Keplerian Astronomy after Kepler: Researches and Problems

See, for example, René Taton , “Sur la diffusion du Copernicanisme et les progrès de l'astronomie aux xvii^me et xviii^me siècles” in De Suzanne (eds), Avant, avec, après Copernic: La représentation de l'univers et ses conséquences épistémologiques (Paris, 1975), 295–308. An example in a popular work is De Timothy , Coming of age in the Milky Way (New York, 1988), chaps. 4, 5.

De Jean , “Gilbert, Bacon, Galilée, Képler, Harvey et Descartes: Leurs relations”, Isis, xxvi (1932), 171–208, p. 201. Pelseneer mistakenly asserted that Kepler's works were even ignored by the Church (p. 201, n. 73). Kepler's Epitome of Copernican astronomy, however, was placed on the Index in 1619, but not because of the planet laws per se ( De Max , Kepler, transl. by Hellman C. Doris (New York, 1959), 298).

De Arthur , The sleepwalkers: A history of man's changing vision of the universe (New York, 1959), 396. Newton did not learn of Kepler's achievements by reading Kepler, nor were they hidden away, as will be shown below. See also Alexandre Koyré , La révolution astronomique: Copernic, Kepler, Borelli (Paris, 1961), 363. For the earlier period, see Montucla Jean Etienne , Histoire des mathématiques (Paris, 1758), ii, 254; Jean-Sylvain Bailly , Histoire de l'astronomie moderne (Paris, 1799), ii, 210–11; De Adam , “The history of astronomy” (written about mid-century), in Wightman W. P. D. (eds), Essays on philosophical subjects … (Oxford, 1980), 33–105, pp. 87–88; Delambre Jean-Baptiste J. , Histoire de l'astronomie moderne (Paris, 1826), i, 360; idem, Histoire de l'astronomie au dix-huitième siècle (Paris, 1827), 61.

Representative examples include De Norbert , Geschichte der Bahnbestimmung von Planeten und Kometen (Leipzig, 1887–94), ii, 218; Pelseneer , op. cit. (ref. 2), 205; De Ernst , Entstehung und Ausbreitung der Coppernicanischen Lehre (Erlangen, 1943), 331; de Santillana Giorgio , The crime of Galileo (Chicago, 1955), 170, n. 11; Koyré , op. cit. (ref. 3), 363–4, 377, n. 5; Stahlman William D. , Foreword to Small Robert , An account of the astronomical discoveries of Kepler (reprint of 1804 edn, Madison, 1963), p. ix; Hall A. Rupert , The revolution in science 1500–1700 (London, 1983), 143; Cohen I. Bernard , Revolution in science (Cambridge, Mass., 1985), 132.

See, for example, the statement from the author of the well-known pejorative term for this historiographical vice — Whig history — That Kepler “has to his credit a collection of discoveries and conclusions … from which we can pick out three that have a permanent importance in the history of astronomy” (Herbert Butterfield, Origins of modern science, 1300–1800 (rev. edn, New York, 1962), 75–76).

Leibniz may be the only figure in the seventeenth century who associated Kepler in general terms with the discovery of the laws of the heavens. Leibniz, in his Tentamen or “Essay on the causes of the motions of the heavenly bodies”, after reading a review of the Principia in the Acta eruditorum of 1689, praised Kepler extravagantly as having been the first “to publish the laws of the heavens”. He there also cited all three of Kepler's “laws” accurately and based his own celestial dynamics on Kepler. See Cohen I. Bernard , “Newton and Keplerian inertia: An echo of Newton's controversy with Leibniz” in Debus Allen G. (ed.), Science, medicine and society in the Renaissance: Essays to honor Walter Pagel (London 1972), 192–211, p. 205, and Aiton Eric J. , The vortex theory of planetary motion (New York, 1972), 127. Kepler himself did not employ the term ‘law’ for the discoveries we now call by that name. Curtis A. Wilson characterized them most appropriately in the title of his important article, “From Kepler's laws, so-called, to universal gravitation: Empirical factors”, Archive for history of exact sciences, vi (1970), 89–170. See also De Nicholas , The birth of history and philosophy of science (Cambridge, 1983), 240. (In deference to established usage, quotation marks will be omitted from further references to the three discoveries known as Kepler's laws.).

