Medieval Europe’s satanic ciphers: on the genesis of a modern myth. (English) Zbl 1466.01005
The article under review begins with a delicious string of titbits quoted from recent idiocies written by mathematicians, science-writers, history-writers (if this neologism emulating the previous category be allowed) and an evangelical. They are then distilled (p. 110) into these theses: (i) the number zero was feared [in the Middle Ages] for being associated with the Devil; (ii) it was therefore outlawed by church or state (or both); (iii) owing to these prohibitions, zero and Hindu-Arabic numerals were for some time only used under the cover of secrecy.
After this appetizer follows a meticulous history of the introduction of Hindu-Arabic numerals into Latin Europe, drawing on the best scholarship on the topic from 1865 until today; it can be recommended as an excellent and very full critical bibliography. Slightly more than the final half of the article then traces the origin of the myth, finding this to be an independent parallel to the 19th-century “warfare between science and religion” thesis, going back to an article by Leo Jordan from 1905, a fanciful combination of quixotic or outright false readings of select passages from medieval documents. At first, Jordan’s theses were taken over uncritically but rather peripherally by otherwise decent historians (David Eugene Smith, Louis Karpinski, Charles Singer, Florence Yeldham, Karl Menninger) and embroidered upon by the mathematician Tobias Dantzig and by the biologist and eminent popularizer Lancelot Hogben. Expansion with added free inventions (as the “sending over-keen algorists to the stake”) occurred from 1987 onward, with highly applauded books by mathematician and philosopher Brian Rotman (1987), mathematics teacher Georges Ifrah (1994, the second edition of his Histoire universelle, source of the words just quoted – not so much the cautious first edition), mathematician Robert Kaplan (1999), journalist Dick Teresi and Charles Seife, professor of journalism.
The reviewer adds to this list of writers of problematic books that they have been published (in original or translation) by Penguin Books, Macmillan and Oxford University Press. So much for the quality of presumed peer reviewing by commercial publishers.
After this appetizer follows a meticulous history of the introduction of Hindu-Arabic numerals into Latin Europe, drawing on the best scholarship on the topic from 1865 until today; it can be recommended as an excellent and very full critical bibliography. Slightly more than the final half of the article then traces the origin of the myth, finding this to be an independent parallel to the 19th-century “warfare between science and religion” thesis, going back to an article by Leo Jordan from 1905, a fanciful combination of quixotic or outright false readings of select passages from medieval documents. At first, Jordan’s theses were taken over uncritically but rather peripherally by otherwise decent historians (David Eugene Smith, Louis Karpinski, Charles Singer, Florence Yeldham, Karl Menninger) and embroidered upon by the mathematician Tobias Dantzig and by the biologist and eminent popularizer Lancelot Hogben. Expansion with added free inventions (as the “sending over-keen algorists to the stake”) occurred from 1987 onward, with highly applauded books by mathematician and philosopher Brian Rotman (1987), mathematics teacher Georges Ifrah (1994, the second edition of his Histoire universelle, source of the words just quoted – not so much the cautious first edition), mathematician Robert Kaplan (1999), journalist Dick Teresi and Charles Seife, professor of journalism.
The reviewer adds to this list of writers of problematic books that they have been published (in original or translation) by Penguin Books, Macmillan and Oxford University Press. So much for the quality of presumed peer reviewing by commercial publishers.
