert of Aurillac, 30; and Darlington, "Gerbert the Teacher," 473. Darlington cites The Life of King Robert The Life of King Robert, written c. 1040. The wandering scholars are quoted by Loren C. MacKinney, Bishop Fulbert and Education at the School of Chartres Bishop Fulbert and Education at the School of Chartres, 12. For his letters on teaching, see Gerbert, 85, 59, 296, 44, 189, 90.

69 Constantine: Constantine: "Sweet solace," Gerbert, 45. Jaeger quotes the ode in "Sweet solace," Gerbert, 45. Jaeger quotes the ode in The Envy of Angels The Envy of Angels, 56-59. Constantine is credited with preserving Gerbert's letter collection by Riche and Callu in their introduction to Gerbert's Correspondences Correspondences , xi-xii. Charles Burnett discusses pen names in "King Ptolemy and Alchandreus the Philosopher," 333. For Constantine's biography, see F. M. Warren, "Constantine of Fleury, 985-1014"; and Thomas Head, , xi-xii. Charles Burnett discusses pen names in "King Ptolemy and Alchandreus the Philosopher," 333. For Constantine's biography, see F. M. Warren, "Constantine of Fleury, 985-1014"; and Thomas Head, Hagiography and the Cult of Saints Hagiography and the Cult of Saints, 217-240, "Letaldus of Micy and the Hagiographic Traditions of the Abbey of Nouaille," and "The Development of the Peace of G.o.d in Aquitaine (970-1005)."

70 friendship: friendship: See Riche, See Riche, Gerbert d'Aurillac Gerbert d'Aurillac, 251-254; Giles Constable, "Dictators and Diplomats in the Eleventh and Twelfth Centuries"; Lattin, "The Letters of Gerbert," in M. Tosi, ed., Gerberto Gerberto, 311-331; Riche, "Le siege de Laon de 988," in Olivier Guyotjeannin and Emmanuel Poulle, Autour de Gerbert d'Aurillac Autour de Gerbert d'Aurillac, 126-133; Focillon, The Year 1000 The Year 1000, 129, 134; and J.-P. Callu, "Les Mots de Gerbert," in Nicole Charbonnel and Jean-Eric Iung, eds., Gerbert l'Europeen Gerbert l'Europeen, 151. For his letters on friendship, see Gerbert, 92, 90 (I have amended Gerbert's reference to "Tully" to the more familiar "Cicero"), 140, 163, 180, 184, 207, 196, 199, 151. Jaeger describes true friendship in Envy of Angels Envy of Angels, 104.

72 "quant.i.ties of sweat": "quant.i.ties of sweat": Richer of Saint-Remy, vol. 2, 51-81; these translations are Darlington's in "Gerbert the Teacher," 464-472. Richer's biography, and the close reading of his autograph ma.n.u.script, are Glenn's, especially 127-129, 176. See also Riche, Richer of Saint-Remy, vol. 2, 51-81; these translations are Darlington's in "Gerbert the Teacher," 464-472. Richer's biography, and the close reading of his autograph ma.n.u.script, are Glenn's, especially 127-129, 176. See also Riche, Gerbert d'Aurillac Gerbert d'Aurillac, 40; and Juste, Les Alchandreana primitifs Les Alchandreana primitifs, 255.

75 What did Gerbert know: What did Gerbert know: Arianna Borrelli discusses the oral transmission of Arabic science in Arianna Borrelli discusses the oral transmission of Arabic science in Aspects of the Astrolabe Aspects of the Astrolabe; also Marco Zuccato (personal communication).

Chapter V: The Abacus.

79 "notorious Gerbert": "notorious Gerbert": Walter Map, Walter Map, De nugis curialium De nugis curialium, edited and translated by M. R. James, C. N. L. Brooke, and R. A. B. Mynors, 351.

79 Gerbert's abacus: Gerbert's abacus: Burnett, "The Abacus at Echternach in ca. 1000 AD" and "Algorismi vel helcep decentior est diligentia: The Arithmetic of Adelard of Bath and His Circle"; he discusses Gerbert's students at Mettlach in "The Abacus at Echternach," 104. Burnett, "The Abacus at Echternach in ca. 1000 AD" and "Algorismi vel helcep decentior est diligentia: The Arithmetic of Adelard of Bath and His Circle"; he discusses Gerbert's students at Mettlach in "The Abacus at Echternach," 104.

