Book Chapter

The spread of Hindu-Arabic numerals among practitioners in Italy and England (13th-16th c.): two moments of a European innovation cycle?

  • Raffaele Danna

Together with introducing a set of key innovations in commercial practices, the merchant-bankers of the commercial revolution of the 13th century were also the first European economic agents to adopt Hindu-Arabic numerals. As practical arithmetic provided the mathematical foundation for commercial innovations, studying its European spread provides a particularly suitable angle to study the diffusion of practical knowledge in the pre-modern period. Italy was the early adopter of these techniques, while in England these practices became widespread at the onset of the little divergence. In this paper, I discuss in comparative perspective the social diffusion of this knowledge in Italy and England, and its wider impact. On the one hand, this analysis makes it possible to show a number of parallels between the trajectories followed by these societies. On the other hand, it allows to observe the complex interactions between practical knowledge and wider economic, institutional, and social changes.

  • Keywords:
  • Knowledge diffusion,
  • learning,
  • economic growth,
  • commercial revolution,
  • little divergence,
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Raffaele Danna

Sant’Anna School of Advanced Studies Pisa, Italy

  1. Abulafia, David. 1997. “The impact of the Orient: Economic interactions between East and West in the medieval Mediterranean.” In Across the Mediterranean frontiers: Trade, politics and religion, 650-1450, ed. Dionisius A. Agius, and Ian Richard Netton, 1-40. International Medieval Research. Turnhout: Brepols Publishers. DOI: 10.1484/M.IMR-EB.4.00002
  2. Acemoglu, Daron, Simon Johnson, and James Robinson. 2005. “The rise of Europe: Atlantic trade, institutional change, and economic growth.” American Economic Review 95, 3: 546-79. DOI: 10.1257/0002828054201305
  3. Allen, Robert C. 2000. “Economic structure and agricultural productivity in Europe, 1300-1800.” European Review of Economic History 4, 1: 1-26.
  4. Allen, Robert C. 2001. “The Great Divergence in European wages and prices from the middle ages to the first World War.” Explorations in Economic History 38, 4: 411-47. DOI: 10.1006/exeh.2001.0775
  5. Allen, Robert C. 2003. “Progress and poverty in early modern Europe.” The Economic History Review 56, 3: 403-43. DOI: 10.1111/j.1468-0289.2003.00257.x
  6. Álvarez‐Nogal, Carlos, and Leandro Prados De La Escosura. 2013. “The rise and fall of Spain (1270-1850).” The Economic History Review 66, 1: 1-37. DOI: 10.1111/j.1468-0289.2012.00656.x
  7. Ambrosetti, Nadia. 2008. L’eredità arabo-islamica nelle scienze e nelle arti del calcolo dell’Europa medievale. Milano: LED.
  8. Amt, Emilie, ed. 2007. Dialogus de Scaccario: The Dialogue of the Exchequer. Oxford Medieval Texts. Oxford: Oxford University Press.
  9. Arrighi, Gino ed. 1970. Piero della Francesca. Trattato d’abaco: dal Codice Ashburnhamiano 280, 359*-291*, della Biblioteca medicea laurenziana di Firenze. Testimonianze di storia della scienza. Pisa: Domus Galilaeana.
  10. Arrighi, Gino. 1966. “Un ‘programma’ di didattica di matematica nella prima metà del quattrocento (dal Codice 2186 della Biblioteca Riccardiana di Firenze).” Atti e Memorie dell’accademia Petrarca di lettere, arti e scienze di Arezzo 38: 117-28.
  11. Barnard, Francis Pierrepont. 1916. The casting-counter and the counting-board: A chapter in the history of numismatics and early arithmetic. Oxford: Clarendon Press.