Reviewer: Jens Høyrup (Roskilde)
References:
| [1] | Allard, André, ‘A propos d’un algorisme latin de Frankenthal: une méthode de recherche’, Janus, 65, 119-141 (1978) · Zbl 0396.01003 |
| [2] | Allard, André; Weijers, Olga, La formation du vocabulaire latin de l’arithmétique médiévale, Méthodes et instruments du travail intellectuel au moyen âge: études sur le vocabulaire, 137-181 (1990), Turnhout: Brepols |
| [3] | Allard, André, The Arabic origins and development of Latin algorisms in the twelfth century, Arabic Sciences and Philosophy, 1, 233-283 (1991) · Zbl 0819.01009 · doi:10.1017/S0957423900001508 |
| [4] | Allard, André, Muhammad Ibn Mūsā al-Khwārizmī: Le calcul indien (Algorismus) (1992), Paris: Blanchard, Paris · Zbl 1067.01536 |
| [5] | Ambrosetti, Nadia, L’eredità arabo-islamica nelle scienze e nelle arti del calcolo dell’Europa medievale (2008), Milan: Edizioni Universitarie di Lettere Economia Diritto, Milan |
| [6] | Ambrosetti, Nadia; Gadducci, Fabio; Tavosanis, Mirko, Algorithmic in the 12th century: the Carmen de Algorismo by Alexander de Villa Dei, History and philosophy of computing, 71-86 (2016), Cham: Springer, Cham |
| [7] | Ball, W. W Rouse, A short account of the history of mathematics (1908), London: Macmillan, London · JFM 20.0001.01 |
| [8] | Beaujouan, Guy, Etude paléographique sur la “rotation” des chiffres et l’emploi des apices du X^e au XII^e siècle, Revue d’histoire des sciences, 1, 4, 301-313 (1948) · Zbl 0031.00202 |
| [9] | Beaujouan, Guy; Guyotjeannin, Olivier; Poulle, Emmanuel, Les chiffres arabes selon Gerbert: l’abaque du Pseudo-Boèce, XIe siècle, Autour de Gerbert d’Aurillac: le pape de l’an mil, 322-328 (1996), Paris: École des Chartes, Paris |
| [10] | Bellos, Alex, Here’s looking at Euclid: a surprising excursion through the astonishing world of maths (2010), New York: Free Press, New York |
| [11] | Berggren, J. Lennart; Contreni, John J.; Casciani, Santa, Medieval arithmetic: Arabic texts and European motivations, Word, image, number: communication in the Middle Ages, 351-365 (2002), Florence: SISMEL, Florence |
| [12] | Bishop, Chris, The “Pear of Anguish”: truth, torture and dark medievalism, International Journal of Cultural Studies, 17, 591-602 (2014) · doi:10.1177/1367877914528531 |
| [13] | Bossuat, Robert, Alain de Lille: Anticlaudianus (1955), Paris: Vrin, Paris |
| [14] | Bubnov, Nicolaus, Gerberti Opera mathematica (972-1003), Berlin: Friedländer, Berlin |
| [15] | Burnam, John M., Palaeographia iberica: fac-similés de manuscrits espagnols et portugais (IX^e-XV^e siècles) avec notices et transcriptions (191225), Paris: Champion, Paris |
| [16] | Burnett, Charles; Folkerts, Menso, Algorismi vel helcep decentior est diligentia: the arithmetic of Adelard of Bath and his circle, Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich, 221-331 (1996), Wiesbaden: Harrassowitz, Wiesbaden |
| [17] | Burnett, Charles; Folkerts, Menso; Lorch, Richard, Latin alphanumerical notation, and annotation in Italian, in the twelfth century: MS London, British Library, Harley, 5402, Sic itur ad astra: Studien zur Geschichte der Mathematik und Naturwissenschaften, 77-90 (2000), Wiesbaden: Harrassowitz, Wiesbaden · Zbl 0991.01004 |
