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Science & Education

, Volume 22, Issue 4, pp 769–788 | Cite as

The Mathematical Courses of Pedro Padilla and Étienne Bézout: Teaching Calculus in Eighteenth-Century Spain and France

  • Mónica Blanco
Article

Abstract

The aim of this paper is to provide a cross-national comparative analysis of the introduction of calculus in Spanish and French military educational institutions through the works of Pedro Padilla y Arcos (1724–1807?) and Étienne Bézout (1730–1783), respectively. Both authors developed their educational work in the context of military schools and academies. Padilla’s Curso Militar de Mathematicas (1753–1756) was the first work published in Spain which introduced the teaching of calculus in formal education. Bézout’s Cours de Mathématiques (1764–1769) was the first work on calculus explicitly addressed to French military students and can be considered a representative of the canonical knowledge on eighteenth-century mathematics, both in France and abroad. Eighteenth-century Spain has traditionally been regarded as a country in the periphery whose scientific culture and education were pervaded by French science and education. This centre-periphery framework is often represented by a static model of one-way transmission from the centre to the periphery. A crossnational comparative analysis can help revisit this monolithic centre-periphery framework. A recent historiographical stream places the emphasis on appropriation, hence moving away from the idea of passive reception. In my paper I focus on the reading and writing of educational books, as practices which contribute actively to the development and circulation of knowledge. To assist the analysis, I explore the differences in communication practices in each case, in contents and approaches, and in particular, I give special attention to their inspiration in mathematical streams other than the French standpoint.

Keywords

Differential Calculus Integral Calculus Military Academy Optional Issue Educational Work 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

I am especially indebted to Josep Simon for his comments on drafts of the manuscript and for his encouragement throughout the period of research. I would like to thank Ana Cardoso de Matos, Antónia Conde, M. Paula Diogo, M. Rosa Massa-Esteve, Carles Puig-Pla and Antoni Roca-Rosell for stimulating discussions on the subject of my study. This paper was written with the support of the Spanish Ministry of Science and Innovation (HP2008-012 and HAR2010-17461).

