Kepler and Bruno on the Infinity of the Universe and of Solar Systems

We omit the starless spheres (the ninth, tenth, eleventh, …) postulated since Ptolemy to account for the different motions of the stars (precession, trepidation, …). In this spirit, the Jesuit astronomer Christoph Clavius had adopted a geocentric universe of eleven spheres. See Lattis J. L. , Between Copernicus and Galileo: Christopher Clavius and the collapse of Ptolemaic cosmology (Chicago and London, 1994). The complete works of the authors studied are quoted according to the following editions: Kepler Johannes , Gesammelte Werke, ed. by Caspar M. (Munich, 1937–), indicated by KGW followed by the volume number; and Giordano Bruno, Opera latine conscripta, ed. by Fiorentino F. (Naples and Florence, 1879–91), indicated by BOL followed by the volume number. Other editions of single works are fully quoted in their first appearance. Italics in the quotations are always ours unless otherwise indicated.

Cf. Caspar Max , Kepler, transl. by Hellman C. Doris , 2nd edn (New York, 1993), 105 ff.

See Koyré A. , From the closed world to the infinite universe (Baltimore, 1957), 38 f; Granada M. A. , “Thomas Digges, Giordano Bruno e il copernicanesimo in Inghilterra”, in Giordano Bruno 1583–1585: The English experience / L'esperienza inglese, ed. by Ciliberto M. Mann N. (Florence, 1997), 125–55, pp. 133–7.

Digges's famous diagram of the universe said about the stars that they were “farr excellinge our sonne both in qvantitye and qvalitye”. See the reproduction in Koyré, op. cit. (ref. 3), 37.

For an exposition of the meaning of this term in Bruno, see Granada M. A. , “Synodus ex mundis”, Bruniana & Campanelliana, xiii (2007), 149–56.

Cf. Mysterium cosmographicum, KGW, i, 9; English translation by Duncan A. M. , The secret of the universe (New York, 1981), 63.

This expression, which identifies the Word of God (Christ) with the universe itself, appears in the Italian dialogue De la causa, principio et uno (1584). See De la causa, principio et uno, in Bruno G. , Oeuvres complètes, iii (Paris, 1996), 207.

Cf. Granada M. A. , “Considerazioni sulla disposizione ed il movimento del sole e delle stelle in Giordano Bruno”, Physis, xxxviii (2001), 257–82. For the cometary theory of Hippocrates of Chios and Aeschylus, two presocratic authors adduced by Bruno (see De immenso, BOL, i/2, 229 f.) against Aristotle's theory, see now Tessicini D. , I dintorni dell'infinito. G. B. e l'astronomia del Cinquecento (Pisa and Rome, 2007), 188 ff., and Wilson M. , “Hippocrates of Chios's theory of comets”, Journal for the history of astronomy, xxxix (2008), 2008–60.

KGW , xiii, 450 (letter no. 272): “ego opinor mundos esse infinitos; unusquisque tamen mundus est finitus sicut Planetarum in cuius medio est centrum Solis. Et quemadmodum tellus non quiescit sic neque Sol; Volvitur namque velocissime in suo loco circa axem suum; quem motum sequuntur reliqui Planetae: In quorum numero Tellurem existimo; sed est tardior unusquisque quo ab eo distat longior. Stellae etiam sic moventur ut Sol; sed non illius vi sicut Planetae circumaguntur; quoniam unaqueque earum Sol est, in non minori mondo hoc nostro Planitarum. Elimentalem mundum nobis proprium et particularem non puto: Nam aer est et inter ipsa corpora: Quae stellas vocamus; per consequens et ignis et aqua et terra…. Planetae vero a Sole eorum lumen assumunt.” Bruce seems to combine here Bruno's cosmology (Bruce's ‘mundus’ being the ‘synodus ex mundis’ of Bruno, who occasionally called it ‘mundus’ as well) with Kepler's conception of the solar system. Thus, the notion of a solar force moving the planets around the Sun with periods proportioned to their respective distances probably derived from the reading of the Mysterium cosmographicum (cf. chap. XX, KGW, i, 68 ff.; English translation, Duncan, op. cit. (ref. 6), 197 ff.).

KGW, xiii, 450: “et mitte has literas ad tuum Vicinum et meum amicum a quo responsum expecto”.

Ibid., xvi, 142 (letter no. 488, 5 April 1608): “Brunum Romae crematum ex Wackherio didici, ait constanter supplicium tulisse”.