  12. Baxandall, Michael. 1972. Painting and experience in fifteenth century Italy: A primer in the social history of pictorial style. Oxford: Clarendon Press.
  13. Baxter, W. 1989. “Early Accounting: The tally and checkerboard.” The Accounting Historians Journal 16: 43.
  14. Bell, Adrian R., Chris Brooks, and Tony K. Moore, ed. 2009. Accounts of the English Crown with Italian merchant societies, 1272-1345. Kew: The List and Index Society.
  15. Bell, Adrian R., Chris Brooks, and Tony K. Moore. 2009. “Interest in medieval accounts: Examples from England, 1272-1340.” History 94 (316): 411-33.
  16. Bell, Adrian R., Chris Brooks, and Tony K. Moore. 2017. “Cambium non est mutuum: Exchange and interest rates in medieval Europe.” The Economic History Review 70, 2: 373-96. DOI: 10.1111/ehr.12374.
  17. Bellosi, Luciano, and Aldo Rossi, ed. 1986. Giorgio Vasari. Le vite de più eccellenti architetti, pittori, et scultori italiani: Da Cimabue insino a tempi nostri. Nell’edizione per i tipi di Lorenzo Torrentino (Firenze, 1550). Torino: Einaudi.
  18. Benoit, P. 1988. “The commercial arithmetic of Nicolas Chuquet.” In Mathematics from manuscript to print, 1300-1600, ed. Cynthia Hay, 96-116. Oxford: Clarendon Press.
  19. Biggs, Norman. 2009. “Mathematics of currency and exchange: Arithmetic at the end of the thirteenth century.” BSHM Bulletin: Journal of the British Society for the History of Mathematics 24, 2: 67-77. DOI: 10.1080/17498430902820861
  20. Black, Robert. 2007. Education and society in Florentine Tuscany. Education and society in the middle ages and Renaissance. Leiden ; Boston: Brill.
  21. Bockstaele, Paul. 1960. “Notes on the first arithmetics Printed in Dutch and English.” Isis 51, 3: 315-21.
  22. Bolton, J. L. 1986. “The city and the Crown, 1456-61.” The London Journal 12, 1: 11-24. DOI: 10.1179/ldn.1986.12.1.11
  23. Bradley, Helen Lesley. 1992. “Italian merchants in London C1350-1450.” Ph.D., Royal Holloway, University of London.
  24. Bradley, Helen, ed. 2012. The views of the hosts of alien merchants, 1440-1444. Suffolk: Boydell Press.
  25. Brett, Colin J, ed. 2020. Thomas Kytson’s “Boke of Remembraunce’, 1529-1540. London: Boydell Press.
  26. Broadberry, Stephen, Bruce M.S. Campbell, Alexander Klein, Mark Overton, and Bas van Leeuwen. 2015. British economic growth, 1270-1870. Cambridge: Cambridge University Press.
  27. Cajori, Florian. 1928. A history of mathematical notations. II vols. Chicago, IL: Open Court Pub. Co.
  28. Cardano, Girolamo. 1663. Opera omnia. 10 vols. Lyon: Jean Antoine Huguetan/Marc Antione Ravaud.
  29. Ceccarelli, Giovanni. 2012. Un mercato del rischio: assicurare e farsi assicurare nella Firenze rinascimentale. Venezia: Marsilio.
  30. Cipolla, Carlo M. 1956. Money, Prices, and civilization in the Mediterranean world: fifth to seventeenth century. Princeton: Princeton University Press.
  31. Clark, Gregory. 2007. A farewell to alms: A brief economic history of the world. Princeton: Princeton University Press.
  32. Connell-Smith, G. 1951. “The ledger of Thomas Howell.” The Economic History Review 3, 3: 363-70. DOI: 10.2307/2599993
  33. Cormack, Lesley B., Steven A. Walton, and John A. Schuster, ed. 2017. Mathematical practitioners and the transformation of natural knowledge in early modern Europe. Cham: Springer. http://ebookcentral.proquest.com/lib/cam/detail.action?docID=4822965.
  34. Crossley, John N., and Alan S. Henry. 1990. “Thus spake Al-Khwārizmī: A translation of the text of Cambridge University Library Ms. Ii.vi.5.” Historia Mathematica 17, 2: 103-31. DOI: 10.1016/0315-0860(90)90048-I
