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Descartes and Jesuit Mathematical Education

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Descartes’s Mathematical Thought

Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 237))

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Abstract

Descartes studied at the Jesuit college of La Flèche, “one of the most famous schools in Europe,”1 for about eight and half years probably from Easter day of 1607 to September 1615. Named after Henri IV, the Collège Henri IV de La Flèche was planned under the king’s patronage in 1603, and actually began to receive students at the end of 1604. The school was built as a college for externs, one of a type designed principally for students not of the Jesuit Order. Both Henri and the Jesuits hoped the students there, chosen from the French elite, would be loyal both to the French monarchy and to the Catholic Church. Descartes, from a family which would qualify as noblesse de robe, was one of those students who were expected to “mount the stage of the theater of the world”2 and play an essential role in French society.

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References

  1. AT, VI, p. 5. G. Rodis-Lewis, Descartes (Introduction, n. 4), p. 25; English translation, p. 8. Adrien Baillet claimed that Descartes was at La Flèche between the years 1604 and 1612. Baillet, La Vie des Monsieur Des-Cartes, t. I (Paris, 1691), pp. 16–17. But this description was corrected by, for example, Charles Adam, Vie et OEuvres de Descartes: Étude historique,Supplément à l’Édition de Descartes (Paris, 1910), pp. 564–565; Karl Six, “Descartes im Jesuitkolleg von La Flèche,” Zeitschrift far katholische Theologie, 38 (1914), pp. 498–508. Both claimed that Descartes was in La Flèche between the years 1606 and 1614. See also Descartes, Discours de la méthode: Texte et commentaire, par Etienne Gilson (Paris, 51967), pp. 103–105. On the other hand, Antonella Romano has ascribed Descartes’s student years at La Flèche to the period between 1605 and 1614. See her La Contre-Réforme mathématique (n. 13, below), p. 483. We have found Rodis-Lewis’s discussion in her aforementioned book to be the most persuasive and will discuss this problem in detail in Chapter 2, § 5. Incidentally, the collège was transformed into a military school by Louis XV, and finally into today’s “Le Prytanée national militaire” by Napoléon Banaparte.

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  2. This expression is taken from the young Descartes’s “Cogitationes privatae,” AT, X, p. 213.

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  3. Gilson, Texte et commentaire (n. 1), p. 181.

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  4. Antonella Romano, La Contre-Réforme mathématique: Constitution et diffusion d’une culture mathématique jésuite à la Renaissance (1540–1640) (Rome, 1999). Descares is discussed in “Le cas Descartes,” pp. 482–491.

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  11. This general characterization has been adopted from Bernhard Jansen, “Die Pflege der Philosophie im Jesuitenorden wärend des 17.-18. Jahrhunderts,” Philosophisches Jahrbuch der Görres-Gesellschaft, 51 (1938), pp. 172–215, 345–66, and 435–456.

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  12. C. de Rochemonteix, Op. cit. (n. 17), t. 1 (Le Mans, Leguicheux, 1889), p. 147. Galileo discovered three satellites around Jupiter in January 1610.

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  16. Monumenta Germaniae Paedagogica, ed. K. Kehrbach, Band II, Ratio Studiorum et Institutiones Scholasticae Soc. J. 1. (Berlin, 1887), pp. 52–54. For the translation see Fitzpatrick, Ibid., pp. 100–101.

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  17. For the following analysis of the Ratio studiorum, see Giuseppe Cosentino, “Le Matematiche nella Ratio Studiorum della Compagnia di Gesù,” Miscellanea Storica Ligure, Anno II (n. s.), N. 2 (1970), pp. 171–213 (I was indebted to the late Professor Charles B. Schmitt, when I visited the Warburg and Courtauld Institutes at the University of London in December 1979 for having obtained a copy of this article. He passed away suddenly in 1986.); Idem, “L’Insegnamento delle matematiche nei collegi gesuitici nell’ Italia settentrinale,” Physis, 13 (1971), pp. 205–217; Romano, Ibid.

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  18. Monumenta Paedagogica Societatis Iesu, ed. L. Lukács, t. I (1540–1556), (Rome, 1965), p. 26: “Praeleget extra ordinem mathematicen, quo tempore commodissimum esse ab ipso Rettore censebitur. Primum aliquot libros Euclidis, donec assuescant demonstrationibus. Deinde practicam arithmeticam Orontii et eiusdem spheram, astrolobium Stoflerini et theoricas Purbachii.” I refer to the editor’s footnotes for the information about the authors and the works cited in the passage.

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  19. bid., p. 515: “Exerpta eorum quae monumenta paedagogia illustrant in Chronico Patris Polanco.”