80 "you alone bear burdens": "you alone bear burdens": For his letters, see Gerbert, 117, 111, 102, 114. For his letters, see Gerbert, 117, 111, 102, 114.

81 first calculator: first calculator: Tim Roufs, "Inter-facing the Inevitable: Appropriate Technology in the 21st Century." Osmo Pekonen accuses Gerbert of "showing off with his supercomputer," in "Gerbert d'Aurillac: Mathematician and Pope," 69. Tim Roufs, "Inter-facing the Inevitable: Appropriate Technology in the 21st Century." Osmo Pekonen accuses Gerbert of "showing off with his supercomputer," in "Gerbert d'Aurillac: Mathematician and Pope," 69.

82 "last number of the abacus": "last number of the abacus": Gerbert, 290. Gerbert, 290.

82 Book of the Abacus: For his letter to Constantine, see Gerbert, 45; for the rules, see Gerbert, Opera mathematica Opera mathematica, 6-22. The discussion of "intervals" is from Gillian R. Evans, "Difficillima et Ardua: Theory and Practice in Treatises on the Abacus, 950-1150," 34. On Ralph of Laon, see Evans, "Schools and Scholars: The Study of the Abacus in English Schools c. 980-c. 1150," 81. Theory and Practice in Treatises on the Abacus, 950-1150," 34. On Ralph of Laon, see Evans, "Schools and Scholars: The Study of the Abacus in English Schools c. 980-c. 1150," 81.

86 "gesticulations of dancers": "gesticulations of dancers": Alexander Murray, Alexander Murray, Reason and Society in the Middle Ages Reason and Society in the Middle Ages, 164.

86 "When you say one": "When you say one": Georges Ifrah cites Bede's instructions, including how to give warnings, in Georges Ifrah cites Bede's instructions, including how to give warnings, in From One to Zero From One to Zero, 72-77, and Arabic insults, 78.

87 Roman numerals: Roman numerals: David King derives Roman numerals from finger counting through Greek letters in David King derives Roman numerals from finger counting through Greek letters in The Ciphers of the Monks The Ciphers of the Monks, 281-290.

88 Arabic numerals: Arabic numerals: Menso Folkerts, "The Names and Forms of the Numerals on the Abacus in the Gerbert Tradition." See also King, Menso Folkerts, "The Names and Forms of the Numerals on the Abacus in the Gerbert Tradition." See also King, The Ciphers of the Monks The Ciphers of the Monks, 309-316. Otto B. Bekken cites Severus Sebokt and al-Khwarizmi in the "Algorismus of Hauksbok: Hauksbok: An Old Norse Text of 1310 on Hindu-Arabic Numeration and Calculation," An Old Norse Text of 1310 on Hindu-Arabic Numeration and Calculation," http://home.hia.no/~ottob/. Ifrah describes the practical handbook of al-Buzjani in The Universal History of Numbers The Universal History of Numbers, 548.

89 "dry" and "dull": "dry" and "dull": Evans discusses Gerbert's influence in " Evans discusses Gerbert's influence in "Difficillima et Ardua" and in "Schools and Scholars," where she quotes William of Malmesbury, 72. Burnett defines "abacist" and abaci doctor abaci doctor in "Abbon de Fleury in "Abbon de Fleury Abaci Doctor Abaci Doctor," in Barbara Obrist, ed., Abbon de Fleury Abbon de Fleury, 129-139. Lattin notes that abacists were called "gerbercists" (she spells it "girbercists") in the introduction to The Letters of Gerbert The Letters of Gerbert, 19.

90 Adelard: Adelard: Burnett, "Algorismi vel helcep." For the connection of Laon to Gerbert, see Evans, "Schools and Scholars," 74. Burnett, "Algorismi vel helcep." For the connection of Laon to Gerbert, see Evans, "Schools and Scholars," 74.

90 took much longer: took much longer: See J. L. Berggren, See J. L. Berggren, Episodes in the Mathematics of Medieval Islam Episodes in the Mathematics of Medieval Islam, 29-39; Burnett, "Arabic Numerals," in Thomas Glick, Steven J. Livesey, and Faith Wallis, eds., Medieval Science, Technology, and Medicine Medieval Science, Technology, and Medicine , 39-40, Burnett, "The Translating Activity in Medieval Spain," in , 39-40, Burnett, "The Translating Activity in Medieval Spain," in Magic and Divination in the Middle Ages Magic and Divination in the Middle Ages, 1040; and King, The Ciphers of the Monks The Ciphers of the Monks, 313-315.