Ibid.: “Religionum omnium vanitatem asseruit, Deum in Mundum in circulos in puncta convertit”.

See Sturlese R. , Bibliografia, censimento e storia delle antiche stampe di Giordano Bruno (Florence, 1987), p. xxxi and ad indicem.

“Nuper enim apud te vidi volumina rerum singularium et rararum.” We quote from Kepler J. , The six-cornered snowflake, transl. by Hardie C. (Oxford, 1966), 4–5.

Of this work, only four copies have survived, none of them connected with Wackher or the Emperor. Cf. Sturlese , op. cit. (ref. 13), 99.

This copy is preserved in the National Library of Prague (Clementinum). On this, see Sturlese , op. cit. (ref. 13), 92 f., and eadem, “Su Bruno e Tycho Brahe”, Rinascimento, 2nd ser., xxv (1985), 309–33. The verdict by Tycho on Bruno's cosmological ideas was strict and summary: “Nullanus nullus et nihil, Conveniunt rebus nomina saepe suis”, he wrote at the end of the volume, making fun of Bruno's denomination ‘Nolanus’, after his birthplace Nola, in the vicinity of Naples.

See Sturlese , op. cit. (ref. 13), 125.

“Ponimus hoc quod summa probabimus evidentia. Duo in universo praecipua primorum corporum genera, Soles nempe atque Tellures”, BOL, i/1, 212 (original edn, Frankfurt, 1591, 159; see reproduction in Figure 2).

Ibid.: “De primo genere fixae (quas appellant) stellae sunt, de quarum singularum loco non maior neque aliter sol iste spectabilis esset quam illae a loco istius solis et a nostris sunt spectabiles regionibus”.

De stella nova, KGW , i, chap. XXI, 253–6. See chaps. XX (“If the matter and the body of the new star existed previously”), 248–51, and XXI (“If this star has been endowed with its motion in the profound ether, and if the sphere of fixed stars extends infinitely”), 251–7.

Ibid., 254. 23–25: “Hoc ego thema cum olim quibusdam proposuissem; qui, ut me exercerent, infinitatis causam, ex ante dictis authoribus susceptam, adversum me propugnabant acriter…”.

On this see Voelkel J. , The composition of Kepler's Astronomia nova (Princeton and Oxford, 2001), 170–210. Fabricius had expounded his theory of the nova's origin and significance in his letter of 14 Jan. 1605 (old style; letter no. 319), KGW, xv, 117 f. Kepler answered critically in his letter of 11 Oct. 1605 (no. 430), KGW, xv, 257 f. In his letter, Kepler referred also to a tract on the nova in Dutch dialect published by Fabricius, which is not extant. Fabricius published three further treatises on the nova, two in German and one in Latin.

“… Deus illas illuminet certis temporibus, ad praesignificanda bona vel mala hominibus”, letter by Fabricius to Kepler, 14 Jan. 1605, KGW , xv, 117. See also De stella nova, KGW, i, 248.

De stella nova, 249 f. Letter to Kepler, 14 Jan. 1605, KGW, xv, 118.

De stella nova, 249: “Itaque lucem producere, creare est”.

Ibid., 251: “Satis opinor patere, causam nullam idoneam esse, cur quis existimet, novas istas stellas prius extitisse, quam viderentur; et postquam extinctae sunt, reservari superstites, ad novam illuminationem”.

Ibid., 248: “non lumen tantum, sed et corpora ipsa in coelo existere repentina et nova”.

Ibid., lines 21–26.

Ibid., 257. 23–24: “Priusquam autem ad creationem [by God], hoc est ad finem omnis disputationis, veniamus, tentanda omnia existimo.” On this point, Bruno agrees with Kepler. Cf. Granada M. A. , “Cálculos cronológicos, novedades cosmológicas y expectativas escatológicas en la Europa del siglo XVI”, Rinascimento, 2nd ser., xxxvii (1997), 357–435, pp. 423 f.

De stella nova , 258 f. Cf. 259. 25–26: “Itaque potius in eo sum; ut credam, coelum undiquaque aptum ad materiam hisce sideribus praebendam”.

Ibid., 251 f. For Tycho, see Astronomiae instauratae progymnasmata, in Brahe Tycho , Opera omnia, ed. by Dreyer J. L. E. (Copenhagen, 1913–29), iii, 78 (for Gemma), 204 (for Dee, Wilhelm IV, Gemma).

