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On the Conceptualization of Force in Johannes Kepler’s Corpus: An Interplay Between Physics/Mathematics and Metaphysics

Abstract

In this chapter, we present the concept of force in Kepler. We follow the development of this concept during Kepler’s scientific career, starting from his early considerations in the Mysterium Cosmographicum (1596) until his ripest conceptions expounded in the Epitome Astronomiae Copernicanae (1618–1621). Kepler tried to supply a dynamical explanation to the planetary movements. This is an important novelty because astronomy was traditionally a kinematical science. Based on the main accredited literature we present a historical account and theoretical/nature of science (NoS) developments: (1) Keplerian forces; (2) physical astronomy; and (3) orbits, force, and the relation between distances and forces. Koyré had a prominent role in the studies on astronomical revolution and on Kepler. He was also important in fully clarifying the pivotal function that physical astronomy—and hence the concept of force—had in Kepler’s system of the world. Indeed, Koyré is still nowadays an almost unavoidable reference point for Kepler’s Forschung. We refer to several interpretations of his. This is the reason why we think appropriate to present this work in homage to Alexandre Koyré.

Keywords

  • Kepler
  • Newton
  • Koyré
  • Concept of force
  • Gravity
  • Virtus Tractoria
  • Virtus Promotoria
  • Relationships Physics–Mathematics
  • Science in context
  • Nature of science
  • Comparative history of physics
  • Historical epistemology of science
  • Mysterium Cosmographicum
  • Astronomia Nova
  • Epitome Astronomiae Copernicanae

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Fig. 16.1
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Fig. 16.7

Notes

  1. 1.

    A terminological specification: while dealing with the term “force” in Kepler, we refer to Latin words used by Kepler as vis and virtus. Kepler spoke of vis and virtus and of their actions, but did not define them. This means, we are free to use the English term “force” to indicate these concepts, without being compromised with Newton’s concept of force . For the sake of brevity, we avoid an introduction about de motu locali related to the genesis of the theory of impetus from Aristotle (384–322 BC)/John Philoponus (490–570) to Jean Buridan (ca. 1300–ca. 1360; see Buridan 1509), Nicolas d’Oresme (1320? 1325?–1382), and so on, until to Niccolò Tartaglia (1499?–1557). Tartaglia also presented contributions to the art of warfare in Nova scientia (Tartaglia 1537, Books I–II) and Quesiti et inventioni diverse (Tartaglia 1554, Books I–III; Pisano and Capecchi 2015).

  2. 2.

    Here, we mention the works to which we have directly referred in the running text and other important contributions which are, at least in part, dedicated to the concept of force in Kepler: Baigre (1990); Baker and Goldstein (1994); Beer and Beer (1975); Blum and Helmchen (1987); Bussotti (2011); Boner (2013); Caspar ([1948] 1962); Cohen (1994, 2011, pp. 161–177); Davis (1981, 1992e), Delambre ([1817] 1965, [1819] 1965, [1821] 1969); Dijksterhuis ([1950] 1961, part IV, C, §§ 36–50, 53–55); Donahue (1981, 1994, 1993, 1996); Dreyer ([1906] 1953, chap. XV); Elena (1983); Gingerich (1975); Goldbeck (1896); Granada (2010); Guidi Itokazu (2006, 2007); Holton (1956); Hoyer (1979); Ihmig (1990); Jaballah (1999. Two volumes. First volume: pp. 60–80; second volume: pp. 39–43, pp. 78–85); Jammer (1957, chap. V); Jardine (1984, 2000); Koyré (1934, 1957, 1961a, b, c); Knobloch (1997, 2012); Krafft (1975, 1991); Martens (2000); Mittelstrass (1972); Petroni (1989); Pisano and Bussotti (2013); Rabin (2005); Radelet–de Grave (1996, 2007, 2009); Schuster (2000, 2013); Small (1804); Stephenson ([1987] 1994a); Treder (1975); Voelkel (1999); and Westfall (1971).

  3. 3.

    Koyré also wrote about the way to develop the history of science. This aspect is significant in his research on astronomical revolution. See Koyré 1963, 1961a; see also 1961b, 1966, 1971, 1973, 1986. His reply to Henry Guerlac’s (1910–1985) talk (Guerlac 1963; see also Crombie 1963) was superb. Recently, on the subject, see Pisano and Bussotti 2015a, b.

