Abstract
Before Newton, Johannes Kepler and Galileo Galilei were the two most important scientists of the Modern Age and their contribution was essential in enabling Newton to achieve his final synthesis.
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Notes
- 1.
Before the birth of modern cosmology in the twentieth century, Johannes Kepler was surely the scientist who most deserves the title of cosmologist.
- 2.
The first edition of the Mysterium was published in 1596, the second edition enriched with several notes in 1621: see KGW, I and VIII, Kepler (1981).
- 3.
The Astronomia Nova was published in 1609: see KGW, III, Kepler (1992).
- 4.
The Harmonice Mundi was published in 1619: see KGW, VI, Kepler (1997).
- 5.
The books I–III were printed in 1617, book IV in 1620 and books V–VII in 1621 (see KGW, VII).
- 6.
A classical work on the Mysterium is Field (1988). There are interesting observations on Kepler’s text by Eric Aiton in Duncan’s English translation of the Mysterium (Kepler, 1981) as well as by Alain Segond in his French translation (Kepler, 1984). For a detailed explanation of the geometrical aspects as well as for references to more modern literature, see Pisano and Bussotti (2012, pp. 121–135).
- 7.
An interesting book dealing with “the dissolution of celestial spheres” is Donahue (1981).
- 8.
Obviously, Kepler’s judgement is arbitrary and subjective, but in his view it corresponded to perfection-criteria inherent in the universe.
- 9.
The term used by Kepler is “concinne” (KGW I, p. 29). We explain in the running text how to interpret the meaning of this term.
- 10.
Our interpretation of property (5) is probable, but not certain, in so far as Kepler stated explicitly that the different regular polyhedra “move more appropriately” around different axes, but did not explain why. We suggest that he had an insight of the most important characteristics of the groups of rotations of the Platonic polyhedra. With regard to the rotational symmetries explained by Kepler in the Mysterium see also Pisano and Bussotti (2012, p. 128).
- 11.
The relations between geometrical and theological aspects in the peculiar cosmological structure of the Mysterium (i.e. the series of spheres and Platonic solids) are encapsulated in this statement by Regier (2013, pp. 152–153): “In the second chapter of the Mysterium, creation is portrayed as an act of divine expression, wherein God communicated his being through geometry. Mainly, God wished to describe his status over his creatures’ by the incommensurability of the curve and the line. In Kepler’s account, the creation of matter was necessary to achieve a medium that might express quantity—and quantity (as in extension or dimension) was essential for the realization of geometric form. Reading between the lines, a priori reasoning must incorporate geometric forms that have a status determined by their nearness to the sphere, that is, by their simplicity and their equalities”. As Regier points out, this topic is also addressed by Hon and Goldstein (2008, pp. 170–176).
- 12.
See the XII Chapter, entitled Investigatio Nodorum Martis, of the second part of Astronomia Nova (KGW III, pp. 130–133). To reach this result Kepler made use of Tycho’s observations of the years 1589, 1590, 1592, 1593, 1594, 1595.
- 13.
These results were achieved in two chapters of the second part of the Astronomia Nova, the XIII and the XIV, entitled respectively Investigatio Inclinationis Planorum Eclipticae et Orbitae Martis (ibidem, pp. 133–140) and Plana Eccentricorum Sunt Ἀταλαντα (ibidem, pp. 140–142).
- 14.
For these details a fundamental text is still the chapter on Kepler in Dreyer’s History of planetary systems from Thales to Kepler (1953, original 1906).
- 15.
For Kepler’s complex argumentations through which he arrived at conceiving the ellipticity of the orbits, see Aiton (1978), Baigre (1990), Donahue (1993), Donahue (1996), Stephenson (1987), Whiteside (1974), Wilson (1968). Caspar’s notes at KGW III, pp. 455–484 are a precious guide to the technical aspects of Kepler’s astronomy as expounded in the Astronomia Nova. For Kepler’s deduction of the ellipticity of the orbits see Koyré (1973, [1961], II, II).
- 16.
