Abstract
The purpose of this article is to investigate the genesis and growth of a historical canard that can be encountered in numerous popular as well as some scholarly publications devoted to the history of mathematics. According to one of the core elements of this story, the number or symbol for zero was the cause of much anxiety in medieval Europe, as its unusual properties caused it to be associated with the Devil or with black magic. This anxiety is supposed to have extended to the entire system of Hindu-Arabic numerals, such that the use of these numerals was banned by the Church or by other powerful institutions. I shall argue that this narrative is false or unsubstantiated at nearly every level of analysis. Some elements arose from an unwarranted interpretation of medieval sources, while others are based on the unbridled imagination of certain modern authors.
Acknowledgements
I should like to thank Joshua Bennett, Charles Burnett, and David Juste for offering helpful comments and advice.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Correction Statement
This article has been corrected with minor changes. These changes do not impact the academic content of the article.
Notes
2 This sort of reconstructive work has been carried out with particular thoroughness for the myth of medieval belief in a flat Earth. See, for example, Russell Citation1991; Wolf Citation2004; Reinhardt Citation2007; Roland Citation2013. See also Classen Citation2007, for the myth of the medieval chastity belt.
3 Wallin’s article is based on the books by Kaplan (Citation1999) and Seife (Citation2000), which will be discussed in section 3.3 of this article.
5 See MS El Escorial, Real Bibliotheca de San Lorenzo, d.I.2, fol. 12v. The numerals on this page were copied into MS El Escorial, Real Biblioteca de San Lorenzo, d.I.1, fol. 9v (from San Millán de la Cogolla), which dates from 994. For transcriptions and facsimiles, see Smith and Karpinski (Citation1911, 138) and Burnam (Citation1912–25, 89–96). On the dating of these codices, see Gómez Pallarès (Citation1987, 29–32).
6 MS El Escorial, Real Bibliotheca de San Lorenzo, d.I.2, fol. 12v, ‘Scire debemus in Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et geometrica et ceteris liberalibus disciplinis concedere’. I here follow the translation in Burnett (Citation2006, 17).
10 These three texts were last edited by Allard (Citation1992). See also the overviews in Allard (Citation1991) and Ambrosetti (Citation2008, 197–212). For the so-called ‘second book’ of the Liber Alchorismi, see Burnett, Zhao and Lampe (Citation2007).
12 John of Sacrobosco’s work was last edited by Pedersen (Citation1983, 174–201). See also Pedersen (Citation1985, 182–183, 195–201). On the Carmen de algorismo, see Ambrosetti (Citation2016), who counts 161 manuscripts. On the topic of thirteenth-century algorithmic writing, see also the texts edited and discussed in Busard (Citation2000) and Burnett (Citation2002b).
14 See, for example, the Liber Algorismi (ed. Folkerts Citation1997, 32–34). For an unconvincing attempt to trace the circular sign for zero back to a ninth-century Carolingian origin, see Stevens (Citation2013–14), whose account is replete with inaccuracies.
15 In Arabic manuscripts prior to 1200, the use of HAN is mostly limited to texts on arithmetic, whereas in astronomical sources the abjad system of notation was preferred. HAN are rarely encountered in texts of a non-scientific nature. See Lemay (Citation1982, 383–384); Kunitzsch (Citation2005, 9–14, 28).
17 See the edition, translation, and discussion of this text by Burnett (Citation1996).
18 Some early examples of this are provided by Burnett (Citation2002a).
19 See Reinher of Paderborn, Compotus emendatus, praef. (ed. Lohr Citation2015, 3, ll. 20–22): ‘In designatione numerorum figuris plerumque utimur aliis quam Latinis propter scribendi et computandi compendium’.
20 See also the quote from this passage in Cunnington (Citation1904, 42).
21 Article no. 102 (Quod nullus de arte scribat in suo libro per abbacum) was lasted edited in Camerani Marri (Citation1955, 72–73). For the appearance of the same article in the revised statutes of 1300, 1313, and 1316, see the comparative table at the end of Camerani Marri (Citation1955, not paginated). For the English translation cited here, see Struik (Citation1968, 292). The passage is translated differently in Murray (Citation1978, 170–171). See the discussion in Lüneburg (Citation2008, 106–110), which offers a valuable criticism of some of the previous literature. On the interpretation of the phrase ‘openly and in full by way of letters’ (aperte et extense scribere per literam), see Nagl (Citation1889, 162–167).