Chapter VI: Math and the Mind of G.o.d.

93 strangely transformed: strangely transformed: John J. Contreni, "The Pursuit of Knowledge in Carolingian Europe," in Richard E. Sullivan, ed., John J. Contreni, "The Pursuit of Knowledge in Carolingian Europe," in Richard E. Sullivan, ed., "The Gentle Voices of Teachers "The Gentle Voices of Teachers , ," 117. 117.

94 ma.s.sive times table: ma.s.sive times table: Abbo of Fleury, Abbo of Fleury, Commentary on the Calculus of Victorius of Aquitaine Commentary on the Calculus of Victorius of Aquitaine, edited by A. M. Peden, xv-x.x.xvi.

94 "your sagacity": "your sagacity": Gerbert, 44; "half-educated philosopher," 46. Lattin identifies him as Abbo, 46n. See also Marco Mostert, "Gerbert d'Aurillac, Abbon de Fleury, et la Culture de l'An Mil," in Nuvolone, ed., Gerbert, 44; "half-educated philosopher," 46. Lattin identifies him as Abbo, 46n. See also Marco Mostert, "Gerbert d'Aurillac, Abbon de Fleury, et la Culture de l'An Mil," in Nuvolone, ed., Gerberto d'Aurillac: Da abate di Bobbio a Papa dell'anno 1000 Gerberto d'Aurillac: Da abate di Bobbio a Papa dell'anno 1000, 405.

94 Abbo: Abbo: Vezin points out that Abbo's and Gerbert's "curiosities" coincided in "La Production et la circulation des livres dans l'Europe du Xe siecle," in Nicole Charbonnel and Jean-Eric Iung, eds., Vezin points out that Abbo's and Gerbert's "curiosities" coincided in "La Production et la circulation des livres dans l'Europe du Xe siecle," in Nicole Charbonnel and Jean-Eric Iung, eds., Gerbert l'Europeen Gerbert l'Europeen, 215. John Nightingale compares Ramsey and Fleury in "Oswald, Fleury, and Continental Reform," in Nicholas Brooks and Catherine Cubitt, eds., St Oswald of Worcester St Oswald of Worcester, 27, 44. Thomas Head cites Abbo's medieval biographer, Aimo of Fleury, in Hagiography and the Cult of Saints Hagiography and the Cult of Saints, 239-240. Pierre Riche describes the reform movement, Les grandeurs de l'an mil Les grandeurs de l'an mil, 114; as does Richard Landes, Relics Relics, Apocalypse, and the Deceits of History Apocalypse, and the Deceits of History, 29. Robert F. Berkhofer discusses charters, Day of Reckoning Day of Reckoning, 73. Teta E. Moehs describes Fleury's privileges, Gregorius V, 996-999 Gregorius V, 996-999, 21, 47.

96 Abbo studied at Reims: Abbo studied at Reims: Aimo of Fleury's comment is translated by Jason Glenn, Aimo of Fleury's comment is translated by Jason Glenn, Politics and History in the Tenth Century Politics and History in the Tenth Century, 62. Charles Burnett identifies Gerbert as Abbo's teacher in "King Ptolemy and Alchandreus the Philosopher," 332; as does David Juste, "La sphere planetaire du ms. Vatican, BAV, Pal. lat. 1356 (XIIe siecle)," 217n. C. H. Lawrence quotes Abbo on writing, Medieval Monasticism Medieval Monasticism, 117. On Abbo's scholarship, see Peden's introduction to Abbo's Commentary Commentary, xiii-xiv; Ron B. Thomson, "Two Astronomical Tractates of Abbo of Fleury," 113-116; Barbara Obrist, "Les tables et figures abboniennes dans l'histoire de l'iconographie des recueils de comput," in Obrist, ed., Abbon de Fleury Abbon de Fleury, 141-186; and Gillian R. Evans, "Schools and Scholars," 73-75. Wesley M. Stevens calls Byrthferth's work "disappointing" in Cycles of Time and Scientific Learning in Medieval Europe Cycles of Time and Scientific Learning in Medieval Europe, 138-139. Discussing Abbo's use of Boethius's On Arithmetic On Arithmetic, Peden notes, "It is the mark, perhaps, of a text well-absorbed, even if only the more elementary sections," x.x.xiv.

