Abstract
Between 1650 and 1651, a special commission appointed by the Ninth General Congregation of the Jesuit Order drew up a list of “Propositiones, quae in scholis societatis non sunt docendae” to be included in the “Ordinatio pro Studiis Superioribus” of 1651. Among those propositions we find the following three:
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25.
The successive continuum and the “intension” of the qualities are made up of indivisibles alone.
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26.
There exist inflated points of which the continuum is composed.
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30.
An infinity in number and magnitude can be contained between two unities or two points.1
Research for this article was made possible through the financial support of the Netherlands Organization for Scientific Research (NWO), grant 200–22–295. I wish to thank Mordechai Feingold and Christoph Luthy for their comments on earlier drafts of this paper.
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Abbreviations
- A.R.S.L.:
-
Archivium Romanum Societatis Iesu.
- C.M.:
-
Correspondance du P. Marin Mersenne, religieux minime, eds. C. De Waard, R. Pintard, B. Rochot, A. Beaulieu, 17 vols. (Paris, 1945–1988).
- G.G.:
-
Le Opere di Galileo Galilei (Edizione Nazionale) ed. A. Favaro, 20 vols. (Florence, 1890–1909).
- P.G.:
-
Petri Gassendi Opera Omnia, 6 vols. (Lyon, 1658).
Notes
“25. Continuum successivum et intensio qualitatum solis indivisibilibus constant. 26. Dantur puncta inflata ex quibus continuum componatur.… 30. Infinitum in multitudine et magnitudine potest claudi inter duas unitates vel duo puncta.” [G. M. Pachtler, Ratio Studiorum et Institutiones Scholasticae Societatis Jesu, 3 vols. (Berlin, 1887–1894), iii. 92].
On the Ordinance of 1651, see M. Hellyer, “‘Because the Authority of my Superiors Commands’: Censorship, Physics and the German Jesuits,” Early Science and Medicine, 3 (1996), 319–54, esp. 325–35.
A. P. Farrel, The Jesuit Code of Liberal Education. Development and Scope of the Ratio Studiorum (Milwaukee, 1938).
“Continuum componitur ex indivisibilibus numero finito” (A.R.S.L, F.G., 656A I, p. 319). The shelfmarks of the various censurae opinionum de compositione continui are indicated in U. Baldini, “Una fonte poco utilizzata per la storia intellettuale: le ‘censurae librorum’ e ‘opinionum’ nell’antica Compagnia di Gesu,” Annali dell’Istituto storico Italo-germanico in Trento, 9 (1985), 16–67.
“Continuum componi ex minimis quibusdam physicis finitis.… tempus componi ex indivisibilibus solis” (A.R.S.L, F.G., 656A II, p. 409).
“Continuum componitur ex indivisibilibus etiam finitis.… permitti quoque non debet, nee etiamsi asseratur indivisibilia esse infinita” (A.R.S.L, F.G. 656A II, p. 462).
“Continuum constat ex punctis individue physicis, quae contrahantur et extendantur ad libitum” (A.R.S.L, F.G. 657, p. 382).
“Christus in Eucharistia existit Unities replicatus, scilicet toties quot sunt indivisibilia quantitatis specierum sacramentalium: ex quibus indivisibilibus quantitas ilia componi” (A.R.S.L, F.G. 656A I, p. 331).
Hellyer, ‘“Because the Authority’,” 326.
“Sometimes, in considering how heat goes snaking among the minimum particles of this or that metal, so firmly joined together, and finally separates and disunites them; and how then, the heat departing, they return to reunite with the same tenacity as before… I have thought that this may come about because of very subtle fire-particles. Penetrating through the tiny pores of the metal… these [fire particles] might, by filling the minimum voids distributed between these minimum particles [of metal], free them from that force with which those voids attract one [particle] against another, forbidding their separation” [G. Galilei, Two New Sciences, translated by S. Drake (Madison, 1974), 27 (= G.G., viii. 66–7)].
Galilei, Two New Sciences, 42 (= G.G., viii. 80).
Galilei, Two New Sciences, 58 (= G.G., viii. 96).
Galilei, Two New Sciences, 28 (= G.G., viii. 68).
On Galileo’s analysis of the Rota Aristotelis problem and of other paradoxes of infinity, see I. E. Drabkin, “Aristotle’s Wheel: Notes on the History of a Paradox,” Osiris, 9 (1950), 161–198;
G. Capone Braga, “Galileo e il metodo degli indivisibili,” Sophia, 18 (1950), 299–337;
M. Clavelin, “Le probleme du continu et les paradoxes de l’infini chez Galilée,” Thalès, 10 (1959), 1–26;
P. Costabel, “La roue d’Aristote et les critiques francaises à l’argument de Galilée,” Revue d’histoire des sciences, 17 (1964), 385–396;
A. Frajese, “Concezioni infinitesimali nella matematica di Galileo,” Archimede, 16 (1964), 241–245;
S. Quan, “Galileo and the Problem of Infinity: A Refutation and a Solution. Part 1: The Geometrical Demonstrations,” Annals of Science, 26 (1970), 115–151;
S. Quan, “Galileo and the Problem of Infinity: A Refutation and a Solution. Part 2: The dialectical arguments, and the solution,” Annals of Science, 28 (1972), 237–284;
F. Palladino, “L’infinito nella scienza di Galilei,” Insegnamento della matematica e delle scienze integrate, 5 (1982), 5–50;
Carla Rita Palmerino, “Galileo’s and Gassendi’s Solutions to the Rota Aristotelis paradox: A Bridge between Matter and Motion Theories,” in Christoph H. Lüthy, John E. Murdoch, and William R. Newman (eds.), Late Medieval and Early Modern Corpuscular Matter Theories (Leiden, 2001), 381–22.