De stella nova, 252. 35–38: “Verum et alias saepe, et nunc iterum abrumpo disputationem, quoties ad absolutam Dei potentiam provocant. Certum enim, nihil nos ad rem dicere posse, quod quicquam in ullam partem habeat momenti, si naturae terminos excesserimus.” For Bruno's similar position, see ref. 29 above.

Ibid., lines 38–39: “Hoc potius illis dicamus, illâ fixarum infinitate seipsos, ceu labyrinthis inexplicabilibus induere”.

Ibid., 252. 40–253. 3.

Ibid., 253. 3–8: “Itaque defendit illam [infinitatem] 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 aliâ re rectius intelligi infinitam Dei potentiam, quam si infinitum mole conderet mundum.” It has been assumed that Kepler here attributes to Gilbert the connection between an infinite universe and an infinite God's power; see e.g. Koyré, op. cit. (ref. 3), 60–61. Gilbert, however, does not affirm in De magnete the infinity of the universe as a corollary of God's infinite power. Moreover, he does not affirm even the infinity of the universe or of the stellar sphere, speaking instead with Copernicus of the immensity of this sphere “cuius finis ignoratur scirique nequit”; see Lerner M.-P. , Le monde des sphères, ii: La fin du cosmos classique (Paris, 1997; 2nd edn, 2008), 148 (in both editions). For this reason, we believe that Kepler's quoted passage in De stella nova can be understood as referring by “religiosum” to Bruno, the sentence speaking explicitly of Gilbert being a kind of parenthesis. It is true that in the Epitome astronomiae copernicanae (KGW, vii, 81) Kepler will attribute the connection to Gilbert, but in this case he probably reads in the other sense the ambiguous text of 1606. In De immenso, chap. III, 1, Bruno rejects the scholastic distinction between God's absolute and ordained power (potentia absoluta et ordinata) and affirms that God's production reflects necessarily his infinite power and essence; see 1st edn, p. 266: “Ubi nihil prohibit ab uno infinito principio (cui non sit plus difficile facere duo quam unum finitum, & innumerabilia quam duo) infinita in eodem genere provenire” (underlining in Wackher's copy; cf. Figure 3). For an analysis of Bruno's conception, see Granada Miguel A. , “‘Blasphemia vero est facere Deum alium a Deo’: La polemica di Bruno con l' aristotelismo a proposito della potenza di Dio”, in Letture bruniane I.II del Lessico Intellettuale Europeo 1996–1997, ed. by Canone E. (Pisa and Rome, 2002), 151–88.

De stella nova, 253. 8–14 (English translation by Koyré).

De immenso , chap. IV, 3, in BOL, i/2, 16.

Ibid., 20: “Propinquiores soles, quorum quoque terras a terris istius synodi minus distare necesse est, sunt astra fixa maiora, quae primae dicuntur et accipiuntur magnitudinis”.

Ibid., 21: “Hinc patet quod si essemus in uno de astris illis primae magnitudinis, sol iste pariter primae magnitudinis astrum videretur.” Cf. also chap. I, 3, 159 (in 1st edn, reproduced as Figure 2), quoted above, ref. 19.

De stella nova, 253. 15–17 (translation by Koyré).

Quoted above.

De immenso, chap. III, 1, in BOL, i/1, 318: “Omnia quippe argumenta quae sunt … ex differentiis sursum & deorsum, extimi & intimi, centri et circumferentiae … sunt passiones finiti, nec probant sed supponunt finitum, petunt finitum, accipiunt finitum” (underlining in Wackher's copy).

Cf. De immenso, chap I, 3, in BOL, i/1, 209 (“lege necesse est naturae, flammas [suns] fomentum sumere ab undis”; and 213 (“opaca planetarum corpora, in quibus elementum aquae dominatur”); chap. IV, 3, in BOL, i/2, 20 (“lymphae”). See also Granada, op. cit. (ref. 5).

De immenso , chap. I, 1, in BOL, i/1, 201 f.: “Intrepidus spacium immensum sic findere pennis / Exorior, neque fama facit me impingere in orbes, / Quos falso statuit verus de principio error, / Ut sub conficto reprimamur carcere vere, / Tanquam adamanteis cludatur moenibus totum…. / Aethereum campumque ex omni parte pererro, /Attonitis mirum et distans post terga relinquo” [“Fearless I rise, endowed with feathers, to traverse infinite space, and fame does not cause me to collide with the orbs established from a false principle by a true error, so that we were truly oppressed in a feigned prison and the whole was closed by diamantine walls…. I run everywhere through the ethereal field, leaving at my back in the distance the glance of the astonished”].