| [18] | Burnett, Charles; Dold-Samplonius, Yvonne; Dauben, Joseph W.; Folkerts, Menso; Dalen, Benno van, Indian numerals in the Mediterranean Basin in the twelfth century, with special reference to the “eastern forms”, From China to Paris: 2000 years transmission of mathematical ideas, 237-288 (2002), Stuttgart: Steiner, Stuttgart · Zbl 1027.01003 |
| [19] | Burnett, Charles, Learning Indian arithmetic in the early thirteenth century, Boletín de la Asociación Matemática Venezolana, 9, 15-26 (2002) · Zbl 1053.01007 |
| [20] | Burnett, Charles, The abacus at Echternach in ca. 1000 A.D, Sciamus, 3, 91-108 (2002) · Zbl 1032.01010 |
| [21] | Burnett, Charles, Fibonacci’s “Method of the Indians”, Bollettino di storia delle scienze matematiche, 23, 2, 87-97 (2003) · Zbl 1202.01027 |
| [22] | Burnett, Charles, The semantics of Indian numerals in Arabic, Greek and Latin, Journal of Indian Philosophy, 34, 15-30 (2006) · Zbl 1149.01010 · doi:10.1007/s10781-005-8153-z |
| [23] | Burnett, Charles, Numerals and arithmetic in the Middle Ages (2010), Farnham: Ashgate, Farnham · Zbl 1228.01005 |
| [24] | Burnett, Charles; Zhao, Ji-Wei; Lampe, Kurt, The Toledan Regule (Liber Alchorismi, part II): a twelfth-century arithmetical miscellany, Sciamus, 8, 141-231 (2007) · Zbl 1213.01001 |
| [25] | Burton, David M., The history of mathematics: an introduction. (First published in 1985) (2011), New York: McGraw-Hill, New York · Zbl 0657.01003 |
| [26] | Busard, H. L L.; Folkerts, Menso; Lorch, Richard, Über zwei Algorismus-Schriften aus dem 13. Jahrhundert, Sic itur ad astra: Studien zur Geschichte der Mathematik und Naturwissenschaften, 91-137 (2000), Wiesbaden: Harrassowitz, Wiesbaden · Zbl 0998.01010 |
| [27] | Caianiello, Eva; Bernard, Alain; Proust, Christine, Leonardo of Pisa and the Liber Abaci: biographical elements and the project of the work, Scientific sources and teaching contexts throughout history: problems and perspectives, 217-246 (2014), Dordrecht: Springer, Dordrecht |
| [28] | Cajori, Florian, A history of mathematics (1894), New York: Macmillan, New York · Zbl 0734.01002 |
| [29] | Camerani Marri, Giulia, Statuti dell’Arte del Cambio di Firenze (1299-1316) (1955), Florence: Olschki, Florence |
| [30] | Cantor, Moritz, Ueber einen Codex des Klosters Salem, Zeitschrift für Mathematik und Physik, 10, 1-16 (1865) |
| [31] | Cherubini, Paolo; Librandi, Rita; Piro, Rosa, Il numero come elemento di disturbo, Lo scaffale della biblioteca scientifica in volgare (secoli XIII-XVI), 313-339 (2006), Florence: SISMEL, Florence |
| [32] | Chrisomalis, Stephen, Numerical notation: a comparative history (2010), Cambridge: Cambridge University Press, Cambridge · Zbl 1229.01002 |
| [33] | Classen, Albrecht, The medieval chastity belt: a myth-making process (2007), New York: Palgrave Macmillan, New York |
| [34] | Cochrane, Louise, Adelard of Bath: the first English scientist (1994), London: British Museum Press, London |
| [35] | Crossley, John N., Old-fashioned versus newfangled: reading and writing numbers, 1200-1500, Studies in medieval and Renaissance history, 10, 79-109 (2013) |
| [36] | Cunnington, Susan, The story of arithmetic: a short history of its origin and development (1904), London: Sonnenschein & Co, London · JFM 35.0071.08 |
| [37] | Dantzig, Tobias, Number: the language of science. (First published in 1930) (2005), New York: Pi Press, New York · JFM 56.0798.10 |