G.G., vi. 331.
Galilei, Two New Sciences, 33 (= G.G., viii. 72).
Galilei, Two New Sciences, 54 (= G.G., viii. 93).
P. Redondi, “Atomi, indivisibili e dogma,” Quaderni Storici, 20 (1985), 529–71, esp. 555–57.
P. Redondi, Galileo Heretic, transl. R. Rosenthal (Princeton, 1987). Concerning the evolution of Galileo’s matter theory,
see also W. Shea, “Galileo’s Atomic Hypothesis,” Ambix, 17 (1970), 13–27;
U. Baldini, “La struttura della materia nel pensiero di Galileo,” De Homine, 57 (1976), 91–164;
A. M. Smith, “Galileo’s Theory of Indivisibles: Revolution or Compromise?,” Journal of the History of Ideas, 37 (1976), 571–88;
H. E. Le Grand, “Galileo’s Matter Theory,” in R. E. Butts and J. C. Pitt (eds.), New Perspectives on Galileo, (Dordrecht, 1978), 197–208; Palmerino, “Galileo’s and Gassendi’s Solutions,” esp. 390–7.
P. Redondi, Galileo Heretic, 26.
“Linea aliqua Mathematica continua componitur ex duobus, tribus, vel quatuor punctis immediatis, aut solum ex punctis simpliciter finitis. Vel tempus est, fuit, vel erit compositum ex instantibus immediatis” [L. Fromondus, Labyrinthus sive de compositione continui liber unus (Antwerp, 1631), 12].
Ibid. Fromondus adds that it is a matter of sheer historical contingency that this view was not definitively banned. For Pope John XXIII, who had presided over this session, was afterwards declared non-canonical by the Council, while Martin V, who was elected Pope in the 41st session, confirmed the condemnation of Wycliff s 45 principal articles, but not of the other 260, which included the proposition concerning the atomist composition of space and time (ibid., 13).
“Subtiliores, inter eos qui continuum ex atomis struxerunt, ex infinitis potius quam finitis composuisse” (ibid., 9).
In the argumentum primum geometricum against physical atomism, Fromondus demonstrates that if geometrical figures were composed out of a finite number of points, “falsum erit hoc Mathematicum principium, A quolibet puncto circumferentiae posse lineam rectam ad centrum duci, et generaliter, falsum quod… postulat sibi sine demonstratione concedi Euclides…a quolibet puncto ad quodlibet aliudpunctum lineam rectam posse duci” (ibid., 32).
Ibid., 43–6.
In the argumentum tertium geometricum, Fromondus observes that “negare vero perfectum isoscelem posse fieri, est una opera universas figuras rectilineas conturbare et evertere: cur enim aliarum potius figurarum areae rectis lineis omnibus claudi possunt, quam area talis trianguli?” (ibid., 40). In the argumentum quintum he tries to show that “circulum nullum ex punctis Epicuri fieri posse” (ibid., 46–50).
“in omni motu tardo pausas et morulas quasdam interiiciunt quibus mobile quiescat, quae in motu celeriori complentur” (ibid., 62).
“Sed ecce, cui non statim hie fucus et falsitas subolet, cum videt illico tarn procul, et usque ad caussam primam eos aufugere? Lapidem autem in aere libero, qui casurus alias omnibus momentis, ut pendulus aliquamdiu ibi haesitet, a Deo subinde per certa intervalla momentorum fraenari?” (ibid., 63).
Aristotle, Physica, IV, 12, 220b; VI, 1, 231b-232a; VI, 2, 232t>-233a
C. R. Palmerino, “Una nuova scienza della materia per la scienza nova del moto. La discussione dei paradossi dell’infinito nella prima giornata dei Discorsi galileiani,” in E. Festa and R. Gatto (eds.), Atomismo e continuo nel XVII secolo (Naples, 2000), 275–319.
Galilei, Two New Sciences, 55.
The first scholar who intuited the existence of a link between Galileo’s analysis of the composition of the continuum and his theory de motu accelerato was T. Settle, Galilean Science: Essays in the Mechanics and Dynamics of the Discorsi, (Ph.D. dissertation, Cornell University, 1966), ch. IV.
A subsequent analysis of the foundational purpose of the paradoxes of the infinite is found in P. Galluzzi, Momento. Studi galileiani, (Rome, 1979), 345–62.