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  20. Monumenta Paedagogica Societatis Iesu, ed. L. Lukács, t. II (1557–1572), (Rome, 1974), p. 256: “Mathematicae disciplinae praeteriri non debent.” “Sed spherae saltem cognitio habenda est […1.”

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  21. Riccardo G. Villoslado, Storia del Collegio Romano dal suo inizio (1551) alla soppressione della Compagnia di Gesu (1773) (Rome, 1954), p. 335, and Ugo Baldi, Legern impone subactis: Studi su filosofia e scienza dei Gesuiti in Italia, 1540–1632 (Rome, 1992), p. 568; Idem,Saggi sulla cultura della Compania di Gesù (secoli XVI-XVIII) (Padova, 2000), p. 55.

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  22. Ugo Baldi, Legern impone subactis(n. 32)pp. 568, 572 & 573.

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  23. Quoted from David Eugene Smith, History of Mathematics,Vol. 1: General Survey of the History of Elementary Mathematics (New York, 1923), p. 334.

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  24. P. Gassendi, “Viri illustris Nicolai Claudii Fabricii de Peresc Senatoris Aquisextiensis vita,” in Opera omnia, t. V (Lyon, 1658; Stuttgart/Bad Cannstatt, 1964), p. 265.

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  25. Farrell, Op. cit. (n. 15), p. 225.

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  26. Monumenta Paedagogica Societatis Jesu, t. V (n. 25), p. 109: A. Romano, Ibid., p. 614; Monumenta Germaniae Paedagogica, Bd. V (n. 25), p. 141: “Illae namque suppeditant atque exponunt poetis ortus occasusque syderum, historicis locorum facies atque intervalla; analyticis solidarum exempla demonstrationum; politicis artes plane admirabiles rerum bene gerendarum domi militiaeque; physicis coelestium conversionum, lucis, colorum, diaphanorum, sonorum formas et discrimina; metaphysicis sphaerarum atque intelligentiarum numerum; theologis praecipuas divini opificii partes; iuri et consuetudini ecclesiasticae accuratas temporum supputationes.”

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  27. Monumenta Paedagogica Societatis Iesu, t. V, p. 109: A. Romano, Ibid., p. 614; Monumenta Germaniae Paedagogica, Bd. V, p. 142: “Ut tantae paucitati ac penuriae medeamur, duobus in Romano Collegio Mathematicis Professoribus indigemus. Quorum unus sesquianno quotidianis lectionibus breve curriculum mathematicarum rerum conficiat a nostris et ab ex-ternis audiendum […].”

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  28. bid.: “[…] Posteriora Analytica, quae sine mathematicis exemplis vix possunt intelligi

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  29. Cosentino, “Le Matematiche nella Ratio Studiorum” (n. 27), p. 209. Cosentino refers to the 1585 edition. But the first edition appeared in 1583. Cf. C. Sommervogel, ed., Bibliothèque de la Compagnie de Jesus, t. II (Bruxelles/Paris, 1891), col. 1215. The Chinese translation of this monograph Tongwen suanzhi was made by Ricci and Li Zhizao, and published in 1614. It is now being discussed among Japanese historians of mathematics whether or not this was imported and had an influence on mathematics of Edo Japan.

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  30. Monumenta Paedagogica Societatis Iesu, t. V, p. 110: A. Romano, Ibid., p. 615; Monumenta Germaniae Paedagogica, Bd. V, p. 143: “Professor alter, qui modo P. Clavius esse posset, constituatur, rerum mathematicarum pleniorem doctrinam conferat in triennium, explicetque privatim nostris otto circiter aut decem, qui mediocri saltem sint ingenio, nec a mathematicis alieno, et philosophiam audierint, qui ex varijs essent convocandi provinciis, unus ex qualibet, si fieri posset.”

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  31. Monumenta Paedagogica Societatis Iesu, t. V (n. 25), p. 234: A. Romano, Ibid., p. 615.

