We are grateful to M. R. Massa-Esteve and A. Roca-Rosell for their support and helpful suggestions. We also thank the Spanish Ministry of Science and Innovation for financial support (HAR 2016-75871-R).
1In 1711, the first Bourbon King of Spain, Philip V (1683-1746) approved the establishment of the Spanish Corps of Military Engineers. This corps was initially conceived by the Engineer General Jorge Próspero de Verboom (1667-1744) to promote the development of scientific activity for the purpose of modernizing Spain supporting the country’s defense and protecting its trade activities [Galland Seguela 2008]. The military engineer was to be a specialist in scientific matters, military engineering, drawing, architecture and mathematics. This new professional could potentially be granted a double promotion, both in the military rank and in their engineering career.
- 1 In general, the teaching of mathematics at the Royal Military Academy of Mathematics of Barcelona (...)
2As a direct consequence, the Royal Military Academy of Mathematics of Barcelona was founded in 1720 to provide scientific knowledge for military and technical training.1 The Military Academy of Mathematics of Barcelona was ruled by royal ordinances [ordenanzas reales] that established the topics to be taught each academic year, outlined how they should be taught and designated the staff overseeing coursework. Two additional military academies were founded in Ceuta (1732) and Oran (1739) along the North coast of Africa, which followed the model of regulations established by what was considered the lead academy in Barcelona.
- 2 On Pedro de Lucuce and his works, see [Ceballos, Núñez et al. 2013].
3Pedro de Lucuce y Ponce (1692–1779) was appointed headmaster of the Military Academy of Mathematics of Barcelona for the periods 1738-1756 and 1760-1779.2 During the time in between, he acted as director of the Royal Military Society of Mathematics, as we will discuss later. In 1739 a royal ordinance established regulations with the view of developing a general course program for military training. The headmaster in Barcelona chose the most useful treatises for each subject to then produce notebooks, which were later dictated by the assistants and copied down by the students, not only in Barcelona, but also in the academies of Ceuta and Oran. Following this ordinance, in 1739 Lucuce began working on his mathematical course for military training [Curso Mathematico para la Instrucción de los Militares]. This course, completed in 1744, consisted of eight treatises on the main fields of Mathematics, including pure mathematics (arithmetic and geometry) and mixed mathematics (cosmography, statics, hydraulics, architecture, artillery, and fortification).
- 3 For a thorough account of such notebooks and, in general, of the mathematical aspects of Lucuce’s (...)
- 4 Between 1751 and 1756 the Marquis of Ensenada had planned to allocate more than 55% of the State b (...)
4While it would never be published, Lucuce’s course was used in the context of the Academy of Barcelona for over forty years and was effectively preserved in the students’ notebooks of the time.3 It is for this reason that in the present work, we consider Lucuce’s course work as a landmark of unity, albeit one that could theoretically be regarded somewhat superficial given the fact that a number of reformist initiatives emerged that were largely supported by Zenón de Somodevilla (1702–1781), the 1st Marquis of Ensenada. First as Minister of Finance, War, Navy and the Indies (1743-1746), chosen by Philip V, and then as Chief Minister (1746-1754) under the kingdom of Ferdinand VI (1713-1759), the Marquis of Ensenada envisaged launching the ambitious project of overhauling and modernizing training programs within both the Army and the Navy.4 As part of his reforms in the 1750’s, several new military academies for scientific and technical training were established, among which are the following two cases that are the focus of this study. The first is the Academy of Mathematics that was founded in 1750 as a part of the Military Academy of Royal Guards (Cuartel de Guardias de Corps). This Academy was governed by the same regulations as the Military Academy of Mathematics of Barcelona until its closure in 1760. During the period from 1753-1756 Pedro Padilla y Arcos (1724-1807?), the Academy’s headmaster until it was dissolved, published a military course of mathematics for the specific use by this Academy. The need for mathematical texts in Spanish for the purpose of military training led to the creation in 1756 of the second case studied in this paper: the Royal Military Society of Mathematics (1757-1760) that was directed by Pedro de Lucuce.
5The aim of this paper is to explore how the study of mathematics was approached in these two transient cases of military training contexts, to compare how each institution integrated Pedro Lucuce’s course into their mathematics curriculum and to identify specific aspects of unity and disunity.