  35. Dalai Emiliani, Marisa, Ottavio Besomi, Carlo Maccagni, Enrico Gamba, Vico Montebelli, Giovanna Derenzini, Enzo Mattesini, Vladimiro Valerio, and Alessandra Sorci, eds. 2012. Piero della Francesca. Trattato d’abaco. III vols. Edizione nazionale degli scritti di Piero della Francesca. Roma: Istituto poligrafico e zecca dello stato.
  36. Dalai Emiliani, Marisa, Ottavio Besomi, Carlo Maccagni, Franca Ela Consolino, Riccardo Migliari, Flavia Carderi, and Alessandra Sorci, ed. 2017. Piero della Francesca. De prospectiva pingendi. Edizione nazionale degli scritti di Piero della Francesca. Roma: Istituto poligrafico e Zecca dello Stato.
  37. Danna, Raffaele. 2019. “Una scienza per la rinascita. Note su Paolo dell’Abaco e la matematica abacistica fiorentina.” Rinascimento 59: 245-69.
  38. Danna, Raffaele. 2021. “Figuring out: The spread of Hindu-Arabic numerals in the European Tradition of practical mathematics (13th-16th centuries).” Nuncius 1: 1-44. DOI: 10.1163/18253911-bja10004
  39. Danna, Raffaele. 2022. “Elaboration and diffusion of useful knowledge in the long run: The case of European practical arithmetic (13th-16th centuries).” Rivista di Storia Economica 1: 57-84. DOI: 10.1410/104038
  40. Day, William R. 2013. “The Cistercian monk and the casting counter.” In Churchmen and urban government in late medieval Italy, c.1200-c.1450: cases and contexts, ed. Frances Andrews, 251-67. Cambridge: Cambridge University Press. DOI: 10.1017/CBO9781107360082.018
  41. De Pleijt, Alexandra M., and Jan Luiten van Zanden. 2016. “Accounting for the “Little Divergence”: What drove economic growth in pre-industrial Europe, 1300-1800?” European Review of Economic History 20, 4: 387-409. DOI: 10.1093/ereh/hew013
  42. De Roover, Raymond. 1944. “Early accounting problems of foreign exchange.” The Accounting Review 19, 4: 381-407.
  43. De Roover, Raymond. 1953. “The Commercial Revolution of the thirteenth century.” In Enterprise and secular change: Readings in economic History, ed. Frederic Chapin Lane, and Jelle C. Riemersma, 80-85. London: Allen and Unwin.
  44. De Roover, Raymond. 1956. “The development of accounting prior to Luca Pacioli According to the account books of medieval merchants.” In Studies in the history of accounting, ed. Basil S. Yamey, and Ananias Charles Littleton, 114-174,. London: Sweet & Maxwell.
  45. De Roover, Raymond. 1967. “The Scholastics, usury, and foreign exchange.” The Business History Review 41, 3: 257-71. DOI: 10.2307/3112192.
  46. De Vries, Jan. 1984. European urbanization 1500-1800. London: Methuen.
  47. Dittmar, Jeremiah E. 2011. “Information Technology and economic change: The impact of the printing press.” The Quarterly Journal of Economics 126, 3: 1133-72. DOI: 10.1093/qje/qjr035.
  48. Djebbar, Ahmed. 2003. “A panorama of research in the history of mathematics in Al-Andalus and the Maghrib between the ninth and sixteenth centuries.” In The enterprise of science in Islam: New perspectives, ed. J. P. Hogendijk, and A.I. Sabra, 309-50. Dibner Institute Studies in the History of Science and Technology. Cambridge, MA-London: MIT Press.
  49. Flegg, Graham. 1988. “Nicolas Chuquet - an Introduction.” In Mathematics from manuscript to print, 1300-1600, ed. Cynthia Hay, 59-72. Oxford: Clarendon Press.
  50. Folkerts, Menso, and Paul Kunitzsch. 1997. Die älteste lateinische Schrift über das indische Rechnen nach al-Ḫwārizmī. Abhandlungen (Bayerische Akademie der Wissenschaften. Philosophisch-Historische Klasse); n.F., Heft 113. München: Verlag der Bayerischen Akademie der Wissenschaften.