References

  1. Alfonsi, L. (2011a). Étienne Bézout (17301783). Mathématicien des Lumières. Paris: L’Harmattan.Google Scholar
  2. Alfonsi, L. (2011b). L’enseignement scientifique et technique au XVIIIe siècle dans les écoles des Gardes de la Marine: le rôle essentiel d’Étienne Bézout (1730–1783). In R. d’Enfert & V. Fonteneau (Eds.), Espaces de l’enseignement scientifique et technique. Acteurs, savoirs, institutions, XVIIe-XXe siècles. Paris: Hermann.Google Scholar
  3. Archivo General de Simancas (AGS). Guerra Moderna, files 3011 and 3778.Google Scholar
  4. Ausejo, E., & Medrano-Sánchez, F. J. (2010). Construyendo la modernidad: nuevos datos y enfoques sobre la introducción del cálculo infinitesimal en España (1717–1787). Llull, 33(71), 25–56.Google Scholar
  5. Bails, B. (1772–1783). Elementos de Matemáticas, in 10 volumes. Madrid: Joaquín Ibarra.Google Scholar
  6. Bertomeu Sánchez, J. R., García Belmar, A., Lundgren, A., & Patiniotis, M. (2006). Introduction: Scientific and technological textbooks in the European periphery. Science & Education, 15(7–8), 657–665.CrossRefGoogle Scholar
  7. Bézout, É. (1764–1769). Cours de mathématiques à l’usage des Gardes du Pavillon et de la Marine, in 6 volumes. Paris: Musier.Google Scholar
  8. Bézout, É. (1770–1772). Cours de mathématiques à l’usage du Corps Royal de l’Artillerie, in 4 volumes. Paris: Imprimerie Royale.Google Scholar
  9. Blanco, M. (2007). Análisis comparativo de la comunicación del cálculo diferencial en el siglo XVIII: la educación militar en Francia y Prusia. Llull, 30(66), 213–229.Google Scholar
  10. Boistel, G. (2003). Inventaire chronologique des œuvres imprimées et manuscrites du père Esprit Pezenas (1692–1776), jésuite, astronome et hydrographe marseillais. Revue d’Histoire des Sciences, 56(1), 221–245.Google Scholar
  11. Boistel, G. (2010). Esprit Pezenas (1692–1776), jésuite, astronome et traducteur: un acteur méconnu de la diffusion de la science anglaise en France au XVIIIe siècle. In R. Fox & B. Joly (Eds.), Échanges franco-britanniques entre savants depuis le XVIIe siècle; Franco-British interactions in science since the seventeenth century. Cahiers de logique et d’épistémologie, 7. Oxford and Lille-3: College Publications.Google Scholar
  12. Broadie, A. (Ed.). (1997). The scottish enlightenment: An anthology. Edinburgh: Canongate.Google Scholar
  13. Brown, G. I. (1991). The evolution of the term «mixed mathematics». Journal of the History of Ideas, 52(1), 81–102.CrossRefGoogle Scholar
  14. Bruneau, O. (2011). Colin Maclaurin ou l’obstination mathématicienne d’un newtonien. Nancy: Presses Universitaires de Nancy.Google Scholar
  15. Cañizares-Esguerra, J. (2004). Iberian science in the renaissance: Ignored how much longer? Perspectives on Science, 12(1), 86–124.CrossRefGoogle Scholar
  16. Capel, H., Sánchez, J. E., & Moncada, O. (1988). De Palas a Minerva. La formación científica y la estructura institucional de los ingenieros militares en el siglo XVIII. Barcelona: CSIC, Ediciones El Serbal.Google Scholar
  17. Carrillo de Albornoz, J. (2008). Ingenieros ilustres del XVIII. Parte II. El Memorial del Arma de Ingenieros (Vol. 80, pp. 69–104). Madrid: Ministerio de Defensa. Available at https://www.060.es. Accessed 12 September 2011.
  18. Cobos, J. M., & Fernández-Daza, C. (1991). El cálculo infinitesimal en los ilustrados españoles: Francisco de Villalpando y Juan Justo García. Cáceres: Servicio de publicaciones de la Universidad de Extremadura.Google Scholar
  19. Cohen, D., & O’Connor, M. (2004). Comparison and history: Europe in cos-national perspective. New York and London: Routledge.CrossRefGoogle Scholar
  20. Cuerpo de Ingenieros del Ejército. (1911). Estudio histórico del Cuerpo de ingenieros del ejército, iniciado al celebrar en 1903 el primer centenario de la creación de su Academia y de sus tropas…., in 2 volumes. Madrid: Estab. Tip. “Sucesores de Rivadeneyra”.Google Scholar