  4. 4.

    On Mysterium Cosmographicum see: Aiton (1977) and Field (1979, 1988). Recently, see also Pisano and Bussotti 2012.

  5. 5.

    “[…] Numerus, Quantitas, et Motus Orbium” (KGW, I, p. 9, line 34; see Fig. 3; authors’ translation).

  6. 6.

    KGW, I, p. 69, lines 1–4. See also Kepler 1981, p. 197.

  7. 7.

    KGW, I, p. 70, lines 21–22. See also Kepler 1981, p. 199.

  8. 8.

    With regard to the force of the Sun or, anyway, to the force acting between the Sun and planets (also considering the occurrences where Kepler speaks of the hypothesis, which he refuses, in which the planets have a soul producing a motive force, too), Kepler uses two words: vis and virtus. The latter is used far more than the former. In particular: vis is used at p. 11, line 11 and at p. 71, line 13, whereas virtus at: p. 11, lines 5 and 21; p. 70, lines 23 and 29; p. 71, line 9; pp. 71–72, lines 38–1; p. 72, lines 1, 11, 12, 27; p. 76, lines 25 and 26; p. 77, lines 4 and 22. The two words are also used with different meanings with which we do not deal.

  9. 9.

    KGW, I, p. 72, lines 1–6. See also Kepler 1981, p. 209.

  10. 10.

    KGW, I, p. 71, lines 13–20. See also Kepler 1981, p. 201.

  11. 11.

    Aiton’s note 7 and 8 to Chapter XX of the Mysterium. See Kepler 1981, p. 249; Aiton 1977, 1978.

  12. 12.

    Cfr. Stephenson [1987] 1994a, pp. 13–14.

  13. 13.

    For this problem, see KGW, I, p. 70, lines 18–34. Quotation, p. 70, lines 25–26.

  14. 14.

    “Sicut igitur fons Lucis in Sole est., et principium circuli in loco Solis, scilicet in centro: ita nunc vita, motus et anima mundi in eundem Solem recidit: vt ita fixarum sit quies, Planetarum actus secundi motuum” (KGW, I, p. 70, lines 24–26).

  15. 15.

    January 4, 1643 according to the Gregorian calendar; December 25, 1642 according to the Julian calendar. Newton’s country switched from the Julian calendar to the Gregorian calendar in September 1752.

  16. 16.

    See, in particular, Propositions I–IX of the third book of the Principia, Newton [1726] [1739–1742] 1822, III, vol. III, pp. 17–40; see also Newton 1687; [1713] 1729. Particularly in Newton’s Geneva edition, our work in progress proposes a critical translation from Latin into English of the whole Newton Geneva Edition (1822) of the Principia (Oxford University Press, 2020, 5 Vols.; see recently Bussotti and Pisano 2014a, b; Pisano and Bussotti 2016).

  17. 17.

    We mainly refer to his studies within Theoremata circa centrum gravitatis solidorum as an appendix of the Discorsi e dimostrazioni matematiche intorno a due nuove scienze attinenti alla meccanica e ai moti locali (1638). On these studies and their relationships with Kepler’s science see Pisano and Bussotti 2012.

  18. 18.

    As to the concept of gravity in Kepler, see Elena (1983), the fundamental Goldbeck (1896), Ihmig (1990) and Trader (1975).

  19. 19.

    “Gravitas est. affectio corporea, mutua inter cognata corpora ad unitionem seu conjunctionem (quo rerum ordine est. et facultas Magnetica).” (KGW, III, p. 25, lines 21–23, author’s parentheses. See also Kepler 1992, p. 55.)

  20. 20.

    “Si Luna et Terra non retinerentur vi animali, aut alia aliqua aequipollenti, quaelibet in suo circuitu; Terra ascenderet ad Lunam quinquagesimaquarta parte intervalli, Luna descenderet ad Terram quinquaginta tribus circiter partibus intervalli: ibique jungerentur: posito tamen, quod substantia utriusque sit unius et ejusdem densitatis.” (KGW, III, p. 25, lines 37–41; see also p. 27, 11–15; Kepler 1992, p. 55.)

  21. 21.

