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

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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.

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Notes

  1. This question was inspired by Munck’s recent picture of Enlightenment as plural and European-wide, and not exclusively radiating from France (Munck 2000).

  2. From the General Archives in Simancas (AGS henceforth), Cuesta Dutari inferred that Padilla was born in 1724 and that he could have died in 1807, although there is no definitive evidence on this point.

  3. On the Military Academy of Oran, see AGS Guerra Moderna 3778.

  4. This kind of agenda was not unusual in eighteenth-century Europe. It could be found, for instance, in England (Stewart 1999, pp. 136–140), Scotland (Bruneau 2011, p. 114; Grabiner 1997, p. 395) and Prussia (Schubring 2005, p. 66). On contemporary self-awareness of the scientific backwardness of Spain, see Navarro-Brotóns (2006, p. 391); Navarro-Brotóns and Eamon (2007).

  5. Although the first ordenanza concerning the Academy of Barcelona dated 1739, this was actually an already existing institution. Founded around 1700 in Madrid, it was transferred to Barcelona some years later, where Verboom sought to establish a military educational institution of high quality (Capel et al. 1988; De Mora and Massa-Esteve 2008).

  6. See Ordenanzas of July 22, 1739, and December 29, 1751 (Portugues 1765, VI, pp. 858–883, 889–925). The mathematics taught at these academies was divided in four courses (3 years). The syllabus supervised by the Academy of Barcelona contained: arithmetic (plain and literal), geometry (speculative and practical), fortification, batallions and squadrons, mechanics, artillery, cosmography, sundials, perspective, civil and military architecture, drawing and navigation. Students had to defend in public expositions the result of their education at the end of their studies. Academies included a headmaster, several teacher assistants (engineers) and a teacher of drawing.

  7. See Ordenanzas of December 21, 1750, September 22, 1751 and November 11, 1755, respectively (Portugues 1765, V, pp. 180–184, 187, 196–199). A set of new academies were founded around 1751. Due to budget cuts, most of them were deemed redundant and, consequently, closed around 1760, except the Academies of Engineers of Barcelona and of Artillery of Cádiz (Capel et al. 1988, pp. 181–182; Lafuente and Peset 1982; Hidalgo 1991).

  8. In the period 1750–1753 Padilla appears to be among the teachers of the Academy of Mathematics of Barcelona as well (Capel et al. 1988, p. 138; Carrillo de Albornoz 2008, p. 79). Oddly enough, as far as I know there is no further references to this fact.

  9. Note that the number of students in the Academy of Mathematics of the Royal Guards was about half the average number of students at the Military Academy of Mathematics of Barcelona (Lafuente and Peset 1982; Hidalgo 1991).

  10. Military Course of Mathematics, about some parts of this science, for the use of the Royal Academy established in the Military Academy of the Royal Guards.

  11. Padilla’s Curso was printed by Antonio Marín (1713–1770), a much acclaimed typographer and printer established in Madrid and closely connected with the Bourbon court. He is considered to have contributed to turn Madrid into a renowned centre for book printing (Villegas 2001).

  12. The Archive of the Military Headquarters of the Bruc in Barcelona keeps the bibliographical collection of the Academy of Mathematics of Barcelona (Segovia 2004). In particular there are two handwritten student notebooks, on Trigonometry and Practical Geometry, and Statics, respectively. A detailed analysis of such notebooks would be enlightening for a better understanding of the teaching of mathematics in eighteenth-century Spain. More case studies like the one described by De Mora and Massa-Esteve (2008) should be carried out for the purpose.

  13. All translations from Spanish and French are by the author of this paper. In the university context, this idea can be found as early as 1732 in the work of Dominique-François Rivard, Élémens de géométrie, avec un Abrégé d’arithmétique et d’algèbre.

  14. In the Archive of the Military Headquarters of the Bruc in Barcelona there are thirty-two brand new copies of a treatise on military camps from 1801, which is approximately the average number of students at that time. It was only natural to infer that the library of the Royal Guards might have purchased around fifteen copies of Padilla’s course for the students of the Academy of Mathematics of the Royal Guards.