With regard to Kepler and the area law see Davis (2003), where the author argues that, though the proof in Astronomia Nova is unsatisfying, the one offered by Kepler in the Epitome is correct. Other works containing remarkable insights on the area law are: Aiton (1973), Davis (1992a), Russell (1964) Stephenson (1987). Stephenson (pp. 94–103) offers an accurate account of how Kepler derived the area law. The first scientist who provided a rigorous proof of the area law was Newton in the second section of Principia’s first book. The area law is the first proposition of Newton’s masterpiece. He also proved (Principia, Book I, Section III, Proposition XVI, Corollary 5) that, given an elliptical orbit in which the centripetal force is located in one of the two foci, the speed of the body in each point P is inversely as the perpendicular drawn from the focus to the tangent at P.
- 17.
The movement of the equant was admitted by Ptolemy, as Koyré recalls: Koyré (1973, [1961], p. 402), note 23 to the chapter II—entitled First Attack upon the Theory of Mars—of the section on Kepler. With regard to the relation between dynamical and kinematical aspects in Kepler a fundamental reference text is Stephenson (1987). See also Petroni (1989), who holds that the influence of Kepler’s dynamics upon his kinematics was negligible. He does not agree with Koyré’s opinion, which is, instead, shared by Bernard Cohen (see Cohen, 1975). Our opinion will be expressed in the running text.
- 18.
The whole reasoning that drove Kepler to guess that the shape of Mars’s orbit is an ellipse and not a more complicated oval—as he had thought for many months—starts with his observation of the particular value assumed by the secant of the maximum Mars’ optical equation (Chapter 56 of Astronomia Nova). From this observation, through a series of arduous—and not always easily reconstructable—deductions Kepler arrived at the ellipticity of Mars’ orbit. This topic is skilfully expounded by Stephenson (1987, pp. 107–110). We think that Kepler noticed that the secant’s value is typical of the ellipse because he worked on the problem of Mars’ orbit for months and months. The connection between the secant’s value and the ellipse was undoubtedly facilitated by Kepler’s almost maniacal research on the possible orbit of Mars.
- 19.
Here Kepler assumed that the velocity of a planet is inversely as its distance from the Sun.
- 20.
- 21.
Kepler (1992, p. 55). Cf. KGW, III, p. 25: “Gravitas est affectio corporea, mutua inter cognata corpora ad unitionem seu conjunctionem (quo rerum ordine est et facultas Magnetica)”.
- 22.
Kepler (1992, p. 55). Cf. KGW III, p. 25: “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”.
- 23.
Kepler (1992, p. 55, note 7). Cf. KGW, XV, p. 241: “non tantum lapis ad Terram eat, sed etiam Terra ad lapidem, diuidantque spacium interjectum in euersa proportione ponderum”.
- 24.
Our translation. Cf. KGW VII, p. 283: “Multiplicata enim mole Saturni 10, in densitatem 5, prodiret copia materiae 50, tantundem scilicet, quantum, si molem lovis 5 in densitatem ejus 10 multiplicasses”.
- 25.
Kepler (1992, p. 56). Cf. KGW III, p. 26: “Si Terra cessaret attrahere ad se aquas suas; aquae marinae omnes elevarentur, 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.” As pointed out in Pisano and Bussotti (2018, p. 309), Kepler’s discovery of the origin of the tides was favored by his profound knowledge of Renaissance literature, in this case of Patrizi’s Nova de universis philosophia (see also Goldbeck, 1896, pp. 15–19), for Patrizi referred to the opinions of the ancients on the tides. The ancient opinions were, in many cases, more advanced than those of Kepler’s contemporaries. Actually, Kepler quoted Patrizi in Astronomia Nova, though in a different context (KGW III, p. 62).
- 26.
Galileo (1967, p. 462). Cf. EN VII, p. 486: “Ma tra tutti gli uomini grandi che sopra tal mirabile effetto di natura [le maree] hanno filosofato, piú mi meraviglio del Keplero che di altri, il quale, d’ingegno libero ed acuto, e che aveva in mano i moti attribuiti alla Terra, abbia poi dato orecchio ed assenso a predominii della Luna sopra l’acqua, ed a proprietà occulte, e simili fanciullezze”. It is not easy to guess which of Kepler’s works Galileo read and how thorough his knowledge was. To be precise, Kepler did not speak of copia materiae in the Astronomia Nova, but in the Epitome.