Honoré Fabri, born around 1607 in Grand-Abergement (Ain), entered the Jesuit order in 1626. He taught grammar at Roanne, philosophy at Aries, and finally logic, physics, metaphysics, and mathematics at Lyons. In 1646 he was sent to Rome, where he became penitentiary of St. Peters. He died in Rome in 1688. For Fabri’s biography, see A. and A. De Backer, Bibliothèque des écrivains de la Compagnie de Jésus, première série, 7 vols. (Liège, 1853–1861), i. 290–24;
C. Sommervogel, Bibliothèque de la Compagnie de Jésus, 12 vols. (Brussels-Paris, 1890–1932), iii. 511–21;
J. Brucker, s.v. Fabri, in A. Vacant and E. Mangenot (eds.) Dictionnaire de Théologie Catholique, 23 vols. (Paris, 1909–1953), v. 2052–55;
H. Beylard, s.v. Fabri, in J. Balteau, M. Barroux, M. Prevost (eds.), Dictionnaire de Biographie Française, 19 vols, to date (Paris, 1929), xiii. 432–44;
E.A. Fellmann, s.v. Fabri, in C. Gillispie, F. Holmes (eds.), Dictionary of Scientific Biography, 18 vols. (New York, 1970–1978), iv. 505–7;
P. de Vrégille, “Un enfant du Bugey. Le Père Honoré Fabri 1607–1688,” Bulletin de la Société Gorini, 3 (1906), 5–15.
For Fabri’s philosophical and scientifical activity, see E. Fellmann, “Die mathematischen Werke von Honoratus Fabry,” Physis, 1 (1959), 6–25;
A. Bohem, “Deux essais de renouvellement de la Scolastique au XVII Siecle. II. L’aristotelisme d’Honoré Fabri (1607–1688),” Révue des sciences religieuses, 39 (1965), 305–360;
E. Caruso, “Honoré Fabri gesuita e scienziato,” Miscellanea seicentesca. Saggi su Descartes, Fabri, White (Milan, 1987), 85–126;
E. Fellmann, “Honoré Fabri (1607–1688) als Mathematiker — Eine Reprise,” in P.M. Harman, A. Shapiro (eds.), The Investigation of Difficult Things: Essays on Newton and the History of the Exact Sciences in Honour of D.T. Whiteside (Cambridge, 1992), 97–112;
Denis Des Chene, “Wine and Water: Honoré Fabri on Mixtures,” in Luthy et al. eds., Late Medieval, 363–79. For Fabri’s theory of acceleration,
see S. Drake, “Impetus Theory and Quanta of Speed before and after Galileo,” Physis, 16 (1974), 47–65;
S. Drake, “Free Fall from Albert of Saxony to Honoré Fabri,” Studies in History and Philosophy of Science, 5 (1975), 347–66;
D.C. Lukens, An Aristotelian Response to Galileo: Honoré Fabri, S.J. (1608–1688) on the Causal Analysis of Motion, (Ph.D. thesis, University of Toronto, 1979);
P. Galluzzi, “Gassendi e Yaffaire Galilée delle leggi del moto,” Giornale critico della filosofia italiana, 72 (1993), 86–119, at 98–103;
P. Dear, Discipline and Experience; The Mathematical Way in the Scientific Revolution (Chicago, 1995), 138–43;
C.R. Palmerino, “Infinite Degrees of Speed: Marin Mersenne and the Debate over Galileo’s Law of Free Fall,” Early Science and Medicine, 4 (1999), 269–328, esp. 313–19.
A. Baillet, La vie de Monsieur Des-Cartes, 2 vols. (Paris, 1691), ii. 300.
E. Caruso, “Honoré Fabri,” 85–6.
H. Fabri, Ad P. Ignatium Gastonem Pardesium epistolae tres de sua hypothesi philosophica (Mainz, 1674), 132–8. By the time the letters were published Pardies was dead. Indeed, the third letter was written in February 1673, two months after Pardies had died.
H. Fabri, Tractatus physicus de motu locali, auctore P. Mousnerio, cuncta excerpta ex praelectionibus R.P.H. Fabri (Lyon, 1646), 76–9.
Ibid., 80–2.
S. Drake, “Impetus Theory Reappraised,” Journal of the History of Ideas, 36 (1975), 27–46, at 38.
A. G. Molland, “The Atomisation of Motion: A Facet of the Scientific Revolution,” Studies in the History and Philosophy of Science, 13 (1982), 31–54, at 48.
“Observabis dictum esse supra instantibus aequalibus, quia temporis natura aliter explicari non potest, quam per instantia finita, ut demonstrabimus in Metaphysica; quidquid sit, voco instans totum illud tempus, quo res aliqua simul producitur… igitur totum illud tempus, quo producitur primus impetus acquisitus, voco instans primum motus” (Fabri, Tractatus, 87–8).
“… dantur instantia Physica; quia datur actio, per quam res est” [H. Fabri, Metaphysica demonstrativa, sive scientia rationum universalium (Lyon, 1648), 371].
Ibid., 367.
Fabri, Tractatus, 89.
Ibid., 88.
Ibid., 88.
Ibid., 105. It is worth pointing out that in Galileo’s work, the identification between the aggregate of the degrees of speed acquired by the body and the space traversed by it appears much more problematic than in Fabri’s account. For Galileo’s indecision regarding this matter, see P. Galluzzi, Momento, 363–7; E. Giusti, Galilei e le leggi del moto, in G. Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze, edited by E. Giusti, (Turin, 1990), IX-LX, esp. LIV-LVII;
P. Damerow, G. Freudenthal, P. Mc Laughlin, and J. Renn, Exploring the Limits of Preclassical Mechanics (New York, 1992), 229–31;
. M. Blay and E. Festa, “Mouvement, continu et composition des vitesses au XVIIe siècle,” Archives Internationales d’Histoire des Sciences, 48 (1998), 65–118, esp. 75–6.
