Kepler’s snow: the epistemic playfulness of geometry in seventeenth-century Europe

Abstract

This paper suggests that layered ontology is important within Kepler’s method, and that it developed at least partially in response to a disciplinary and religious crisis. As such, and despite Platonic allegiances, it places him in a longue durée of geometrical constructivism in seventeenth-century Europe. After introducing the pivotal role of Mysterium cosmographicum (1596) and how Kepler’s career may be seen in the context of courtly bricolage, the exposition realigns De nive sexangula (1611) with the mathematical communities of its time and argues in Reviel Netz’s tradition that its cognitive practices enact a ludic style of demonstration. Kepler’s essay on crystallography is an epistemological improvement on previous types of natural jokes.

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No potential conflict of interest was reported by the author(s).

Notes

2 Pointing to a demarcation of playful and serious, Vassányi (2020) observes how Kepler remained ostensibly jovial, yet intrinsically devastated by the attempts made to influence his theological persuasion.

3 JKGW, 1, 82–84; on Maestlin’s prefatory letter ‘Candido lectori’ see (Omodeo 2014, 234–254). This, and the reissue of Rheticus, emphasizes the underlying, neo-Melanchthonian character of the collective volume, including, I suggest, the place of geometry in the edifice of natural philosophy.

4 A full review of the scholarship is beyond the present scope. Andrews (2017, 294–300) denies Kepler’s agency, making Tabula III an opportunistic move at the printer’s behest, but fails to present evidence that Kepler’s organizational system would be threatened or undermined by ludic metamorphoses. Aït-Touati (2011, 17–44) is correct about the mutual disciplinary borrowings of the cosmological discourse and offers an intervention on Kepler’s Mysterium that is focused on the nested solids as a fictional, interlocking experiment. Shank (2020) uses turned ivories to discuss the unilateral claims of art and science on Kepler’s astrophysics. Crowther and Barker (2013) recall the Augustinian tradition of ‘reading’ in the mind’s eye and note how cosmic cross-sections played an essential role in the pedagogical process of Sphaera and Theorica texts, which Kepler intends to borrow and replace in a natural continuation of astronomical illustration.

5 By 1619, with the complete list of rediscovered solids in Harmonices Mundi, Kepler completed a survey of polyhedra that consciously looked back to Luca Pacioli and Dürer’s fold studies; see (Malkevitch 2013).

6 Conceptualizing the intellectual relations between Kepler and the artisans he was in touch with as a ‘trading zone’ à la Pamela Long is unsatisfactory because it overstates the ‘trading’ needs of the parties involved; it also strains the sources, which indicate, at best, a state of mutual awareness and thus an ‘interactive zone’. It is quite clear that Kepler acted as legal adjudicator of mechanical inventions and techniques; see Section 3.

7 As Burnyeat (2000, 26) notes, recreational math is regarded in Plato’s Laws 819a–c as an Egyptian import.

8 ‘Non aspernor hanc de anima et virtute motrice speculationem. Verum metuo ne nimis subtilis sit, si nimium extendatur. Qualis illa ipsa est, quam de Luna moves. Vereor profecto, si ultra modum nimis specialis fiat ne iacturam vel certe ruinam totius Astronomiae post se trahat. Existimo omnino parce et valde moderate hac speculatione utendum. Et ut vere dicam quod sentio: Non aspernor, at profecto languidus est meus assensus, plurima enim contraria mihi obstant’ (JKGW, 13, no 63, ll 127–138).

9 ‘Solque suos etiam dimittere languidus ignis / tempore cum certo […]’ (Lucretius, DRN 5.758–759). On Lucretius’ account of solar eclipses see (Bakker 2016, 48–49). An echo of Maestlin’s choice is in his remark on astronomy awakening from a previous neglect, ‘ex recentioribus […] languoribus’ (JKGW, 1, 84).

10 ‘Languidus mathematicus’ (appended to Reinhold 1562, f. biiv), a copy at the Herzog August Bibliothek in Wolfenbüttel, marked as H: N 104.

11 Among many examples: ‘ex Geometria invento’, ‘ex Geometria depromptis’ (JKGW, 1, 82).

12 While there is no doubt that the door to a priori astronomy stood open, as Regier (2013, 151) nicely put it, Kepler’s means of exploring morphological regularities, at least up until 1611, were still purely geometrical; for Kepler’s later attention to the motion of bodies and the post 1619, ‘Proclean’ period of his research, see (Chen-Morris 2009; Claessens 2011).

13 Rothman (2009) notes Kepler’s shrewd judicial strategy in Contra Ursum (1601), in which astronomical hypotheses themselves are treated like a client to be defended against defamation.

14 JKGW, 1, no 19, ll 40–42, and no 246, ll 18–24. Regardless of his personal sensitivity for issues of priority (Thoren 1990, 441; Mosley 2007, 237), Tycho’s stiff protocols looked back at early modern craft secrecy (Newman and Principe 1998; Davids 2005; Vermeir 2012).