22 See, for example, Smith and Ginsburg (Citation1937, 17); Neill Wright (Citation1952, 126); Menninger (Citation1969, 426–427); Tucci (Citation1989, 556); Burton (Citation2011, 280); King (Citation2001, 315); Cherubini (Citation2006, 332); Chrisomalis (Citation2010, 123). According to Gazalé (Citation2000, 48) the guild ‘ruled that the old figures could not be as easily falsified as the new ones, which could be turned into different figures without difficulty.’ It should be emphasized that no such comment appears in the source text.
23 According to Struik (Citation1968, 292–293) the ordinance reflects a power struggle between merchant guilds in the context of the ongoing conflict between Guelphs and Ghibellines in Florence.
24 As already noticed by Crossley (Citation2013, 92n42), who gives the example of Chrisomalis (Citation2010, 123–124) citing Berggren (Citation2002, 361), who cites Pullan (Citation1968, 34). To give but two additional ones: Caianiello (Citation2014, 229n61) cites Cherubini (Citation2006, 331–332n55), who cites Tabarroni (Citation1983, 148n5) citing Horn d’Arturo (Citation1925, 211), who refers to Taylor (Citation1883, 263), whose sources are unknown. Also, Ambrosetti (Citation2008, 247) cites Pellegrini (Citation1972, 27), who in turn cites Pareja (Citation1951, 693), who fails to provide a source, as do many others, including Haskins (Citation1904, 154–155) and King (Citation2001, 316).
25 See the edition by Denifle (Citation1892, 453, ll. 25–28): ‘Ponat eciam in libro venali extrinsecus et in evidenti loco et claris litteris non [per] zyphras nomen venditoris cum ipsius congnomine [sic] et precium libri, sub pena viginti solidorum Universitati solvenda et per rectoris exigenda’. Murray (Citation1978, 171), for unclear reasons, dates these statutes to 1305.
26 Statuta dominorum artistarum Achademiae Patauinae, [sine loc.]: Pasquino di Roma, [after 1500], fol. 34v (lib. III, c. 24). These statutes were already cited by Kirchhoff (Citation1853, 30).
28 Some results of this article are summarized in Jordan (Citation1906, 64–68).
29 Jordan’s idea of two opposing camps of calculators was to have a remarkably long afterlife, as seen from the popular articles by Stone (Citation1972) and Reynolds (Citation1993). As recently as 2010 a serious history of mathematics (Chrisomalis Citation2010, 123) declared that Leonardo of Pisa’s Liber abbaci ‘sparked an important debate between two camps, the abacists, those who preferred computation with the medieval abacus, and the algorithmists, who preferred pen-and-paper calculations using the Western ciphered-positional numerals. The history of this debate is well documented, as it involved many important commercial families, renowned mathematicians and clergymen, and even state authorities (Menninger Citation1969, 422–445; Evans Citation1977b; Murray Citation1978, 163–175)’. None of the studies cited here provide robust documentation of an actual debate between two factions. See also Caianiello (Citation2014, 229).
30 William of Malmesbury, Gesta regum Anglorum, 2.167.3, ed. Mynors (Citation1998, 280): ‘Abacum certe primum a Saracenis rapiens, regulas dedit quae a sudantibus abacistis uix intelliguntur’. For the English translation see ibid., 281.
31 The notion that William regarded Gerbert’s mathematics as ‘dangerous Saracen magic’ was introduced into the literature by Cochrane (Citation1994, 7, 43), who masked her incorrect interpretation of the text as a quote taken from it. From Cochrane’s book, the fake quote passed into Lyons (Citation2012, 86) and Kaplan (Citation1999, 66, 75), who repeated it in an appearance on BBC Radio 4’s In Our Time (13 May 2004).
33 William of Malmesbury, Gesta regum Anglorum, 2.167.5, ed. Mynors Citation1998, 282, ‘Sed haec uulgariter ficta crediderit aliquis, quod soleat populus litteratorum famam ledere, dicens illum loqui cum demone quem in aliquo uiderint excellentem opere’. English translation ibid., 283.
34 Alain de Lille, Anticlaudianus, 2.436–438 (ed. Bossuat Citation1955, 85); Alain de Lille, De planctu naturae, 2.192–193 (ed. Häring Citation1978, 816).