| [38] | Dauben, Joseph, The universal history of numbers and The universal history of computing, Notices of the American Mathematical Society, 49, 211-216 (2002) |
| [39] | Denifle, Heinrich, Die Statuten der Juristen-Universität Padua vom Jahre 1331, Archiv für Literatur- und Kirchengeschichte des Mittelalters, 6, 309-562 (1892) |
| [40] | Evans, Gillian R., Difficillima et ardua: theory and practice in treatises on the abacus, 950-1150, Journal of Medieval History, 3, 21-38 (1977) · doi:10.1016/0304-4181(77)90038-0 |
| [41] | Evans, Gillian R., From abacus to algorism: theory and practice in medieval arithmetic, British Journal for the History of Science, 10, 114-131 (1977) · Zbl 0358.01003 · doi:10.1017/S0007087400015375 |
| [42] | Evans, Gillian R., Schools and scholars: the study of the abacus in English schools c. 980-c. 1150, English Historical Review, 94, 71-89 (1979) · doi:10.1093/ehr/XCIV.CCCLXX.71 |
| [43] | Evans, Gillian R., The saltus Gerberti: the problem of the “leap”, Janus, 67, 261-268 (1980) · Zbl 0471.01005 |
| [44] | Folkerts, Menso; Franci, Raffaella; Pagli, Paolo; Rigatelli, Laura Toti, Frühe Darstellungen des Gerbertschen Abakus, Itinera mathematica: studi in onore di Gino Arrighi per il suo 90^o compleanno, 23-43 (1996), Siena: Centro studi sulla matematica medioevale, Università di Siena, Siena · Zbl 0871.01003 |
| [45] | Folkerts, Menso, Die älteste lateinische Schrift über das indische Rechnen nach al-Ḫwārizmī (1997), Munich: Verlag der Bayerischen Akademie der Wissenschaften, Munich · Zbl 0949.01005 |
| [46] | Folkerts, Menso; Folkerts, Menso; Lorch, Richard, Frühe westliche Benennungen der indisch-arabischen Ziffern und ihr Vorkommen, Sic itur ad astra: Studien zur Geschichte der Mathematik und Naturwissenschaften, 216-233 (2000), Wiesbaden: Harrassowitz, Wiesbaden · Zbl 0993.01007 |
| [47] | Folkerts, Menso, Early texts on Hindu-Arabic calculation, Science in Context, 14, 13-38 (2001) · Zbl 1193.01021 · doi:10.1017/S0269889701000023 |
| [48] | Folkerts, Menso; Nuvolone, Fabio G., The names and forms of the numerals on the abacus in the Gerbert tradition, Gerberto d’Aurillac, da Abate di Bobbio a Papa dell’Anno 1000, 245-265 (2001), Bobbio: Associazione culturale Amici di Archivum Bobiense, Bobbio |
| [49] | Folkerts, Menso; Hughes, Barnabas; Katz, Victor J., The Latin mathematics of medieval Europe, Sourcebook in the mathematics of medieval Europe and North Africa, 4-223 (2016), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 1359.01004 |
| [50] | Franci, Raffaela, ‘Leonardo Pisano e la trattatistica dell’abaco in Italia nei secoli XIV e XV’, Bollettino di storia delle scienze matematiche, 23, no. 2 (2003), 33-54. · Zbl 1202.01028 |
| [51] | Fröhlich, Jürgen; Schmitz, Ulrich; Wenzel, Horst, Meßkram oder Die Einwanderung der Null in den modernen Schaltkreislauf über das spätmittelalterliche Rechnungsbuch, Wissen und neue Medien: Bilder und Zeichen von 800 bis 2000, 135-158 (2003), Berlin: Schmidt, Berlin |
| [52] | Frova, Carla, Le opere aritmetiche di Gerberto d’Aurillac, Studi sul medioevo cristiano offerti a Raffaelo Morghen per il 90^o anniversario dell’Istituto Storico Italiano (1883-1973), Vol. 1, 323-353 (1974), Rome: Istituto Storico Italiano, Rome |
| [53] | Gazalé, Midhat, Number: From Ahmes to Cantor (2000), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 1062.00001 |