- 5 On the creation and organization of the Academy of Mathematics of the Royal Guards, see [Blanco 20 (...)
6In 1717 Philip V established the Military Academy of the Royal Guards of Madrid, mirroring the French garde du corps du roi. Intended mainly for noblemen, it was an elitist institution where all members held the rank of officers and benefitted from substantial privileges.5 Towards the end of 1750, an Academy of Mathematics was created within the Academy of the Royal Guards under the patronage of the Marquis of Ensenada. Chief Minister at the time, Ensenada undertook several initiatives to improve and standardize the level of scientific knowledge integrated into military training. Among other projects, the Marquis of Ensenada encouraged the production of educational works in Spanish that would raise the level of education in a more uniform manner across all military academies [Capel, Sánchez et al. 1988, 161].
- 6 It is worth mentioning that, upon the closure of the Academy of Mathematics of the Royal Guards, p (...)
- 7 In fact, a number of new academies were created around 1751, which, due to budget cuts, were deeme (...)
7Despite being a rather small academy (with around fifteen students per year, one headmaster and two assistants), the financial budget of the Academy of Mathematics was much higher than that granted to other larger academies, a fact made evident by its extensive library.6 Closure of the Academy in 1760 was most likely due to budget cuts.7
- 8 On Pedro Padilla, see [Capel, Sánchez et al. 1988], [Cuesta Dutari 1985, 136–137], [Garma 2002].
8Up until that point, the Academy was under the direction of Pedro Padilla,8 who in 1753 began publishing his Curso Militar de Mathematicas, sobre partes de esta ciencia, para uso de la Real Academia establecida en el Cuartel de Guardias de Corps [Military Course of Mathematics, which in part addresses the mathematical sciences, for use by the Royal Academy established as a part of the Military Academy of the Royal Guards], and dedicated to the King Ferdinand VI (Figure 1). Padilla’s Curso can be regarded as an outcome of the reforms carried out at the time by the Marquis of Ensenada.
Figure 1: Title page of the Military Course of Mathematics [Padilla 1753-1756].
![Figure 1: Title page of the Military Course of Mathematics [Padilla 1753-1756].](docannexe/image/2185/img-1.jpg)
Source: Biblioteca Digital Hispánica (Biblioteca Nacional de España).
9The preface of his first volume shows that Padilla’s primary aim was to show that understanding the basic principles of each branch of mathematics could be useful to not only infantry and cavalry regiments, but could serve engineers, artillery and navy personnel [Padilla 1753-1756, Preface].
- 9 All the translations are ours unless otherwise indicated.
10Through his work, Padilla introduced what represented a pivotal change in the pedagogical methods thus far [Blanco 2013, 772]. As mentioned above, Lucuce’s course was dictated by assistants and copied down by students at the Military Academy of Mathematics of Barcelona, hence following the royal ordinances [ordenanzas reales]. By making printed versions of his Curso available, Padilla aimed to relieve students “from the annoyance of writing, incompatible with their daily duties”, so that they could “make greater progress in the studies” [Padilla 1753-1756, Dedication to the King].9
- 10 Although the first treatise was reprinted in 1807, the project of reprinting the remaining volumes (...)
11Table 1 displays the contents of the treatises of Padilla’s course, according to the index in the first volume. Of the twenty mathematical treatises that Padilla originally intended to develop, only the first five would be published in the end (in four volumes).10
| Part I |
Arithmetic and Geometry |
|
1. Ordinary arithmetic; 2. Elementary (Euclidean) geometry; 3. Elementary algebra; 4. Higher geometry, or geometry of curves; 5. Differential and integral calculus, or the method of fluxions; 6. Logarithms; 7. Plane trigonometry; 8. Spherical trigonometry |
| Part II |
Mechanics |
|
9. General principles of mechanics; 10. Statics or solid mechanics; 11. Hydraulics or fluid mechanics |
| Part III |
Sphere |
|
12. General principles of astronomy; 13. Geography; 14. Chronology; 15. Gnomonics |
| Part IV |
War |
|
16. Fortification and military buildings; 17. Artillery; 18. Tactics |
| Part V |
Military Drawing |
|
19. Perspective; 20. Plans, sections and military elevations |
- 11 On the division of mathematics, see [Blanco 2013, 774], [Blanco & Puig-Pla 2014], [Puig-Pla 2002].