- 27.
In the next pages we will discuss in details the action and the nature of such species immateriata.
- 28.
Our translation. Cf. KGW VII, p. 306: “Deinde si quis dubitat, an magneticae, h. e. terrestres facultates in coelo sint, et an terra, grave corpus, de loco in locum transponi possit a specie immateriata Solis: is Lunam intueatur, quam Terrae cognatam, videt circumire nullo substrato sollido orbe. Valere vero ad inferendum motum species corporum mutuo commeantes, patet in eadem Luna, quae per emissam speciem penes nos movet maria”.
- 29.
Kepler (1992, p. 57). Cf. KGW III, p. 27: “Sequitur enim, si virtus tractoria Lunae porrigitur in Terras usque, multo magis virtutem tractoriam Telluris porrigi in Lunam et longe altius […]”.
- 30.
For an account of the forces (3) and (4) we refer to Pisano and Bussotti (2018, pp. 330–332), where the reader can also find bibliographical indications.
- 31.
This adjective is not used in Astronomia Nova, but in the letter to Fabricius mentioned above in note 23. In this book our aim is not to analyse all the difficult technical details and interpretations of the solar virtue, but to present its main features in order to show the connection between the dynamical and the kinematical aspects of Kepler’s cosmology. For more specific details on the solar virtue, see: Davis (1992b), Granada (2009), Guidi Itokazu (2006, 2007), Holton (1956), Hoyer (1979), Krafft (1991), Rabin (2005), Pisano and Bussotti (2018), Stephenson (1987).
- 32.
Kepler (1992, p. 376). Cf. KGW, III, p. 236: “[…] quo longius abest Planeta a puncto illo, quod pro centro mundi assumitur, hoc debilius illum incitari circa illud punctum. […] Ut hic intentio et remissio motus, cum accessu et recessu a centro mundi, in proportione perpetuo coincidit”.
- 33.
We used the expression “mass” rather than mass to signify that Kepler in Epitome spoke of copia materiae as equal to the volume multiplied by density (KGW, VII, p. 283), but then identified this quantity with weight (ibid., p. 306).
- 34.
The action of this force is explained in the Astronomia Nova (KGW III, pp. 348–364) but more clearly in the Epitome Astronomiae Copernicanae (KGW VII, pp. 337–342). For more specific details on how this force acts see Caspar’s Nachbericht (KGW VII, pp. 598–600), Davis (2003), Pisano and Bussotti (2018, pp. 322–330), Stephenson (1987, pp. 154–172).
- 35.
For a complete treatment of this topic, we refer to the accurate analysis by Stephenson (1987, pp. 146–172).
- 36.
As a matter of fact, the situation is more complicated because Kepler introduced the concept of “magnetic fibers”. However, for our present aims, it is enough to refer to the magnetic axis. For Kepler’s concept of magnetic fibers, see Stephenson’s book mentioned in the previous note.
- 37.
- 38.
For the complete reasoning see KGW VII, Book V, Part II, Chapters IV–V, pp. 390–396. See also Caspar’s Nachbericht (KGW VII, pp. 598–600) and Stephenson (1987, pp. 154–172).
- 39.
In Principia’s Book I, Section III, Proposition XI, Problem VI Newton solved the direct problem of central forces for an ellipse because he proved that, given an elliptical orbit where the centripetal force is in one of the foci, the inverse square law holds. In the following Proposition XII, he proved the same property for a hyperbola and in Proposition XIII for a parabola in which the force is located in the parabola’s focus. In the fundamental Proposition XLI, Problem XXVIII (Book I, Section VIII), Newton solved the inverse problem of centripetal forces: given the force, to find the trajectories and the times of the motions in the trajectories. As a particular case, it follows that if the force is as the inverse square distance between the moving point and the point in which the force is applied, then the trajectory is a conic and the force is in one of the foci.
- 40.
For our purposes, the expression “magnetic fibers” can be replaced with “magnetic axis of the planet”.