35 Gautier de Coinci, Miracles de Nostre Dame, II Mir 32, ll. 224–227 (ed. Koenig Citation1970, 426–427).
36 Compare, for example, the Modern Hebrew slang term dapar efes or the German insult Du Null! Gazalé (Citation2000, 46) notes that ‘[i]n contemporary Egypt, an individual despised by his or her peers, is metaphorically referred to as sifr ‘ala al-yassaar, literally meaning “zero on the left-hand side”, or as we would say today, “nonsignificant zero”’.
37 The questionable value of this argument becomes clear once one considers that that Alexandre de Villedieu authored (or is believed to have authored) equally popular didactic poems on the ecclesiastical calendar (Massa compoti) and grammar (Doctrinale puerorum). Would anyone claim that their verse form was indicative of a ‘popular disinclination’ towards these subjects?
38 Morrison and Morrison (Citation1999, 546), even saw fit to include it among ‘100 or so books that shaped a century of science’.
39 Asked to corroborate the reference in an email exchange of 1 June 2019, Kaplan responded to me with the following: ‘Rather than misquoting me, let’s say [Teresi] well exercised his considerable powers of summing up’.
Pernoud, Régine, Pour en finir avec le Moyen Âge, Paris: Éditions du Seuil, 1977. Harris, Stephen J., and Grigsby, Byron Lee (eds), Misconceptions about the Middle Ages, New York: Routledge, 2008. Weill-Parot, Nicolas, and Sales, Véronique (eds), Le vrai visage du Moyen Âge: au-delà des idées reçues, Paris: Vendémiaire, 2017. Numbers, Ronald L (ed), Galileo goes to jail and other myths about science and religion, Cambridge, Mass.: Harvard University Press, 2009. Russell, Jeffrey Burton, Inventing the flat Earth: Columbus and modern historians, New York: Praeger, 1991. Wolf, Jürgen, Die Moderne erfindet sich ihr Mittelalter—oder wie aus der ‘mittelalterlichen Erdkugel’ eine ‘neuzeitliche Erdscheibe’ wurde (Akademie der Wissenschaften und der Literatur in Mainz, Abhandlungen der geistes- und sozialwissenschaftlichen Klasse, 2004.5), Stuttgart: Steiner, 2004. Reinhardt, Thomas, ‘Die Erfindung der flachen Erde: Der Mythos Kolumbus und die Konstruktion der Epochenschwelle zwischen Mittelalter und Neuzeit’, Paideuma, 53 (2007), 161–180. Roland, Bernhard, ‘Der Eingang des “Mythos der flachen Erde” in deutsche und österreichische Geschichtsschulbücher im 20. Jahrhundert’, Geschichte in Wissenschaft und Unterricht, 64 (2013), 687–701. Classen, Albrecht, The medieval chastity belt: a myth-making process, New York: Palgrave Macmillan, 2007. Kaplan, Robert, The nothing that is: a natural history of zero, London: Allen Lane, 1999. Seife, Charles, Zero: the biography of a dangerous idea, London: Souvenir, 2000. Lemay, Richard, ‘The Hispanic origin of our present numeral forms’, Viator, 8 (1977), 435–477. doi: 10.1484/J.VIATOR.2.301575 Lemay, Richard, ‘Arabic numerals’, in Joseph R. Strayer (ed), Dictionary of the Middle Ages, Vol. 1, Aachen–augustinism, New York: Charles Scribner’s Sons, 1982, 382–398. Allard, André, ‘La formation du vocabulaire latin de l’arithmétique médiévale’, in Olga Weijers (ed), Méthodes et instruments du travail intellectuel au moyen âge: études sur le vocabulaire, Turnhout: Brepols, 1990, 137–181. Folkerts, Menso, ‘Early texts on Hindu-Arabic calculation’, Science in Context, 14 (2001a), 13–38. doi: 10.1017/S0269889701000023 Berggren, J Lennart, ‘Medieval arithmetic: Arabic texts and European motivations’, in John J. Contreni and Santa Casciani (eds), Word, image, number: communication in the Middle Ages, Florence: SISMEL, 2002, 351–365. Kunitzsch, Paul, Zur Geschichte der ‘arabischen’ Ziffern (Sitzungsberichte der Bayerischen Akademie der Wissenschaften, phil.