The geometers of God: Mathematics and reaction in the Kingdom of Naples

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Volumn 89, Issue 4, 1998, Pages 674-701

  Mazzotti, Massimo ^a  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

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EID: 0042137893     PISSN: 00211753     EISSN: 15456994     Source Type: Journal    
DOI: 10.1086/384160     Document Type: Article

References (117)

1. In the social history of mathematics, the pioneering work is Henk J. M. Bos and Herbert Mehrtens, "The Interactions of Mathematics and Society in History: Some Exploratory Remarks," Historia Mathematica, 1977, 4:7-30.
2. See also the essays in Mehrtens, Bos, and Ivo Schneider, eds., Social History of Nineteenth-Century Mathematics (Boston: Birkhäuser, 1981);
3. Donald MacKenzie, Statistics in Britain, 1865-1930: The Social Construction of Scientific Knowledge (Edinburgh: Edinburgh Univ. Press, 1981);
4. Joan Richards, Mathematical Visions: The Pursuit of Geometry in Victorian England (San Diego, Calif.: Academic, 1988);
5. Steven Shapin, "Robert Boyle and Mathematics: Reality, Representation, and Experimental Practice," Science in Context, 1988, 2:23-58;
6. Mario Biagioli, "The Social Status of Italian Mathematicians (1450-1600),"History of Science, 1989, 27:41-69;
7. and Mehrtens, Moderne - Sprache - Mathematik: Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjeckts formaler Systeme (Frankfurt am Main: Suhrkamp, 1990).
8. For an overview of both the social history of mathematics and recent empiricist trends in philosophy of mathematics see Richards, "The History of Mathematics and L'esprit humain: A Critical Reappraisal," Osiris, N.S., 1995, 10:122-135.
9. For the epigraph see Gioacchino Ventura, "Elogio di Nicola Fergola" (1824), in Raccolta di elogi funebri e lettere necrologiche (Genoa: G. D. Rossi, 1852), p. 84. Here and throughout the essay, translations are mine unless otherwise indicated.
10. An "image" of mathematical knowledge is the set of fundamental beliefs, metaphysical and philosophical, that inform the practice of a community of mathematicians.
11. The official denomination of the kingdom, after the Bourbon Restoration (1815), was "Kingdom of the Two Sicilies," because it included both the former Kingdom of Naples and the former Kingdom of Sicily. I will employ the denomination "Kingdom of Naples" even when referring to the age of the Restoration. For an overview of mathematics in the Italian states in the nineteenth century see Umberto Bottazzini, Va' Pensiero: Immagini della matematica nell'Italia dell'ottocento (Bologna: II Mulino, 1994).
12. Among the students and followers of Fergola were Felice Giannattasio (1759-1849), Giuseppe Scorza (1781-1843), Giuseppe Sangro (ca. 1775-ca. 1835), Vincenzo Flauti (1782-1863), Francesco Bruno (?-1862), and Ferdinando De Luca (1783-1869). Among the more active members of the analytic school were Ottavio Colecchi (1773-1847), Salvatore De Angelis (1789-1850), Francesco Paolo Tucci (1790-1875), and Fortunato Padula (1815-1881).
13. Ottavio Colecchi, "Riflessioni sopra alcuni opuscoli che trattano della funzioni fratte e del loro risolvimento in funzioni parziali," Memorie della Biblioteca Analitica, 1810, 2:249-269, 329-376.
14. "Le goût de cette Géométrie, qui a donné tout d'éclat aux sciences mathématiques jusques il y a près d'un siècle, surtout dans la patrie de Newton, s'est affaibli depuis, et aurait presque disparu, si les géomètres italiens ne lui fussent restés fidèles. On doit, de nos jours, au célèbre Fergola, et à ses disciples, MM. Bruno, Flauti, Scorza, plusieurs écrits importants sur l'Analyse géométrique des Anciens, qui s'y trouve rétablie dans sa pureté originarire": Michel Chasles, Aperçu historique sur l'origine et le développement des méthodes en géométrie (1835; Paris: Gauthier-Villars, 1875), p. 46.
15. For Loria's survey see Gino Loria, Nicola Fergola e la scuola che lo ebbe a duce (Genoa: Tipografia del Regio Istituto Sordo-Muti, 1892).