De stella nova, 253. 20–27.

Ibid., 253–6; Koyré, op. cit. (ref. 3), 61–72.

Cf. De immenso, chap. I, 3, in BOL, i/1, 209: “Ut solem hunc circa Tellus, Luna, aliger Hermes, / Saturnus, Venus et Mavors, et Juppiter errant, / Et numerus fasso major, nam caetera turba / Partim pro vicibus, partim non cernitur unquam, / Sic circum fit quemque alium” (English translation given above, Section 1).

Galileo's work was printed in March, Kepler's Dissertatio in May. On 19 April, Kepler presented to the Florentine ambassador in Prague a letter to Galileo, which was the written answer requested by Galileo. The Dissertatio is the printed version of this letter. For the slight divergencies between both versions see KGW, iv, 506 f.

Kepler's Conversation with Galileo's Sidereal Messenger, translation, introduction and notes by Rosen E. (New York and London, 1965), 25 f.

Ibid., 26. This indicates that Kepler's decision to write the Somnium (whose first idea goes back to a failed dissertation in the Tübingen years in defence of the movement of the Earth) occurred in the course of this intense and enduring discussion with Wackher on Bruno's cosmology and metaphysics. For the Somnium, see the edition by Rosen E. : Kepler's Somnium: The Dream, or posthumous work on lunar astronomy (Madison, Milwaukee and London, 1967). The composition of the Strena (later dedicated to Wackher as a New Year's gift for 1611) at the beginning of 1610 also indicates a serious discussion with Wackher on philosophical matters regarding the minimum, that is, Bruno's conception of minimal particles. In this case, the discussion revolved around Bruno's De minimo — Present, as I said, in Wackher's library — Probably in connexion with Bruno's Articuli adversus mathematicos. For an account of Kepler's debt to Bruno in his thoughts on snow particles, see Lüthy Ch. , “Bruno's Area Democriti and the origins of atomist imagery”, Bruniana & Campanelliana, iv (1998), 1998–92, especially pp. 79–83. Kepler's debt to Bruno has been recognized also by M.-L. Heuser-Kessler in her essay “Maximum und Minimum: Zu Brunos Grundlegung der Geometrie in den Articuli adversus mathematicos und ihre weiterführende Anwendung in Keplers Neujahrsgabe oder Von sechseckigen Schnee“, in Die Frankfurter Schriften Giordano Brunos und ihre Voraussetzungen, ed. by Heipcke K. Neuser W. Wicke E. (Weinheim, 1991), 181–97. As rightly indicated by Lüthy, Kepler's mocking words in Strena to himself (“My endeavour to give almost Nothing [Nihil] almost comes to nothing! From this almost Nothing I have almost formed the all-embracing Universe itself!”, Kepler, op. cit. (ref. 14), 39) seem to be addressed actually to Bruno, possibly a pun played on Bruno's patronymic Nolanus, as Tycho had done previously. See ref. 16 above. Wackher's copy of De minimo (preserved in Prague's Clementinum) bears some traces of reading according to Sturlese; see Bibliografia, censimento e storia delle antiche stampe di Giordano Bruno (ref. 13), 113 f. (no. 102). Nevertheless, these traces do not correspond to the chapters pertaining to the problem discussed by Kepler. I am grateful to Petr Hadrava and Alena Hadravova for having verified this point for me.

Kepler's Conversation with Galileo's Sidereal Messenger (ref. 49), 11.

Ibid., 10.

Ibid., 11.

Ibid., 34.

Ibid., 36 f.

Ibid., 39. Cf. the Latin text: “At putabant fixas stellas esse quae sic circumirentur; causam etiam dixit Brunus cur esset necesse: Fixas quippe Solaris et igneae esse Naturae, Planetas aqueae; et fieri lege Naturae inviolabili, ut diversa ista combinentur, neque Sol Planetis, ignis aqua sua, neque vicissim haec illo carere possit” (KGW, iv, 305. 13–17; italics ours).