  15. There is no current evidence as to why Padilla did not fully accomplish his Curso, but the abovementioned budget cuts could have accounted for his unfulfilled work (AGS Guerra Moderna 3011; Hidalgo 1991). Low student registration at the Academy of Mathematics might also have made it not worthwhile financially to print the whole Curso. According to the index in the first volume, the contents of the remaining treatises were: logarithms; plane and spherical trigonometry; general principles of mechanics; statics or solid mechanics; hydraulics or fluid mechanics; general principles of astronomy; geography; chronology; gnomonics; fortification and military buildings; artillery; land strategy; perspective; plans, sections and military elevations.

  16. I am referring here to the système figuré, concerning the mathematics, wherein algebra is divided into elementary algebra and infinitesimal algebra. In Gilain (2010) there is a thorough discussion on the complexity and confusion of the status of the analysis and the calculus in D’Alembert’s classification of mathematics throughout the Encyclopédie.

  17. It is true, though, that Padilla’s views differed from D’Alembert’s in that military architecture was found under Geometry (within pure mathematics) in D’Alembert’s classification, whereas Padilla placed it in mixed mathematics (Padilla 1753–1756, preface, §30). On the meaning of “mixed mathematics”, see Brown (1991) and Puig-Pla (2002). On “useful science” in eighteenth-century Spain, see Sánchez-Blanco Parody (1991, pp. 65-ff).

  18. On the reception of the Encyclopédie in Spain, see Sánchez-Blanco Parody (1991, p. 82).

  19. On the history of the introduction of calculus in Spain, see Ausejo and Medrano-Sánchez (2010), Cobos and Fernández-Daza (1991) and Cuesta Dutari (1985).

  20. For a review on learned societies in eighteenth-century Spain, see Garma (2002).

  21. Conclusiones Mathematicas, sobre los tratados de Arithmetica, Geometrìa Elementàr, Trigonometrìa, Geometrìa Práctica, Algebra, Geometrìa Sublime, y Calculos Diferencial, è Integràl. Defendidas en el Quartel de Guardias de Corps de Madrid. Madrid: Antonio Marín, 1752 (AGS Guerra Moderna, 3778). The title of these examinations refers to a treatise on trigonometry, which is not included among the printed treatises of Padilla’s Curso. Here I can speculate on the existence of a number of handwritten treatises, used by Padilla in his teaching, which were eventually not printed.

  22. It is worth mentioning that similar public examinations on calculus were not held in the French military context until the 1780s (Hahn 1986a, p. 534).

  23. See Garcia Hourcade and Valles Garrido (1989, p. 186). Regarding its use for private study, evidences are but rare and a survey of these would shed some light on the circulation of Padilla’s work, however modest.

  24. A lot has been written on Bézout’s career and on his influence on the French educational system. See for instance Alfonsi (2011a, b), Hahn (1986a, b) and Schubring (2005, pp. 122–126, 217–220).

  25. For a thorough account of the écoles militaires in eighteenth-century France, see Alfonsi (2011a, pp. 31–33) and Hahn (1986a) for artillery, Alfonsi (2011a, pp. 33–35) and Hahn (1986b) for navy, and Taton (1986) for military engineers at the École de Mézières. See also Schubring (2005, pp. 121–126).

  26. For the three naval schools there was no entry exam before 1786 (Hahn 1986b, p. 551).

  27. Blanco (2007) includes a comparative analysis of the artillery educational systems in eighteenth-century France and Prussia, based on the works on the calculus written by Bézout and Tempelhoff.

  28. For further details on the production of the artillery course, see Alfonsi (2011a, pp. 225–234).

  29. Alfonsi (2011a, pp. 236–238) indicates that the number of students enrolled in the naval schools was about 320. In addition, one could estimate that a similar number of students were using Bézout’s Cours each year in the context of artillery.

  30. Several authors, such as Peyrard, Reynaud, Lacroix and Garnier, published modified editions of the entire Cours, or parts of it, after Bézout’s death. There is a detailed list of all these editions in Alfonsi (2011a, pp. 342–345).

  31. For a thorough account of the translations of Bézout’s Cours de mathématiques see Alfonsi (2011a, pp. 345–348).

  32. Faria (1774) was reprinted in 1793, 1801, 1818, 1825 and 1827. For Bails see main text below.

  33. Bézout’s presentation of the foundations of calculus, the same for both versions, is discussed in Schubring (2005, pp. 218–220). It is beyond the scope of this paper to provide a thorough account of the introduction and circulation of the calculus in France, given the amount of outstanding studies devoted to the subject, like Grattan-Guinness (1990), Mancosu (1989), Robinet (1960) or Schubring (2005), just to mention a few.