Fabri, Tractatus, 105.
Ibid., 89.
Ibid., 115. As Peter Dear has pointed out, Fabri was nonetheless convinced that Galileo’s law did not properly correspond to experimental data. As the Jesuit wrote in his Philosophiae tomus primus, “the name experimentum ought to exclude all that does not fall under the senses,” and distances and times were precisely the kind of magnitudes whose exact measure eluded the senses. H. Fabri, Philosophiae tomus primus: qui complectitur scientiarum methodum sex libris explicatam (Lyons, 1646), 88,
discussed in P. Dear, Discipline & Experience: The Mathematical Way in the Scientific Revolution (Chicago, 1995), 141.
“Igitur haec esto clavis huius difficultatis; progressio simplex principium physicum habet, non experimentum; progressio numerorum imparium experimentum non principium; utramque cum principio et experimento componimus; prima enim si assumantur partes temporis sensibiles transit in secundam, secunda in primam, si ultima assumantur instantia” (Fabri, Tractatus, 108).
CM., xii. 291–2. It is worth mentioning that a few years before Fabri, some Doctores Salmanticenses had also tried to reconcile atomism with Euclidean geometry. The Salmanticenses argued that matter was composed of physical atoms, each of which occupied a space divisible ad infinitum. See R. Gatto, Tra scienza e immaginazione. Le matematiche presso il collegio gesuitico napoletano (1552–1670 ca.), (Florence, 1994), 231–2.
“Idem dico de intensione qualitatum; nulla enim agnosco indivisibilia, seu puncta Mathematica, nisi terminantia et copulativa, quae sunt merae negationes, licet admittam minima physica, divisibilia potentia in infinitum” (Fabri, Epistolae, 133).
“… si tempus constet ex inflnitis actu partibus… non potest esse alia progressio, in qua fiat acceleratio motus naturalis, quam ilia Galilei iuxta hos numeros 1. 3. 5. 7….; si vero tempus constat ex finitis instantibus aequalibus, nulla datur progressio motus naturaliter accelerati; quia motus accelerari non potest;… si tempus constat ex finitis instantibus actu, et infinitis potentia, non potest esse alia progressio huius accelerationis, quam haec nostra iuxta numeros… 1. 2. 3. 4. 5” (Fabri, Tractatus, 131).
J. E. Murdoch, “Atomism and Motion in the Fourteenth Century,” in E. Mendelsohn (ed.), Transformation and Tradition in the Sciences. Essays in Honor of I. B. Cohen, (Cambridge, 1984), 45–66, esp. 52.
“Quod si roges, cur potius in hac parte quam in ilia [lapis] detineatur, cum utrobique sit eadem gravitas lapidis & resistentia aëris; facile respondeo ignis dum calefacit… aliquando cessat, et aliquando operatur: item papyrus levis descendens ex alto, aliquando immoratur etsi brevissime, aliquando descendit.… Respondeo… ad Deum pertinere determinationem individuorum, & consequenter qua parte immoretur lapis qui ex se petit immorari… eo quod habet limitatam virtutem utrum autem in hac vel ilia, pertinet ad Deum” [R. Arriaga, Cursus philosophicus (Paris, 1632), 489b].
Fabri, Tractatus, 109.
“… nam equidem fateor instanti mathematico nihil esse posse minus; secus vero instanti physico, quod est divisibile potentia, ut dicemus alias” (ibid., 110).
Fabri, Metaphysica, 395–6.
Ibid., 425.
Ibid., 375.
Ibid.
Ibid., 413.
Fabri, Metaphysica, 375. It is worth pointing out, however, that in the Metaphysica demonstrativa the hypothesis of the discontinuity of natural motions is criticized only on the basis of physical reasons. Fromondus’ argument that such a hypothesis was to be rejected because it entailed a need for a continuous divine intervention in the course of all natural events was certainly not shared by Fabri, for whom it was in fact God qua first cause to guarantee the conservation of the impetus previously acquired. For the theologically alluring idea that the deficit of atomism could be used to introduce God more directly into the course of natural events — in substitution of the secondary causes offormae — see C.H. Lüthy, “Thoughts and Circumstances of Sebastien Basson. Analysis, Micro-History, Questions,” Early Science and Medicine, 2 (1997), 1–73, esp. 15–18.
“Facile iuxta hanc hypothesim, omnia quae pertinent ad quantitatem explicantur; Primo motus velocitas et tarditas.… Secundo rarefactio, condensatio, compressio, dilatatio; quia quodlibet punctum potest habere, modo maiorem, modo minorem extensionem.” (Fabri, Metaphysica, 414).
“Si aer constat ex punctis mathematicis, non potest explicari, quomodo rarescat, vel densetur, vel comprimatur, contra post. Nee enim punctum mathematicum potest esse maius, vel minus: nee est quod Arriaga explicet condensationem per extrusionem corpusculorum, & rarefactionem per intrusionem, quippe hoc manifestae experientiae repugnat” (Fabri, Metaphysica, 397). Arriaga’s explanation of rarefaction and condensation is also criticized in ibid., 424.