16. Information on the debate between the synthetic and analytic schools in Naples can be found in Federico Amodeo, Vita matematica napoletana, 2 vols., Vol. 1 (Naples: Giannini, 1905), Vol. 2 (Naples: Tipografia dell'Accademia Pontaniana, 1924).
17. See also Massimo Galluzzi, "Le scuole del Fergola e del Ruffini," in Storia d'Italia, Suppl. 3: Scienza e tecnica nella cultura e nella società dal Rinascimento a oggi, ed. Gianni Micheli (Turin: Einaudi, 1980), pp. 1008-1019.
18. Recently, the early work of Fergola has been studied in Giovanni Ferraro and Franco Palladino, "Sui manoscritti di Nicolò Fergola," Bollettino di Storia delle Scienze Matematiche, 1993, 13(2):147-197;
19. and Ferraro and Palladino, Il calcolo sublime di Eulero e Lagrange eposto col metodo sintetico nel progetto di Nicolò Fergola (Naples: La Città del Sole, 1995).
20. On the notions of "analysis" and "synthesis" in mathematics see Michel Otte and Marco Panza, eds., Analysis and Synthesis in Mathematics: History and Philosophy (Dordrecht: Kluwer, 1997).
21. For the explanation based on "mindedness" see Loria, Nicola Fergola; for the generational explanation see Galluzzi, "Scuole del Fergola e del Ruffini."
22. On methods in geometry see Julian L. Coolidge, A History of Geometrical Methods (New York: Dover, 1963);
23. Wilbur Knorr, The Ancient Tradition of Geometric Problems (Boston: Birkhäuser, 1986);
24. and Carl Boyer, History of Analytic Geometry (New York: Scripta Mathematica, 1956).
25. On the history of analysis in the eighteenth and nineteenth centuries see Ivor Grattan-Guinness, The Development of the Foundations of Mathematical Analysis from Euler to Riemann (Cambridge, Mass.: MIT Press, 1970);
26. and Umberto Bottazzini, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass (New York: Springer, 1986).
27. On the eighteenth century see also Henk Bos, "Calculus in the Eighteenth Century: The Role of Applications," Bulletin of the Institute of Mathematics and Its Applications, 1977, 13:121-227;
28. and Craig Fraser, "The Calculus as Algebraic Analysis: Some Observations on Mathematical Analysis in the Eighteenth Century "Archive for History of Exact Sciences, 1989, 39:317-336.
29. A more philosophical investigation of analysis is Jaako Hintikka and Unto Remes, The Method of Analysis: Its Geometrical Origins and Its General Significance (Boston: Reidel, 1974).
30. J.-L. Lagrange, "Solutions analytiques de quelques problèmes sur les pyramides triangulaires" (1773), in Oeuvres de Lagrange, ed. J. A. Serret, 14 vols., Vol. 3 (Paris: Gauthier-Villars, 1869), pp. 661-692;
31. and Lagrange, Mécanique analytique (1788),
32. Lagrange, Mécanique analytique (ibid., Vol. 11 (Paris: Gauthier-Villars, 1888).
33. Vincenzo Flauti, "Programma destinato a promuovere e comparare i metodi per l'invenzione geometrica presentato a' matematici del Regno delle Due Sicilie nell'aprile 1839," in Produzioni relative al programma di tre quistioni geometriche proposto da un nostro professore (Naples, 1840), pp. 3-16, on pp. 12, 13.
34. The synthetic school allowed the use of algebraic methods to solve certain problems but stipulated that in these cases the solving equation should always be "constructed" - that is, a geometrical construction had to be performed, yielding line segments with lengths equal to the roots of the equation. On this practice see Henk J. M. Bos, "Arguments on Motivation in the Rise and Decline of a Mathematical Theory: The 'Construction of Equations,' 1637-ca. 1750," Arch. Hist. Exact Sci., 1984, 30:331-380. Bos links its decline with the general shift from a "geometrical" to an "analytic" conception of mathematics and places its disappearance around 1750. The reinstatement of this theory was part of the Fergolian restoration of the supremacy of geometry over analysis. "Cramer's problem" called for inscribing in a given circle a triangle whose sides pass through three given points. It had been proposed by the Swiss mathematician Gabriel Cramer (1704-1752) to the Tuscan Castillon (Giovanni F. M. Salvemini [1704-1791]). Castillon presented a synthetic solution for this problem at the Berlin Academy in 1776. Later, Lagrange presented an analytic solution. According to Flauti, Lagrange's solution was incomplete because it lacked the final "construction of the solving equation."