-hist. Kl., 2005.3.), Munich: Verlag der Bayerischen Akademie der Wissenschaften, 2005. Burnett, Charles, ‘The semantics of Indian numerals in Arabic, Greek and Latin’, Journal of Indian Philosophy, 34 (2006), 15–30. doi: 10.1007/s10781-005-8153-z Ambrosetti, Nadia, L’eredità arabo-islamica nelle scienze e nelle arti del calcolo dell’Europa medievale, Milan: Edizioni Universitarie di Lettere Economia Diritto, 2008. Schärlig, Alain. Du zéro à la virgule: les chiffres arabes à la conquête de l’Europe, 1143–1585. Lausanne: Presses polytechniques et universitaires romandes. Wedell, Moritz, ‘Numbers’, in Albrecht Classen (ed), Handbook of medieval culture: fundamental aspects and conditions of the European Middle Ages, Vol. 2, Berlin: de Gruyter, 2015, 1205–1260. Folkerts, Menso, and Hughes, Barnabas, ‘The Latin mathematics of medieval Europe’, in Victor J. Katz (ed), Sourcebook in the mathematics of medieval Europe and North Africa, Princeton, NJ: Princeton University Press, 2016, 4–223. Burnett, Charles, Numerals and arithmetic in the Middle Ages, Farnham: Ashgate, 2010. Smith, David Eugene, and Karpinski, Louis Charles, The Hindu-Arabic numerals, Boston, Mass.: Ginn and Company, 1911. Burnam, John M, Palaeographia iberica: fac-similés de manuscrits espagnols et portugais (IXe–XVe siècles) avec notices et transcriptions. Paris: Champion, 1912–25. Gómez Pallarès, Juan, ‘Sobre manuscritos latinos de cómputo en escritura visigótica’, Hispania Sacra, 39 (1987), 25–48. Burnett, Charles, ‘The semantics of Indian numerals in Arabic, Greek and Latin’, Journal of Indian Philosophy, 34 (2006), 15–30. doi: 10.1007/s10781-005-8153-z Beaujouan, Guy, ‘Etude paléographique sur la “rotation” des chiffres et l’emploi des apices du Xe au XIIe siècle’, Revue d’histoire des sciences, 1, no. 4 (1948), 301–313. Beaujouan, Guy, ‘Les chiffres arabes selon Gerbert: l’abaque du Pseudo-Boèce, XIe siècle’, in Olivier Guyotjeannin and Emmanuel Poulle (eds), Autour de Gerbert d’Aurillac: le pape de l’an mil, Paris: École des Chartes, 1996, 322–328. Evans, Gillian R, ‘Difficillima et ardua: theory and practice in treatises on the abacus, 950–1150’, Journal of Medieval History, 3 (1977a), 21–38. doi: 10.1016/0304-4181(77)90038-0 Evans, Gillian R, ‘Schools and scholars: the study of the abacus in English schools c. 980–c. 1150’, English Historical Review, 94 (1979), 71–89. doi: 10.1093/ehr/XCIV.CCCLXX.71 Gibson, Craig A, and 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 (1995), 293–335. doi: 10.1484/J.MS.2.306437 Folkerts, Menso, ‘Frühe Darstellungen des Gerbertschen Abakus’, Raffaella Franci, Paolo Pagli, and Laura Toti Rigatelli (eds), Itinera mathematica: studi in onore di Gino Arrighi per il suo 90o compleanno, Siena: Centro studi sulla matematica medioevale, Università di Siena, 1996, 23–43. Folkerts, Menso, ‘Frühe westliche Benennungen der indisch-arabischen Ziffern und ihr Vorkommen’, in Menso Folkerts and Richard Lorch (eds), Sic itur ad astra: Studien zur Geschichte der Mathematik und Naturwissenschaften, Wiesbaden: Harrassowitz, 2000, 216–233. Folkerts, Menso, ‘The names and forms of the numerals on the abacus in the Gerbert tradition’, in Fabio G. Nuvolone (ed), Gerberto d’Aurillac, da Abate di Bobbio a Papa dell’Anno 1000, Bobbio: Associazione culturale Amici di Archivum Bobiense, 2001b, 245–265. Modified reprint as Chapter 6 of Folkerts, Essays on early medieval mathematics: the Latin tradition, Aldershot: Ashgate, 2003. Burnett, Charles, ‘The abacus at Echternach in ca. 1000 A.D.’, Sciamus, 3 (2002c), 91–108. Reprinted as Chapter 1 in Burnett 