Ibid., 422.
“porro certuni est [punctum physicum corporeum] omnem figuram habere posse; ac proinde posset esse punctum sphaericum, cubicum etc. imo potest dari punctum, quod semper eamdem figuram retineat, atque adeo sit maxime siccum; potest etiam dari punctum, quod figuram mutet, et facile conformari possit; atque adeo sit maxime humidum” (Ibid., 395).
H. Fabri, Physic a, id est scientia rerum corporearum, 4 vols. (Lyons, 1669–1670), i. 201–10; 333; 363.
Ibid., iii. 154, 188.
Ibid., m. 115, 138–9.
Fabri, Epistolae, 134—7.
Fabri, Tractatus, 112. 12 Ibid., 114.
Lukens, An Aristotelian, 204.
“Datur aliquis series numerorum irrationabilium, seu surdorum minorum, & minorum; quorum primus ita superet secundum, secundus tertium, tertius quartum, etc.” (Fabri, Tractatus, 113).
See Le Tenneur’s letter to Gassendi of 16 January 1647 (CM., xv. 49).
“Unde constat non posse haec duo simul conciliari, to turn simul acquiri & successive acquiri (quod tamen asserit adversarius, dum ait instans esse totum illud tempus quo res aliqua simul producitur, & tamen illud instans componi ex multis instantibus minoribus) nisi forte dicatur tempus, & velocitatem rarefied et condensari. Quod quidem quis non fateatur absurdorum absurdissimum, risuque & cachinnis excipiendum?” (J. A. Le Tenneur, De motu naturaliter accelerato tractatus physico-mathematicus (Paris, 1649), 57).
Ibid., 56.
Ibid., 59–60.
Palmerino, “Infinite Degrees,” 295–6 and 319–24, where it is shown that not only Le Tenneur, but Theodore Deschamps and Christiaan Huygens as well recognized in the property of the scalar invariance a sign of the superiority of Galileo’s laws over its rivals.s
Fabri, Metaphysica, 612.
Ibid., 623.
Ibid., 623.
Ibid., 636.
“nempe effectus, & operatio naturae, non est spatium, nee enim sit spatium quod decurritur; sed est ipse motus, & velocitas, & impetus; atque velocitas revera crescit uniformiter, secundum hos numeros, in temporibus aequalibus, 1. 2. 3. 4. &c.” (Fabri, Metaphysica, 648).
Drake, “Free Fall,” 362; Lukens, An Aristotelian, 219.
“nee est quod dicas, inde sequi accelerationem non esse continuam, sed discretam et interruptam; nam censeri debet continua, modo singulis temporibus, primo instanti aequalibus, nova fiat velocitatis accessio” (Fabri, Metaphysica, 623).
Drake, “Impetus Theory Reappraised,” 38.
“Animadversiones quaedam circa propositiones quae propositae sunt ut censurae subjicerentur” (A.R.S.I., Cong. 20e, ff. 234r-235r). The document is discussed in Hellyer, “Because the authority,” 330–332. Pierre Le Cazre, born in 1589 at Rennes, entered the Jesuit order in 1608. He taught humanities, philosophy, mathematics, and theology. He was Rector of the Colleges of Metz, Dijon, and Nancy, Provincial of Champagne, and finally Assistant of France. Sommervogel, Bibliothèque, ii. 934–5.
“Talis videri potest 5ā propositio philosophica (Non datur materia Iā) quis enim tarn crude hoc dicat? Sed multi sunt et sententia valde communis est, non dari talem materiam primam qualem descripsit Aristoteles, sed Elementa ipsa esse materiam primam qua prior non detur. Et haec opinio fere est eorum omnium qui censent Elementa manere actu in mixto, et esse ingenerabilia et incorruptibilia, qua opinio multis experientiis, et valdissimis rationibus fulcitur. Propositio 8ā et 9ā videntur esse consectaria 5ā si praesertim 8ā intelligeretur de forma substantiali et 9āde accidentali. Qui enim dicunt Elementa esse materiam primam, eaque esse ingenerabilia et incorruptibilia (quod consequenter dicendum est) illi censent Elementa esse corpora simplicia, et nullo modo composita esse ex materia et forma, eo modo quo plures putant Aristotelem sensisse de corpore coelesti. Ergo in horum sententia, Elementa extra mixta carent omni forma substantiali. Aliunde Elementa a proprio statu distracta, se ipsa reducunt ad nativum statum, et sine dubio effective: Ergo materia prima effective concurrit ad quarundam saltern formarum accidentalium productionem. Et haec sententiae valde sunt communes.… Propositio 37ā conjuncta est cum 5ā, de qua supra iam dictum est et est valde communis in aliquibus locis, et magis in Scholis externis quam in nostris praesertim in quantum Elementa dicuntur non esse transmutabilia ad invicem” (A.R.S.I. Cong. 20e, f. 234 r-v.)