35. Fortunato Padula, Risposta di Fortunato Padula al programma destinato a promuovere e comparare i metodi per l'invenzione geometrica presentato a' matematici del Regno delle Due Sicilie (Naples: Stamperia del Fibreno, 1839), pp. 7, 45-46.
36. Fortunato Padula, Raccolta di problemi di geometria risoluti con l'analisi algebrica (Naples: Stamperia del Fibreno, 1838).
37. Flauti himself had published an influential textbook of descriptive geometry: Vincenzo Flauti, Elementi di geometria descrittiva (Rome: Perego Salvioni, 1807).
38. Padula, Raccolta, p. 13.
39. Padula, Risposta (cit. n. 11), pp. xxvi, xxxi.
40. Vincenzo Flauti, preface to Nicola Fergola, Della invenzione geometrica: Opera postuma di Nicola Fergola ordinata, compiuta e corredata d'importanti note dal Prof. V. Flauti: Parte 1 (1807) (Naples: Stamperia private dell'Autore [V. Flauti], 1842), pp. xii, xxii. In this context, the verb inventare (to invent), and the related terms inventore (inventor), invenzione (invention), and metodo di invenzione (method of invention), refer to the Latin verb invenire (to discover, to find). Thus, to say that the geometer "invents" geometrical truths means that he discovers them. In fact, Fergola's posthumous book Della invenzione geometrica [On geometrical invention] was originally titled L'arte euristica in sistema scientifico ridotta [Heuristic art in the form of a scientific system]; Flauti's new title was inspired by Cicero's On Rhetorical Invention.
41. Fergola, Invenzione geometrica, p. 196. Flauti quotes Edmond Halley to explain what elegance is. "Analisi brevissima et simul perspicua, Synthesi concinna et minime operosa":
42. Vincenzo Flauti, "Considerazioni generali su tre difficili problemi e sul modo di risolverli, lette alla R.A. Accademia delle Scienze di Napoli in agosto 1839," in Produzioni (cit. n. 10), p. 21.
43. The quotation is from Halley's preface to Apollonii Pergaei de sectione rationis ... (Oxford, 1706), unpaginated (last page).
44. Flauti, preface to Fergola, Invenzione geometrica, pp. xxiii, xxvi; and Flauti, "Considerazioni," p. 20. On the construction of the equation see note 10, above.
45. On so-called Reactionary Catholicism in the Italian states see Luigi Salvatorelli, Il pensiero politico italiano (Turin: Einaudi, 1935), Chs. 4, 5, 6;
46. Daniele Menozzi, "Tra riforma e restaurazione: Dalla crisi della società cristiana al mito della cristianità medievale (1758-1848)," in Storia d'Italia, Suppl. 9: La chiesa e il potere politico dal medioevo all'età contemporanea (Turin: Einaudi, 1986), pp. 767-806;
47. and Menozzi, "Intorno alle origini del mito della cristianità," Cristianesimo nella Storia, 1984, 5:523-562.
48. References about authors linked to Reactionary Catholicism are in Letteratura Italiana Laterza, 65 vols., Vol. 46:
49. Vito Lo Curto and Mario Themelly, Gli scrittori cattolici dalla Restaurazione all'Unità (Bari: Laterza, 1976).
50. Gioacchino Ventura, "Elogio di Pio VII," in Raccolta (cit. n. 1), p. 15. In fact, the reaction of the Roman Catholic Church against the values of the Enlightenment dates back to the antimodern turn of Pope Clement XIII (1758). But until the 1790s the Church was isolated in its reactionary position, as is clear from the expulsion of the Jesuits from many European countries (in Naples it took place in 1767) and from the official suppression of the Company in 1773.
51. Karl Mannheim, "Conservative Thought" (1927), in Essays on Sociology and Social Psychology (London: Routledge, 1953);
52. and Mannheim, Ideology and Utopia (1929; London: Routledge, 1936).