98 "Take number away": "Take number away": Isidore of Seville, quoted by Faith Wallis, "'Number Mystique' in Early Medieval Computus Texts," in Teun Koetsier and Luc Bergmans, eds., Isidore of Seville, quoted by Faith Wallis, "'Number Mystique' in Early Medieval Computus Texts," in Teun Koetsier and Luc Bergmans, eds., Mathematics and the Divine Mathematics and the Divine, 187.

98 most common math problem: most common math problem: Stevens, Stevens, Cycles of Time Cycles of Time, 44-45.

100 800 Anno Domini: 800 Anno Domini: See Landes, See Landes, Relics, Apocalypse, and the Deceits of History Relics, Apocalypse, and the Deceits of History, 290. Hraban Maur's computus computus is discussed by Wesley M. Stevens, "Compotistica et Astronomica in the Fulda School," 32-34. For Abbo's recalculations, see Arno Borst, is discussed by Wesley M. Stevens, "Compotistica et Astronomica in the Fulda School," 32-34. For Abbo's recalculations, see Arno Borst, The Ordering of Time The Ordering of Time, 52-54; Johannes Fried, "Awaiting the End of Time Around the Turn of the Year 1000," in Richard Landes, Andrew Gox, and David C. Van Meter, eds., The Apocalyptic Year 1000 The Apocalyptic Year 1000, 21, 37; and Landes, "The Fear of an Apocalyptic Year 1000," in the same work, 250-252.

100 "unchangeable and true": "unchangeable and true": Abbo of Fleury, summarized by Peden, xvii-xviii. Abbo of Fleury, summarized by Peden, xvii-xviii.

101 "I did not restrict myself": "I did not restrict myself": Boethius, Boethius, De inst.i.tutione arithmetica De inst.i.tutione arithmetica, in Boethian Number Theory Boethian Number Theory, edited and translated by Michael Masi, 67. "The original pattern" is from Joseph V. Navari, "The Leitmotiv in the Mathematical Thought of Gerbert of Aurillac," 140-141.

101 Boethius's Boethius's On Arithmetic On Arithmetic: Gerbert's gold and purple copy is described by Rosamond McKitterick, "Charles the Bald (823-877) and His Library: The Patronage of Learning," 31. For his letters, see Gerbert, 296, 39. Menso Folkerts finds the problems "completely irrelevant for practical calculation" in Gerbert's gold and purple copy is described by Rosamond McKitterick, "Charles the Bald (823-877) and His Library: The Patronage of Learning," 31. For his letters, see Gerbert, 296, 39. Menso Folkerts finds the problems "completely irrelevant for practical calculation" in The Development of Mathematics in Medieval Europe The Development of Mathematics in Medieval Europe, Chap. 1, 16. For the meaning of the exercises, see Alison White, "Boethius in the Medieval Quadrivium," in Margaret Gibson, ed., Boethius: His Life, Thought, and Influence Boethius: His Life, Thought, and Influence, 169-170; Navari, 141-142; and Wallis, in Koetsier and Bergmans, 196.

103 organ pipes: organ pipes: See Flusche, See Flusche, The Life and Legend of Gerbert of Aurillac The Life and Legend of Gerbert of Aurillac, 147- 156; Christian Meyer, "Gerbertus musicus: Gerbert et les fondements du systeme acoustique," in Charbonnel and Iung, 183-192.

104 musica humana: C. Stephen Jaeger describes moral music in Envy of Angels Envy of Angels , 165. , 165.

105 "sweet mixture": "sweet mixture": Bukofzer, in "Speculative Thinking in Mediaeval Music," discusses the history of chant, 168-172; he cites the "deeper and divine reason" controlling harmony, from the anonymous Bukofzer, in "Speculative Thinking in Mediaeval Music," discusses the history of chant, 168-172; he cites the "deeper and divine reason" controlling harmony, from the anonymous Musica enchiriadis Musica enchiriadis (c. 860), 173. (c. 860), 173.

105 master of master of musica musica: Richer of Saint-Remy, vol. 2, 53-59. "But the difficulties" is my translation. "Music, previously unfamiliar to the Gauls," is that of Glenn, 50; he also discusses the evidence that Richer was a cantor, 49-52. Richer of Saint-Remy, vol. 2, 53-59. "But the difficulties" is my translation. "Music, previously unfamiliar to the Gauls," is that of Glenn, 50; he also discusses the evidence that Richer was a cantor, 49-52.