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  32. Modus quo disciplinae mathematicae in scholis Societatis possent promoveri“: Monumenta Paedagogica Societatis Iesu, t. VII, Collectanea de Ratio Studiorum Societatis Iesu (1588–1616) (Rome, 1992), pp. 115–116; Monumenta Paedagogica Societatis Jesu Quae primam Ratiomem Studiorum anno 1586 editam praecessere (Madrid, 1901), pp. 471–472: ”Primum deligendus erit magister eruditione atque authoritate non vulgari. Alterutra enim si absit, discipuli, ut experientia docet, non videntur ad disciplinas enim si absit, discipuli, ut experientia docet, non videntur ad disciplinas mathematicas allici posse. […] [P]ersuadeant sibi coniunctas esse, ut vere sunt, philosophiam scientiasque mathematicas; praesertim, quia hactenus discipuli contempsisse videntur fere has scientias hac una adducti ratione, quod putent eas non haberi in pretio, imo inutiles esse […]. Necessarium etiam videntur, ut praeceptor habeat inclinationem quandam et propensionem ad has scientias praelegendas, et non sit multis aliis occupationibus distentus; alias vix discipulos iuvare potent. Ut autem Societas semper habere possit idoneos harum scientiarum professores, eligi deberent aliquot ad hoc munus obeundum apti et [12] idonei, qui in privata academia instituentur in variis rebus mathematicis […]. Omitto philosophiam naturalem sine disciplinis mathematicis mancam esse et imperfectam […].“ A discussion on these sentences appears in Robert Schwickerath, Jesuit Education: Its History and Principles (St. Louis, 1903), pp. 133–135. Cf. Edward C. Phillips, ”The Proposals of Father Clavius, S. J., for Improving of the Teaching of Mathematics,“ Bulletin of the American Association of the Jesuit Scientists, Eastern Section, Vol. 18, No. 4 (1941), pp. 203–204. The translation is taken from Roger Ariew, John Cottingham, and Tom Sorell, eds., Descartes’ Meditations: Background Source Materials (Cambridge, 1998), pp. 25–26.

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  33. bid., p. 116; p. 472: “Secundo ergo loco, necesse est, ut discipuli intelligant, has scientias esse utiles et necessarias ad reliquam philosophiam recte intelligendam, et simul magno eas ornamento esse omnibus aliis artibus, ut perfectam eruditionem quis acquirat. Immo vero tantam inter se habere affinitatem haste scientias et philosophiam naturalem, ut nisi se mutuo iuvent, tueri dignitatem suam nullo modo possint. Quod ut fiat, necessarium erit primo, ut auditores physices audiant simul disciplinas mathematicas. Qui mos hactenus in scholis Scietatis semper fuit. Nam si alio tempore praelegentur hae scientiae, existimarent philosophiae auditores, neque immerito, eas nullo modo esse necessarias ad physicam; atque adeo pausissimi eas intelligere vellent […].” Cf. Phillips, p. 204; R. Ariew et al., p. 26.

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  34. bid.: “[…1 de numero et motu orbium caelestium, de multitudine intelligentiarum, de effectibus astrorum, qui pendent ex varus coniunctionibus, oppositionibus et reliquis distantiis inter sese, de divisione quantitatis continuae in infinitum, de fluxu et refluxu maris, de ventis, de cometis, iride, halone et aliis rebus meteorologicis, de proportione motuum, qualitatum, actionum, passionum et reactionum etc., de quibus multa scribunt calculatores.” Cf. Phillips, pp. 204–205; R. Ariew et al., p. 26.

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  35. See, for example, John E. Murdoch and Edith E. Sylla, “The Science of Motion,” in David Lindberg, ed., Science in the Middle Ages (Chicago, 1978), pp. 206–264, esp. pp. 223231; E. D. Sylla, “The Oxford Calculators,” in Norman Kretzmann, Anthony Kelley, and Jan Pinborg, eds., The Cambridge History of Later Medieval Philosophy (Cambridge, 1982), pp. 540–563; William A. Wallace, “The Calculatores in the Sixteenth Century,” in his Prelude to Galileo (Dordrecht, 1981), pp. 78–90.

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  36. Monumenta Paedagogica Societatis Jesu, t. VII (n. 42), p. 116; Monumento Paedagogica Societatis Jesu (n. 42), pp. 472–473: “Omitto infinita exempla in Aristotele, Platone et eorum interpretibus illustrioribus, quae nulla ratione intelligi possunt sine mediocri scientiarum mathematicarum cognitione. Immo propter earum ignorationem nonnulli philosophiae professores saepissime multos errores, eosque gravissimos, commiserunt, et (quod peius est) scriptis etiam mandarunt; quorum aliquos in medium proferre non esset difficile.” Cf. Phillips, p. 205; R. Ariew et al., pp. 26–27.

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  37. bid., p. 116; p. 473: “[…] docent scientias mathematicas non esse scientias, non habere demonstrationes, abstrahere ab ente et bono etc.” Cf. Phillips, p. 205; Ariew et al., p. 27. For “the common professor” whom Clavius is supposed to have referred to, see Chapter 7, Chapter 7, § 2. One of them was clearly B. Pereira.