53. Lauberg's first published work is [Carlo Lauberg], Riflessioni sulle operazioni dell'umano intendimento (Naples, n.d. [between 1786 and 1789]).
54. The king's recognition of Fergola is noted in [Luigi Telesio], Elogio di Niccolò Fergola scritto da un suo discepolo (Naples: Trani, 1830), p. 199.
55. On the pressure to exclude Lauberg see Amodeo, Vita matematica napoletana (cit. n. 6), Vol. 2, p. 60.
56. Carlo Lauberg, Memoria sull'unità dei principi della meccanica (Naples, n.d. [1789]); quotations are from the unpaginated dedication.
57. Ibid., p. 4.
58. The phenomenon of the studi privati (private studios) was typical of the educational system in the Kingdom of Naples during the eighteenth and nineteenth centuries. Given the extreme conservativism of the university curricula, students who wanted to study contemporary authors, both in the natural sciences and in the humanities, had to attend such private schools. Often university professors also ran private studios. These schools were generally tolerated by the authorities, except in particularly critical situations such as the years of the Jacobin plots (1794-1799).
59. Benedetto Croce, La Rivoluzione Napoletana del 1799 (Bari: Laterza, 1912), p. 210.
60. Among the Neapolitan Jacobins at Lauberg's studio was the radical thinker Francesco Mario Pagano (1748-1799). Liberal and physiocratic in economics, he became a well-known lawyer in the 1770s. In 1785 he was given the chair of criminal rights at the Royal University of Naples. His most famous book is Considerazioni sul processo criminale (1787), in which he suggested reforms of Neapolitan judicial procedures and proposed the establishment of a universal legal code for all the countries of the world. In 1799 he was a member of the Provisional Government of the Republic, responsible for the preparation of the Neapolitan Constitution. At the return of the Bourbons he was sentenced to death and hanged.
61. Annibale Giordano and Carlo Lauberg, Principii analitici delle matematiche (Naples: Giaccio, 1792), pp. 1-2.
62. Annibale Giordano and Carlo Lauberg, Principii analitici delle matematiche (Ibid., p. 2.
63. On this point, the link between Neapolitan Jacobinism and the Neapolitan Enlightenment is evident: Genovesi's aim was, indeed, to identify the true "laws of politics and economics." "Politics, like economics, has its own certain and eternal principles: thus, it has its own theorems and its own problems": Antonio Genovesi, Logica per gli giovinetti (1766; Turin: Einaudi, 1977), p. 232.
64. On the basis of these alleged laws, Genovesi criticized the feudal social setting of the kingdom as well as its juridical system, which openly favored the aristocracy in its quarrels with local communities. The extreme complexity of the old laws, he showed, in fact functioned to maintain a status quo that originated from an illegal action - namely, the acquisition of feudal rights. While Genovesi approached the creation of the new sciences of economics, politics, and law from a reformist perspective, Lauberg and Giordano held that the principles of these sciences justified revolutionary action. In the metaphor of Francesco Lomonaco, a Jacobin who defended Naples in 1799: "The great tree of the sciences ... will form leafy branches, which will provide a restful shadow for a humiliated mankind." Francesco Lomonaco, Rapporto al cittadino Carnot (1800),
65. in Vincenzo Cuoco, Saggio storico sulla rivoluzione napoletana del 1799 (1801; Bari: Laterza, 1913), p. 288.
66. Croce, Rivoluzione Napoletana (cit. n. 25), p. 213;
67. and Amodeo, Vita matematica napoletana (cit. n. 6), Vol. 2, p. 71.
68. On the conspiracy of 1794 see Tommaso Pedio, Massoni e giacobini nel Regno di Napoli: Emanuele De Deo e la congiura del 1794 (Bari: Levante, 1986).
69. In fact, at the university, only a few courses in geometry and arithmetic were given. Traditionally, the faculties of medicine and law attracted the majority of the students and resources. The social and academic prestige of disciplines such as medicine and law was decidedly higher than that enjoyed by natural philosophy and mathematics. On teaching at the University of Naples in the eighteenth and early nineteenth centuries see Michelangelo Schipa, "Il secolo decimottavo," in Storia dell'università di Napoli (Naples: Ricciardi, 1924), pp. 433-466;
70. and Alfredo Zazo, "L'ultimo periodo borbonico," Storia dell'università di Napoli (ibid., pp. 467-588.