  51. Folkerts, Menso. 2001. “Early texts on Hindu-Arabic calculation.” Science in context 14, 1-2: 13-38. DOI: 10.1017/S0269889701000023
  52. Folkerts, Menso. 2003. Essays on early medieval mathematics: The latin tradition. Aldershot: Ashgate Variorum.
  53. Folkerts, Menso. 2004. “Leonardo Fibonacci’s knowledge of Euclid’s Elements and of other mathematical texts.” Bollettino di Storia delle Scienze Matematiche 24, 1: 93-113. DOI: 10.1400/17070
  54. Foster, Brian, ed. 1963. The Local Port Book of Southampton for 1435-36. Southampton Records Series; Vol. 7. Southampton: Southampton University Press.
  55. Franci, Raffaella. 2002. “Il Liber Abaci di Leonardo Fibonacci 1202-2002.” Bollettino dell’Unione Matematica Italiana, 8, 5, 2: 293-328.
  56. Gärtner, Barbara. 2000. Johannes Widmanns “Behende vnd hubsche Rechenung’: die Textsorte “Rechenbuch” in der Frühen Neuzeit. Reihe Germanistische Linguistik ; Bd. 222. Tübingen: Niemeyer.
  57. Giusti, Enrico, and Paolo D’Alessandro, ed. 2020. Leonardi Bigolli Pisani Vulgo Fibonacci Liber Abbaci. Biblioteca Di «Nuncius». Firenze: Leo S. Olschki.
  58. Giusti, Enrico, and Raffaella Petti, ed. 2002. Un ponte sul Mediterraneo: Leonardo Pisano, la scienza araba e la rinascita della matematica in Occidente. Firenze: Polistampa.
  59. Goetzmann, William N. 2003. “Fibonacci and the Financial Revolution.” SSRN Scholarly Paper ID 461740. Rochester, NY: Social Science Research Network. https://papers.ssrn.com/abstract=461740 (2023-31-01)
  60. Goldthwaite, R. A. 1972. “Schools and teachers of commercial arithmetic in Renaissance Florence.” Journal of European Economic History 1, 2: 418-33.
  61. Goldthwaite, Richard A. 2009. The economy of Renaissance Florence. Baltimore: Johns Hopkins University Press.
  62. Goldthwaite, Richard A. 2015. “The practice and culture of accounting in Renaissance Florence.” Enterprise & Society 16, 3: 611-47. DOI: 10.1017/eso.2015.17.
  63. Gransden, Antonia. 2007. A history of the Abbey of Bury St Edmunds, 1182-1256. Woodbridge: Boydell Press.
  64. Gransden, Antonia. 2015. A history of the Abbey of Bury St Edmunds, 1257-1301. Woodbridge: Boydell Press.
  65. Grendler, Paul F. 1989. Schooling in Renaissance Italy: Literacy and learning, 1300-1600. Baltimore: Johns Hopkins University Press.
  66. Hanham, Alison, ed. 1975. The Cely letters, 1472-1488. Early English text society. London: Oxford University Press for the Early English text society.
  67. Hanham, Alison. 1985. The Celys and their world: An English merchant family of the fifteenth century. Cambridge: Cambridge University Press.
  68. Holmes, George A. 1960. “Florentine merchants in England, 1346-1436.” Economic History Review 13, 2: 193-208. DOI: 10.1111/j.1468-0289.1960.tb02115.x
  69. Holmes, George A. 1961. “The ‘Libel of English Policy’.” The English Historical Review LXXVI, CCXCIX: 193-216. DOI: 10.1093/ehr/LXXVI.CCXCIX.193
  70. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.282053. (2023-31-01)
  71. Ifrah, Georges. 2000. The universal history of numbers. Translated by David Bellos. III vols. London: Harvill Press.
  72. Jenkinson, Hilary. 1926. “The use of Arabic and Roman numerals in English archives.” The Antiquaries Journal 6, 3: 263-75. DOI: 10.1017/S0003581500056481
  73. Johnson, Charles, ed. 1950. The Course of the Exchequer by Richard, Son of Nigel, Treasurer of England and Bishop of London. Translated from the Latin, with Introd. and Notes by Charles Johnson. London-New York: T. Nelson.
  74. Johnson, Francis. 1942. “Thomas Hood’s inaugural address as First Mathematical Lecturer of London (1588).” Journal of the History of Ideas 3, 1: 94-106.
