The Archaeology of the Inverse Square Law: (1) Metaphysical Images and Mathematical Practices

Hooke Robert , Micrographia (London, 1665; facsimile reprint, Brussels, 1966), 227, brackets in the original.

Hooke , op. cit. (ref. 1), 227.

Cf. Lohne Johannes , “Hooke versus Newton: An analysis of the document in the case of free fall and planetary motion”, Centaurus, vii (1960), 6–52.

Newton , “On circular motion”, in Herivel John , The background to Newton's Principia (Oxford, 1965), 192–8.

The relevant paragraph and its meaning were already referred to in Gal Ofer , Meanest foundations and nobler superstructures: Hooke, Newton and the compounding of the celestial motions of the planets (Dordrecht, 2002), 169.

Hooke , op. cit. (ref. 1), 225.

Hooke , op. cit. (ref. 1), 227.

Hale Mathew , An essay, touching the gravitation and non-gravitation of fluid bodies, and the reasons thereof (London, 1673), 18.

Birch Thomas , The history of the Royal Society (4 vols, London, 1756–57; facsimile reprint, New York, 1968), ii, 272: 6 July 1663.

Gal , op. cit (ref. 5), 34–57.

Hooke , op. cit. (ref. 1), 219. For a discussion of Hooke's development of ‘inflection’ see Gal , op. cit. (ref. 5), chap. 1.

Hooke , op. cit. (ref. 1), 225.

Birch , op. cit. (ref. 9), i, 141–4.

Birch , op. cit. (ref. 9), i, 142.

“Sicut se habent sphaericae superficies, quibus origo lucis pro centro est, amplior ad angustiorem: Ita se habet fortitudo seu densitas lucis radiorum in angustiori, ad illam in laxiori sphaerica superficie, hoc est, conuersim. Nam per 6. 7 tantundem lucis est in angustiori sphaerica superficie, quantum in fusiore, tanto ergo illic stipatior & densior quam hic. Si autem radii linearis alia atque alia esset densitas, pro situ ad centrum (quod Prop. 7 negatum est) res aliter se haberet.” Kepler , Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur (Frankfurt, 1604; facsimile reprint, Brussels, 1968), 10.

Henceforth Ad Vitellionem for the original and Optics for William Donahue's translation: Kepler Johannes , Optics: Paralipomena to Witelo and optical part of astronomy, transl. by Donahue William H. (Santa Fe, NM, 2000).

“Sphaericum Lucis (adeoque mundi) archetypus”, Kepler , op. cit. (ref. 15), 7.

Kepler , Optics (ref. 16).

“Hinc Centri punctum, est sphaerici quaedam quasi origo, superficies puncti intimi imago, & via ad id inueniendum, quaeque infinito puncti egressu ex se ipso, vsque ad quondam omnium egressuum aequalitatem, gigni intelligitur, puncto se in hanc amplitudinem communicante, sicvt punctum & superficies, densitatis cum amplitudine commutata proportione, sint aequalia. Hinc est vndique punctum inter & superficiem absolutissa aequalitas, arctissima vnio, pulcherrima conspiratio, connexus, relatio, proportio, commensus.” Kepler , op. cit. (ref. 15), 6.

Kepler Johannes , Mysterium cosmographicum: The secret of the universe, transl. by Duncan A. M. (Norwalk, CT, 1981), 59/167.

Grosseteste Robert , “De luce”, in Die philosophischen Werke des Robert Grosseteste, Bischofs von Lincoln, ed. by Baur Ludwig (Beitrage zur Geschichte der Philosophie des Mittelalters, ix; Münster, 1912), 51–59. Translated as On light, by Riedl Clare C. (Milwaukee, 1942), 10.

Grosseteste , op. cit. (ref. 20), 12.

Grosseteste , op. cit. (ref. 20), 16.

Kepler's indebtedness to Grosseteste is frequently suggested (e.g. by Aiton in their annotations to Kepler's The harmony of the world, transl. by Aiton E. J. Duncan A. M. Field J. V. (Philadelphia, 1997), p. xli and 292, n. 12) but Lindberg's remark that “it is generally impossible to identify the specific sources of Kepler's thought on light, since … he does not cite them” ( Lindberg David C. , “The genesis of Kepler's theory of light metaphysics from Plotinus to Kepler”, Osiris, 2nd ser., ii (1986), 5–42, p. 29) is especially pertinent in the case of Grosseteste, who is never (to our knowledge) mentioned by him. This is somewhat of a mystery, since Kepler was usually generous with his sources on the one hand, and the similarity between him and Grosseteste, on the other hand, is too obvious to deny. Concerning Grosseteste's theory of light see Lindberg , Theories of vision from al-Kindi to Kepler (Chicago, 1976), 94–102, and Introduction to Bacon Roger , Roger Bacon's philosophy of nature, ed. by Lindberg David C. (Oxford, 1983), pp. xlix–lv.

Scholem G.G. , Major trends in Jewish mysticism (New York, 1941), 20.

Cf. Marrone Steven P. , William of Auvergne and Robert Grosseteste: New ideas of truth in the early 13th century (Princeton, NJ, 1983), 167–88.