T. Gregory, “Studi sull’atomismo del ‘600.1. Sebastiano Basson,” Giornale critico della filosofia italiana, 18 (1964), 38–65;
Sebastiano Basson, “Studi sull’atomismo del ‘600. II. David Van Goorle e Daniel Sennert,” Giornale critico della filosofia italiana, 20 (1966), 44–63;
C. Lüthy, “Thoughts and Circumstances,” esp. 12–14. For the seventeenth-century debate over the permanence of the elements in the mixtio, see, among the others, A. Maier, Die Struktur der materiellen Substanz, in An der Grenze von Scholastik und Naturwissenschaft. Studien zur Naturphilosophic des 14. Jahrhunderts (Essen, 1943), 7–140; eadem, “Kontinuum, Minima und aktuell Unendliches,” in Die Vorläufer Galileis im 14. Jahrhundert. Studien zur Naturphilosophic der Spätscholastik, (Rome, 1949), 155–215;
W. Subow, “Zur Geschichte des Kampfes zwischen dem Atomismus und dem Aristotelismus im 17. Jahrhundert (Minima naturalia und mixtio),” in G. Harig (ed.), Sowjetische Beiträge zur Geschichte der Wissenschaft (Berlin, 1960), 161–91;
N. Emerton, The Scientific Reinterpretation of Form (Ithaca, 1984).
“Propositio 41a adhuc communior est, levitatem scilicet nihil esse aliud nisi minorem gravitatem; idque experientia multa convincunt” (A.R.S.I. Cong. 20e, f. 234r, quoted in Hellyer, “Because the authority,” 332).
“Propositio 25 ortum duxit e Collegio Romano, atque ita propagata est, ut nam difficile sit hoc malum extinguere. Si prohibenda est, primum Roma prohibeatur, et omnes intelligant earn Roma non tolerari.” (A.R.S.I. Cong. 20 e, f. 234r).
“Cum intelligam quosdam in Societate esse qui Zenoni secuti dicant in cursu philosohiae, quantitatem ex meris punctis componi, significo nolle me hanc sententiam convalescere, utpote aperte (velut ejus sequaces fatentur) contrariam Aristoteli. Et cum Romae P. Sforza Pallavicinus eandem docuisset, in eodem cursus iussus et retractare” quoted in Pachtler, Ratio, iii. 76.
P.G., vi. 448–52. On Cazre’s polemic with Gassendi, see P. Galluzzi, “Gassendi,” 90–97; C. R. Palmerino, Atomi, meccanica, cosmologia. Le lettere galileiane di Pierre Gassendi (Ph.D. thesis, Florence, 1998), 150–94 and 227–84.
“perveuem proinde ipsum ostendisses, atque adeo subindicasses, qua ergo alia proportione accelerationem decidentium fieri, aut experiundo notaveris, aut deduxeris demonstrando. Certe non satis intelligo quamobrem censueris, sive haec proportio, sive alia sit, earn nihil referre ad meum institutum: quippe si alia fuerit, quam quae supposita a me est, frustra est tota ratiocinatio” (P.G., iii. 626a).
P. Le Cazre, Physica demonstratio qua ratio, mensura, modus, ac potentia, accelerationis motus in naturali descensu gravium determinantur. Adversus nuper excogitatam a Galilaeo Galilaei Florentino Philosopho ac Mathematico de eodem Motu Pseudo-scientiam (Paris, 1645).
In the translation of Stillman Drake, Galileo’s argument reads: “When speeds have the same ratio as the spaces passed or to be passed, those spaces come to be passed in equal times; if therefore the speeds with which the falling body passed the space of four braccia were the doubles of the speeds with which it passed the first two braccia, as one space is double the other space, then the times of those passages are equal; but for the same moveable to pass the four braccia and the two in the same time cannot take place except in instantaneous motion. But we see that the falling heavy body makes its motion in time; and passes the two braccia in less [time] than the four; therefore it is false that its speed increases as the space.” [Galileo, Two New Sciences, 160 (= G.G., viii. 203)]. In the Physica demonstratio, Cazre gives a very unfaithful Latin translation of the passage just quoted; he replaces Salviati’s numerical example with a geometrical one and, much more importantly, transforms the plural “velocities” into the singular “velocity.” As Stillman Drake has pointed out, Cazre’s mistake was repeated by many modern translators and commentators of the Two New Sciences, such as von Oettingen, Crew and De Salvio, and Koyre. What Salviati wanted to express was that if the speed of fall grew in proportion to the space traversed, then all the velocities, that is to say, all the degrees of speed, acquired by a body in the space of four braccia should have been double of all the velocities acquired in a space of two braccia. By substituting a plural with a singular, Galileo’s interpreters gave the wrong impression that “Galileo, in his published argument against proportionality of velocity to space traversed in uniform acceleration relied on some concept of average speed in free fall, and on a naive assumption that such average speed would obey the rule applying to uniform motion.” (S. Drake, “Uniform Acceleration, Space, and Time (Galileo Gleanings XIX),” The British Journal for the History of Science, 5 (1970), 21–43, 29).
In this article, Drake takes issue with the interpretation of Galileo’s argument proposed by I. B. Cohen (“Galileo’s Rejection of the Possibility of Velocity Changing Uniformly with Respect to Distance,” The British Journal for the History of Science, 47 (1956), 231–5)
and A. R. Hall (“Galileo’s Fallacy,” The British Journal for the History of Science, 49 (1958), 342–6). Cohen and Hall both believed that Galileo had based his argument on the erroneous assumption that the mean-speed theorem, which was valid in the case of the proportionality between speed and time, was also valid in the case of a proportionality between speed and space. Drake was instead convinced that Galileo wanted to call the reader’s “attention to the varying velocities with which the falling body moved, not to any uniform velocity that might represent them” (S. Drake, “Uniform Acceleration,” 33).