12His approach to the general division of mathematics, which he discusses at length in the preface, appears to be a direct echo of D’Alembert’s système figuré in the Discours Préliminaire of the Encyclopédie [D’Alembert & Diderot 1751, I]. Hence, for instance, according to Padilla, algebra could be divided into elementary algebra and infinitesimal algebra, the latter in turn being divided into differential calculus and integral calculus [Padilla 1753-1756, Preface, § 21], as displayed in the système figuré.11
- 12 See [Blanco 2013] for a discussion about Padilla’s views on pedagogy.
13Fully aware of the elementary nature of his course, Padilla encouraged those readers willing to progress in the study of mathematics to complete and improve his Curso [Padilla 1753-1756, V, § 86]. The Curso was regarded as auxiliary to the teaching program, which placed greater emphasis on lectures and discussions conducted in the classroom [Padilla 1753-1756, Preface, § 17].12 This approach contrasts radically with the teaching regulations in the military academies in Barcelona, Ceuta and Oran where teaching assistants were obliged to rely exclusively on the notebooks provided and developed by the headmaster, and to not deviate from their content without the headmaster’s approval [Portugues 1765, VI, 907–911]. Contrary to these regulations, Padilla’s work was presented more as a guide, an introduction to all branches of mathematics that could be expanded and completed, if necessary.
14The Spanish mathematician Jorge Juan (1713-1773) expressed his approval of all the published treatises included in Padilla’s Curso. Juan praised the clarity, extension and order with which Padilla treated the different branches of mathematics and consequently he recommended the work as a useful addition to teaching curriculums. This is particularly remarkable given Juan’s interest in improving science teaching in Spain, and the substantial level of institutional support he had received to further this mission, notably from the Marquis of Ensenada. A scientist, mariner and naval officer, Juan was one of the most worldly Spanish nationals of the time. In 1752, he took charge of the Academy of Marine Guards at Cadiz, the Spanish military school for naval officers, where he promoted modernizing mathematics by means of producing and publishing scientific and technical works, among other initiatives [Ausejo & Medrano Sánchez 2015], [Garma 2002].
- 13 As mentioned above, the Academy of Oran relied heavily on courses offered by the Academy of Barcel (...)
15Since Padilla began his studies at the Military Academy of Mathematics in Oran,13 and later became an engineer in 1744, his own course was no doubt influenced by the mathematics course by Pedro de Lucuce [Blanco & Puig-Pla 2014], [Blanco & Massa-Esteve 2018]. Padilla’s treatises generally correspond to the books included in Lucuce’s course, although he sometimes presented them in a different order.
16However, when comparing the contents, the most remarkable difference concerns calculus, which was not included in Lucuce’s course. In fact, Padilla’s treatise (V) was the first educational book on calculus to be written in Spanish, see [Ausejo & Medrano Sánchez 2010], [Blanco 2013], [Cuesta Dutari 1985].
17There is yet another difference concerning the structure of the courses or, rather, their views on the division of mathematics. While Lucuce included only two chapters dealing with algebra in Book II (On the literal algorithm) of his treatise on arithmetic (Chapter 1: “On basic operations regarding literal quantities”, and Chapter 2: “On literal fractions”), arithmetic and algebra were presented in two different treatises in Padilla’s course. Treatise III of Padilla’s course focused exclusively on algebra, was broader in scope and included the study of series in Section III.
18As Ausejo & Medrano-Sánchez [2015, 157–158] pointed out, Padilla’s course was, however, not enough to compensate for the overall lack of works published in Spanish designed for scientific and technical training, specifically works that addressed the military context. To promote and enhance the production of a concrete mathematical treatise in Spanish, in 1757 Pedro Pablo Abarca de Bolea y Ximenez de Urrea (1719-1794), 10th Count of Aranda, started organizing what would become the Royal Military Society of Mathematics, which we examine in the next section.