Cf. De immenso, chap. I, 3, in BOL, i/1, 209–13, especially 209 f.: “Ut solem hunc circa Tellus, Luna, aliger Hermes, / Saturnus, Venus et Mavors, et Juppiter errant, / Et numerus fasso major, nam caetera turba / Partim pro vicibus, partim non cernitur unquam, / Sic circum fit quemque alium: nam lege necesse est / Naturae, flammas fomentum sumere ab undis…. Sic circum unumquemque Phoebum cytharoedum / Plures discurrunt Nymphae … / Quas vegeto sensu, ac clara ratione videmus, / Quando unam ad normam venit abstans atque propinquum, / Nec variat numerus primorum principiorum” (italics ours). Later in the Conversation, Kepler speaks of the Sun as “truly an Apollo, the term frequently used by Bruno” (45). If in the passage just quoted Apollo is designated as “Phoebum Cytharoedum”, Bruno does speak frequently of Apollo in this sense. See e.g. De immenso, chap. IV, 3, in BOL, i/2, 20: “hinc circa unum medium plures nymphae seu Musae, inde unus intra plures nymphas Apollo”.

Kepler's Conversation with Galileo's Sidereal Messenger (ref. 49), 36 f.

See Kepler J. , Narratio de observatis a se quatuor Iovis satellitibus erronibus, KGW , iv, 317–25. Cf. also the recent edition with French translation by Pantin I. Kepler J. , Dissertatio cum Nuncio sidereo: Narratio de observatis Jovis satellitibus (Paris, 1993).

Kepler's Conversation with Galileo's Sidereal Messenger (ref. 49), 14.

Ibid., 47.

Ibid., 39. Cf. the Latin text: “Primum esto ut fixa quaelibet Sol sit, nullae illas [fixas] Lunae hucusque circumcursitare visae sunt: Hoc igitur in incerto manebit, quoad aliquis subtilitati observandi mira instructus, et hoc detexerit: Quod quidem hic successus tuus, iudicio quorundam nobis minatur“, KGW, iv, 305. 18–22 (italics ours).

Kepler's Conversation with Galileo's Sidereal Messenger (ref. 49), 137, note 340 to this passage. For the variant in the letter sent to Galileo, cf. KGW, iv, 506.

A few lines below, the “recent gathering of certain philosophers”, where it was observed that the new moons of Jupiter are inhabited (Conversation, 39), is said in the letter to Galileo to have been held “at Wackher's dinner table” (ibid., 138, note 345). Cf. KGW, iv, 506: “quod nuperrime in mensa nostri Vakherii iucunde motum”.

More precisely, Kepler could concede that every fixed star was like the Sun an igneous body, shining by itself with a proper light (see ref. 54 above), but not that it was indifferent to the Sun in size and brightness.

Kepler's Conversation with Galileo's Sidereal Messenger (ref. 49), 34 f.

Ibid., 35. As has been repeatedly observed, Kepler here formulates the so-called ‘Olbers's Paradox’. On this see Harrison E. , Darkness at night: A riddle of the universe (Cambridge, MA, 1987), 49 f. on Kepler.

Kepler's Conversation with Galileo's Sidereal Messenger (ref. 49), 35 f.

Ibid., 36.

Ibid., 43.

Ibid.

See Mysterium cosmographicum, KGW , i, 24. 1–6: “Cum igitur Idaeam mundi Conditor animo praeconceperit … ut forma futuri operis et ipsa fiat optima: Patet quod … nullius rei Idaeam pro constituendo mundo suscipere potuerit, quam suae ipsius essentiae” [“Since, then, the Creator conceived the Idea of the universe in his mind … so that the Form of future creation may itself be the best: It is evident that … the only thing of which he could adopt the Idea for establishing the universe is his own essence”, English translation, op. cit. (ref. 6), 93]. Cf. the later formulation in the Harmonice mundi, KGW, vi, 223. 32–34: “Geometria ante rerum ortum Menti divinae coaeterna, Deus ipse (quid enim in Deo, quod non sit Ipse Deus) exempla Deo creandi mundi suppeditavit, et cum imagine Dei transivit in hominem” [“Geometry, which before the origin of things was coeternal with the divine mind and is God himself (for what could there be in God which would not be God himself?), supplied God with patterns for the creation of the world, and passed over to Man along with the image of God”, The Harmony of the World by Johannes Kepler, transl. by Aiton E. J. Duncan A. M. Field J. V. (Philadelphia, 1997), 304].

Mysterium cosmographicum, KGW , i, 13 (Preface to the reader), 25 f. (chap. II) [English translation, 67 f., 97 f.]. See Field J. V. , Kepler's geometrical cosmology (London, 1988), chap. 3.

Kepler's Conversation with Galileo's Sidereal Messenger (ref. 49), 43 f.