15 Describing in these terms his double persona as a humanist and scientist, but perhaps overemphasizing the influence of Cardano, Grafton (1992, 563) had in mind Alexandre Koyré’s famous description of Kepler as a ‘veritable Janus’ (Jardine 2000), for retaining traits of Aristotelian physics in his Platonic grounding of the cosmos. This two-faced impression is also generated by genuine entrepreneurial and technological features. Van Nouhuys (1998, esp 566–579) found the expression in the Dutch cometary discourse of the late sixteenth century, as a marker of political insecurity and within the long tradition of the scholastic quaestio.

16 Some patronized and disseminated his work; his mother practiced herbal folk medicine (Rublack 2015).

17 A major achievement of the DNS lies in its passing from stationary geometry to the genesis of organic life, making snowflakes the equivalent of planetary orbits and light rays. Lipking (2020) leaps over centuries, and regrettably much of Kepler’s text, to argue that a novel biotech emphasis on ecosystems fulfills Kepler’s vision.

18 The key passage is: ‘consentaneum est, figuratas potius recipere quantitates, quam rudes’ (1611, 20).

19 By contrast, Descartes (1637, Discours VI) regards hexagonal shapes as flat. Kepler cites rhombic solids, in reference to Archimedes and with a clear sense of having discovered some of them, previously unknown, in a letter of November 1599 to Maestlin: (JKGW, 14, no 142, ll 21–22; see Field 1988, 202).

20 ‘sive Cosmopoecticis [sic] figuris cognatum est’ (1611, 19).

21 ‘Non enim omnes plantae ex semine, pleraque ex automato primum ortae, et si sese porro seminent. Facultas enim terrae, quae seipsa una est et eadem, dividit sese in corpora et cum corporibus, inque ea inolescit et pro cuiusque materiae conditione interna externisve aliud atque aliud architectatur’ (1611, 19); on generatio spontanea, see (Hasse 2007; Enenkel 2014). Kepler’s use of the inchoative inolescit expresses his documented attention for the language of habitus, see (Sakamoto 2008, 72).

22 ‘Ipsa sua per se sponte omnia dis agere expers’ (Lucretius, DRN 2.1092).

23 ‘[…] totum tota possederat anima’ (1611, 19); see (Boner 2008).

24 ‘Docti mediocriter, quo sunt prudentiores, hoc cautius sese immiscent hisce mathematicorum litibus; quinimo fascinari possunt, quod expertus loquor, authoritate matheseos peritorum’ (JKGW, 13, no 76, ll 32–35).

25 This argument runs in parallel with research on the bureaucratic rationalities of textual communities, where institutional ties are stronger than individual contributions (eg Berrey 2015), and also with trends on Kepler, which focus, respectively, on legal expertise, the backing of mathematicians, and the authority of knowledge as based on distance: see (Serjeantson 2006, 145–149; Rothman 2009; Biagioli 2006, 26).

26 JKGW, 16, no 618, ll 21–24.

27 It is from this network that the Swiss instrument-maker Jost Bürgi came to Prague and managed to collaborate with Kepler on a MS of logarithmic tables that was protected by secrecy (Staudacher 2014).

28 Rheinberger underlines that there are local, social, institutional, technical and instrumental factors at play, but the premium of an experimental system, at least in his approach (see Klein and Spary 2010), is geared toward the creation of ‘epistemic things’; once something shifts from the ordinary and becomes epistemic, it generates genuine surprises and displacements. See also (Mosley 2007).

29 ‘Even while dying she nevertheless took much forethought to fall honorably’ (Hecuba, 568–569).

30 Calculated flourishes aside, the presence of skeletal mathematical shapes is not necessarily to be connected to the realm of Platonic ideas; recognizing abstract proportions was required in Peripatetic science and, more specifically, a topology of strata and boundaries was important in Aristotelian biology (Papadopoulos 2019).

31 Dornavius (1619) included the DNS in his compendium of rhetorical disputations, but while Kepler played repeatedly around the assonances of nihil and nix (1611, 3–4), with a hint to Jean Passerat’s poem, and followed the generic conventions of ‘serious jokes’, the DNS is neither a paradoxical praise nor a forerunner of radical skepticism. It does, however, recast cross-disciplinary academic discussions to press its targets: a defence of geometrical concinnitas against physical void and atomism. See (Ossola 2007; Archdeacon 2020).

32 This is a point perceptively made by Chen-Morris (2016, 171), who refers to Dürer’s melancholic angel.

33 For the importance of Thomas Harriot’s correspondence with Kepler as a preparation to the DNS see (Henry 2012); more recently, Regier (2021) has proposed Anselmus Boëtius de Boodt’s treatise on mineralogy (1609) as a major influence for Kepler’s work.

34 JKGW, 16, no 606, l 5.

35 Tessicini (2021, 23) detects a degree of condescension in Kepler’s critical attitude towards his friend; for more on Kepler’s battle against corpuscular visions, see also (Omodeo 2021).