19In 1756, a unification of the two military corps (artillery and engineers) took place under the command of a general director, the Count of Aranda. Only the year before the same initiative aimed at uniting both bodies was attempted in France. Finally, King Louis XV of France ordered the joining together of the two bodies (December 8, 1755). They did so under the denomination of “Corps Royal de l’Artillerie et du Génie”. Their union would be short lived, since a new ordinance (May 5, 1758) separated them a second time [Daniel 1773, 172–173]. The unification of artillery and engineers aimed to reduce costs, to promote efficiency and to allow the Minister of War to take over the artillery by avoiding the appointment of a new Grand Master of the artillery [Galland Seguela 2008].
20The Count of Aranda had undertaken military studies in Italy. After the king appointed him Field Marshal, he visited several European courts (France, Prussia...) to further his training. In France he met D’Alembert, Diderot and Voltaire, among others. It is not unlikely that the merging of the two military bodies in France gave rise to the idea of undertaking a similar initiative in Spain. In the end, the decision was made to change the structure of the engineering corps and combine it with the artillery corps, to have both then fall under the command of a man who belonged to neither, the Count of Aranda. His mission was to rationalize the artillery’s organization by reducing the number of departments and increasing the number of engineers in the corps to match the number of gunners [Galland Seguela 2008].
21On September 21, 1756, the Count of Aranda proposed to both King Ferdinand VI and the Minister of War, Sebastián de Eslava (1684-1759) that this could be achieved by creating a Society of gunners and engineer officers in Madrid under the direction of Pedro de Lucuce.
22The main objective of this Society, according to Aranda, would be “the elaboration of a Course of Mathematics, adapted for military training, extensive and critical, to have in our language, all what the foreigners and the nationals wrote, thus improving the teaching of the military schools in Spain” [AGS, bundles 3005 & 3011]. In his extensive proposal, he developed his ideas. He claimed that: “Spain has never excelled in mathematics”; “the books are those who form men” [AGS, bundle 3005]. Aranda was convinced that “Mathematics is the instructive and necessary science for war” and defended in his proposal with “it is necessary to write an extensive and critical course in which is approved what is well founded and reproved what is wrongly produced” [AGS, bundle 3005].
- 14 Carlos Lemaur or Charles Le Maur (Montmirail 1721 – Madrid 1785) was a French engineer with expert (...)
23His project was to create a team of engineers and gunners—“the best in Mathematics”—equipped with textbooks, mathematical instruments and a cabinet with models and machines for instruction. The following month (October 23, 1756) the King approved the appointment of five engineers and five artillery officers for that purpose. Shortly after, on November 1, 1756, a team was appointed under the direction of Pedro de Lucuce. This team consisted of four engineers and four gunners: Charles Le Maur (or Carlos Lemaur), engineer; John Garland (or Juan Garland), ordinary engineer; Antonio de Córdova (or Cordoba), extraordinary engineer; Bernardo Fillera, drafting engineer; Francisco Cardoso, ordinary commissioner of Artillery; Lorenzo Laso (or Lasso), extraordinary commissioner of Artillery; Manuel de Rueda, extraordinary commissioner of Artillery and José Dátoli, provincial commissioner of Artillery.14
24The eight officers under the command of Lucuce arrived in Madrid in March 1757 [Cuesta Dutari 1985, 207]. Members of the Society were authorized by the Inquisition to read books that had previously been banned. They contacted Gregorio Mayans (1699-1781), a Spanish historian and linguist, for the purpose of developing a scientific vocabulary in Castilian. Initially, in theory, the members of the Society were asked to dedicate 5 hours of study per day (3 hours in the morning and 2 hours in the afternoon).
25The yearly handwritten reports by Pedro de Lucuce (1757, 1758 & 1760), found in the General Archive of Simancas (AGS) in Valladolid, are the most revealing source for understanding how the Society of Mathematics evolved and progressed (Figure 2).
Source: General Archive of Simancas [AGS, bundle 3004].
Figure 2: Examen de la verdad (or Review of the truth) handwritten report drafted by Pedro de Lucuce for the Minister of War, Sebastián de Eslava (December 2, 1758)
Source: General Archive of Simancas [AGS, bundle 3004].