“accelerationem illam fieri per subdivisionem primi cuiuslibet temporis, in partes simper minores, pro multitudine et ratione spatiorum aequalium, quae motu decurruntur” (Le Cazre, Physica demonstration 30).
Palmerino, Atomi, 242–65.
“Nihil est opus, ut desudem ad ostendendum non increvisse velocitatem aequabiliter, eodemve tenore ex C in D, quo incoeperat, perrexeratque usque in D; ut fecisset enim, oporteret descriptum esse non quadrangulum LD constans ex duobus triangulis; sed trapezion CN constitutum ex tribus. Eadem autem ratione manifestum est, si ad DE aptentur tria triangula, defutura duo; si ad EF quatuor, defutura tria, et ita deinceps… ut proinde intelligamus totidem deesse ad accelerationis aequabilitatem velocitatis gradus, quot numerare licet triangulos ad laevam e regione cuiusque partis, complendo summam traingulorum APB. Constare ergo videtur Motum aequabiliter acceleratum definiri non posse ilium Qui aequabilibus spatiis aequalia celeritatis augmenta acquirat; sed potius ilium, Qui acquirat aequalia aequalibus temporibus.” (P.G., iii. 567b).
Ibid., 577b-578b.
P. Le Cazre, Vindiciae demonstrationis physicae de proportione qua gravia decidentia accelerantur. Ad Clarissimum Petrum Gassendum (Paris, 1645). The text of the Vindiciae was entirely reproduced by Gassendi in the Epistola secunda de proportione qua gravia decidentia accelerantur (P.G., iii. 588b-625b).
“Ingenue enim fatendum est, in mentem mihi nunquam incidisse, ut inquirerem, an globus ex unius diametri altitudine decidens, posset ultra aequilibrium, & pondus sibi aequale amplius aliquid attollere” (P.G., iii. 604a).
P.G., iii. 604b.
“Motus enim, eiusque acceleratio effectus Physicus est, qui propterea (ut in caeteris rebus Physicis accidit) per parteis sensibileis… metiendus videatur. Accedit, quod tecum, et cum Galileo non sentio corpus grave descendens statim a quiete, atque in omnibus, ac singulis indivisibilibus momentis accelerare motus” (P.G., iii. 616a-b).
“Ad quod sine dubio requiritur pars temporis, ac spatij non Mathematice minima, sed Physice notabilis, quae cum pro gravium diversitate, & varia resistentia mediorum modo maior sit, modo minor… Haec porro, si cum tuis, ac Galilei decretis minus forte conveniant; principiis certe Physicis apte congruent” (P.G., iii. 616 b).
“Igitur cum primum momentum accipio, minimum intelligo, in quo unus, et simplex ictus per attractionem imprimatur, peragaturque minimum spatium, motu exsistente simplici, et cui deinceps accedere, ex repetitis ictibus, gradus celeritatis possint.” (P.G., iii. 497b).
“Nam fac unicam esse causam, exempli gratia attractionem; concipies quidem… radij magnetici… motum, sive impetum lapidi imprimunt… in primo momento, qui non deleatur, sed perseveret in secundo, in quo alius similis imprimitur… adeo ut impetus ex continua ilia adiectione continuo increscat, motusque semper velocior fiat. Verum facile erit pervidere consequi ex hac adiectione incrementuum celeritas secundum unitatum seriem; nempe ita ut in primo momento sit unus velocitatis gradus, in secundo sint duo, in tertio tres, in quarto quatuor.” (P.G., iii. 497a).
Galilei, Two New Sciences, 167 (= G.G., viii. 210).
C. R. Palmerino, “Infinite Degrees,” 304–7.
G. Galilei, Dialogue Concerning the Two Chief World Systems, translated by S. Drake, 2d ed. (Berkeley, 1967), 228–9 (= G.G., vii. 255); I have slightly modified Drake’s translation.
P.G., iii. 621b.
Ibid., 566a
“Declaratum certe est quoque iam ante et infinitatem illam partium in continuo, et insectilitatem Mathematicam in rerum natura non esse, sed Mathematicorum hypothesin esse, atque idcirco non oportere argumentari in Physica ex iis quae natura non novit.” (P.G., i. 341b).
Cf. Palmerino, “Galileo’s and Gassendi’s solutions,” esp. 413–20. In the Syntagma, Gassendi explains that the only really continuous movement found in nature is the rectilinear uniform motion of atoms which move all at a speed of one minimum per space per minimum of time. The motions of the res concretae, which are slower than those of the atoms, are all discontinuous, that is to say, they are periodically interrupted by moments of rest (P.G., i. 341b). Since according to Gassendi’s theory, the atoms possess an innate principle of motion, the res concretae can be encountered in a state of rest only if the speeds of their constituent atoms mutually annul each other. The first to observe the radical inconsistency between the principle of inertia stated by Gassendi in his Epistolae and the theory of discontinuous motion set forth in the Syntagma philosophicum was A. Koyre, “Pierre Gassendi: le savant,” in Centre International de Synthese, Pierre Gassendi, 1592–1655, Sa vie et son oeuvre (Paris, 1955), 59–70 and 108–115, esp. 109.