  75. Johnston, Stephen. 1991. “Mathematical practitioners and instruments in Elizabethan England.” Annals of Science 48, 4: 319-44. DOI: 10.1080/00033799100200321.
  76. Kaeuper, Richard W. 1973. Bankers to the Crown: The Riccardi of Lucca and Edward I. Princeton: Princeton University Press.
  77. Karpinski, Louis C. 1921. “Two twelfth century algorisms.” Isis 3, 3: 396-413. DOI: 10.1086/357950.
  78. Kohn, Meir. 1999. “Bills of exchange and the money market to 1600.” SSRN Scholarly Paper ID 151849. Rochester, NY: Social Science Research Network. DOI: 10.2139/ssrn.151849
  79. Kool, Marjolein. 1988. “What could we learn from Master Christianus van Varenbraken?” In Mathematics from manuscript to print, 1300-1600, ed. Cynthia Hay, 147-55. Oxford: Clarendon Press.
  80. Lane, Frederic Chapin, and Reinhold C. Mueller. 1985. Money and banking in medieval and Renaissance Venice. Baltimore; London: Johns Hopkins University Press.
  81. Latham, Robert, and William Matthews, eds. 1996. Pepys’s Diary, selected and edited by Robert Latham from the Diary of Samuel Pepys, a new and complete transcription. London: Folio Society.
  82. Lee, Geoffrey A. 1977. “The coming of age of double Entry: The Giovanni Farolfi ledger of 1299-1300.” The Accounting Historians Journal 4, 2: 79-95.
  83. Long, Pamela O., David McGee, Alan M. Stahl, and Franco Rossi, ed. 2009. The Book of Michael of Rhodes: A fifteenth-century maritime manuscript. 3 vols. Cambridge, MA-London: MIT Press.
  84. Lopez, Robert S. 1976. The Commercial Revolution of the middle ages, 950-1350. Cambridge-New York: Cambridge University Press.
  85. Maher, David W., and John F. Makowski. 2001. “Literary evidence for Roman arithmetic with fractions.” Classical Philology 96, 4: 376-99. DOI: 10.1086/449557
  86. Malinowski, Mikołaj, and Jan Luiten van Zanden. 2017. “Income and Its Distribution in Preindustrial Poland.” Cliometrica 11, 3: 375-404. DOI: 10.1007/s11698-016-0154-5
  87. Mallett, Michael Edward. 1967. The Florentine galleys in the fifteenth century. With the Diary of Luca di Maso degli Albizzi, captain of the galleys 1429-1430. Oxford: Clarendon Press.
  88. Marinoni, Augusto, ed. 1973. Leonardo da Vinci. Il codice atlantico. Edizione in facsimile dopo il restauro dell’originale conservato nella Biblioteca ambrosiana di Milano. Edizione nazionale dei manoscritti e dei disegni di Leonardo da Vinci. Firenze: Giunti-Barbèra.
  89. Martines, Lauro. 1963. The social world of the Florentine humanists, 1390-1460. London: Routledge & Kegan Paul.
  90. Melis, Federigo, and Bruno Dini. 1975. Origini e sviluppi delle assicurazioni in Italia: (secoli 14-16). Roma: Istituto Nazionale delle Assicurazioni.
  91. Meskens, Ad. 2013. Practical mathematics in a commercial metropolis: Mathematical Life in late 16th century Antwerp. Dordrecht: Springer.
  92. Mokyr, Joel. 2005. “The Intellectual origins of modern economic growth.” The Journal of Economic History 65, 2: 285-351. DOI: 10.1017/S0022050705000112.
  93. Mokyr, Joel. 2011. The Enlightened economy: Britain and the Industrial Revolution 1700-1850. London: Penguin.
  94. Mokyr, Joel. 2017. A culture of growth: The Origins of the modern economy. Graz Schumpeter Lectures. New Jersey: Princeton University Press.
  95. Mueller, Reinhold C. 1997. The Venetian money market: Banks, panics, and the public debt, 1200-1500. Baltimore-London: Johns Hopkins University Press.
  96. Murano, Giovanna. 2015. ““Nella invenzione della quale molto pensai”. Libri e autografi del matematico ed astronomo fiorentino Paolo Dell’Abbaco († 1367).” Archives Internationales d’Histoire des Sciences 65, 174: 55-75.