| [54] | Gherardi, Alessandro, Statuti della università e Studio Fiorentino dell’anno MCCCLXXXVII (1881), Florence: Vieusseux, Florence |
| [55] | Gibson, Craig A.; Newton, Francis, Pandulf of Capua’s De calculatione: An illustrated abacus treatise and some evidence for the Hindu-Arabic numerals in eleventh-century South Italy, Mediaeval Studies, 57, 293-335 (1995) · doi:10.1484/J.MS.2.306437 |
| [56] | Gómez Pallarès, Juan, Sobre manuscritos latinos de cómputo en escritura visigótica, Hispania Sacra, 39, 25-48 (1987) |
| [57] | Green, Ronald, Nothing matters (2011), Alresford: Iff Books, Alresford |
| [58] | Häring, Nikolaus M, ‘Alan of Lille: De planctu Naturae’, Studi medievali, 3rd ser., 19 (1978), 797-879. |
| [59] | Harris, Stephen J.; Grigsby, Byron Lee, Misconceptions about the Middle Ages (2008), New York: Routledge, New York |
| [60] | Haskins, Charles Waldo, Business education and accountancy (1904), New York: Harper & Brothers, New York |
| [61] | Herreman, Alain, La mise en texte mathématique: une analyse de l’“Algorisme de Frankenthal”, Methodos, 1 (2001) · doi:10.4000/methodos.45 |
| [62] | Hitchens, Christopher, God is not great: the case against religion (2007), London: Atlantic, London |
| [63] | Hogben, Lancelot, Mathematics for the million: a popular self educator (1936), London: George Allen & Unwin, London |
| [64] | Horn-d’Arturo, Guido, Numeri arabici e simboli celesti, Memorie della Società astronomica italiana, n.s, 3, 195-212 (1925) · JFM 51.0005.01 |
| [65] | Høyrup, Jens, Jacopo da Firenze’s Tractatus algorismi and early Italian abbacus culture (2007), Basel: Birkhäuser, Basel · Zbl 1126.01006 |
| [66] | Ifrah, Georges, From one to zero: a universal history of numbers (1985), New York: Viking Penguin, New York · Zbl 0589.01001 |
| [67] | Ifrah, Georges, The universal history of numbers: from prehistory to the invention of the computer (1998), London: Harvill Press, London · Zbl 0955.01002 |
| [68] | Jordan, Leo, ‘Materialien zur Geschichte der arabischen Zahlzeichen in Frankreich’, Archiv für Kulturgeschichte, 3 (1905), 155-195. |
| [69] | Jordan, Leo; Stollreither, E., Wortgeschichtliches, Festschrift zum XII. Allgemeinen deutschen Neuphilologentage in München, Pfingsten 1906, 61-80 (1906), Erlangen: Junge, Erlangen |
| [70] | Kaplan, Robert, The nothing that is: a natural history of zero (1999), London: Allen Lane, London · Zbl 1001.01500 |
| [71] | Karpinski, Louis C., Two twelfth-century algorisms, Isis, 3, 396-413 (1921) · doi:10.1086/357950 |
| [72] | King, David A., The ciphers of the monks: a forgotten number-notation of the Middle Ages (Boethius, 44) (2001), Stuttgart: Steiner, Stuttgart · Zbl 0982.01002 |
| [73] | Kirchhoff, Albrecht, Die Handschriftenhändler des Mittelalters, Leipzig: Kirchhoff, 2nd ed., 1853. |
| [74] | Kline, Morris, Mathematical thought from ancient to modern times (1972), New York: Oxford University Press, New York · Zbl 0277.01001 |
| [75] | Koenig, V. Frederic, Les Miracles de Nostre Dame par Gautier de Coinci, Vol. 4 (1970), Geneva: Droz, Geneva |
| [76] | Krantz, Steven G., The proof is in the pudding: the changing nature of mathematical proof (2011), New York: Springer, New York · Zbl 1318.00005 |
| [77] | Kunitzsch, Paul, Zur Geschichte der ‘arabischen’ Ziffern (2005), Munich: Verlag der Bayerischen Akademie der Wissenschaften, Munich |