  34. In Cuesta Dutari (1985, pp. 124–127) there is a description of the likely sources upon which Padilla relied to elaborate his 4th and 5th treatises.

  35. L’Hospital’s Analyse des infiniment petits (1696) has been traditionally regarded as the first systematic educational work on differential calculus. It was certainly based upon Johann Bernoulli’s work on the topic, one of the most salient supporters of the Leibnizian calculus. Yet, from an educational point of view, L’Hospital’s was the first attempt to popularize the teaching of calculus.

  36. See an instance in Juan and Ulloa (1748, pp. 85–86).

  37. In the late 1620s Pierre de Fermat (1601–1665) produced his work on the rule for maxima and minima, published later on as Methodus ad disquirendam maximam et minima, followed by De tangentibus linearum curvarum (1679). It is also worth reminding that, in their seminal works on the calculus, both Newton and Leibniz applied it to find maxima and minima to show its usefulness.

  38. As a consequence of the publication in 1734 of Berkeley’s The Analyst, Maclaurin published his two-volume A Treatise of Fluxions (1742). A key figure of Scottish Enlightenment, Maclaurin’s goal was to defend Newton’s calculus against Berkeley’s attack, and his treatise was an attempt to provide Newton’s calculus with systematic rigorous foundations. In his first book Maclaurin relied largely on Greek geometry and kinematic approach to fulfil his aim. But the second part of the second book is devoted to physico-mathematical applications, in which he used the algorithms of calculus. In A Philosophical and Mathematical Dictionary (1815) Hutton refers to Maclaurin’s treatise as the most complete treatise on fluxionary method up to that moment. See Grabiner (1997, 2002, 2004), Bruneau (2011). On Scottish Enlightenment see Broadie (1997).

  39. On the relevance of Maclaurin regarding the determination of maxima and minima see Grabiner (1997), Katz (2009, pp. 614 and 624).

  40. In the context of French universities, one of the first textbooks to introduce the study of calculus was the Leçons élémentaires des mathématiques by the Abbé Nicolas Louis de La Caille (1713–1762). Originally published in 1741, it contained merely algebra and geometry. Yet, from 1756 onwards the textbook incorporated a section on differential and integral calculus. In particular, in §824 La Caille dealt with the characterisation of maxima and minima, albeit not using the series expansion as Maclaurin did. On La Caille’s approach to calculus, see Schubring (2005, pp. 213–214).

  41. The teaching of mathematics in seventeenth- and eighteenth-century Spain was markedly Jesuit. Upon the expulsion of the Jesuits in 1767, their educational institutions were converted into civil or military institutions (Navarro-Brotóns 2002, 2006). On the subject of the transmission of Newtonianism by the Jesuits, see Grabiner (1997), Guicciardini (1989) and Russell (1991). In particular, on the role of the Jesuits in the circulation of the method of fluxions in Spain see Ausejo and Medrano-Sánchez (2010). For a study of Pezenas, his works and his role in the diffusion of British science in eighteenth-century France, see Boistel (2003, 2010).

  42. On Bails and his Elementos see Ausejo and Medrano-Sánchez (2010, pp. 28–33) and Garma (2002, pp. 336–339). In contrast with Padilla’s Curso, Bails’s work was reprinted several times. It would be enlightning to compare the printing and publishing matters regarding Padilla’s Curso with those regarding Bails’s Elementos.

  43. To my knowledge, this is the only occasion when Bézout refers to a “function” in the part on differential calculus. A rather nominal use of the term, he nevertheless proceeded to use “quantities” and “variable quantities”, as it has been discussed above.

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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).

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  1. Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, ESAB, Campus del Baix Llobregat, Esteve Terradas 8, 08860, Castelldefels (Barcelona), Spain

    Mónica Blanco

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Blanco, M. The Mathematical Courses of Pedro Padilla and Étienne Bézout: Teaching Calculus in Eighteenth-Century Spain and France. Sci & Educ 22, 769–788 (2013). https://doi.org/10.1007/s11191-012-9537-6

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