  97. Nobes, Christopher, ed. 1984. The development of double entry: Selected essays. New York; London: Garland.
  98. Nothaft, C. Philipp E. 2014. “The reception and application of Arabic science in twelfth-century computistics: New evidence from Bavaria.” Journal for the History of Astronomy 45, 1: 35-60.
  99. Omodeo, Pietro D. 2017. “Utilitas Astronomiae in the Renaissance: The rhetoric and epistemology of astronomy.” In The Structures of practical knowledge, ed. Matteo Valleriani, 307-31. Cham, Switzerland: Springer.
  100. Otis, Jessica. 2017. ““Set them to the cyphering schoole”: Reading, writing, and arithmetical education, circa 1540-1700.” Journal of British Studies 56, 3: 453-82. DOI: 10.1017/jbr.2017.59
  101. Padgett, John F., and Paul D. McLean. 2006. “Organizational invention and elite transformation: The birth of partnership systems in Renaissance Florence.” American Journal of Sociology 111, 5: 1463-568. DOI: 10.1086/498470
  102. Padgett, John F., and Walter W. Powell. 2012. The emergence of organizations and markets. Princeton: Princeton University Press.
  103. Parker, Robert Henry, and Basil Selig Yamey, ed. 1994. Accounting History: Some British Contributions. Oxford: Clarendon Press.
  104. Patterson, Louise Diehl. 1951. “Recorde’s Cosmography, 1556.” Isis 42, 3: 208-18.
  105. Pomeranz, Kenneth. 2000. The Great Divergence: China, Europe, and the Making of the modern world economy. Princeton: Princeton University Press.
  106. Ramsay, G.D. 1973. “The undoing of the Italian mercantile colony in sixteenth century London.” In Textile history and economic history: Essays in honour of miss Julia de Lacy Mann, ed. N.B. Harte and Kenneth G. Ponting, 22-49. Manchester: Manchester University Press.
  107. Rawcliffe, Carole. 1991. “’That kindliness should be cherished more, and discord driven out’: The settlement of commercial disputes by arbitration in later medieval England.” In enterprise and individuals in fifteenth-century England, ed. Jennifer Kermode, 99-117. Stroud: Alan Sutton.
  108. Richeson, A.W. 1947. “The first arithmetic printed in English.” Isis 37, 1-2: 47-56.
  109. Roberts, Gareth, and Fenny Smith, ed. 2012. Robert Recorde: The life and times of a Tudor mathematician. Cardiff: University of Wales Press.
  110. Roberts, Gordon. 2016. Robert Recorde: Tudor scholar and mathematician. Cardiff: University of Wales Press.
  111. Romano, Ruggiero, Alberto Tenenti, and Francesco Furlan, ed. 1994. Leon Battista Alberti. I libri della famiglia. Torino: Einaudi (Nuova universale Einaudi, 102).
  112. Rose, Susan, ed. 1982. The navy of the Lancastrian Kings: Accounts and inventories of William Soper, keeper of the King’s ships, 1422-1427. London: Allen & Unwin for the Navy Records Society (Publications of the Navy Records Society, 123).
  113. Rubin, Jared. 2010. “Bills of Exchange, interest bans, and impersonal exchange in Islam and Christianity.” Explorations in Economic History 47, 2: 213-27. DOI: 10.1016/j.eeh.2009.06.003
  114. Ruddock, Alwyn A. 1946. “Alien hosting in Southampton in the fifteenth century.” The Economic History Review 16, 1: 30-37. DOI: 10.2307/2590578.
  115. Ruddock, Alwyn A. 1951. Italian merchants and shipping in Southampton 1270-1600. Southampton-Oxford: University College; Blackwell.
  116. Sapori, Armando, 1944. “La compagnia dei Frescobaldi in Inghilterra.” Archivio Storico Italiano 102, 389: 3-81.
  117. Sapori, Armando, 1955. “La Compagnia Frescobaldi in Inghilterra.” In Studi di storia economica (secoli XIII, XIV, XV), 3a ed. accresciuta, 579-646. Firenze: Sansoni.
