From valla to viète: The rhetorical reform of logic and its use in early modern algebra

Early Science and Medicine

Volumn 11, Issue 4, 2006, Pages 390-423

  Cifoletti, Giovanna ^a  

^a ÉCOLE DES HAUTES ÉTUDES EN SCIENCES SOCIALES   (France)

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EID: 77949533495     PISSN: 13837427     EISSN: 15733823     Source Type: Journal    
DOI: 10.1163/157338206778915152     Document Type: Article

References (80)

1. For Quintilian oratory included invention, disposition, elocution, memory, pronunciation (Institutiones Oratoriae I, 22).
2. Nancy Siraisi, "Oratory and Rhetoric in Renaissance Medicine," Journal of the History of Ideas, 65 (2004), 191-211, at 204.
3. According to Aristotle's Rhetoric I, 3, epideictic is the genre of rhetorical discourse in which the public (as opposed to the judge - judiciary - or the town - deliberative) judges the talent of an orator.
4. I use 'emic' in the sense of Nick Jardine, "Etics and Emics (Not to Mention Anemics and Emetics) in the History of the Sciences," History of Science, 42 (2004), 261-278, as discussed in the Introduction to this fascicle.
5. See in particular G. Cifoletti, Mathematics and Rhetoric. Jacques Peletier, Guillaume Gosselin and the Making of the French Algebraic Tradition (PhD thesis, Princeton University, 1992)
6. and "La question de l'algèbre. Mathématiques et rhétorique des hommes de droit dans la France du XVIe siècle," Annales. Histoire, Sciences sociales, 6 (1995), 1385-1416.
7. See S. Camporeale, Lorenzo Valla. Umanesimo e teologia (Florence, 1972)
8. See S. Camporeale, Umanesimo, riforma e controriforma (Rome, 2002)
9. This crucial point has been made for Ramus' grammar by P. Magnard, "L'enjeu philosophique d'une grammaire," Revue des sciences philosophiques et théologiques, 70 (1986), pp.3-14.
10. The background to Magnard seems to be Ernst Cassirer's Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik (Berlin, 1910).
11. G. Cifoletti, "Matematica e fatti scientifici: polarità e simmetrie. Il ruolo degli algebristi francesi del Cinquecento nella costruzione dei fatti scientifici," Quaderni storici, 108 (2003), 771-797.
12. Carneades, who flourished in Athens in the second century B.C., was the main representative of the sceptic Academy.
13. See C.B. Schmitt, "'Prisca Theologia' e 'Philosophia Perennis': due temi del Rinascimento italiano e la loro fortuna," in Il pensiero italiano del Rinascimento e il tempo nostro (Florence, 1970), 211-236.
14. Proclus, who flourished in the fifth century A.D. at the Academy in Athens, was an extremely productive author, contributing forcefully to the spreading of neo-Platonic philosophy.
15. We quote the standard English translation, Proclus, A Commentary on the First Book of Euclid's Elements, transl. intr. and notes by G. R. Morrow (Princeton, 1970), 151-52.
16. The classical work on Greek geometrical analysis as an analytical corpus, a logical procedure and a 'toolbox' for problem-solving is M.S. Mahoney, "Another Look at Greek Geometrical Analysis," Archive for the History of Exact Sciences, 5 (1968), 318-348.
17. G. Cifoletti , "The Creation of the History of Algebra in the Sixteenth Century," in C. Goldstein, J. Gray, J. Ritter (eds.), L'Europe mathématique (Paris, 1996), pp.121-141.
18. G. Cifoletti, "Quaestio sive aequatio, la nozione di problema proposta nelle Regulae," Da Democrito a Collingwood. Studi di storia della filosofia, ed. A. Ingegno (Florence, 1991), 43-79. The appendix to my Mathematics and Rhetoric contains an English version of this.