    We remark that in a letter to Maestlin on March 5, 1605 (KGW, XV, pp. 170–176) Kepler defined the force of the Sun that determines the movements of the planets as virtus promotoria because it induces the movement through a mechanism different from gravitational attraction (virtus tractoria). In the Astronomia nova Kepler gave such virtus promotoria the name of virtus motrix.

  22. 22.

    KGW, VII, p. 79. Original Latin text: “Iovis certe corpus umbram jacit, ut Terra et Luna, Veneris corpus parte a Sole aversa lumine caret, ut Terra et Luna.”

  23. 23.

    KGW, VII, Book IV, Part I, chapter entitled “ De raritate et densitate horum sex globorum. Quid tenendum?” pp. 283–284. See, for example, Koyré 1961a, b, c, pp. 354–356.

  24. 24.

    With regard to the difficult problems concerning the relations among these three aspects of Kepler’s thought, we mention, without any claim to be exhaustive: Barker-Goldstein (2001); Bialas (2003); Boner (2006, 2008, 2009, 2011, 2013); Bruhn (2005); Escobar (2008); Fabbri (2009); Field (2009); Gingerich (2011); Granada (2009); Grössing (2003, 2005); Haase (1998); Juste (2010); Menschl (2003); Rabin (1997); Schwaetzer (1997); Stephenson (1994b); Voltmer (1998); and Westman (2001). On the structure of science Nagel (1961) is always of interest (for our aims).

  25. 25.

    To the relations between Sun–planets–fixed stars and the Holy Trinity is dedicated the Primariae Demonstrationis Delineatio, second chapter of the Mysterium Cosmographicum (KGW, I, pp. 23–27). See also Pisano Bussotti 2012, pp. 126–127; Peuerbach 1473; Pichler 2003.

  26. 26.

    Cfr.: Davis 1992e, p. 181; Stephenson [1987] 1994a, pp. 4–7.

  27. 27.

    “Punctum mathematicum, sive centrum mundi sit sive non, nequit movere gravia neque effective neque objective.” (KGW, III, p. 24, lines 37–38. See also Kepler 1992, p. 54.)

  28. 28.

    “Leve vero nihil est. absolute, quod corporea materia constat, sed comparate levius est., quod rarius est. sive natura sua, sive ex accidente calore.” (KGW, III, p. 27, lines 16–17. See also Kepler 1992, p. 57.)

  29. 29.

    “Aliter ego definio gravitatem, seu illam vim, quae intrinsece movet lapidem, vim magneticam coagmentantem similia, quae eadem numero est. in magno et parvo corpore, et dividitur per moles corporum accipitque dimensiones easdem cum corpo re. Itaque si lapis aliquis esset pone Terram positus in notabili aliqua proportione magnitudinis ad molem Telluris, et casus daretur, utrumque liberum esse ab omni alio motu: tunc ego dico futurum, ut non tantum lapis ad Terram eat, sed etiam Terra ad lapidem, dividantque spatium interjectum in eversa proportione ponderum.” (KGW, III, p. 456, Nachbericht.)

  30. 30.

    We have no room here to deal with the difference between masses and weights (namely weight-force). As to this problem, also in connection with the relationship between physics and mathematics, see: Pisano 2011, 2013a; Pisano and Capecchi 2013; Gillispie and Pisano 2014; Dhombres 1978, 2013; Alvarez and Dhombres 2011. On foundations of mathematics, see an interesting essay by Heinzmann (2009).

  31. 31.

    “Nam si quis hoc sequeretur, is peccaret jam in aliam varietatis legem, introducens copiam materiae non inaequalem, sed eandem per omnes planetas. Multiplicata enim mole Saturni 10, in densitatem 5, prodiret copia materiae 50, tantundem scilicet, quantum, si molem Iovis 5 in densitatem ejus 10 multiplicasses.” (KGW, VII, p. 283, lines 31–34.)

  32. 32.

    Kepler speaks of Jupiter in the mentioned lines (KGW, VII, pp. 283–285).

  33. 33.

    “Si Terra cessaret attrahere ad se aquas suas; aquae marinae omnes eleverentur, et in corpus Lunae influerent. Orbis virtutis tractoriae, quae est. in Luna, porrigitur usque ad Terras, et prolectat aquas sub Zonam Torridam, quippe in occursum suum quacunque in verticem loci incidit, insensibiliter in maribus inclusis, sensibiliter ibi ubi sunt latissimi alvei Oceani, aquisque spaciosa reciprocationis libertas.” (KGW, III, p. 26, lines 1–7; see also Kepler 1992, p. 56.)