  118. Sapori, Armando, ed. 1934. I libri di commercio dei Peruzzi. Milano: Fratelli Treves.
  119. Sapori, Armando. 1926. La crisi delle compagnie mercantili dei Bardi e dei Peruzzi. Firenze: Leo S. Olschki.
  120. Scriba, Christoph J. 1970. “The autobiography of John Wallis, F.R.S.” Notes and Records of the Royal Society of London 25, 1: 17-46.
  121. Shapin, Steven, and Simon Schaffer. 1985. Leviathan and the Air-Pump: Hobbes, Boyle, and the experimental life. Princeton: Princeton University Press.
  122. Shapin, Steven. 2010. Never pure: Historical studies of science as if it was produced by people with bodies, situated in time, space, culture, and society, and struggling for credibility and authority. Baltimore: Johns Hopkins University Press.
  123. Spufford, Peter. 1988. Money and its use in medieval Europe. Cambridge-New York: Cambridge University Press.
  124. Tanzini, Lorenzo, and Sergio Tognetti, ed. 2012. ‘Mercatura è arte’: uomini d’affari toscani in Europa e nel Mediterraneo tardomedievale. Roma: Viella.
  125. Taylor, Eva Germaine Rimington. 1954. The mathematical practitioners of Tudor & Stuart England. Cambridge: Cambridge University Press.
  126. Thick, Anne, ed. 1995. The Southampton Steward’s Book of 1492-93 and the Terrier of 1495. Stroud: Alan Sutton (Southampton Records Series, 38).
  127. Thick, Anne. 1999. “Accounting in the late medieval town: The account books of the Stewards of Southampton in the fifteenth century.” Accounting, Business & Financial History 9, 3: 265-90. DOI: 10.1080/095852099330214
  128. Thomas, Keith. 1987. “Numeracy in early modern England: The Prothero Lecture.” Transactions of the Royal Historical Society 37: 103-32. DOI: 10.2307/3679153
  129. Tognetti, Sergio. 1995. “Prezzi e salari a Firenze nel tardo medioevo: un profilo.” Archivio Storico Italiano 153: 263-333.
  130. Tognetti, Sergio. 2014. “Nuovi documenti sul fallimento della compagnia Frescobaldi in Inghilterra.” In Città e campagne del basso medioevo: studi sulla società italiana offerti dagli allievi a Giuliano Pinto, 135-58. Firenze: Leo S. Olschki.
  131. Tognetti, Sergio. 2015. “Le compagnie mercantili-bancarie toscane e i mercati finanziari europei tra metà XIII e metà XVI secolo.” Archivio Storico Italiano 173, 4, 646: 687-718.
  132. Tognetti, Sergio. 2018. “Notai e mondo degli affari nella Firenze del Trecento.” In Notariorum itinera. Notai toscani del basso Medioevo tra routine, mobilità e specializzazione, ed. Giuliano Pinto, Lorenzo Tanzini, and Sergio Tognetti, 127-61. Firenze: Leo S. Olschki.
  133. Ulivi, Elisabetta. 1996. “Per una biografia di Antonio Mazzinghi, maestro d’abaco del XIV Secolo.” Bollettino di Storia delle Scienze Matematiche 16, 1: 101-50.
  134. Ulivi, Elisabetta. 2002a. “Benedetto da Firenze (1429-1479) un maestro d’abaco del XV secolo, con documenti inediti e con un’appendice su abacisti e scuole d’abaco a Firenze nei secoli XIII-XVI.” Bollettino di Storia delle Scienze Matematiche 22, 1.
  135. Ulivi, Elisabetta. 2002b. “Scuole e maestri d’abaco in Italia tra medioevo e Rinascimento.” In Un ponte sul Mediterraneo: Leonardo Pisano, la scienza araba e la rinascita della matematica in Occidente, ed. Enrico Giusti, and Raffaella Petti, 121-59. Firenze: Polistampa.
  136. Ulivi, Elisabetta. 2004. “Maestri e scuole d’abaco a Firenze: la bottega di Santa Trinita.” Bollettino di Storia delle Scienze Matematiche 24, 1: 43-91.
  137. Ulivi, Elisabetta. 2008. “Scuole d’abaco e insegnamento della matematica.” In Il Rinascimento italiano e l’Europa, ed. G. L. Fontana and Luca Molà, 5: 403-20. Treviso-Vicenza: Angelo Colla Editore.