| [78] | Lebsanft, Franz; Christmann, Hans Helmut; Hausmann, Frank-Rutger; Briegel, Manfred, Ein deutsch-jüdisches Schicksal: Der Philologe und Linguist Leo Jordan (1874-1940), Deutsche und österreichische Romanisten als Verfolgte des Nationalsozialismus, 157-175 (1989), Tübingen: Stauffenburg, Tübingen |
| [79] | Lemay, Richard, The Hispanic origin of our present numeral forms, Viator, 8, 435-477 (1977) · doi:10.1484/J.VIATOR.2.301575 |
| [80] | Lemay, Richard; Strayer, Joseph R., Arabic numerals, Dictionary of the Middle Ages, Vol. 1, Aachen-augustinism, 382-398 (1982), New York: Charles Scribner’s Sons, New York |
| [81] | Lohr, Alfred, Opera de computo saeculi duodecimi (2015), Turnhout: Brepols, Turnhout |
| [82] | Lohr, Alfred; Nothaft, C. Philipp E., Robert Grosseteste’s Compotus (2019), Oxford: Oxford University Press, Oxford |
| [83] | Lüneburg, Heinz, Von Zahlen und Größen: Dritthalbtausend Jahre Theorie und Praxis, Vol. 1 (2008), Basel: Birkhäuser, Basel · Zbl 1158.00003 |
| [84] | Lyons, Jonathan, Islam through Western eyes: from the Crusades to the war on terrorism (2012), New York: Columbia University Press, New York |
| [85] | Malagola, Carlo, Statuti delle università e dei collegi dello Studio Bolognese (1888), Bologna: Zanichelli, Bologna |
| [86] | Mann, Charles C., 1491: new revelations of the Americas before Columbus (2005), New York: Knopf, New York |
| [87] | Menninger, Karl, Number words and number symbols: a cultural history of numbers (1969), Cambridge: Mass.: M.I.T. Press, Cambridge · Zbl 0191.00202 |
| [88] | Morrison, Philip; Morrison, Phylis, 100 or so books that shaped a century of science, American Scientist, 87, 542-544 (1999) |
| [89] | Murray, Alexander, Reason and society in the Middle Ages (1978), Oxford: Clarendon Press, Oxford |
| [90] | Mynors, R. A B., William of Malmesbury: Gesta regum Anglorum; the history of the English kings, Vol. 1, completed by R. M. Thomson and M. Winterbottom (1998), Oxford: Clarendon Press, Oxford |
| [91] | Nagl, Alfred, Ueber eine Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande, Zeitschrift für Mathematik und Physik, Hist.-lit. Abt, 34, 161-170 (1889) |
| [92] | Neill Wright, G. G., The writing of Arabic numerals (1952), London: University of London Press, London |
| [93] | Nothaft, C. Philipp E., The reception and application of Arabic science in twelfth-century computistics: new evidence from Bavaria, Journal for the History of Astronomy, 45, 35-60 (2014) · doi:10.1177/002182861404500103 |
| [94] | Numbers, Ronald L., Galileo goes to jail and other myths about science and religion (2009), Cambridge: Mass.: Harvard University Press, Cambridge |
| [95] | Otisk, Marek, ‘The interpretations and applications of Boethius’s Introduction to the arithmetic II, 1 at the end of the 10^th century’, GERBERTVS, 2 (2011), 33-56. |
| [96] | Otisk, Marek, ‘Descriptions and images of the early medieval Latin abacus’, Średniowiecze Polskie i Powszechne, 11 (2015), 13-35. |
| [97] | Pareja, Felix M., Islamologia (1951), Rome: Orbis Catholicus, Rome |
| [98] | Pedersen, Fritz Saaby, Petri Philomenae de Dacia et Petri de S. Audomaro Opera quadrivialia, Vol. 1, Opera Petri Philomenae (1983), Copenhagen: Gad, Copenhagen |
| [99] | Pedersen, Olaf, In quest of Sacrobosco, Journal for the History of Astronomy, 16, 175-221 (1985) · doi:10.1177/002182868501600302 |