71. [Telesio], Elogio di Fergola (cit. n. 21), p. 32.
72. Among Fergola's early works on applications of calculus see Nicola Fergola, "Risoluzione di alcuni problemi ottici" [read in 1780], in Atti della Reale Accademia delle Scienze e delle Belle-Lettere di Napoli dalla fondazione sino all'anno 1787 (Naples: Campo, 1788), pp. 1-14;
73. and Fergola, "La vera misura delle volte a spira" [read in 1783], Atti della Reale Accademia delle Scienze e delle Belle-Lettere di Napoli dalla fondazione sino all'anno 1787 (ibid., pp. 65-84.
74. On the turn toward the "ancients" see the introductory note to Fergola, "Nuovo metodo da risolvere alcuni problemi di sito e di posizione" [read in 1786], Atti della Reale Accademia delle Scienze e delle Belle-Lettere di Napoli dalla fondazione sino all'anno 1787 (ibid., pp. 119-138.
75. Amodeo, Vita matematica napoletana, Vol. 2, p. 141;
76. and [Luigi Telesio], Appendicetta all'Elogio di Niccolò Fergola (Naples, 1836), pp. 26-27.
77. See Louis Châtellier, La religion des pauvres (Paris: Aubier, 1993).
78. On the social role of Catholicism and of the Church in the Kingdom of Naples see Giuseppe Galasso and Carla Russo, eds., Per la storia sociale e religiosa del mezzogiorno d'Italia, 2 vols. (Naples: Guida, 1980, 1982).
79. Nicola Fergola, Teorica de' miracoli (Naples: Flauti, 1839). A second edition was printed in Milan in 1842. It presents a selection from Fergola's manuscripts. It contains an introduction by the editor (Vincenzo Flauti), a biography of Fergola, the essay on miracles, the "Discorso-apologetico sul miracolo di San Gennaro" [Apologetical speech on Saint January's miracle], and an anthology of Fergola's aphorisms on philosophy and religion. The third edition, printed in Naples in 1843, also contains a proof of the spirituality of the soul by Flauti: "Il sentimento ed il pensiero essere incompatibili alla materia" [Feelings and thoughts are incompatible with matter].
80. [Nicola Fergola], Prelezioni sui Principi matematici della filosofia naturale del cavalier Isacco Newton per uso dell'Università interna del Real Convitto del Salvatore, 2 vols., Vol. 1 (Naples: Porcelli, 1792), Vol. 2 (Naples: Di Bisogno, 1793),
81. Nicola Fergola], Prelezioni sui Principi matematici della filosofia naturale del cavalier Isacco Newton per uso dell'Università interna del Real Convitto del Salvatore, Vol. 1, pp. 24, 25;
82. and Lauberg, Principi della meccanica (cit. n. 22), p. 25.
83. A similar elaboration of empiricist themes can be found in the writings of Johan Georg Hamann (1730-1788). Hamann provided an "irrational" and antimaterialist reading of Hume to oppose the intrusion of quantitative science into the sphere of social and individual life. He claimed that the artificial conceptual systems of the philosophes, their material and causal explanations, cannot capture the real life of the universe. This is accessible only through a series of intuitive insights. See Isaiah Berlin, The Magus of the North: J. G. Hamann and the Origins of Modern Irrationalism (London: Fontana, 1994).
84. On the visit see [Telesio], Elogio di Fergola (cit. n. 21), pp. 59-60;
85. on Caracciolo see Benedetto Croce, Uomini e cose della vecchia Italia (Bari: Laterza, 1956), pp. 82-111.
86. For Lagrange's reply to the invitation see Lagrange, Oeuvres, ed. Serret (cit. n. 9), Vol. 14 (Paris: Gauthier-Villars, 1892), pp. 279-282.
87. Luigi Telesio, one of Fergola's pupils and later his biographer, describes an episode that occurred in the late 1780s. Some officers of the French army visited Fergola's school and questioned the students in order to assess their mathematical knowledge. According to Telesio, the officers were surprised by the students' competence in classical methods and by their ability to use those methods, rather than the more common analytic procedures, to solve geometrical problems. See [Telesio], Elogio di Fergola, p. 78.