106 His solution: His solution: Constantino Sigismondi, "Gerberto e la misura delle canne d'organo," 392-393. Constantino Sigismondi, "Gerberto e la misura delle canne d'organo," 392-393.

106 "full of accurate observations": "full of accurate observations": Quotes from Gerbert's geometry are translated by Navari, 141, 147. See also Menso Folkerts, "The Importance of the Pseudo-Boethian Geometria During the Middle Ages," in Quotes from Gerbert's geometry are translated by Navari, 141, 147. See also Menso Folkerts, "The Importance of the Pseudo-Boethian Geometria During the Middle Ages," in The Development of Mathematics in Medieval Europe The Development of Mathematics in Medieval Europe, Chap. 7, 190-201. Stevens cites Folkerts's work as proof of the "quite unexpected amount" of Euclid, in "Compotistica et Astronomica," 40-42. On Gerbert's letter to Adalbold (Gerbert, 299), see G. A. Miller, "The Formula a (a + 1) for the Area of an Equilateral Triangle" and "Gerbert's Letter to Adelbold."

111 experimental science: experimental science: See Folkerts, "The Importance of the Latin Middle Ages for the Development of Mathematics," in See Folkerts, "The Importance of the Latin Middle Ages for the Development of Mathematics," in The Development of Mathematics in Medieval Europe The Development of Mathematics in Medieval Europe, Chap. 1, 4; see also his "Review of De Verhandeling over de Cirkelkwadratuur van Franco van Luik van Omstreeks 1050 De Verhandeling over de Cirkelkwadratuur van Franco van Luik van Omstreeks 1050 by A. J. E. M. Smeur," 272-274. Loren C. MacKinney continues the story of Rodolf and Ragimbold, by A. J. E. M. Smeur," 272-274. Loren C. MacKinney continues the story of Rodolf and Ragimbold, Bishop Fulbert and Education at the School of Chartres Bishop Fulbert and Education at the School of Chartres, 14-15, 29-30; as does Michael Mahoney, "Mathematics," in David Lindberg, ed., Science in the Middle Ages Science in the Middle Ages, 149.

111 prayer book: prayer book: Reviel Netz and William Noel, Reviel Netz and William Noel, The Archimedes Codex The Archimedes Codex.

Chapter VII: The Celestial Sphere.

113 "giant silvery cloud": "giant silvery cloud": Joe Sharkey, "Helping the Stars Take Back the Night," Joe Sharkey, "Helping the Stars Take Back the Night," New York Times New York Times, Aug. 30, 2008. For how Gerbert's peers considered the stars, see C. Stephen Jaeger, Envy of Angels Envy of Angels, 175; David King, Ciphers of the Monks Ciphers of the Monks, 355; Stephen C. McCluskey, Astronomies and Cultures in Early Medieval Europe Astronomies and Cultures in Early Medieval Europe, 3; and Ron B. Thomson, "Two Astronomical Tractates of Abbo of Fleury," 113-133.

114 Gregory of Tours: Gregory of Tours: Arno Borst, Arno Borst, The Ordering of Time The Ordering of Time, discusses temporal and equinoctial hours and the use of the alarm glocke glocke, 31, 42.

115 "it would take too long": "it would take too long": Richer of Saint-Remy, translated by Darlington, "Gerbert the Teacher," 467. Richer of Saint-Remy, translated by Darlington, "Gerbert the Teacher," 467.

115 "We have sent no sphere": "We have sent no sphere": Gerbert, 172. "Your good will," Gerbert, 184. Gerbert, 172. "Your good will," Gerbert, 184.

116 Reims and Trier: Reims and Trier: See Thomas Head, "Art and Artifice in Ottonian Trier," 65-82; Dominique Alibert, "Majeste ottonienne: L'hommage des nations a l'empereur," in Olivier Guyotjeannin and Emmanuel Poulle, eds., See Thomas Head, "Art and Artifice in Ottonian Trier," 65-82; Dominique Alibert, "Majeste ottonienne: L'hommage des nations a l'empereur," in Olivier Guyotjeannin and Emmanuel Poulle, eds., Autour de Gerbert d'Aurillac Autour de Gerbert d'Aurillac, 82-87; and Pierre Riche, Les grandeurs de l'an mil Les grandeurs de l'an mil, 191.