  21. Cuesta Dutari, N. (1985). Historia de la invención del análisis infinitesimal y de su introducción en España. Salamanca: Universidad de Salamanca.Google Scholar
  22. de L’Hospital, G. F. A. (1696). Analyse des infiniment petits pour l’intelligence des lignes courbes. Paris: Imprimerie Royale (reprinted by ACL-Éditions, Paris, 1988).Google Scholar
  23. De Mora, M., & Massa-Esteve, M. R. (2008). On Pedro de Lucuce’s mathematical course: sources and influences. In H. Hunger, F. Seebacher, & G. Holzer (Eds.), Proceedings of the third international conference of the European society for the history of science (pp. 869–878). Vienna: Austrian Academy of Sciences.Google Scholar
  24. D’Alembert, J. le R., & Diderot, D. (1751–1772). In R. Morrissey (Ed.), Encyclopédie, ou dictionnaire raisonné des sciences, des arts et des métiers, par une société de gens de lettres. University of Chicago: ARTFL Encyclopédie Project (Spring 2011 Edition). http://encyclopedie.uchicago.edu/.
  25. Euler, L. (1748). Introductio in analysin infinitorum, in 2 volumes. Lausanne. English version by J. D. Blanton (1988), Introduction to Analysis of the Infinite. New York: Springer.Google Scholar
  26. Faria, J. J. (1774). Elementos de Analyse Mathematica de Mr Bézout. Coimbra: Imprensa da Universidade.Google Scholar
  27. Farrar, J. (1824). First principles of the differential and integral calculus, or, The doctrines of fluxions: Taken chiefly from the mathematics of Bézout. And translated from the French for the use of the students of the university at Cambridge. New England: Printed by Hilliard and Metcalf (second edition, 1836).Google Scholar
  28. Feigenbaum, L. (1985). Brook Taylor and the Method of Increments. Archive for History of Exact Sciences, 34(1–2), 1–140.CrossRefGoogle Scholar
  29. Folkes, M., Stanhope, C., Elliott, J., Robins, B., & Watson, W. (1749). Certificate of election and candidature of Jorge Juan. Available at http://www2.royalsociety.org/DServe/dserve.exe?dsqIni=Dserve.ini&dsqApp=Archive&dsqDb=Catalog&dsqSearch=RefNo==‘EC%2F1749%2F14’&dsqCmd=Show.tcl. Accessed 25 January 2011.
  30. Galland, M. (2005). Los ingenieros militares españoles en el siglo XVIII. In A. Cámara (Ed.), Los ingenieros militares de la monarquía hispánica en los siglos XVII y XVIII. Madrid: Ministerio de Defensa- Asociación Española de Amigos de los Castillos- Centro de Estudios Europa Hispánica.Google Scholar
  31. García Hourcade, J. L., & Valles Garrido, J. M. (1989). Catálogo de la biblioteca diociochesca del Real Colegio de Artillería de Segovia. Fondos de los siglos XVI, XVII y XVIII hasta 1808. Vol. I, Libros científicos. Segovia: Academia de Artillería de Segovia.Google Scholar
  32. Garma, S. (2002). La enseñanza de las matemáticas. In J. L. Peset Reig (Ed.), Historia de la ciencia y de la técnica en la Corona de Castilla (Vol. IV). Junta de Castilla y León.Google Scholar
  33. Gilain, C. (2010). La place de l’analyse dans la classification des mathématiques: de l’Encyclopédie à la Méthodique. Recherches sur Diderot et sur l’Encyclopédie, 45, 109–128.CrossRefGoogle Scholar
  34. Grabiner, J. V. (1997). Was Newton’s calculus a dead end? The continental influence of maclaurin’s treatise of fluxions. American Mathematical Monthly, 104(5), 393–410.CrossRefGoogle Scholar
  35. Grabiner, J. V. (2002). Maclaurin and Newton: The newtonian style and the authority of mathematics. In C. W. J. Withers & P. Wood (Eds.), Science and medicine in the scottish enlightenment. East Linton: Tuckwell Press.Google Scholar
  36. Grabiner, J. V. (2004). Newton, Maclaurin and the authority of mathematics. American Mathematical Monthly, 111(10), 841–852.CrossRefGoogle Scholar
  37. Grattan-Guinness, I. (1990). Convolutions in French mathematics, in 3 volumes. Basel: Birkhäuser.Google Scholar