In this formulation, ‘mundus’ means ‘heavenly body’. This is the most usual meaning of ‘mundus’ in Bruno. Nevertheless, he often uses also the word in the sense of ‘universe’ and of ‘planetary system’. See Granada , “Synodus ex mundis” (ref. 5), 149–56.

Kepler's Conversation with Galileo's Sidereal Messenger (ref. 49), 44.

Ibid.: “Suppose those infinite worlds are unlike ours. Then they will be supplied with something different from the five perfect solids. Hence they will be less noble than our world”.

De stella nova, KGW , i, 252. 30–253. 3. For a contemptuous judgement in Bruno of contemporary astronomers, see his Articuli adversus mathematicos, dedicated to the Emperor and probably known to Kepler: “Sphaerae ergo mundanae corporum ordo, qualem fingunt et pingunt pauperes isti [mathematici, i.e. astronomers], nusquam est”, BOL, i/3, 77.

Huygens C. , Cosmotheoros , in Oeuvres complètes (The Hague, 1888–1950), xxi, 810 f. (our translation). Huygens, who nevertheless describes Kepler as “cet Homme si génial, qui fut le grand instaurateur de l'Astronomie” (ibid., 812), conceded to the nearest star to our Sun (Sirius) a distance of 27,664 a.u. (ibid., 816). Huygens accepted for any pair of stars a distance separating them of at least this amount (ibid.). Thus, he arrived at a view similar to that of Bruno, except that he considered the infinite number of stars as unsure (ibid.). On Huygens, see Dick S. J. , Plurality of worlds: The origins of the extraterrestrial life debate from Democritus to Kant (Cambridge, 1982), 127–35, and now Seidengart J. , Dieu, l'univers et la sphère infinie: Penser l'infinité cosmique à l'aube de la science classique (Paris, 2006), 554–60. Huygens owned in his library Bruno's Italian dialogue De l'infinito, universo e mondi and also the Latin philosophical poems De monade and De minimo. On this, see Sturlese , op. cit. (ref. 13), 57, 122.

KGW , i, 12–13: “Quantitas enim initio cum corpore creata; coeli altero die” [“For quantity was created in the beginning along with matter, but the heavens on the second day”, English translation, 67]. Cf. the note to this page in the second edition (1621), KGW, viii, 30: “Imo Ideae quantitatum sunt erantque Deo coaeternae, Deus ipse” [“Rather the Ideas of quantities are and were coeternal with God, and God himself”, English translation, 73].

English translation, 93 [“Hac enim una re divinus mihi Cusanus, aliique videntur: Quod Recti, Curvique ad invicem habitudinem tanti fecerunt, et Curvum Deo, Rectum creaturis ausi sint comparare”, KGW, i, 23].

Ibid.: “By a most perfect Creator it was absolutely necessary that a most beautiful work should be produced” [“a Conditore perfectissimo necesse omnino fuit, ut pulcherrimum opus constitueretur”, KGW, i, 23]. Kepler appeals for this to the authority of Plato in the famous passage in Timaeus 29d–30a through the translation by Cicero. If theology affirmed traditionally that God had created the world through the Word by appealing to the exemplary ideas in it, Kepler considered that the mathematical archetypes constituting God's very essence were the model. On this, see Marion J.-L. , Sur la théologie blanche de Descartes: Analogie, création des vérités éternelles et fondement (Paris, 1981), 178–203. According to Marion, there is in Kepler “une transcription de l'exemplarisme en termes de mathématiques, ou, plus exactement, d'une attribution aux mathématiques du statut, des propriétés et des fonctions jusqu'alors reconnues aux idées divines. Les idées divines régressent au rang des ideae quantitatum“, ibid., 180. Moreover, “Dieu ne pratique pas tant les mathématiques qu'il ne consiste en elles…. Sans doute, Dieu tout entier ne s'épuise-t-il pas dans les vérités mathématiques, mais toutes les vérités mathématiques s'inscrivent en Dieu comme Dieu même…; [les vérités mathématiques] non seulement se soustraient elles-mêmes à la création, mais président à la création de toutes autres choses”, ibid., 182 f. For this reason, Marion considers it very probable that Descartes formulated his doctrine of the creation by God of the eternal mathematical truths as a reaction against the necessitarianism implicit in Kepler's conception.

The secret of the universe, 93 [Mysterium cosmographicum, KGW, i, 23: “Dei triuni imago in sphaerica superficie, Patris scilicet in centro, Filii in superficie, Spiritus in aequalitate inter punctum et ambitum”]. Kepler found this analogy in the Complementum theologicum of Nicholas of Cusa and in De harmonia mundi (1525) of the Venetian Platonist Francesco Giorgio Veneto (c. 1460–1540). See Mahnke D. , Unendliche Sphäre und Allmittelpunkt (Halle, 1937), 141 f. and 106 f.