| [100] | Pellegrini, Giovan Battista, Gli arabismi nelle lingue neolatine con speciale riguardo all’Italia (1972), Brescia: Paideia, Brescia |
| [101] | Pernoud, Régine, Pour en finir avec le Moyen Âge (1977), Paris: Éditions du Seuil, Paris |
| [102] | Principe, Lawrence M.; Hardin, Jeff; Numbers, Ronald L.; Binzley, Ronald A., The warfare thesis, The warfare between science and religion: the idea that wouldn’t die, 6-26 (2018), Baltimore, Md: Johns Hopkins University Press, Baltimore, Md |
| [103] | Pullan, J. M., The history of the abacus (1968), London: Hutchinson, London · Zbl 0224.01008 |
| [104] | Rauhaus, Alfred, Kleine Kirchenkunde: Reformierte Kirchen von innen und außen (2007), Göttingen: Vandenhoeck & Ruprecht, Göttingen |
| [105] | Reinhardt, Thomas, Die Erfindung der flachen Erde: Der Mythos Kolumbus und die Konstruktion der Epochenschwelle zwischen Mittelalter und Neuzeit, Paideuma, 53, 161-180 (2007) |
| [106] | Reynolds, Barbara E, ‘The algorists vs. the abacists: an ancient controversy on the use of calculators’, The College Mathematics Journal, 24 (1993), 218-223. · Zbl 1266.01010 |
| [107] | Ricklin, Thomas, Der Philosoph als Nekromant: Gerbert von Aurillac (Silvester II.) und Vergil im europäischen Hochmittelalter, Interfaces, 1, 236-264 (2015) |
| [108] | Roland, Bernhard, Der Eingang des “Mythos der flachen Erde” in deutsche und österreichische Geschichtsschulbücher im 20. Jahrhundert, Geschichte in Wissenschaft und Unterricht, 64, 687-701 (2013) |
| [109] | Rotman, Brian, Signifying nothing: the semiotics of zero (1987), Basingstoke: Macmillan, Basingstoke |
| [110] | Russell, Jeffrey Burton, Inventing the flat Earth: Columbus and modern historians (1991), New York: Praeger, New York |
| [111] | Sanford, Vera, A short history of mathematics (1930), London: George G. Harrap & Co., London · Zbl 0002.32303 |
| [112] | Schärlig, Alain., Du zéro à la virgule: les chiffres arabes à la conquête de l’Europe, 1143-1585, Lausanne: Presses polytechniques et universitaires romandes, Lausanne · Zbl 1223.01002 |
| [113] | Seife, Charles, Zero: the biography of a dangerous idea (2000), London: Souvenir, London · Zbl 0984.01002 |
| [114] | Smith, David Eugene; Ginsburg, Jekuthiel, Numbers and numerals: a story book for young and old (1937), Washington, DC: National Council of Teachers of Mathematics, Washington, DC |
| [115] | Smith, David Eugene; Karpinski, Louis Charles, The Hindu-Arabic numerals (1911), Boston, Mass.: Ginn and Company, Boston, Mass. · Zbl 1147.01008 |
| [116] | Spiesser, Maryvonne, Questions sur la diffusion du Liber Abbaci en France au XV^e siècle à travers l’étude des traités commercaux, Bollettino di storia delle scienze matematice, 24, 1, 115-135 (2004) · Zbl 1202.01041 |
| [117] | Stevens, Wesley M., A small circle in the Tabula Paschalis of Murbach, A.D. 814-820, Physis, n.s., 49, 1-19 (201314) |
| [118] | Stone, Williard E., Abacists versus algorists, Journal of Accounting Research, 10, 345-350 (1972) · doi:10.2307/2490013 |
| [119] | Struik, Dirk J, ‘The prohibition of the use of Arabic numerals in Florence’, Archives internationales d’histoire des sciences, 21 (1968), 291-294. · Zbl 0186.29502 |
| [120] | Sulowski, Jan, Radulphus de Longo Campo: In Anticlaudianum Alani commentum (1972), Wrocław: Zakład Narodowy im. Ossolińskich, Wrocław |