36 ‘ne consuescant ad meum dolium venire’ (JKGW, 16, no 546, ll 5–6).

37 The letter is a treasury of hints to courtly playfulness: ‘tecumque jocandum, serio consedi’, ‘[…] speculo armatus, adversarium habes Wacherium’, ‘ludendi necessitatem […] meam festivitatem’ (JKGW, 16, no 600, ll 14, 82, 99–100).

38 On Kepler’s little-studied treatise in German, the Messekunst Archimedis, see (Pastorino 2020).

39 Presumably, in Kepler’s mind, the same held true for Aristarchus’ heliocentric measurement.

40 ‘[…] perpetuo cadant, sexangulae, villosis, ut pennulae, senis radiis’ (1611, 5). The expression ‘glomerata multitudine’, a few lines below, describing the irregular drifting of snow is virtually unique in Latin, and may have been inspired by a winter landscape in Leandro Alberti’s chorography of Italy (1561), where he talks about the frozen Lombard lowlands of the Garda lake: ‘hymato lacu fluctibus glomeratae anguillae voluuntur in tantum mirabili multitudine’ (1561, Ddd 4v).

41 Compare ‘Talia dum meditans anxie, pontem transeo, confusus’ (1611, 5; emphasis mine, to signal that the first part of this sentence actually has a hexametrical cadence) with ‘Ibam forte via Sacra, sicut meus est mos, nescio quid meditans nugarum’ (Hor. Sat. 1.9). Horace’s nugae transfer into Kepler’s ludic style.

42 ‘an ex sua natura, puta cui congenitus sit vel archetypus pulchritudinis quae est in sexangulo, vel finis notitia, ad quem ista forma conducat? […] utamur exemplis nobilibus, sed geometrice descriptis’ (1611, 6).

43 Kepler ends his DNS hoping in future chemical research, while one of the major innovations by Plato was precisely to transform the chemistry of Empedoclean elements (Giacomo 2021); for a reconstruction of Plato’s mathematics, see (Fowler 1987, esp 106–157; Cleary 2014; Echterling 2018).

44 In the DNS, there is a direct citation of Virgil’s expression ‘consortia tecta’ (Geor. 4, 153) and an admirable consideration of the cognitive proficiency of the bees, who seem to follow an ‘instinctual routine’ and a plan that enables them to build their masterpiece of animal architecture: ‘quare ipsa apis natura hunc instinctum habet ex proprietate sua, ut hac potissimum figura aedificet’ (1611, 10; see Gallo and Chittka 2018).

45 ‘Sed non est haec adaequata figurae causa, neque enim hoc praestat fructui ex formali proprietate, sed adiuvatur necessitate materiali’ (1611, 8).

46 ‘hic inquam locum habet speculatio pulchritudinis aut proprietatis figurae, quae animam harum plantarum characterisavit’ (1611, 12). See (Van der Schoot 2001; Gielis 2017).

47 ‘ad huius rei confirmationem iucundissima contemplatione possent adduci’ (1611, 12).

48 As is the case with post-Archimedean geometry, Pappus is interested in alternative constructions as a way to reach mathematical truth (Knorr 1978, 89); on mechanical apodeixis, see (De Groot 2009). I maintain that Kepler’s means of explanation look beyond the Platonic tradition, while obviously being influenced by it, and that our overdependence on Plato’s harmony results from a neglect of Aristotelian problem-solving and mathematical ‘puzzles’; if snowflakes are ensouled, as the DNS suggests, their premium is on taxis, a position to which Aristotle commits himself in De anima II, 3

49 ‘Etenim obiicere possis: […] ibi ordo, ibi nullus casus, ibi mera mens, mera ratio; in nivis vero formatione finem nullum spectari posse, neque fieri per figuram sexangulam, ut nix perduret’ (1611, 18).

50 ‘Respondeo, rationem formatricem non tantum agere propter finem sed et propter ornatum, non solum tendere ad corpora naturalia efficienda sed et solere ludere in fluxis, quod multis fossilium exemplis patet. Quorum ego universorum rationem a ludicro (dum dicimus naturam ludere)’ (1611, 18).

51 ‘Vidi enim Dresdae in aede Regia cui Stabulo nomen, exornatum abacum aere argentoso, ex quo quasi efflorescebat dodecahedron avellanae parvae magnitudine’ (1611, 23). On Kepler and Kunstkammer games, see (Dupré 2008; Dupré and Korey 2009); on metallic generation, see (Smith 2017). Zorach (2020, 357) persuasively recalls how ‘thorny puzzles of natural philosophy’ were important for Albertus Magnus, and includes naturalists such as Conrad Gessner and Ulisse Aldrovandi in the larger discourse on geometric systems that is presupposed in the DNS.

Additional information

Funding

Support for this research came from the project ‘TacitRoots’, PI Giulia Giannini, funded by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (GA n. 818098).