88. Mannheim himself, in describing the "conservative style of thought" in political writers, philosophers, historians, and novelists, excludes the sciences from his analysis. Empirical sciences and mathematics are presented as part of the "liberal style of thought." There is no space, in his view, for a "conservative style of thought" in science.
89. See Francesco Colangelo, Riflessioni storico-politiclie su la Rivoluzione accaduta a Napoli nel 1799 (Naples: Orsino, 1799);
90. Colangelo, Apologia della religione cristiana, compilata dalle risposte degli antichi Padri della Chiesa alle accuse fatte al cristianesimo (Naples: Orsino, 1818);
91. Colangelo, Saggio di alcune considerazioni sull'opera di G. B. Vico intilolata Scienza nuova (Naples: Trani, 1822);
92. and Colangelo, Storia dei filosofi e matematici napoletani e delle loro dottrine da' Pitagorici al secolo XVIII dell'era volgare, 3 vols. (Naples: Trani, 1833-1834).
93. Francesco Colangelo, L'irreligiosa libertà di pensare, nemica del progresso delle scienze (Naples: Orsino, 1804).
94. Francesco Colangelo, L'irreligiosa libertà di pensare, nemica del progresso delle scienze (Ibid., p. 289.
95. Isaac Newton is one of Fergola's resources in the theological discussion on miracles. Fergola quotes largely from the "Scholium generale" of the Principia, where God is said to rule "non ut anima mundi, sed ut universorum dominus. Et propter dominioum suum, dominus deus áíôïêñÜôùñ, id est Imperator universalis, dici solet": Isaac Newton, Philosophiae naturalis Principia mathematica (1687), 2 vols., Vol. 2 (Cambridge: Cambridge Univ. Press, 1972), p. 760.
96. Fergola remarks: "The Lord rules heavens and nature as a sovereign. He didn't write the destinies of things like constitutional laws of the universe, to which He is not subject. He is not the god of the Stoics, quid scripsit fata, sed sequitur qui semel iussit et semper paret." Fergola, Teorica de' miracoli (cit. n. 34), p. 54.
97. The political implications of the theological debates over God's will and God's wisdom have been pointed out in Steven Shapin, "Of Gods and Kings: Natural Philosophy and Politics in the Leibniz-Clarke Disputes," Isis, 1981, 72:187-215.
98. Colangelo, Irreligiosa libertà di pensare, p. 357;
99. and Francesco Colangelo, Galileo come guida per la gioventù studiosa (Naples: Orsino, 1815), pp. 6-7.
100. See also [Fergola], Prelezioni sui Principia matematica (cit. n. 35), Vol. 2, p. 319.
101. What I call "apologetical empiricism" has some similarities with Popkin's "mitigated scepticism." See Richard H. Popkin, The History of Scepticism from Erasmus to Spinoza (Berkeley: Univ. California Press, 1979), pp. 129-150. It should be noted that the two positions were elaborated with opposite aims (i.e., against the "scientism" of Condillac, Condorcet, and the Neapolitan Jacobins, and in response to the crise pyrrhonienne). Moreover, the stress on the metaphysical relevance of mathematics and on the necessity of integrating scientific investigations with religious values (e.g., "humility") seems to differentiate "apologetical empiricism" from earlier phenomenalistic conceptions of scientific knowledge. See also note 35, above.
102. Ventura, "Elogio di Fergola" (cit. n. 1). On Ventura see Paolo Cultrera, Della vita e delle opere del Rev. P. D. G. Ventura (Palermo, 1877);
103. and Mario Tesini, Gioacchino Ventura: La chiesa nell'età delle rivoluzioni (Rome: Studium, 1988).
104. These arguments - the opposition between "abstract" laws and "concrete" practice, for example - are common among conservative European authors. The Neapolitan reactionary intelligentsia made reference, in particular, to works by Edmund Burke (1729-1797); Nicola Spedalieri (1740-1795), theorist of the theocratic society; Pietro Tamburini (1737-1827), who stressed the empirical nature of political knowledge; Karl Ludwig von Haller (1768-1854); and Augustine Barruel (1741-1820), one of the founders of the theory of the "great philosophical plot" against religion and society. They also appealed to the "Traditionalists" Joseph de Maistre (1753-1821), Luis de Bonald (1754-1840), and Félicité de Lamennais (1782-1854).