| [121] | Szalay, Jessie, ‘Who invented zero?’, Live Science, 18 September 2017, https://www.livescience.com/27853-who-invented-zero.html. Accessed on 10 June 2019. |
| [122] | Szombathely, Marino, Statuti di Trieste del 1350 (1930), Trieste: Cappelli, Trieste |
| [123] | Tabarroni, Giorgio, La matematica occidentale dopo il Mille: sua interazione con la vita quotidiana e la cultura, ‘Imago mundi’: la conoscenza scientifica nel pensiero basomedievale, 141-153 (1983), Todi: Presso l’Accademia tudertina, Todi |
| [124] | Taylor, Isaac, The alphabet: an account of the origin and development of letters, Vol. 2, Aryan alphabets (1883), London: Kegan Paul, London |
| [125] | Teresi, Dick, Lost discoveries: the ancient roots of modern science—from the Babylonians to the Maya (2002), New York: Simon & Schuster, New York |
| [126] | Tropfke, Johannes, Geschichte der Elementar-Mathematik in systematischer Darstellung mit besonderer Berücksichtigung der Fachwörter, Vol. 1, Rechnen (1921), Berlin: de Gruyter, Berlin · JFM 56.0001.02 |
| [127] | Truitt, Elly R., ‘Celestial divination and Arabic science in twelfth-century England: the history of Gerbert of Aurillac’s talking head’, Journal of the History of Ideas, 73, 201-222 (2012) · doi:10.1353/jhi.2012.0016 |
| [128] | Tucci, Ugo, Il documento del mercante, Civilità comunale: libro, scrittura, documento; atti del convegno, Genova, 8-11 novembre 1988, 541-565 (1989), Genova: Società ligure di storia patria, Genova |
| [129] | Ulivi, Elisabetta, Maestri e scuole d’Abaco a Firenze: la Bottega di Santa Trinita, Bollettino di storia delle scienze matematice, 24, 1, 43-91 (2004) · Zbl 1202.01049 |
| [130] | Ulivi, Elisabetta, Gli abacisti fiorentini delle famiglie ‘del maestro Luca’, Calandri e Micceri e le loro scuole d’abaco (secc. XIV-XVI) (2013), Florence: Olschki, Florence |
| [131] | Van Egmond, Warren, Practical mathematics in the Italian Renaissance: a catalog of Italian abbacus manuscripts and printed books to 1600, Florence: Istituto e museo di storia della scienza, Florence |
| [132] | Wallin, Nils-Bertil, ‘The history of zero’, YaleGlobal Online, 19 November 2002, https://yaleglobal.yale.edu/history-zero. Accessed on 10 June 2019. |
| [133] | Wedell, Moritz; Classen, Albrecht, Numbers, Handbook of medieval culture: fundamental aspects and conditions of the European Middle Ages, Vol. 2, 1205-1260 (2015), Berlin: de Gruyter, Berlin |
| [134] | Weill-Parot, Nicolas; Sales, Véronique, Le vrai visage du Moyen Âge: au-delà des idées reçues (2017), Paris: Vendémiaire, Paris |
| [135] | Weiss, Ittay, ‘Nothing matters: how the invention of zero helped create modern mathematics’, The Conversation, 20 September 2017, https://theconversation.com/nothing-matters-how-the-invention-of-zero-helped-create-modern-mathematics-84232. Accessed on 10 June 2019. |
| [136] | Westmaas, Reuben, ‘The concept of zero is newer than you’d expect’, Curiosity.com, 7 September 2017, https://curiosity.com/topics/the-concept-of-zero-is-newer-than-youd-expect-curiosity. Accessed on 10 June 2019. |
| [137] | Wolf, Jürgen, Die Moderne erfindet sich ihr Mittelalter—oder wie aus der ‘mittelalterlichen Erdkugel’ eine ‘neuzeitliche Erdscheibe’ wurde, Stuttgart: Steiner, Stuttgart |
| [138] | Yeldham, Florence A., The story of reckoning in the Middle Ages (1926), London: George G. Harrap & Co., London |
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