116 "Continual torrents": "Continual torrents": Gerbert, 149. He mentions the cross in two other letters; see Gerbert, 145, 147. Gerbert, 149. He mentions the cross in two other letters; see Gerbert, 145, 147.

117 Gerbert's student: Gerbert's student: For Remi, see Darlington, "Gerbert the Teacher," 470- 471. For Leofsin, see Charles Burnett, "The Abacus at Echternach in ca. 1000 AD," 104. For Remi, see Darlington, "Gerbert the Teacher," 470- 471. For Leofsin, see Charles Burnett, "The Abacus at Echternach in ca. 1000 AD," 104.

118 "the Celestial Sphere": "the Celestial Sphere": Bruce Eastwood cites Martia.n.u.s Capella in Bruce Eastwood cites Martia.n.u.s Capella in Ordering the Heavens Ordering the Heavens, 189. For Al-Battani's "egg" and its appearance in Catalonia, see Julio Samso, "Battani, al-," in Thomas Glick, Steven J. Livesey, and Faith Wallis, eds., Medieval Science, Technology, and Medicine Medieval Science, Technology, and Medicine, 79-80. Al-Sufi's silver sphere is described by E. S. Kennedy and Marcel Des...o...b..s in "Introduction to Kitab al'Amal bil Asturlab Kitab al'Amal bil Asturlab," in E. S. Kennedy et al., eds., Studies in the Islamic Exact Sciences Studies in the Islamic Exact Sciences, 405-406.

118 Gerbert may have learned: Gerbert may have learned: On Gerbert's technique, see David Juste, "La sphere planetaire du ms. Vatican, BAV, Pal. lat. 1356 (XIIe siecle)," 205- 221; and Marco Zuccato, "Gerbert's Islamicate Celestial Globe," 167- 188. Emilie Savage-Smith quotes the Arabic text translated in Spain, On Gerbert's technique, see David Juste, "La sphere planetaire du ms. Vatican, BAV, Pal. lat. 1356 (XIIe siecle)," 205- 221; and Marco Zuccato, "Gerbert's Islamicate Celestial Globe," 167- 188. Emilie Savage-Smith quotes the Arabic text translated in Spain, Islamicate Celestial Globes Islamicate Celestial Globes, 81-82. For the 1518 printed text, see R. Lorch, "The Sphaera Solida Sphaera Solida and Related Instruments," 156. and Related Instruments," 156.

119 two ways of thinking: two ways of thinking: See Savage-Smith, 3, and Wesley M. Stevens, "The Figure of the Earth in Isidore's See Savage-Smith, 3, and Wesley M. Stevens, "The Figure of the Earth in Isidore's De natura rerum De natura rerum," 275-277. Martia.n.u.s Capella called the planets "confusers"; see Eastwood, Ordering the Heavens Ordering the Heavens , 323. "There is no inconstancy in divine acts" is Calcidius, quoted by Eastwood, "Calcidius's Commentary on Plato's , 323. "There is no inconstancy in divine acts" is Calcidius, quoted by Eastwood, "Calcidius's Commentary on Plato's Timaeus Timaeus in Latin Astronomy of the Ninth to Eleventh Centuries," 176. in Latin Astronomy of the Ninth to Eleventh Centuries," 176.

120 "something divine": "something divine": Richer of Saint-Remy, translated by Zuccato, "Gerbert's Islamicate Celestial Globe," 169. "Differ from organ pipes" is from Gerbert, 36. "So well contrived" is from Richer of Saint-Remy, translated by Darlington, "Gerbert the Teacher," 467. Richer of Saint-Remy, translated by Zuccato, "Gerbert's Islamicate Celestial Globe," 169. "Differ from organ pipes" is from Gerbert, 36. "So well contrived" is from Richer of Saint-Remy, translated by Darlington, "Gerbert the Teacher," 467.

121 most sophisticated astronomical instrument: most sophisticated astronomical instrument: See Juste, "La sphere planetaire," 205-221. According to Richer's numbering system, which Juste uses, the armillary sphere is the third sphere, not the fourth. See Juste, "La sphere planetaire," 205-221. According to Richer's numbering system, which Juste uses, the armillary sphere is the third sphere, not the fourth.