19. Jacques Peletier du Mans In Euclidis Elementa Geometrica Demonstrationum Libri sex (Lyon, 1577).
20. See in particular Cifoletti, "La question de l'algèbre."
21. Peletier du Mans, L'Algèbre, 1: "L'algèbre apprend à discourir, et à chercher tous les points nécessaires pour résoudre une difficulté."
22. See P. Riccardi, Saggio di una bibliografia euclidea (Bologna, 1887-1893); repr. Hildesheim, 1974)
23. C. Thomas-Stanford, Early Editions of Euclid's 'Elements' (London, 1926)
24. M. Steck and M. Folkerts, Bibliographia Euclideana: die Geisteslinien der Tradition in den Editionen der 'Elemente' (Stoicheia) des Euklid (um 365-300): Handschriften, Inkunabeln, Frühdrucke (16. Jahrhundert)
25. textkritische Editionen des 17.-20. Jahrhunderts; Editionen der Opera minora (16.-20. Jahrhundert) (Hildesheim, 1981).
26. Geometricorum elementorum libri XV, ed. J. Faber Stapulensis (Paris, 1516).
27. Euclid of Alexandria, author of the Elements, was often confused with Euclid of Megara, Socrates's disciple and the founder of Stoic dialectic. Both aspects agreed with the sixteenth century's dialectical project.
28. Simon Grynaeus, Euclidis Elementorum libri XV ex Theonis coloquiis. In primum ejus librum commentariorum Procli libri IV Graece (Basel, 1533).
29. I am preparing a fuller account of this text.
30. See N.Z. Davis, "Sixteenth-century French Arithmetics on the Business Life," Journal of the History of Ideas, 21 (1960), 18-48, at 31.
31. J. Scheubel's Euclidis Megarensis, philosophi et mathematici excellentissimi, sex libri priores (Basel, 1550) must also have been important for Peletier, who had read Scheubel's Algebrae compendiosa facilisque descriptio (Paris, 1552).
32. On this issue, see in particular M. N. Fried and S.Unguru, Apollonius of Perga's Conica: Text, Context, Subtext (Leiden, 2001)
33. A. Bernard, "Ancient Rhetoric and Greek Mathematics: A Response to a Modern Historiographical Dilemma," Science in Context, 16 (2003), 391-412.
34. Only two years later, in 1559, Jean Borrel maintained that the proofs were indeed Euclid's own.
35. The only available edition of Proclus was Grynaeus', of 1533 (see fn. 20). It was replaced by Franciscus Barocius' version in the Padua edition of 1560.
36. The role of Cicero in Peletier's work is discussed in my Mathematics and Rhetoric. On Peletier's role in the literary tradition and especially in connection with astronomy
37. see I. Pantin, La Poésie du ciel en France dans la seconde moitié du seizième siècle (Genève, 1995) and eadem, «La représentation des mathématiques chez Jacques Peletier du Mans: Cosmos hiéroglyphique ou ordre rhétorique?» Rhetorica 20 (2002), 375-389.
38. On Cicero's philosophy, see James M. May (ed.), Brill's Companion to Cicero: Oratory and Rhetoric (Leiden, 2002).
39. See the classical works by G. Crapulli, Mathesis universalis (Rome, 1969)
40. and C. Sasaki, Descartes' Mathematical Thought (Dordrecht, 2004).
41. Jacques Peletier du Mans, In Euclidis Elementa geometrica demonstrationum libri sex (Lyon, 1557), 12: «Hypothesis, quantum huc attinet, est subiectio, seu fundamentum ratiocinationis. Hanc re existere non est necessarium : probabilem esse satis est :eamque legem duntaxat habet, nequid absurdi importet. Ut quum convenerit Figuram Figurae esse aequalem, aut maiorem, atque ex hoc deducatur argumentatio: ea aequalitas aut inaequalitas, Hypothesis est. A qua discedere, constante argumentatione, non licet.» My translation takes into account the French translation contained in Les six premiers livres des Eléments géométriques d'Euclide, avec les démonstrations de Jaq. Peletier... traduit en François (Geneva, 1611).