- 41.
Kepler (1997, p. 411). Cf. KGW VI, chapter V, p. 302: “Sed res est certissima exactissimaque quod proportio quae est inter binorum quorumcunque Planetarum tempora periodica, sit praecise sesquialtera proportionis mediarum distantiarum, id est Orbium ipsorum” (italics in the text). On the Harmonice and on the role of Kepler’s third law in his system see: Field (2009), Gingerich (1975), Haase (1998), Knobloch (1992, 1995), Stephenson (1994), Vijaya (2019).
- 42.
Kepler maintained the denomination “spheres”, though he obviously knew that they were not spheres in the strict sense of the term.
- 43.
Kepler (1997, p. 411). Cf. KGW VI, chapter V, p. 302: “Rursum igitur hic aliqua pars mei Mysterij Cosmographici, suspensa ante 22. annos, quia nondum liquebat, absolvenda, et huc inferenda est. Inventis enim veris Orbium intervallis, per observationes BRAHEI, plurimi temporis labore continuo; tandem, tandem, genuina proportio Temporum periodicorum ad proportionem Orbium - - sera quidem respexit inertem, Respexit tamen et longo post tempore venit; eaque si temporis articulos petis, 8.Mart. hujus anni millesimi sexcentesimi decimi octavi animo concepta, sed infoeliciter ad calculos vocata, eoque pro falsa rejecta, denique 15. Maji reversa, novo capto impetu, expugnavit Mentis meae tenebras; tanta comprobatione et laboris mei septendecennalis in Observationibus Braheanis, et meditationis hujus, in unum conspirantium; ut somniare me, et praesumere quaesitum inter principia, primo crederem” (italics in the text). The lines “sera quidem respexit inertem, Respexit tamen et longo pòst tempore venit” are drawn from Virgilius’ Eclogues, I, 27–29. The complete sentence is “Libertas, quae sera tamen respexit inertem, / candidior postquam tondenti barba cadebat, / respexit tamen et longo post tempore venit /[…]”. The meaning is that, when Kepler was already old, the inspiration (libertas) finally arrived after a long time and enabled him to complete his cosmological theory with the addition of his third law.
- 44.
Translation from Stephenson (1987, p. 70). Cf. KGW III, p. 240: “Rursum lux rectis effluit orbiculariter, virtus movens rectis quidem sed circulariter; hoc est in unam tantum plagam mundi ab occasu in ortum nititur, non contra, non ad polos.ˮ (Our italics).
- 45.
The exact way in which the virtus motrix is spread is a very problematic question. Until Stephenson (1987), scholars thought that it was spread only in proximity of the Ecliptic. Stephenson (ibid., pp. 67–75) proved this to be false. He proposed a different spread-mechanism which, though very ingenious, was criticized by Guidi Itokazu (2006, 2007). Pisano and Bussotti (2018) proposes a solution of which we have here offered the conclusions, namely that the virtus motrix is spread in circles belonging to planes parallel to the Ecliptic’s. For a complete historiographical account of the interpretations of this problematic and technical step of Kepler’s theory see Pisano and Bussotti (2018, pp. 318–322).
- 46.
Our translation. Cf. KGW I, p. 268: “Etenim **uedo omnis, et omne inflammabile, omne in materiam lucidam mutabile, videtur opus esse cujsdam architectonicae naturalis facultatis, quae nativi caloris in suo corpore sit propagatrix, eoque calore instruatur, ad alterandam et sibi assimilandam eam materiam, in quam incumbit”.
- 47.
On the other hand, we will see that Kepler disregarded completely the approach of those thinkers like Fludd, who also belonged to the Renaissance tradition and who developed speculative and fanciful views without any analysis of the phenomena.
- 48.
Our translation. Cf. KGW I, pp. 241–242: “Placet SCALIGERO, et ipse vehementer approbo, naturalem esse in sideris corpore vim luminis sese spargentis non sine motu […] Spiritus enim vitalis corporeum quippiam est: lumen, sincerissima qualitas, crassa ista impulsione non indiget. Sed tamen analogus aliquis intelligatur in stella motus, non localis, sed alterationis, secundum eam qualitatem, qua stella lucidum est corpus. Imo vero vivum propono hujus alterationis exemplum, in vivis carbonibus; qui ab hoc ipso motu, et quasi vigore alternante, nomen vivorum sunt adepti.” (Italics in the text).