2010. Otisk, Marek, ‘Descriptions and images of the early medieval Latin abacus’, Średniowiecze Polskie i Powszechne, 11 (2015), 13–35. Burnett, Charles, ‘Algorismi vel helcep decentior est diligentia: the arithmetic of Adelard of Bath and his circle’, in Menso Folkerts (ed), Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich, Wiesbaden: Harrassowitz, 1996, 221–331. Reprinted as Chapter 3 in Burnett 2010. Burnett, Charles, ‘The abacus at Echternach in ca. 1000 A.D.’, Sciamus, 3 (2002c), 91–108. Reprinted as Chapter 1 in Burnett 2010. Burnett, Charles, ‘Fibonacci’s “Method of the Indians”’, Bollettino di storia delle scienze matematiche, 23, no. 2 (2003), 87–97. Reprinted as Chapter 11 in Burnett 2010. Burnett, Charles, ‘The semantics of Indian numerals in Arabic, Greek and Latin’, Journal of Indian Philosophy, 34 (2006), 15–30. doi: 10.1007/s10781-005-8153-z Folkerts, Menso, ‘Frühe westliche Benennungen der indisch-arabischen Ziffern und ihr Vorkommen’, in Menso Folkerts and Richard Lorch (eds), Sic itur ad astra: Studien zur Geschichte der Mathematik und Naturwissenschaften, Wiesbaden: Harrassowitz, 2000, 216–233. Folkerts, Menso (ed), Die älteste lateinische Schrift über das indische Rechnen nach al-Ḫwārizmī (Abhandlungen der Bayerischen Akademie der Wissenschaften, phil.-hist. Kl., n.s., 113.), Munich: Verlag der Bayerischen Akademie der Wissenschaften, 1997. Allard, André (ed), Muhammad Ibn Mūsā al-Khwārizmī: Le calcul indien (Algorismus), Paris: Blanchard, 1992. Allard, André, ‘The Arabic origins and development of Latin algorisms in the twelfth century’, Arabic Sciences and Philosophy, 1 (1991), 233–283. doi: 10.1017/S0957423900001508 Ambrosetti, Nadia, L’eredità arabo-islamica nelle scienze e nelle arti del calcolo dell’Europa medievale, Milan: Edizioni Universitarie di Lettere Economia Diritto, 2008. Burnett, Charles; Zhao, Ji-Wei, and Lampe, Kurt, ‘The Toledan Regule (Liber Alchorismi, part II): a twelfth-century arithmetical miscellany’, Sciamus, 8 (2007), 141–231. Reprinted as Chapter 8 in Burnett 2010. Pedersen, Fritz Saaby (ed), Petri Philomenae de Dacia et Petri de S. Audomaro Opera quadrivialia, Vol. 1, Opera Petri Philomenae. Copenhagen: Gad, 1983. Pedersen, Olaf, ‘In quest of Sacrobosco’, Journal for the History of Astronomy, 16 (1985), 175–221. doi: 10.1177/002182868501600302 Ambrosetti, Nadia, ‘Algorithmic in the 12th century: the Carmen de Algorismo by Alexander de Villa Dei’, in Fabio Gadducci and Mirko Tavosanis (eds), History and philosophy of computing, Cham: Springer, 2016, 71–86. Busard, H L L, ‘Über zwei Algorismus-Schriften aus dem 13. Jahrhundert’, in Menso Folkerts and Richard Lorch (eds), Sic itur ad astra: Studien zur Geschichte der Mathematik und Naturwissenschaften, Wiesbaden: Harrassowitz, 2000, 91–137. Burnett, Charles, ‘Learning Indian arithmetic in the early thirteenth century’, Boletín de la Asociación Matemática Venezolana, 9 (2002b), 15–26. Reprinted as Chapter 9 in Burnett 2010. Lemay, Richard, ‘The Hispanic origin of our present numeral forms’, Viator, 8 (1977), 435–477. doi: 10.1484/J.VIATOR.2.301575 Evans, Gillian R, ‘From abacus to algorism: theory and practice in medieval arithmetic’, British Journal for the History of Science, 10 (1977b), 114–131. doi: 10.1017/S0007087400015375 Allard, André, ‘La formation du vocabulaire latin de l’arithmétique médiévale’, in Olga Weijers (ed), Méthodes et instruments du travail intellectuel au moyen âge: études sur le vocabulaire, Turnhout: Brepols, 1990, 137–181. Burnett, Charles, ‘Algorismi vel helcep decentior est diligentia: the arithmetic of Adelard of Bath and his circle’, in Menso Folkerts (ed), Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich, Wiesbaden: Harrassowitz, 1996, 221–331. Reprinted as Chapter 3 in Burnett 2010. Ambrosetti, Nadia, ‘Algorithmic in the 12th century: the Carmen de Algorismo by Alexander de Villa Dei’, in Fabio Gadducci and Mirko Tavosanis (eds), History and philosophy of computing, Cham: Springer, 2016, 71–86. Tropfke, Johannes, Geschichte der Elementar-Mathematik in systematischer Darstellung mit besonderer Berücksichtigung der Fachwörter, Vol. 1, Rechnen, Berlin: de Gruyter, 2nd ed., 1921. Folkerts, Menso (ed), Die älteste lateinische Schrift über das indische Rechnen nach al-Ḫwārizmī (Abhandlungen der Bayerischen Akademie der Wissenschaften, phil.-hist. Kl., n.s., 113.), Munich: Verlag der Bayerischen Akademie der Wissenschaften, 1997. Stevens, Wesley M, ‘A small circle in the Tabula Paschalis of Murbach, A.D. 814–820’, Physis, n.s., 49 (2013–14), 1–19. Lemay, Richard, ‘Arabic numerals’, in Joseph R. Strayer (ed), Dictionary of the Middle Ages, Vol. 1, Aachen–augustinism, New York: Charles Scribner’s Sons, 1982, 382–398. Kunitzsch, Paul, Zur Geschichte der ‘arabischen’ Ziffern (Sitzungsberichte der Bayerischen Akademie der Wissenschaften, phil.-hist. Kl., 2005.3.), Munich: Verlag der Bayerischen Akademie der Wissenschaften, 2005. Burnett, Charles, ‘Latin alphanumerical notation, and annotation in Italian, in the twelfth century: MS London, British Library, Harley, 5402’, in Menso Folkerts and Richard Lorch (eds), Sic itur ad astra: Studien zur Geschichte der Mathematik und Naturwissenschaften, Wiesbaden: Harrassowitz, 2000, 77–90. Reprinted as Chapter 10 in Burnett 2010. Burnett, Charles, ‘The semantics of Indian numerals in Arabic, Greek and Latin’, Journal of Indian Philosophy, 34 (2006), 15–30. doi: 10.1007/s10781-005-8153-z Berggren, J Lennart, ‘Medieval arithmetic: Arabic texts and European motivations’, in John J. Contreni and Santa Casciani (eds), Word, image, number: communication in the Middle Ages, Florence: SISMEL, 2002, 351–365. King, David A, The ciphers of the monks: a forgotten number-notation of the Middle Ages (Boethius, 44), Stuttgart: Steiner, 2001. Kunitzsch, Paul, Zur Geschichte der ‘arabischen’ Ziffern (Sitzungsberichte der Bayerischen Akademie der Wissenschaften, phil.-hist. Kl., 2005.3.), Munich: Verlag der Bayerischen Akademie der Wissenschaften, 2005. Burnett, Charles, ‘Algorismi vel helcep decentior est diligentia: the arithmetic of Adelard of Bath and his circle’, in Menso Folkerts (ed), Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich, Wiesbaden: Harrassowitz, 1996, 221–331. Reprinted as Chapter 3 in Burnett 2010. Burnett, Charles, ‘Indian numerals in the Mediterranean Basin in the twelfth century, with special reference to the “eastern forms”’, in Yvonne Dold-Samplonius, Joseph W. Dauben, Menso Folkerts, and Benno van Dalen (eds), From China to Paris: 2000 years transmission of mathematical ideas, Stuttgart: Steiner, 2002a, 237–288 . Reprinted as Chapter 5 in Burnett 2010. Lohr, Alfred (ed), Opera de computo saeculi duodecimi (Corpus Christianorum Continuatio Mediaevalis, 272), Turnhout: Brepols, 2015. Cunnington, Susan, The story of arithmetic: a short history of its origin and development, London: Sonnenschein & Co, 1904. Camerani Marri, Giulia (ed), Statuti dell’Arte del Cambio di Firenze (1299–1316), Florence: Olschki, 1955. Camerani Marri, Giulia (ed), Statuti dell’Arte del Cambio di Firenze (1299–1316), Florence: Olschki, 1955. Struik, Dirk J, ‘The prohibition of the use of Arabic numerals in Florence’, Archives internationales d’histoire des sciences, 21 (1968), 291–294. Murray, Alexander, Reason and society in the Middle Ages, Oxford: Clarendon Press, 1978. Lüneburg, Heinz, Von Zahlen und Größen: Dritthalbtausend Jahre Theorie und Praxis, Vol. 1, Basel: Birkhäuser, 2008. 