  34. 34.

    “Omnis substantia corporea, quatenus corporea, apta nata est. quiescere omni loco, in quo solitaria ponitur, extra orbem virtutis cognate corporis.” (KGW, III, p. 25, lines 9–10. See also Kepler 1992, p. 55.) Several references to Kepler’s concept of inertia are available in a recent Leibnizian book of one of us (Bussotti 2015, Chap. 3). Particularly on Leibniz, on the occasion of his anniversary, see Leibniz and the Dialogue between Sciences, Philosophy and Engineering, 1646–2016 (Pisano, Fichant, Bussotti and Oliveira 2017; Bussotti and Pisano 2017).

  35. 35.

    Clearly, this conception of inertia is different from Newton’s, but also from Galilei’s and Descartes’. See also De Gandt 1995; Pute and Mandelbrote 2011; Bussotti and Pisano 2013; Shank 2008.

  36. 36.

    “Sequitur enim, si virtus tractoria Lunae porrigitur in Terras usque, multo magis virtutem tractoriam Telluris porrigi in Lunam et longe altius, ac proinde nihil eorum quod ex terrena materia quomodocunque constat, inque altum subvehitur, complexum hunc fortissimum virtutis tractoriae unquam effugere.” (KGW, III, p. 27, lines 11–15. See also Kepler 1992, p. 57.)

  37. 37.

    Kepler called “libration” the approaching and moving away of a planet to/from the Sun. The libratory or librating force is the force responsible for the radial motion.

  38. 38.

    This adjective is not used in Astronomia Nova, but in the letter to Fabricius mentioned in Footnote 31.

  39. 39.

    KGW, III, Chapter XXXIII, p. 236, lines 12–16. Original Latin text: “quo longius abest Planeta a puncto illo, quod pro centro mundi assumitur, hoc debilius illum incitari circa illud punctum: necessarium est. igitur, ut causa hujus debilitationis insit aut in ipso Planetae corpore, eique insita vi motrice, aut in ipso suscepto mundi centro.” See also Kepler 1992, p. 376.

  40. 40.

    KGW, III, Chapter XXXIII, p. 236, lines 20–21. Original Latin text: “Ut hic intentio et remissio motus, cum accessu et recessu a centro mundi, in proportione perpetuo coincidit.” See also Kepler 1992, p. 376.

  41. 41.

    We use here the word speed, high or low speed, to indicate what Kepler calls “intentio et remissio motus.” This is the almost universally accepted translation. However, in the next section we show that not all scholars agree this is the right translation.

  42. 42.

    KGW, III, Chapter XXXIII, p. 237, lines 14–16. Original Latin text: “Quod si itaque elongatio centri mundi a corpore Planetae, praestat Planetae tarditatem, appropinquatio velocitatem; fons itaque virtutis motricis in illo suscepto mundi centro insit necesse est.” See also Kepler 1992, p. 377.

  43. 43.

    On this question Koyré 1961a, b, c, pp. 202–205 and the connected note 18, p. 408, is exhaustive.

  44. 44.

    We deal with the exact meaning of the word immateriata in the next section while providing the interpretations of Kepler’s thought. In this section we wish only to present Kepler’s theory without interpretation as far as this is possible.

  45. 45.

    Ibidem, Chapter XXXIII, p. 241, lines 4–6. Original Latin text: “Videntur autem pugnantia, materia carere, et tamen dimensionibus Geometricis subjacere: diffundi per mundi amplitudinem, et tamen nuspiam esse nisi ubi est mobile.” See also Kepler 1992, p. 382.

  46. 46.

    Ibidem, Chapter XXXIII, p. 241, lines 7–10. Original Latin text: “Respondetur autem sic: quamvis virtus motrix non sit materiale quippiam, quia tamen materiae hoc est. corpori Planetae vehendo destinatur, non liberam esse a legibus Geometricis, saltem ob hanc materialem actionem transvectionis” (Ivi, p. 241, lines 7–10). See also Kepler 1992, p. 383.

  47. 47.