Grosseteste , op. cit (ref. 20), 13.

Nicholas of Cusa, De ludo globi: The game of spheres, transl. by Watts P. M. (New York, 1986), 67, fol. CLV.

Nicolas of Cusa, op. cit. (ref. 27), 93, fol. CLXI.

Nicolas of Cusa, op. cit. (ref. 27), 95, fol. CLXII.

Nicolas of Cusa, op. cit. (ref. 27), 57, fol. CLII.

Nicolas of Cusa, op. cit. (ref. 27), 101, fol. CLXIII.

Smith A. M. , Introduction to Witelo, Witelonis perspectivae liber quintus: Book V of Witelo's Perspectiva ( Studia Copernicana , xxiii; Wroclaw, 1983), 20 ff.

Cited in Lindberg, Theories of vision (ref. 23), 63.

“Lux & color reflexi sunt debiliores luce & colore primis: Fortiores autem secundis, cum quibus ab eodem aequaliter distant”, Ibn al-Haytham Alhazen , De aspectibus in Opticae thesaurus Alhazeni arabis libri septem, ed. by Risner Friedrich (Basel, 1572), p. iv, c. 5, 103 (italics in original).

“Poterit aliquis dicere, non esse debilitatem formarum in reflexione, nisi ex elongatione earum a sua origine. Sed explanabitur, quod licet ab ortu aequaliter elongatur lux directa & lux reflexa: Tamen debilior erit reflexa” (ibid.).

Bacon , op. cit. (ref. 23), 207.

“species in parte priori est equalis vel in aliqua proportione respectu speciei in parte sequente secundum aliquam speciorum maioris inequalitatis”, Bacon , op. cit. (ref. 23), 216–17.

Grosseteste , op. cit. (ref. 20), 10.

“In puncto propinquiori fortior est lux unius corporis quam in remotiori”, Bacon , op. cit. (ref. 23), 208–9.

“Quapropter distantia in quantum huiusmodi non videtur esse causa cum tali expositione”, Pecham John , John Pecham and the science of optics: Perspectiva communis, ed. by Lindberg David C. (Madison, 1970), 94–95.

Witelo , Witelonis perspectivae liber secundus et liber tertius: Books II and III of Witelo's Perspectiva, ed. by Unguru S. ( Studia Copernicana , xxviii; Wroclaw, 1991), 61.

Strenuously argued in Prop. 6, see Unguru's commentary (ref. 41), 190.

Witelo , op. cit. (ref. 41), 60.

Alhazen Ibn al-Haytham , The optics of Ibn al-Haytham, transl. by Sabra A. I. (London, 1989), 20.

Cited in Lindberg, op. cit. (ref. 23), 28.

Lindberg , op. cit. (ref. 23), 26–30.

Cited in Lindberg, op. cit. (ref. 23), 159.

Lindberg , op. cit. (ref. 23), 18–32.

Dee read both Grosseteste and Cusanus ( Clulee Nicholas H. , John Dee's natural philosophy: Between science and religion (London and New York, 1988), 55, 153).

Josten C. H. , “A translation of John Dee's Monas hieroglyphica (Antwerp 1564), with an introduction and annotations”, Ambix, xii (1964), 112–221.

Witelo , op. cit. (ref. 42), 62.

Cited by Unguru in Witelo, op. cit. (ref. 42), 15, fn. 10, from the dedicatory epistle to William of Moerbeke.

Smith , in Witelo, op. cit. (ref. 32), 39.

Smith , in Witelo, op. cit. (ref. 32), V, 23–44; Lindberg , op. cit. (ref. 23), 66.

Sylla Edith D. , The Oxford Calculators and the mathematics of motion, 1320–1350 (New York, 1991), 142–4, 247–52; Crombie A. C. , Robert Grosseteste and the origins of experimental science, 1100–1700 (Oxford, 1953), 184–5.

Crombie , op. cit. (ref. 55), 184.

“aget difformiter et remissus ad puncta remota quam ad propinqua propter distantiam”, Crombie , op. cit. (ref. 55), 185, fn. 5.

“Debilitatur lux distans a sole”, Crombie , op. cit. (ref. 55), 185, fn. 3.

Sylla , op. cit. (ref. 55), 251.

Kepler , Mysterium cosmographicum (ref. 19), 76/201, italics added.

Witelo , op. cit. (ref. 41), 62.

Cf. Lindberg David C. , “Laying the foundations of medieval optics: Maurolico, Kepler, and the medieval tradition”, in Lindberg David C. Cantor Geoffrey (ed.), The discourse of light from the Middle Ages to the Enlightenment (Los Angeles, 1985), 3–65, p. 51.

Kepler , Mysterium cosmographicum (ref. 19), 76/201.

For Kepler's critique of medieval perspectivist tradition see Chen-Morris Raz D. , “Optics, imagination, and the construction of scientific observation in Kepler's new science”, The monist, lxxxiv (2001), 453–86.

Kepler , Optics (ref. 16), 19, and Ad Vitellionem (ref. 15), 6.

Kepler , Ad Vitellionem (ref. 15), 10.