  138. Ulivi, Elisabetta. 2015. “Masters, questions and challenges in the abacus schools.” Archive for History of Exact Sciences 69, 6: 651-70. DOI: 10.1007/s00407-015-0160-1
  139. Van Egmond, Warren. 1977. “New light on Paolo Dell’Abbaco.” Annali dell’Istituto e Museo di Storia della Scienza di Firenze 2, 2: 3-21.
  140. Van Egmond, Warren. 1980. Practical mathematics in the Italian Renaissance: A catalog of Italian abbacus manuscripts and printed books to 1600. Istituto e Museo di Storia della Scienza. Firenze: Giunti Barbera.
  141. Van Zanden, Jan Luiten, and Bas van Leeuwen. 2012. “Persistent but not consistent: The growth of national income in Holland 1347-1807.” Explorations in Economic History 49, 2: 119-30. DOI: 10.1016/j.eeh.2011.11.002
  142. Van Zanden, Jan Luiten. 1999. “The development of agricultural productivity in Europe, 1500-1800.” In Land productivity and agro-systems in the North Sea area (middle ages - 20th century). Elements for comparison, ed. Bas J.P. Van Bavel, and Erik Thoen, 2: 357-75. Turnhout: Brepols Publishers (Comparative Rural History of the North Sea Area, 2). DOI: 10.1484/M.CORN-EB.4.00114
  143. Van Zanden, Jan Luiten. 2009. “The skill premium and the ‘Great Divergence’.” European Review of Economic History 13, 1: 121-53. DOI: 10.1017/S1361491609002408
  144. Vogel, Kurt. 1963. Mohammed ibn Musa Alchwarizmi’s Algorismus: das früheste Lehrbuch zum Rechnen mit indischen Ziffern. Aalen: Otto Zeller.
  145. Williams, Jack. 2011. Robert Recorde: Tudor polymath, expositor and practitioner of computation. London: Springer Science & Business Media.
  146. Williams, Travis D. 2012. “The earliest English printed arithmetic books.” The Library, 7, 13, 2: 164-84. DOI: 10.1093/library/13.2.164
  147. Winjum, James O. 1971. “Accounting and the rise of capitalism: An accountant’s view.” Journal of Accounting Research 9, 2: 333-50 DOI: 10.2307/2489937.
  148. Yamey, Basil S., and Ananias C. Littleton, ed. 1956. Studies in the history of accounting. London: Sweet & Maxwell.
  149. Yeldham, Florence A. 1927. “Notation of fractions in the earlier middle ages.” Archeion 8, 3: 313-29.
  150. Zilsel, Edgar. 1942. “The sociological roots of science.” American Journal of Sociology 47, 4: 544-62. DOI: 10.1086/218962
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  • Pages: 59-87
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Chapter Information

Chapter Title

The spread of Hindu-Arabic numerals among practitioners in Italy and England (13th-16th c.): two moments of a European innovation cycle?

Authors

Raffaele Danna

Language

English

DOI

10.36253/979-12-215-0092-9.06

Peer Reviewed

Publication Year

2023

Copyright Information

© 2023 Author(s)

Content License

CC BY 4.0

Metadata License

CC0 1.0

Bibliographic Information

Book Title

L’economia della conoscenza: innovazione, produttività e crescita economica nei secoli XIII-XVIII / The knowledge economy: innovation, productivity and economic growth, 13th to 18th century

Editors

Giampiero Nigro

Peer Reviewed

Number of Pages

456

Publication Year

2023

Copyright Information

© 2023 Author(s)

Content License

CC BY 4.0

Metadata License

CC0 1.0

Publisher Name

Firenze University Press

DOI

10.36253/979-12-215-0092-9

ISBN Print

979-12-215-0091-2

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979-12-215-0092-9

Series Title

Datini Studies in Economic History

Series ISSN

2975-1241

Series E-ISSN

2975-1195

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