See Epitome astronomiae Copernicanae, KGW , vii, 47.: “Mundi Archetypus Deus ipse est, cujus nulla figura similar est … quam sphaerica superficies. Nam uti Deus est ens Entium, antecedens omnia, ingenitum, simplicissimum, perfectissimum, immobile, sibi ipsi creaturisque omnibus sufficientissimum, creans et sustentans omnia, unus essentia, in personis trinus: Sic sphaericum etiam easdem rudi quodam modo proprietates habet inter figuras caeteras”.

Ibid., 51: “In Sphaerico tria sunt, Centrum, superficies, et aequalitas intervalli; quorum uno negato caetera corruunt, suntque distincta inter se, ut unum non sit alterum. Centrum est quasi Origo Sphaerici…. Centrum seipso est invisibile et impervestigabile; monstratur vero undique flexu aequabilissimo superficiei, mediante aequabilitate intervalli. Itaque superficies est character et imago centri, et quasi fulgor ab eo, et via ad id; et qui superficiem videt, is eo ipso videt et centrum, non aliter. Intervallum resultat ex comparatione Centri cum superficie, et sic procedit ab utroque.” The implicit references to the relations between the persons in the Trinity are obvious: The intervallum (‘aura etherea’ where the six planets are disposed according to the distances determined by the five polyhedra) proceeds, as the Holy Spirit “proceedeth from the Father and the Son” according to the Creed, “from both the Centre and the surface”; for Christ as way to the Father, see John 14: 6–7: “I am the way, the truth, and the life: No man cometh unto the Father, but by me. If ye had known me, ye should have known my Father also.” Cf. Hübner J. , Die Theologie Johannes Keplers (Tübingen, 1975), 191–2; Hallyn F. , La structure poétique du monde: Copernic, Kepler (Paris, 1987), 187 ff.

Field , op. cit. (ref. 73), 18.

Ibid., 53.

Cf. De stella nova, chap. XXI, KGW , i, 256 f; Epitome, KGW, vii, 45 f. See also Seidengart , Dieu, l'univers et la sphère infinie (ref. 79), 356 f., 367–73, and Segonds A. , “Kepler et l'infini”, in Monnoyeur F. (ed.), Infini des philosophes, infini des astronomes (Paris, 1995), 21–40, especially pp. 23, 33.

Epitome of Copernican astronomy & Harmonies of the world , transl. by Wallis C. G. (Amherst, NY, 1995), 45 [KGW, vii, 287 f.: “Cum haec tria corpora sint analoga centro, superficiei sphaericae, et intervallo, tribus Symbolis trium in SS. Trinitate personarum: Credibile est tantundem esse materiae in uno, quantum in uno quolibet duorum reliquorum; sic ut tertia pars materiae totius universi compacta sit in corpus Solis, quamvis id sit respectu amplitudinis mundi angustissimum: Tertia item pars materiae extenuata et explicata per immensum mundi spacium…: Tertia denique pars materiae expansa in orbem, et mundo exterius pro moenibus circumjecta”].

Epitome of Copernican astronomy , 46; KGW, vii, 289.

Ibid.

This formulation constitutes the second definition of God in the medieval treatise attributed to Hermes Trismegistus, Liber XXIV philosophorum: “Deus est sphaera infinita cuius centrum est ubique, circumferentia nusquam.” Cf. Liber viginti quattuor philosophorum , ed. by Hudry F. (Corpus Christianorum, Continuatio Mediaevalis, 143 A; Brepols, 1997), 7–8.

Cf. the citation by Koyré, op. cit. (ref. 3), 17: “Thus, the fabric of the world (machina mundi) will quasi have its center everywhere and its circumference nowhere, because the circumference and the center are God, who is everywhere and nowhere”.

See Mahnke , op. cit. (ref. 83), 130–3, 141–3, and more recently Meier-Oeser S. , Die Präsenz des Vergessenen: Zur Rezeption der Philosophie des Nicolaus Cusanus vom 15. bis zum 18. Jahrhundert (Münster, 1989), 286, 290f.

Cf. De stella nova, chap. XXI, 256f.: “omnis figura finibus quibusdam est circumscripta, hoc est finita vel finiens”; Epitome, KGW, vii, 45. 36–40.