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 (1889), 129–146, 161–170. Smith, David Eugene, and Ginsburg, Jekuthiel, Numbers and numerals: a story book for young and old, Washington, DC: National Council of Teachers of Mathematics, 1937. Neill Wright, G G, The writing of Arabic numerals, London: University of London Press, 1952. Menninger, Karl, Number words and number symbols: a cultural history of numbers (trans Paul Broneer), Cambridge, Mass.: M.I.T. Press, 1969 (original work published 1934). Tucci, Ugo, ‘Il documento del mercante’, in Civilità comunale: libro, scrittura, documento; atti del convegno, Genova, 8–11 novembre 1988, Genova: Società ligure di storia patria, 1989, 541–565. Burton, David M, The history of mathematics: an introduction, New York: McGraw-Hill, 7th edition, 2011 (First published in 1985). King, David A, The ciphers of the monks: a forgotten number-notation of the Middle Ages (Boethius, 44), Stuttgart: Steiner, 2001. Cherubini, Paolo, ‘Il numero come elemento di disturbo’, in Rita Librandi and Rosa Piro (eds), Lo scaffale della biblioteca scientifica in volgare (secoli XIII–XVI), Florence: SISMEL, 2006, 313–339. Chrisomalis, Stephen, Numerical notation: a comparative history, Cambridge: Cambridge University Press, 2010. Gazalé, Midhat, Number: From Ahmes to Cantor, Princeton, NJ: Princeton University Press, 2000. Struik, Dirk J, ‘The prohibition of the use of Arabic numerals in Florence’, Archives internationales d’histoire des sciences, 21 (1968), 291–294. Crossley, John N, ‘Old-fashioned versus newfangled: reading and writing numbers, 1200–1500’, Studies in medieval and Renaissance history, 3rd ser., 10 (2013), 79–109. Chrisomalis, Stephen, Numerical notation: a comparative history, Cambridge: Cambridge University Press, 2010. Berggren, J Lennart, ‘Medieval arithmetic: Arabic texts and European motivations’, in John J. Contreni and Santa Casciani (eds), Word, image, number: communication in the Middle Ages, Florence: SISMEL, 2002, 351–365. Pullan, J M, The history of the abacus. London: Hutchinson, 1968. Caianiello, Eva, ‘Leonardo of Pisa and the Liber Abaci: biographical elements and the project of the work’, in Alain Bernard and Christine Proust (eds), Scientific sources and teaching contexts throughout history: problems and perspectives, Dordrecht: Springer, 2014, 217–246. Cherubini, Paolo, ‘Il numero come elemento di disturbo’, in Rita Librandi and Rosa Piro (eds), Lo scaffale della biblioteca scientifica in volgare (secoli XIII–XVI), Florence: SISMEL, 2006, 313–339. Tabarroni, Giorgio, ‘La matematica occidentale dopo il Mille: sua interazione con la vita quotidiana e la cultura’, in ‘Imago mundi’: la conoscenza scientifica nel pensiero basomedievale (Convegni del Centro di studi sulla spiritualità medievale, Università degli studi di Perugia, 22), Todi: Presso l’Accademia tudertina, 1983, 141–153. Horn-d’Arturo, Guido, ‘Numeri arabici e simboli celesti’, Memorie della Società astronomica italiana, n.s., 3 (1925), 195–212. Taylor, Isaac, The alphabet: an account of the origin and development of letters, Vol. 2, Aryan alphabets, London: Kegan Paul, 1883. Ambrosetti, Nadia, L’eredità arabo-islamica nelle scienze e nelle arti del calcolo dell’Europa medievale, Milan: Edizioni Universitarie di Lettere Economia Diritto, 2008. Pellegrini, Giovan Battista, Gli arabismi nelle lingue neolatine con speciale riguardo all’Italia, Brescia: Paideia, 1972. Pareja, Felix M, Islamologia, Rome: Orbis Catholicus, 