26Lucuce assigned the task of writing on a topic (i.e., drafting a treatise) to all of the members except Dátoli, who was in charge of directing the construction of models for all machinery (see Table 2).
| Member |
Treatises |
| Carlos Lemaur |
Treatise of Mechanics |
| Juan Garland |
Treatise of Military Architecture |
| Antonio de Córdova |
Treatise of Algebra |
| Bernardo Fillera |
Treatise of Speculative and Practical Geometry |
| Francisco Cardoso |
Treatise of Artillery |
| Manuel de Rueda |
Treatise of Arithmetic |
| Lorenzo Laso |
Treatise of Cosmography |
| José Dátoli |
Direction of the construction of machines and models |
- 15 The fifteen sections were the following 1) Four first quantity rules; 2) Four first rules of irrat (...)
27A remarkable difference between Lucuce’s Mathematical Course and the Society’s project was that the latter contained a treatise specifically dedicated to algebra. That is, again, algebra and arithmetic can be found in two different volumes. In his first report (1757) Lucuce claimed that Córdova was preparing a treatise on algebra containing fifteen sections15 including series and exponential calculus [AGS, bundle 3005], [Cuesta Dutari 1985, 195]. From this report, we can infer that this treatise was intended to cover a broader scope than Lucuce’s course. In addition, according to Lucuce’s report, Córdova had prepared several notes on differential and integral calculus. It is worth pointing out that, here again, calculus seems to be connected with algebra.
- 16 The second report of Lucuce, under the title Relation of what the individuals of Royal Society of (...)
28In 1758, according to Lucuce’s second report16, Córdova “studied works of the Académie Royale des Sciences of Paris and one part of the Leipzig Acts, and other authors” [AGS, bundle 3005], [Cuesta Dutari 1985, 201].
29Apparently, the progress of the work commissioned was slow. Aranda was faced with the challenge of presenting ideas that were initially difficult for both the engineers and the gunners to understand. He also faced certain opposition to his authority from the Minister of War, Sebastián de Eslava, who interfered with his orders. The combination of these two dynamics eventually led to Aranda’s resignation on February 4, 1758.
30Jaime Masones de Lima (1696-1778) was appointed for the post of general director [Capel, Sánchez et al. 1988, 180]. However, since he was ambassador in Paris at that time, Marshal Maximilien de La Croix, a gunner and the most senior officer of two corps, served as interim general director until April 20, 1761 [Salas 1831, 67]. La Croix introduced a change of orientation in the Society of Mathematics. While Aranda envisaged a Society that could prepare manuals and develop technical activities, La Croix believed that the scientific society should be expanded to include men of science and men of letters. In fact, he was worried about the future of the specialist corps and he wanted to transform the Society’s regulations with a view of simplifying the office depending on the corps under his command.
31Prompted by La Croix’s views, the Minister of War, Sebastián de Eslava, proposed a reform due to the fact that very few results were being produced by the Society. La Croix developed a complementary regulation to that of Aranda’s stipulating that the role of Lucuce be considerably reduced to that of a mere “dean” without authority or initiative. Lucuce criticized the new regulation as being illegal and forgave the delay in completing the tasks entrusted to the Society.
32In his last global report (October 14, 1760) Lucuce commented on the conflictual nature of his interactions with the temporary command of Maximilien de La Croix. Among other things, he said that La Croix’s ideas “have nothing to do with the ideas of Mr. Count of Aranda” and that “he tried, under his own authority, to change everything” [AGS, bundle 3011].
- 17 The members of the Society were assigned to different positions. Pedro de Lucuce became director a (...)
33Finally, it was decided by a Royal decree (November 17, 1760) first, to close down the Society of Mathematics (December 1, 1760), secondly, to reassign its members17 and, thirdly, to distribute its books and instruments (and those of the extinct Military Academy of the Royal Guards) to Barcelona and Cadiz Academies [Cuesta Dutari 1985, 217].
- 18 Academies of Mathematics in Barcelona, Orán and Ceuta; Academy of Artillery of Barcelona; Academy (...)