  38. Grattan-Guinness, I. (1994). France. In I. Grattan-Guinness (Ed.), Companion encyclopedia of the history and philosophy of the mathematical sciences. London: Routledge.Google Scholar
  39. Guicciardini, N. (1989). The development of Newtonian calculus in Britain 1700–1800. Cambridge: Cambridge Universtiy Press.CrossRefGoogle Scholar
  40. Guillem-Llobat, X. (2008). Science in the periphery. In J. Simon, N. Herran, T. Lanuza-Navarro, P. Ruiz-Castell, & X. Guillem-Llobat (Eds.), Beyond borders: Fresh perspectives in history of science. Cambridge: Cambridge Scholars Press.Google Scholar
  41. Hahn, R. (1986a). L’enseignement scientifique aux écoles militaires et d’artillerie. In R. Taton (Ed.), Enseignement et diffusion des sciences en France au dix-huitième siècle. Paris: Hermann.Google Scholar
  42. Hahn, R. (1986b). L’enseignement scientifique des gardes de la marine au XVIIIe siècle. In R. Taton (Ed.), Enseignement et diffusion des sciences en France au dix-huitième siècle. Paris: Hermann.Google Scholar
  43. Hidalgo, E. (1991). El Aula de Matemáticas de los Guardias de Corps (1750–1761). In M. Valera & C. López Fernández (Eds.), Actas del V Congreso de la Sociedad Española de Historia de las Ciencias y de las Técnicas (Vol. II). Murcia: DM-PPU.Google Scholar
  44. Juan, J., & Ulloa, A. (1748). Observaciones astronómicas y físicas hechas en los reinos del Perú. Madrid: Juan de Zúñiga.Google Scholar
  45. Kaiser, D. (2005). Pedagogy and the practice of science: Historical and contemporary perspectives. Cambridge, MA: MIT Press.Google Scholar
  46. Katz, V. (2009). A history of mathematics: An introduction (3rd edn). Boston: Addison-Wesley.Google Scholar
  47. Lafuente, A., & Peset, J. L. (1982). Las academias militares y la inversión en ciencia en la España ilustrada (1750–1760). Dynamis, 2, 193–209.Google Scholar
  48. Maclaurin, C. (1742). A Treatise of fluxions (in two books). Printed by T. W. and T. Edinburgh: Ruddimans. French version by Esprit Pezenas (1749), Traité des fluxions. Paris: Jombert.Google Scholar
  49. Mancosu, P. (1989). The metaphysics of the calculus: A foundational debate in the Paris Academy of Sciences, 1700–1706. Historia Mathematica, 16, 224–248.CrossRefGoogle Scholar
  50. Munck, T. (2000). The enlightenment. In T. Munck (Ed.), The enlightenment. A comparative social history 17211794. London: Oxford University Press.Google Scholar
  51. Navarro-Brotóns, V. (2002). Tradition and scientific change in modern Spain: The role of the jesuits. In M. Feingold (Ed.), Jesuit science and the republic of letters. Cambridge, MA/London: The MIT Press.Google Scholar
  52. Navarro-Brotóns, V. (2006). Science and enlightenment in eighteenth-century Spain: The contribution of the Jesuits before and after the expulsion. In J. W. O’Malley, G. A. Bailey, S. J. Harris, & T. F. Kennedy (Eds.), The jesuits II: Cultures, sciences and the arts, 15401773. Toronto: University of Toronto Press.Google Scholar
  53. Navarro-Brotóns, V., & Eamon, W. (2007). Spain and the scientific revolution: Historiographical questions and conjectures. In V. Navarro-Brotóns & W. Eamon (Eds.), Más allá de la Leyenda Negra: España y la Revolución Científica/beyond the black legend: Spain and the scientific revolution. Valencia: Instituto de Historia de la Ciencia y Documentación López Piñero.Google Scholar
  54. Nieto-Galan, A. (2008). The sistory of science in Spain. A critical overview. Nuncius, XXIII, 2, 211–236.Google Scholar
  55. Olesko, K. M. (1991). Physics as a calling: Discipline and practice in the Königsberg Seminar for Physics. Ithaca and London: Cornell University Press.Google Scholar
  56. Olesko, K. M. (2006). Science pedagogy as a aategory of historical analysis: Past, present and future. Science & Education, 15, 863–880.CrossRefGoogle Scholar
  57. Padilla y Arcos, P. (1753–1756). Curso militar de mathematicas, sobre partes de esta ciencia, para uso de la Real Academia establecida en el Cuartel de Guardias de Corps, in 4 volumes. Madrid: Antonio Marín.Google Scholar