42. Ibid., "Demonstrationem vero appellant Dialectici, Syllogismum qui faciat scire: nempe qui ex probatissimis concludat. Atque haec a geometria ortum habet. Immo omnis quae ad verum perducit probatio, Geometrica est. Ut verissime dictum sit, neminem scire verum a falsum distinguere, cui Euclides non fuerit familiaris. Quod siquis attentius sciscitabitur, quur in Demonstrandis Propositionibus non eluceat forma Syllogismi, sed tantum concise quaedam membra Syllogismorum appareant: is sic habeat, praeter dignitatem esse, quae in scholis docentur, ea quum in rem praesentem ventum sit, ex praescriptis formulis observare. Neque enim Orator, quum ad forum accedit, quae a Rhetore excepit dictate, in digitis collocate: immo id agit, ut quum praeceptorum maxime meminit, nihil minus quam Rhetoricen cogitare videatur. Sic in opera Geometrico, quum nil aliud spectemus quam ut scopum exquisite assequamur. Syllogismi figuram omnino dissimulamus. Quae tamen si exigatur, e probationibus Geometricis ad vivum exprimitur. Sed nos ea resecamus, quae repetita non modo taedium, sed etiam obscuritatem parerent. Quod inter Demonstrationes facile perspicient qui iudicio praediti erunt. Demontrationum autem conclusiones, sunt Problemata et Theoremata."
43. Incidentally, their arguments have some affinity with those of Orna Harari, Knowledge and Demonstration: Aristotle's Posterior Analytics (Dordrecht, 2005).
44. J. Peletier In Euclidis Elementa (end of the Praefatio): "Ut in mente divina ab aeterno infixam fuisse Mundi constitutionem: sic disciplinas, caelestia quadam semina esse: quae in nobis insita, et pro rata cuiusque portione exculta, fructum edunt."
45. See Philip Melanchthon, Liber de anima in Philippi Melanchthonis Opera ed. C. G. Bretschneider (Braunschweig, 1846), XIII: 138, "De potentia rationali seu mente." In fact, for Melanchthon, the lumen naturae proves the existence of the mind qua architectatrix sapiens.
46. Peletier du Mans, In Euclidis Elementa, 40: «Duorum Parallelogrammorum circa Dimetientem maioris Parallelogrammi consistentium, Supplementa sunt æqualia.»
47. Ibid, 41: "In hac Propositione demonstranda, structuram ab aliis aliquantum variavi, non novitatis studio, sed ut totum negotium Supplementorum et integri Parallelogrammi evidentius exponerem."
48. Ibid.: "Vix enim usquam in toto opere Geometrico occurrit Figuratio magis fœcunda quam hic Gnomica : hoc est, qui uno parallelogrammo et gnoma conflatur. Ut hoc loco, si ABCD parallelogrammum sumamus, Figura illa HFCD, qui cum ABCD perficit totum BHFK Parallelogrammum, Gnoma seu Gnomon vocatur."
49. Ibid. : "Nam hic Gnomonis explicandi locus est maxime oportunus: licet Euclides ad secundum librum distulerit."
50. Ibid. : "Hanc ego Figuram mysticam soleo vocare: Ex ea enim, veut ex locuple-tissimo promptuario, innumerabiles exeunt Demonstrationes."
51. C. Clavius, Opera Mathematica (Mainz, 1612), 28-9. I expect to publish shortly, with Samuel Gessner, Algebra in Euclid, a detailed comparison between the various interpretations of book II of the Elements by sixteenth-century algebraist such as Nunes, Tartaglia and Clavius.
52. Petrus Ramus, Euclidis Elementa mathematica (Paris, 1549).
53. Ibid., page 2 of the dedicatory letter: "(Pythagoras and Plato) in anima tam excellentes notitias ab intelligentiae primae ingenitis et aeternis exemplis insitas, et ingeneratas crediderunt: eique propterea nomen ipsum matheseos, quasi reminiscentiae, recordationisque (ut Proclus author est) affinxerunt."
54. Pierre de la Ramée, Dialectique (Paris, 1555). Texte modernisé par Nelly Bruyère (Paris, 1996), 16.