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  38. bid., p. 117; pp. 473–474: “Praeterea, ad haec studia maxime incitabuntur scholastici, si singulis mensibus omnes philosophi in unum aliquem locum conuenirent, ubi unus disciplulorum habeat brevem commendationem disciplinarum mathematicarum; deinde cum uno aut altero explicet problema aliquod geometricum vel astronomicum, quod et iucumdum esset auditoribus et utile rebus humanis; qualia problemata plurima reperiri poterunt; vel declaret locum aliquem mathematicum ex Aristotele vel Platone, qualia loca apud ipsos non pauca sunt; vel etiam afferat novas demonstrationes quarumdam propositionum Euclidis a se excogitatas. Ubi laudari possent ii, qui melius problema propositum solvissent […].” Cf. Phillips, p. 205; Ariew et al., p. 27.

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  39. See F. Purnell, Jr., “Jacopo Mazzoni and Galileo,” Physis, 14 (1972), pp. 272–294.

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  40. “De re mathematica instructio”: Monumenta Paedagogica Societatis Iesu, t. VII (n. 42), p. 117; Monumenta Paedagogica Societatis Jesu (n. 42), p. 474: “[…] mathematicae studia, quae pene iam negligebantur […].” Cf. Phillips, p. 206.

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  41. bid., p. 118; p. 476: “[…] poterunt a septimo initium audiendi facere usque ad duodecimum inclusive; tum vero addere Theodosii sphaerica elementa, et aliqua ex cognitis Apollonii. Quod satis commode uno anno fieri posset […].” Cf. Phillips, p. 207.

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  42. bid.: “[…] exercentur […] theoricas planetarum, gnomonicen, astrolabium, aliquid ex Archimede et ex algebra […] audire […].”

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  43. W R. Laird, “`Archimedes among the Humanists,” Isis, 82 (1991), pp. 629--638.

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  44. “Ordo servandus in addiscendis disciplinis mathematicis”: Monumenta Paedagogica Societatis Iesu, t. VII (n. 42), p. 112.

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  45. A. Romano, Ibid., p. 181 “Du centre romain à la périphérie française (Seconde moitié du XVI’ siècle)”.

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  46. Quoted from François de Dainville, L’Éducation des jésuites, XVI’-XVIII’ siècles (Paris, 1978), p. 326.

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  47. As for England, see Lawrence Stone, “The Educational Revolution in England, 15601640,” Past and Present, No. 28 (July 1964), pp. 41–60; and about the Continent, see Henry Kamen, The Iron Century (London, 1971).

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  48. See, for example, Émile Durkheim, L’Évolution pédagogique en France (Paris, 1938; 21969), Deuxième partie, Chapitres 5–7, pp. 261–303; Eugenio Garin, L’Éducation de l’homme moderne: La Pédagogique de la Renaissance (1400–1600), tr. de l’italien par J. Humbert (Paris, 1968), “Les Contre-Reforme et les jésuites,” pp. 183–189; Roger Chartier, Dominique Julia et Marie-Madeleine Compère, L’Éducation en France du XVI’ au XVIII’ siècles (Paris, 1976), Ch. V, “Naissance du Collège,” pp. 147–173. In relation with the Counter Reformation, see A. D. Wright, The Counter-Reformation: Catholic Europe and the Non-Christian World (London, 1982), Ch. 3, “Scholasticism and Science,” pp. 84–120.

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  49. Farrell, Op. cit. (n. 15), pp. 219–220.

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  50. bid., p. 220.

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  51. For characteristic aspects of the medieval quadrivium, see articles in David L. Wagner, ed., The Seven Liberal Arts in the Middle Ages (Bloomington, 1983).

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  52. For a general characterization of mathematics in the Middle Ages, see Michael S. Mahoney’s contributions in David Lindberg, Op. cit. (n. 45), Ch. 5, “Mathematics,” pp. 145–178, and “Mathematics,” in Dictionary of the Middle Ages, 8 (1987), pp. 205–222.

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  53. See F. de Dainville, “L’Enseignement des mathématiques,” XVII’ siècles: Bulletin de la Société d’Étude du XVII siècle, no. 30 (1956), pp. 62–68; R. Hooykaas, Humanisme, science, et réforme: Pierre de la Ramée (Leiden, 1958); Jean-Claude Margolin, “L’Enseignement des mathématiques en France (1540–1570): Charles de Bovelles, Fine, Peletier, Ramus,” in Peter Scharratt, ed., French Renaissance Studies (Edinburgh, 1976), pp. 109–155.

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  54. See Mordechai Feingold, The Mathematicians’ Apprenticeship: Science, Universities and Society in England, 1560–1640 (Cambridge, 1984).