  58. Portugues, J. A. (1765). Colección General de las ordenanzas militares, sus innovaciones, y Aditamentos, dispuesta en diez tomos y con separación de clases, Vols. V-VI. Madrid: Imprenta de Antonio Marín.Google Scholar
  59. Puig-Pla, C. (2002). Sobre el significat del concepte matemàtiques: Matemàtiques pures i mixtes en els segles XVIII i XIX. In J. Batlló Ortiz, P. Bernat López, & R. Puig Aguilar (Eds.), Actes de la VI Trobada d’Història de la Ciència i de la Tècnica. Barcelona: Societat Catalana d’Història de la Ciència i de la Tècnica.Google Scholar
  60. Roberts, L. (2005). Circulation promises and challenges. In The circulation of knowledge and practices: The low countries as an historical laboratory. Workshop, Woudschoten, 27–28 May. Available at http://www.gewina.nl/werkgroep/. Accessed 25 January 2011.
  61. Robinet, A. (1960). Le groupe malebranchiste introducteur du Calcul infinitésimal en France. Revue d’Histoire des Sciences et de leurs applications, XIII, n., 2, 287–308.CrossRefGoogle Scholar
  62. Russell, C. A. (1991). The reception of Newtonianism in Europe. In D. Goodman & C. A. Russell (Eds.), The rise of scientific Europe (pp. 1500–1800). Kent: The Open University.Google Scholar
  63. Sánchez-Blanco Parody, F. (1991). Europa y el pensamiento español del siglo XVIII. Madrid: Alianza Editorial.Google Scholar
  64. Schubring, G. (1996). Changing cultural and epistemological views on mathematics and different institutional contexts in nineteenth-century Europe. In C. Goldstein et al. (Eds.), Mathematical Europe. Myth, history, identity. Paris: Éditions de la Maison des sciences de l’homme.Google Scholar
  65. Schubring, G. (2005). Conflicts between generalization, rigor, and intuition. New York: Springer.Google Scholar
  66. Segovia, F. (2004). Los fondos bibliográficos de la Academia de Matemáticas. In J. M. Muñoz Corbalán (Ed.), La Real Academia de Matemáticas de Barcelona, el legado de los Ingenieros Militares. Madrid: Ministerio de Defensa.Google Scholar
  67. Sellés, M. (2002). Rodear continentes y surcar mares. In J. L. Peset Reig (Ed.), Historia de la ciencia y de la técnica en la Corona de Castilla, Vol. IV. Junta de Castilla y León.Google Scholar
  68. Simon, J. (2009). Circumventing the ‘elusive quarries’ of popular science: the communication and appropriation of Ganot’s physics in nineteenth-century Britain. In F. Papanelopoulou, A. Nieto-Galan, & E. Perdiguero (Eds.), Popularising science and technology in the European periphery (pp. 1800–2000). Aldershot: Ashgate.Google Scholar
  69. Simon, J., & Herran, N. (2008). Introduction. In J. Simon, N. Herran, T. Lanuza-Navarro, P. Ruiz-Castell, & X. Guillem-Llobat (Eds.), Beyond borders: Fresh perspectives in history of science. Cambridge: Cambridge Scholars Press.Google Scholar
  70. Stewart, L. (1999). Other centres of calculation, or, where the royal society didn’t count: commerce, coffee-houses and natural philosophy in early modern London. The British Journal for the History of Science, 32, 133–153.CrossRefGoogle Scholar
  71. Taton, R. (1986). L’École Royale du Génie de Mézières. In R. Taton (Ed.), Enseignement et diffusion des sciences en France au dix-huitième siècle. Paris: Hermann.Google Scholar
  72. Taylor, B. (1715). Methodus directa et inversa. London: Innys.Google Scholar
  73. Villegas, M. (2001). J. Ibarra, el grabado y las artes impresoras en el Madrid del siglo XVII (doctoral dissertation). Madrid: E-prints Universidad Complutense de Madrid.Google Scholar
  74. Warwick, A. (2003). Masters of theory: Cambridge and the rise of mathematical physics. Chicago: Chicago University Press.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IIIUniversitat Politècnica de CatalunyaCastelldefels (Barcelona)Spain

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