- 49.
Cf. Kepler (1606, p. 267): “Primus igitur ego sententiam dicam: ut habeant caeteri materiam dicendi tanto copiosiorem”.
- 50.
- 51.
Our translation. Cf. Kepler (1606), in KGW I, p. 251: “Sed est alia philosophantium secta, eorum, qui […] non initium ratiocinationis ex sensibus deducunt, neque causas rerum ad experimenta accommodant: sed qui ex abrupto, et quasi quodam Enthusiasmo, concipiunt et de**unt intra sui cerebri parietes, aliquam de Mundi constitutione opinionem”.
- 52.
Our translation. Cf. KGW I, p. 253: “Itaque defendit illam infelix ille JORDANUS BRUNUS: nec obscure asseruit, specie dubitantis, et GULIELMUS GILBERTUS, libro de Magnete, caetera praeclarissimo, religiosum tamen affectum eo demonstravit, quod existimaret non alia re rectius intelligi infinitam Dei potentiam, quam si infinitum mole conderet mundum. Sed BRUNUS ita infinitum facit mundum, ut quot sunt stellae fìxae, tot mundos, et hanc nostram regionem mobilium, unum ex innumerabilibus mundis faciat, nulla fere nota a caeteris circumpositis distinctam: adeoque si quis in stella Canis esset […], ei faciem eandem mundi inde esse apparituram, quae nobis hinc apparet, ex nostro mundo fìxas suspectantibus. Itaque secundum illos, hoc novum sidus, novus aliquis mundus fuerit. Quae sola cogitatio, nescio quid horroris occulti prae se fert; dum errare sese quis deprehendit in hoc immenso; cujus termini, cujus medium, ideoque et certa loca, negantur”.
- 53.
Our translation. Cf. KGW VI, p. 374: “Videas etiam, ipsum plurimum delectari rerum aenigmatibus tenebrosis, cum ego res ipsas obscuritate involutas in lucem intellectus proferre nitar. Illud quidem familiare est Chymicis, Hermeticis, Paracelsistis; hoc proprium habent Mathematici”.
- 54.
Kepler (1981, p. 85). Cf. KGW I, p. 15: “Atque hoc loco nunquam assentiri potui illis, qui freti exemplo accidentariae demonstrationis, quae ex falsis praemissis necessitate Syllogistica verum aliquid infert. Qui, inquam, hoc exemplo freti contendebant, fieri posse, ut falsae sint, quae COPERNICO placent hypotheses, et tamen ex illis vera φαινόμενα tanquam ex genuinis principijs sequantur. Exemplum enim non quadrat. Nam ista sequela ex falsis praemissis fortuita est, et quae falsi natura est, primum atque alij rei cognatae accommodatur, seipsam prodit: nisi sponte concedas argumentatori illi, ut infinitas alias falsas propositiones assumat, nec unquam in progressu, regressuque sibijpsi constet”.
- 55.
- 56.
Simon has made several valuable contributions to the study of Kepler, the most important of which is his Ph.D. dissertation (Simon, 1976) that was partly published in the celebrated (Simon, 1979). The rest has been published recently (Simon, 2019). On the concept of “structure of thought”, see also Simon (1996, 2003). Simon’s studies are always refined, profound and informative. We only intend to express doubts on this particular category if interpreted as a rigid historiographical pattern.
- 57.
For the considerations developed in this paragraph, we refer to Regier (2013, pp. 158–160).
- 58.
Regier (2013, p. 158).
- 59.
KGW VIII, p. 65.
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Bussotti, P., Lotti, B. (2022). Kepler: The Cosmographer Par Excellence. In: Cosmology in the Early Modern Age: A Web of Ideas. Logic, Epistemology, and the Unity of Science, vol 56. Springer, Cham. https://doi.org/10.1007/978-3-031-12195-1_3
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