34Why did the Society fail? To begin with, certain tensions had developed between gunners and engineers, which the unification of the two Corps by Aranda failed to resolve. The fact is that Lucuce himself was an engineer and engineers outnumbered gunners within the Society. Although the Royal Treasury made a considerable financial investment by allocating 100,000 reals per year or 60.97% of the overall budget provided by the Treasury for the teaching of mathematics to army officers and cadets enrolled in Spanish military academies18, the results were almost null [Puell 1990, 451]. Disputes that had erupted between La Croix and Lucuce, as well as the conflicts between Lucuce and Lemaur and the affinity of La Croix and Lemaur (both French) all contributed in important ways to the Society’s failure as an institution and its ultimate closure.
35On December 6, 1760, La Croix went further by proposing that Lemaur and Cardoso remain in Madrid to write the mathematics course that should serve as a model for the mathematics courses offered by the military academies. His proposal, however, was eventually rejected.
36As would happen later to other Spanish military academies established over the course of the 18th century (such as those in Avila and Ocaña), the Society ultimately dissolved when its greatest promoter, the Count of Aranda, fell from power. Count Aranda’s fall from grace necessarily involved the failure of the initiatives he had championed; developing the Society of Mathematics represented a project that relied heavily on personal perspective or vision, and handing it off to those who did not share the initial vision of the project would lead to its closure [Abián 2017].
- 19 Already in 1757, works were acquired through the booksellers Angel Corradi, Juan Barthelemy, Josep (...)
- 20 These volumes were transferred to the Academies of Barcelona and Cadiz.
37Despite the fact that the Society of Mathematics failed to succeed, it did produce a number of noteworthy results. Its very creation highlighted the need for modern mathematical texts produced in Spanish and Lucuce did effectively succeed in building a substantial scientific and technical library, which served to supply future military libraries [AGS, bundle 3004]. Lucuce’s inventory (September 30, 1760) of the books that were purchased19 by the Society since its creation shows no less than 1,278 volumes (249 works).20 Among them there were several mathematics courses (by Bélidor, Camus, Dechales, Wolff and others), works of Descartes, Newton, Maclaurin, Ozanam, Maupertuis, Tosca, Zaragozá or Deidier, and books of relevant authors such as: D’Alembert, Agnesi, Acevedo, Bernoulli, Bion, Boscowich, Le Clerc, Clairaut, Cramer, Chafrion, Dedier, Dechales, Desaguliers, Euler, L’Hôpital, Maupertuis, Mariotte, Napier, Rolle, Varignon, Vauban, and many others.
38The two transient cases selected here illustrate two contrasting approaches towards developing more effective course curriculums for teaching mathematics in the context of Spanish military instruction during the 1750’s. While an integrative approach can be seen in beginning stages of both cases, specific aspects of each underline how they would evolve very differently. A context of unity is perhaps best illustrated by Lucuce’s course developed following the Royal Ordinance of 1739 and its use at the Academy of Barcelona, whereas a more polarized approach can be seen in Padilla’s course and the position of the Society of Mathematics.
39The first publication of Padilla’s course as an early initiative was done so during a time when the tradition of dictating was still very strong. While dictation as a teaching method was used for more than forty years for teaching Lucuce’s course, Lucuce himself was also involved with the Society of Mathematics and contributed to its endeavour of producing a printed version of a mathematics course in Spanish. In this way, the two cases presented here show how two pedagogical systems or teaching strategies existed at the same time: teaching through dictation and teaching using printed texts.
40Similarly, studying these cases allows us to identify key differences concerning the contents and structure of the course in question. The first being the very different role algebra appears to play in the two courses. Lucuce’s course contained just a few pages on algebra as a part of his treatise on arithmetic. By contrast, Padilla and the Society of Mathematics included a treatise on algebra that was presented independently of the treatise on arithmetic, both of which were broader in scope than what Lucuce’s course included. The second striking difference is that Padilla’s course, unlike Lucuce’s course, included a treatise on calculus. That said, there is evidence to suggest that the Society of Mathematics intended the treatise on algebra to include at least some sections on calculus, which tells us that Padilla and the Society of Mathematics appear to have shared the same viewpoint regarding the division of mathematics, in particular, the idea that calculus should be considered as a branch of algebra.
41Finally, examining these two cases brings to light two lines of conflict between military engineers and gunners, notably divergences between the Navy and the Corps of Military Engineers, and approaches to teaching mathematics.
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