    See, for example, Dreyer [1906] 1951, pp. 387–388; Caspar [1948] 1962, p. 143; and Dijksterhuis [1950] 1961, IV, 44. Koyré is quite prudent on this aspect and, although he seems to agree with Dreyer’s, Caspar’s, and Dijksterhuis’s interpretation, he does not underestimate the difficulty of an exegesis of Kepler’s assertions (Koyré 1961a, b, c, pp. 210–214).

  48. 48.

    Stephenson [1987] 1994a, pp. 73–74 and see the figure on p. 73.

  49. 49.

    See our quotation from Astronomia Nova in the previous section (KGW, III, p. 241, lines 7–10).

  50. 50.

    As far as the difference among the magnetic fibers is concerned, a good explanation is given by Koyré 1961a, b, c, Chapter L’Epitome, in particular Note 43 and connected text.

  51. 51.

    KGW, VII, pp. 594–595. With permission of the Bayerische Akademie der Wissenschaften.

  52. 52.

    KGW, VIII, Book V, Part II, Chapters IV–V, pp. 390–396. See also Caspar’s Nachbericht (KGW, VII, pp. 598–600) and Stephenson (Stephenson [1987] 1994a, pp. 154–172).

  53. 53.

    We have no room to deal with the problem of the relationships between physics and mathematics inside physics, in particular with regard to the definition of the forces. We suggest to the reader Pisano 2011; Barbin and Pisano 2013; and Pisano 2014.

  54. 54.

    “Nel processo di costruzione delle equazioni non interviene mai alcuna considerazione dinamica: un tolemaico avrebbe potuto accettare senza grossi problemi il modo di procedere di Keplero” (Petroni 1989, p. 192, pp. 190–196).

  55. 55.

    Davis (2003) expounds and tries to prove a well-defined historiographical thesis: it is improper to speak of speed or velocity in Kepler. Concepts such as those of movements or change of movements cannot be equated, in a Keplerian perspective, with our modern concept of velocity or speed. In the paper a series of interesting considerations on the use of Euclidean geometry in Kepler is developed.

  56. 56.

    For this problem, see Bussotti 2015, Section 6.1.2.1, pp. 121–125.

  57. 57.

    Cfr.: Stephenson [1987] 1994a, pp. 130–137.

  58. 58.

    See also Epitome Astronomiae Copernicanae (KGW, VII) discussed above.

  59. 59.

    This discussion also refers to other metaphysical aspects that, for the sake of brevity, we avoid facing. For example: the Aristotelian conviction that the series of natural numbers is potentially infinite, but it cannot be considered as an actually infinite entity (Aristotle 1999, Physics, Book III, Chapter VI; Id., (1801), Metaphysics, Book IX, Chapter VI); the problematization of infinity by Plato (also Pythagoreans), where the universe could be represented by a finite arrangement of natural numbers (Aristotle 1999, Physics, Book III, Chapter IV).

  60. 60.

    Kepler [1606] 1859, p 687 (KGW, I, Chap. XXI, pp. 251–252).

  61. 61.

    Ivi, p 688 (KGW, I, Chapter XXI, pp. 252–253).

  62. 62.

    Ivi, p 689 (KGW, I, Chapter XXI, p. 253).

  63. 63.

    “[…] denique quicquid fere librorum Astronomicorum ex illo tempore edidi, id ad unum aliquod capitum, hoc libello propositorum, referre potuit, cuius aut illustrationem aut integrationem contineret” (KGW, VIII, p. 9, lines 25–28).

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Acknowledgments

We want to express our warm gratitude to Daniel Di Liscia for his valuable comments, to English proofreaders and to anonymous referees for their good blind peer–reviewed job. Their comments were of great help. We also want to express our sincere thankfulness and appreciation to Peter Michael Schenkel (Kepler-Kommission) who on behalf of the Bayerische Akademie der Wissenschaften gave us permission to use images from the official and prestigious KGW edition.

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Pisano, R., Bussotti, P. (2018). On the Conceptualization of Force in Johannes Kepler’s Corpus: An Interplay Between Physics/Mathematics and Metaphysics. In: Pisano, R., Agassi, J., Drozdova, D. (eds) Hypotheses and Perspectives in the History and Philosophy of Science. Springer, Cham. https://doi.org/10.1007/978-3-319-61712-1_16

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