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  55. Epistolae S. Francisci Xavierii, t. II (Rome, 1945), p. 265: “São tam curiosos e emportunos em preguntar, tao desejosos de saber, que numqua acabäo de pregumtar e de falar aos outras as coussas que lhes respomdemos aas suas pregumtas. Nom sabiao eles ho mundo ser redomdo, nem sabiao ho cursso do sol; pergumtando eles por estas coussas e per outras tomo das cometas, relampagos, chuva, e neve, e outras semelhamtes, a que nos respomdendo e declaramdo-lhas, ficavao muito contemtes e satisffeytos, temdo-nos por homens doctos, o que ajudou nao pouquo pera darem credito a nossas palavras.”

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  56. bid., p. 373: “También es necesario que tiengan letras para responder a las muchas preguntas que hazen los gipones. Seria bueno que fuesem buenos artistas; y no perdirían nadie [!] que fuesem ssofistas para en las disputas tomar los gipones un contradición; que supiesen alguna cosa de la esphera, porque huegan en grande manera los gipones en saber los movimiendel cielo, lo[s] eclipsis del sol, mengoar y creser la luna; como se engendra el agua de la lluvia, la nieve y piedra, trovanes, relanpagos, cometas y otras cosas ansi naturales. Mucho aprovecha Ila declaración destas cosas para ganar la voluntad al pueblo. Esta information de la gente de Gipón me pareció ser cosa conveniente escrivir a vuestra santa Charidad, para que esté al cabo de las virtudes que han de tener los Padres que alla an de yr.”

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  57. Relaçâo Anual das coisas que fizeram os Padres da Companhia de Jesus nas suas Missióes, Nos Anos de 1600 a 1609, Pelo P. Fernao Guerreiro, Tomo Secunda (Coimbra, 1931), p. 280: “Imitamno também nisto alguns senhores grandes da côrte, e de outros reinos quando veem ao Meaco, que é ordinariamente cada ano; os quais muitas vezes väo a casa dos padres; uns por desejo de ouvir as coisas de Deus, outros levados por curiosidade de ouvir coisas novas, e principalmente de matematica, astrologia e mais segredos naturais que os padres lhes declaram, de que ficam por extremo mara vilhados, e conhecendo a ignorância dos seus bonzos, rindo-se das patranhas e disparates que sôbre estas mesmas coisas lhes diziam. E como säo de agudo engenho e caem bem pelas demonstraçôes e clareza com que os padres lhes explicam, inferem bem daqui que, pois os padres pestas coisas naturais lhes falam tanta verdade, descobrindo-lhes o que até agora näo sabiam nem entendiam, rido poderâo deixar de também lha falar no que lhes pregam de Deus e da salvaçäo; e assim por este meio ficam muitos na rêde de Cristo.”

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  58. Archivum Romanum Societatis Iesu, Japonica—Sinica 36, f. 145v: “La Matematica mi serve molto per insinuarmi nella famigliarità di questi Toni principali, I quali si dilettano assai di cotale scienza; e per questa eziandio il Dayri e il Xougunsama hanno di me notizia, e mi hanno fatta chiamare. Anzi per essere da Giapponesi stimato è la cosa più necessaria, sicchè atteso che sapea la Matematica, è etato meglio che venissi nel Giappone; e coloro che sono per venirci se la sapranno saranno stimati. Una cosa mi duole, ed è, che abbiamo libri; e siccome ho smarriti quei portati d’Italia, insieme cogli scritti di quanto aveva letto in 3 anni a Milano, non mi ricordo più di molte curiosità, le quali senza dubbio farebbero stupire questi Giapponesi. Il perchè la prego per carità di voler mandarmi alcuni acritti e libri moderni o sieno composti dai nostril padri, o dagli esterni, di quei che sopravvanzano nei Collegi di costi, ancorchè sieno nella ligua italiana, per esempio qualche Aritmetica copiosa (chè la piccolo del P. Clavio, colla Sfera, de Holologiis, e de Astrolabio ho qui), qualche trattato di misurare i campi ecc., di machine diverse, di architettura, prospettiva, pittura, ed altri. Qualunque libro di siffalte materie ora farà qui più frutto, che non altri libri di Teologia.” I am indebted to Prof. MIYAZAKI Kentaró of Nagasaki Junshin Catholic University for providing a photocopy of this letter which has not been published before.

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Sasaki, C. (2003). Descartes and Jesuit Mathematical Education. In: Descartes’s Mathematical Thought. Boston Studies in the Philosophy of Science, vol 237. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1225-5_2

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