105. Fergola presents a division between different levels of reality, and a corresponding hierarchy among the human faculties, which reflects the Thomistic theory of knowledge. According to Fergola, faith, intellect, and sensibility deal, respectively, with theology (metaphysics), pure mathematics, and the empirical sciences. This threefold structure is analogous to that presented in the fifth question of Aquinas's comments on the De Trinitate of Boethius, which deals with the division of the theoretical sciences. See Thomas von Aquin, In libruin Boethii De Trinitate quaestiones quinta et sexta, ed. Paul Wyser (Fribourg: Société Philosophique, 1948), pp. 23-51.
106. Étienne Bonnot de Condillac (1714-1780) aimed to investigate morals and metaphysics with the exactness proper to geometry. He considered the analytic method the most appropriate mode of reasoning because it is simpler and easier than the synthetic one. Algebra became, for Condillac, the universal language that can be employed in any field of human experience. Analytic reasoning creates the proper link between simple sensations and complete knowledge, avoiding the pernicious metaphysical systems of past philosophies. See Étienne Bonnot de Condillac, Traité des sensations (1754), in Oeuvres philosophiques de Condillac, ed. George Le Roy, 4 vols., Vol. 1 (Paris: Presses Univ. France, 1947), pp. 218-314;
107. and Condillac, La langue des calculs (1798), Oeuvres philosophiques de Condillac, ibid., Vol. 2 (Paris: Presses Univ. France, 1948), pp. 417-529. Marie Jean Antoine, marquis de Condorcet (1743-1794), introduced the notion of "social mathematics." This was a new science that aimed to apply mathematical tools to the human sciences. Its immediate objectives were to describe society by means of statistics and to establish scientific economic theory. Condorcet was explicitly attacked by Fergola and other Reactionary Catholic authors; see, e.g., Ventura, "Elogio di Fergola" (cit. n. 1), p. 97. Particularly important from our point of view are
108. Marie Jean Antoine, marquis de Condorcet, "Tableau général de la science qui a pour object l'application du calcul aux sciences physiques et morales" (1795), in Oeuvres de Condorcet, ed. M. F. Arago, 12 vols., Vol. 1 (Paris: Firmin Didot Frères, 1847), pp. 539-573;
109. and Condorcet, Essai sur l'application de l'analyse à la probabilité des decisions rendues à la pluralité des voix (Paris, 1785), where Condorcet applied analysis to the voting process.
110. See also Gilles Granger, La mathématique sociale du Marquis de Condorcet (Paris: Presses Univ. France, 1956);
111. and Keith M. Baker, Condorcet: From Natural Philosophy to Social Mathematics (Chicago: Univ. Chicago Press, 1975).
112. Flauti, preface to Fergola, Invenzione geometrica (cit. n. 15), p. xi.
113. When Padula was writing, in the 1830s, there was yet another interesting component: the debate about the social status of engineers (like Padula) and the limits of their authority. Behind this debate lay the fundamental question of control of the territory of the kingdom, which was contested between the central government and local elites. The controversy over the new professional figure of the engineer and his mathematical education is treated in Massimo Mazzotti, "Engineers and Romantic Painters: Science, Art, and Administration in the Kingdom of Naples (1830-1840)," unpublished MS, 1998.
114. The term ibrido (hybrid) appeared in the Italian language during the eighteenth century to indicate plants or animals obtained from the reproduction of exemplars of different species. But the classic moral connotation (Latin, hybrida; Greek, hy'bris) echoes in Flauti's words when he refers to the analytics' work as "hybrid advancements [ibridi progressi]": Flauti, "Programma" (cit. n. 10), p. 10. See Dizionario etimologico italiano (Florence: Barbera, 1952).
115. On "scientistic hubris" see Friedrich A. Hayek, The Counter-Revolution of Science: Studies on the Abuse of Reason (Glencoe, Ill.: Free Press, 1952), pp. 105-116.
116. "Happy to study the ancients, more happy/to surpass all the modern mathematicians": [Telesio], Elogio di Fergola (cit. n. 21), p. 4.
117. For an overview on scientific thought in Naples in the late seventeenth and early eighteenth centuries see Fabrizio Lomonaco and Maurizio Torrini, eds., Galileo e Napoli (Naples: Guida, 1987).