Koyre’s argument has been further developed by M.H. Carre, “Pierre Gassendi and the New Philosophy,” Philosophy, 33 (1958), 112–120;
P.A. Pav, “Gassendi’s Statement of the Principle of Inertia,” Ms, 57 (1966), 24–34;
W. Detel, “War Gassendi ein Empirist?” Studia Leibnitiana, 6 (1974), 178–221;
B. Brundell, Pierre Gassendi. From Aristotelianism to a New Natural Philosophy (Dordrecht, 1987), 79.
Disagreement with Koyre’s criticism has been voiced by O. R. Bloch, La philosophie de Gassendi. Nominalisme, materialisme et metaphysique (The Hague, 1971), 226–7, who claims that the theory of the discontinuity of motion plays only a passing role in the Syntagma, being nothing else than an ad hoc hypothesis introduced so as to account for the paradoxes of motion.
But Bloch’s interpretation has been convincingly refuted by M. Messeri, Causa e spiegazione. Lafisica di Pierre Gassendi (Milan, 1985), 86–93.
For the relationship between the method of proof employed by Galileo in the Third Day of the Two New Sciences and the traditional method of the mixed-mathematical sciences see, among others, W. L. Wisan, “Galileo’s Scientific Method: A Reexamination,” in Butts and Pitt (eds.), New Perspectives, 1–57; Ernan McMullin, “The Conception of Science in Galileo’s Work,” in ibid., 209–57; Dear, Discipline, esp. 124–9.
Fabri, Philosophiae, 88–9 and the analysis in Dear, Discipline, 138–43.
Dear, Discipline, 43–4.
Ibid., 67–71.
Cazre, Physica demonstratio, 18–25. Dear wrote that “Galileo’s problem was that a true science had to rely on evident and universally acceptable premises; in having to adduce specialized, contrived experiences, Galileo admitted failure” (Dear, Discipline, 111). If this is true, then even more total is Cazre’s failure. This Jesuit uses the experiment of the balance to confirm the validity of his definition of accelerated motion, which he describes as that “qui aequalibus spatiis, aequalia celeritatis augmenta acquirit.” As for Galileo, in the Two New Sciences he proves the validity of his own definition of accelerated motion, which he describes as “that which, abandoning rest, adds on to itself equal momenta of swiftness in equal times,” by demonstrating on logical grounds that the hypothesis that speed increases as space does entail a contradiction. As Wisan has observed, it is only later on that Galileo invokes experiments, the results of which he claims to be “in agreement with properties of motion demonstrated from his definition” (Wisan, “Galileo’s Scientific Method,” 40).
Drake, “Free Fall,” 365.
It is important to clarify that the first conclusion would not have been shared by Galileo, who in a marginal note in his personal copy of the Dialogue wrote that if “a massive body” which moves “with any given velocity” encounters “any body at rest… the former body meeting the latter, can never confer upon it immediately its own velocity” [Galilei, Dialogue, 21 (= G.G. vii. 45)]. I wish to thank Sophie Roux for bringing this passage to my attention]. The second conclusion, however, seems to be in accordance with what Galileo says in various passages of his work, e.g. G.G. viii. 108–9.
Palmerino, “Infinite Degrees,” 282–95; 324–327.
Molland, “The Atomization”, 48.
The Correspondence of Henry Oldenburg, ed. A. R. Hall and M. Boas Hall, 13 vols. (Madison, Wisconsin, 1965–1986), viii. 458 (italics added). I wish to thank Moti Feingold for drawing my attention to this letter.
That Pardies believed in the validity of Galileo’s law of fall and in the underlying assumption that the falling body passes through infinite degrees of speed is clear from the preface of his La statique ou la science des forces mouvantes (Paris, 1673). Here the Jesuit announces the publication of a Discours du mouvement des corps pesans, anticipating that “on y voit la raison de cette augmentation ou diminution merveilleuse de vitesse des corps, qui passent en montant et en descendant par tous les degrez imaginables de lenteur. Galilée n’a montre ces proprietez, qu’en supposant une definition qu’on luy a contestee. Baliani a voulu donner une autre progression au mouvement de ces corps [i.e. the progression of natural numbers].” By using an argument very similar to the one we have found in the letter to Oldenburg, Pardies maintains that the falling body has to pass through infinite degrees of speed and that, therefore, neither Baliani’s law nor Cazre’s can be valid.
It is worth mentioning that, in his De motu, Le Tenneur argued, against Fabri, that the Galilean law of fall could still be considered valid even if one denied the actual composition of the continuum out of an infinity of mathematical instants. For although Galileo had admitted the existence of mathematical instants and points, his theory of acceleration “could be explained and defended equally well by means of ever divisible parts.” (Le Tenneur, De motu, 90–1). This was the same as saying that the Galilean theory could be easily reconciled with the Aristotelian theory de compositione continui.
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Palmerino, C.R. (2003). Two Jesuit Responses to Galileo’s Science of Motion: Honore Fabri and Pierre le Cazre. In: Feingold, M. (eds) The New Science and Jesuit Science: Seventeenth Century Perspectives. Archimedes, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0361-1_4
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