55. Pierre de la Ramée, Dialectique, 1996, Ibid., 18.
56. We should remember that the logic of the Stoics stood at the cradle of the modus ponens, a syllogism which makes a conclusion valid if there is a connection between the premise and the conclusion and the premise is valid.
57. de la Ramée, Dialectique, 85.
58. de la Ramée, Dialectique, Ibid., 59.
59. de la Ramée, Dialectique, Ibid., 57.
60. de la Ramée, Dialectique, Ibid., 58.
61. de la Ramée, Dialectique, Ibid.
62. de la Ramée, Dialectique, Ibid.
63. de la Ramée, Dialectique, Ibid.
64. de la Ramée, Dialectique, Ibid.
65. Algebra f. 14 : «Primus Canon deduciturque e 4.2 Elem. Nam 1 q - 81 aequatur gnomoni, ideoque & 65, eidem aequantur : at 1 q est gnomonis quadratus: ergo 8 L sunt duo plani aequales & 41 planus alter aequalium planorum , cuius longitudo 1 l est latus quadrati jam positi: latitudo autem 4, est latus reliqui quadrati. Quare 16 erit quadratus ipse reliquus, additusque gnomoni 65, explebit totum quadratum 81, a cuius latere 9 si tollatur 4 latus secundi quadrati, restabit 5 latus primi. Sed Canonis exempla pleniora statuantur.»
66. Jakob Schegk Hyperaspites responsi ad quatuor epistolas P. Rami contra se aeditas (Tübingen, 1566). Jakob Schegk was the author of a number of commentaries on Aristotle, but especially of the great De demonstratione libri XV (Basel, 1564).
67. About Schegk, Ramus and Kepler, see G. Cifoletti, "Kepler's 'De Quantitatibus'," Annals of Science, 43 (1986), 213-238
68. as well as André Robinet, Aux sources de l'esprit cartésien (Paris, 1996).
69. Galen also contributed to the success of Stoic logic or dialectic in the sixteenth century. Galen could be used as an authority who followed Aristotelian ontology in logic, while also taking into account some crucial Stoic results. Notice that doctors needed mathematics, not least because of their reliance on astrology.
70. On Ramus and mathematics, see the R. Goulding, "Method and Mathematics
71. Peter Ramus's Histories of the Sciences," Journal of the History of Ideas 67, (2006), 63-85.
72. Ramus, Scholae mathematicae, 196. Notice that Peletier had translated proposition II, 4 differently: "Si recta linea secetur utcunque, Quadrata quae fiunt ex segmentis cum eo quod bis sub ipsis segmentis comprehenditur Rectangulo, aequalia sunt ei quod a tota fit Quadrato."
73. Ibid.: "Haec propositio comparat speciem cum genere et ipsamet specie. Itaque praecedere debuit 2 & 3. Theon vero etiam figuras fabricatur, et eas demonstrat, tacet de positae aequalitatis quaestione. At hic etiam consectarium est e 43 p I, quae generalis ad istam fuerit. Parallelogrammum si quidem aequatur quatuor eiusmodi particularibus parallelogrammis, et diagonalia quadrati totius esse quadrata 24 p 6 demonstrat, quae hanc praecedere debuit, generalis quippe de parallelogrammo, ut nobis item praecedit."
74. Peletier du Mans, In Euclidis Elementa, 41: «Hanc ego Figuram mysticam soleo vocare: Ex ea enim, velut ex locupletissimo promptuario, innumerabiles exeunt Demonstrationes.»
75. Ramus, Scholae Mathematicae, 80.
76. Ramus, Dialectique, 85.
77. Ramus, Scholae, 81: «Demonstrativa methodus, inquit Proclus, est transitus a principiis ad quaesita.»
78. Guillaume Gosselin, De Arte magna, seu de Occulta parte numerorum que algebra et almucabala vulgo dicitur (Paris, 1577), 54-55.
79. Guillaume Gosselin, De Arte magna, seu de Occulta parte numerorum que algebra et almucabala vulgo dicitur (Paris, 1577), Ibidem, pp. 57-58
80. In his version of Viète's Zététiques; see J.L. Vaulezard, La nouvelle Algèbre de M. Viète (anastatic edition; Paris, 1986).