1951. Haskins, Charles Waldo, Business education and accountancy, New York: Harper & Brothers, 1904. King, David A, The ciphers of the monks: a forgotten number-notation of the Middle Ages (Boethius, 44), Stuttgart: Steiner, 2001. Denifle, Heinrich, ‘Die Statuten der Juristen-Universität Padua vom Jahre 1331’, Archiv für Literatur- und Kirchengeschichte des Mittelalters, 6 (1892), 309–562. Murray, Alexander, Reason and society in the Middle Ages, Oxford: Clarendon Press, 1978. Kirchhoff, Albrecht, Die Handschriftenhändler des Mittelalters, Leipzig: Kirchhoff, 2nd ed., 1853. Lebsanft, Franz, ‘Ein deutsch-jüdisches Schicksal: Der Philologe und Linguist Leo Jordan (1874–1940)’, in Hans Helmut Christmann, Frank-Rutger Hausmann, and Manfred Briegel (eds), Deutsche und österreichische Romanisten als Verfolgte des Nationalsozialismus, Tübingen: Stauffenburg, 1989, 157–175, 287–288. Jordan, Leo, ‘Wortgeschichtliches’, in E. Stollreither (ed), Festschrift zum XII. Allgemeinen deutschen Neuphilologentage in München, Pfingsten 1906, Erlangen: Junge, 1906, 61–80. Stone, Williard E, ‘Abacists versus algorists’, Journal of Accounting Research, 10 (1972), 345–350. doi: 10.2307/2490013 Reynolds, Barbara E, ‘The algorists vs. the abacists: an ancient controversy on the use of calculators’, The College Mathematics Journal, 24 (1993), 218–223. Chrisomalis, Stephen, Numerical notation: a comparative history, Cambridge: Cambridge University Press, 2010. Menninger, Karl, Number words and number symbols: a cultural history of numbers (trans Paul Broneer), Cambridge, Mass.: M.I.T. Press, 1969 (original work published 1934). Evans, Gillian R, ‘From abacus to algorism: theory and practice in medieval arithmetic’, British Journal for the History of Science, 10 (1977b), 114–131. doi: 10.1017/S0007087400015375 Murray, Alexander, Reason and society in the Middle Ages, Oxford: Clarendon Press, 1978. Caianiello, Eva, ‘Leonardo of Pisa and the Liber Abaci: biographical elements and the project of the work’, in Alain Bernard and Christine Proust (eds), Scientific sources and teaching contexts throughout history: problems and perspectives, Dordrecht: Springer, 2014, 217–246. Mynors, R A B (ed), William of Malmesbury: Gesta regum Anglorum; the history of the English kings, Vol. 1, completed by R. M. Thomson and M. Winterbottom, Oxford: Clarendon Press, 1998. Cochrane, Louise, Adelard of Bath: the first English scientist, London: British Museum Press, 1994. Lyons, Jonathan, Islam through Western eyes: from the Crusades to the war on terrorism. New York: Columbia University Press, 2012. Kaplan, Robert, The nothing that is: a natural history of zero, London: Allen Lane, 1999. 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 (2012), 201–222. doi: 10.1353/jhi.2012.0016 Ricklin, Thomas, ‘Der Philosoph als Nekromant: Gerbert von Aurillac (Silvester II.) und Vergil im europäischen Hochmittelalter’, Interfaces, 1 (2015), 236–264. Mynors, R A B (ed), William of Malmesbury: Gesta regum Anglorum; the history of the English kings, Vol. 1, completed by R. M. Thomson and M. Winterbottom, Oxford: Clarendon Press, 1998. Bossuat, Robert (ed), Alain de Lille: Anticlaudianus, Paris: Vrin, 1955. Häring, Nikolaus M, ‘Alan of Lille: De planctu Naturae’, Studi medievali, 3rd ser., 19 (1978), 797–879. Koenig, V. Frederic (ed), Les Miracles de Nostre Dame par Gautier de Coinci, Vol. 4, Geneva: Droz, 1970. Gazalé, Midhat, Number: From Ahmes to Cantor, Princeton, NJ: Princeton University Press, 2000. Morrison, Philip, and Morrison, Phylis, ‘100 or so books that shaped a century of science’, American Scientist, 87 (1999), 542–544, 546, 549–550, 553.