Bruno G. , On the infinite universe and worlds , transl. by Singer D. Waley in eadem, Giordano Bruno: His life and thought, with annotated translation of his work On the infinite universe and worlds (New York, 1968), 260. Cf. De l'infinito, universo e mondi, in Bruno G. , Oeuvres complètes, iv (Paris, 2006), 85: “Perché volete che quel centro della divinità, che può infinitamente in una sfera (se cossì si potesse dire) infinita amplificarse, come invidioso, rimaner più tosto sterile che farsi comunicabile, padre fecondo, ornato e bello?” The impossible ‘envy’ in God towards His creation of the best (infinite, according to Bruno) world alludes to Plato's Timaeus, 29e. For Bruno's adoption of the ‘principle of plenitude’, see the classic study by Lovejoy A. O. , The Great Chain of Being: A study of the history of an idea (New York, 1960), 116–21. For Bruno's logic of omnipotence, see Granada M. A. , “Il rifiuto della distinzione fra potentia absoluta e potentia ordinata di Dio e l'affermazione dell'universo infinito in Giordano Bruno”, Rivista di storia della filosofia, xlix (1994), 1994–532.

De immenso, chap. II, 9, in BOL, i/1, 291: “Hoc [the universe] est quod sphaeram definivit Xenophanes infinitam, cuius centrum est ubique, circumferentia nusquam…. Sic infinitum nusquam est circumferentialiter, ubique est centraliter.” Cf. Articuli adversus mathematicos, BOL, i/3, 72: “For us [Bruno says against contemporary mathematicians or astronomers] the universal sphere is one continuous universe, infinite and immovable, where there are infinite spheres or particular worlds [Nobis sphaera universalis est unum continuum universum infinitum immobile, seu in quo consistentia sunt numero infinitae sphaerae seu particulares mundi]”. Cf. Mahnke, op. cit. (ref. 83), 49–59, and 53 note 1 for a list of references.

This is probably the reason for Kepler's statement, in a letter to Brengger, that Bruno “transformed God in the world, in circles, in points”. See ref. 12 above, and the recent study by A. Del Prete: “‘Une sphère infinie dont le centre est partout et la circonférence nulle part’: L'omnicentrisme chez Giordano Bruno”, in Tinguely F. (ed.), La Renaissance décentrée (Geneva, 2008), 33–47.

See Granada M. A. , “Aristotle, Copernicus, Bruno: Centrality, the principle of movement and the extension of the universe”, Studies in history and philosophy of science, xxxv (2004), 91–114, and Tessicini , op. cit. (ref. 8).

Bruno G. , The Ash Wednesday Supper , transl. and ed. by Gosselin E. A. Lerner L. S. (Hamden, CT, 1977), 203 [La cena de le ceneri, in Oeuvres completes, ii (Paris, 1994), 233: “Non è più degno d'esser chiamato ottava sfera dove à la coda de l'Orsa, che dove è la terra, nella quale siamo noi”].

Newton I. , MS. Add. 4005, fols. 21–22, in Unpublished scientific papers of Isaac Newton: A selection from the Portsmouth Collection in the University Library Cambridge , ed. by Hall A. R. Hall M. Boas (Cambridge, 1962), 374 f.

“And if the fixed Stars are the centers of similar systems, they will all be constructed according to a similar design and subject to the dominion of One…. And so that the systems of the fixed stars will not fall upon one another as a result of their gravity, he has placed them at immense distances from one another”, Newton Isaac , The Principia: Mathematical principles of natural philososophy, transl. by Cohen I. Bernard Whitman Anne (Berkeley, Los Angeles and London, 1999), 940.

See Hoskin Michael , “Newton, Providence and the universe of stars”, Journal for the history of astronomy, viii (1977), 77–101, and now idem, “Gravity and light in the Newtonian universe of stars”, Journal for the history of astronomy, xxxix (2008), 2008–64. Needless to say, God's providence is in Newton very different from Bruno's concept of it, since the Italian philosopher identifies it with the very same law of nature related to his pantheism. Bruno's concept, which included the principle of motion proper to matter, formed the basis for John Toland's attack against the Newtonian ideology promulgated from the pulpit in the Boyle lectures. For this, see Jacob M. C. , “John Toland and the Newtonian ideology”, Journal of the Warburg and Courtauld Institutes, xxxii (1969), 1969–31; and eadem, The Newtonians and the English Revolution, 1689–1720 (Cornell, 1976), chap. 6.