For material published before 1975, see Aiton Eric J. , “Johannes Kepler in the light of recent research”, History of science , xiv (1976), 77–100. Recent efforts include idem, “Kepler's path to the construction of his first oval orbit for Mars”, Annals of science, xxxix (1978), 173–90; Cohen I. Bernard , The Newtonian revolution (Cambridge, 1980); Brackenridge J. Bruce , “Kepler, elliptical orbits and celestial circularity: A study in the persistence of metaphysical commitment”, Annals of science, xxxix (1982), 117–43; De Bruce , Kepler's physical astronomy (Berlin and New York, 1987); De Yasukatsu , “Kepler's hypothesis vicaria”, Archive for history of exact sciences, xli (1990), 53–92; Davis A. E. L. , several articles forming the entire issue of Centaurus, xxxv (1992), 97–191; Donahue William H. , “Kepler's first thoughts on oval orbits”, Journal for the history of astronomy, xxiv (1993), 71–100; idem, “Kepler's invention of the second planetary law”, The British journal for the history of science, xxvii (1994), 89–102; De Peter Goldstein Bernard R. , “Distance and velocity in Kepler's astronomy”, Annals of science, li (1994), 59–73.

This is denied in the account by Davis A. E. L. , “Kepler's resolution of individual planetary motion”, Centaurus, xxxv (1992), 97–102. But see Voelkel James R. , “The development and reception of Kepler's physical astronomy 1593–1609”, unpubl. Ph.D. diss., Indiana University, 1994.

See the comments of De James Ellis Robert L. (eds), The works of Francis Bacon, Baron of Verulam, Viscount St. Albans, and Lord High Chancellor of England (London, 1887–1901), iii, 511, 723–6. For Bacon's scepticism regarding the Copernican system, see his Novum organum, Book ii, Art. xxxvi. Pascal refused to commit himself to Copernicanism, possibly out of religious scruples ( Léon Brunschvicq De Pierre (eds), Oeuvres de Blaise Pascal, première serie (Paris, 1908), ii, 100).

De William (ed.), The English works of Thomas Hobbes of Malmesbury (London, 1839–45), i, p. viii;vii, 101.

de Waard Cornelis (ed.), Journal de Isaac Beeckman tenu de 1604 à 1634 (La Haye, 1939–53), iii, 65–66; Hooykaas Reijer J. , “Isaac Beeckman”, in Gillispie Charles C. (ed.), Dictionary of scientific biography (New York, 1970–80; hereafter DSB), i, 566–8, p. 568a; De Robert , Mersenne, ou la naissance du mécanisme (Paris, 1943), 12; Léon Auger , Un savant méconnu: Giles Personne de Roberval (1602–1675) (Paris, 1962), 106–7. De William (The dissolution of the celestial spheres 1595–1650 (New York, 1981), 291) notes that Mersenne's reference to Kepler's physical mechanism was taken from Hobbes rather than from Kepler.

De Johannes , Prodromus dissertationem cosmographicarum continens mysterium cosmographicum (Tübingen, 1596). A useful reprint with English translation is Mysterium cosmographicum: The secret of the universe , transl. by Duncan A. M. with an introduction by Aiton Eric J. (New York, 1981). The reception of the work merits further detailed study. An important beginning has been made by Voelkel James R. , op. cit. (ref. 8). See also Schofield Christine J. , Tychonic and semi-Tychonic world-systems (New York, 1981), 234–5; Donahue , Dissolution (ref. 11), 177.

De Jeremiah , Opera posthuma (in some copies titled Opuscula astronomica), ed. by De John (London, 1672, 1673, 1678), 10.

Tycho to Mästlin, 21 April 1598; Tycho to Kepler , 9 December 1599, in De Johannes , Gesammelte Werke , ed. by von Dyck Walther (Munich, 1937-; hereafter cited as Kepler , GW), xiii, 204–5; xiv, 94.

Caspar , Kepler (ref. 2), 68; Donahue , Dissolution (ref. 11), 164; Wilson Curtis A. , “The inner planets and the Keplerian revolution”, Centaurus, xvii (1972), 205–48, p. 243; De Christopher , An astrological discourse (London, 1650), 82–85, 96. Kepler, aware of these objections, omitted his third law from the Rudolphine tables in calculating planetary distances ( Wilson Curtis A. , “Horrocks, harmonies and the exactitude of Kepler's third law” in De Erna (eds), Science and history: Studies in honor of Edward Rosen (Studia Copernicana, xvi; Wroclaw, 1978), 235–58, pp. 238, 240).

Kepler (Mysterium cosmographicum (ref. 12), 41) writes that booksellers, friends and natural philosophers had been pressing for a reissue.

Quoted from the “Tentamen de motuum coelestium causis”, Acta eruditorum , Feb. 1689, by Cohen I. Bernard , “Newton and Keplerian inertia” (ref. 6), 205. See also Meli Domenico Bertoloni , “Public claims, private worries: Newton's Principia and Leibniz's theory of planetary motion”, Studies in history and philosophy of science, xx (1991), 415–49, p. 424, where it is pointed out that there were “three main areas in which Kepler was important for Leibniz, namely astronomy and the laws of planetary motion, the order, regularity and essentially harmony of nature; and the role of theology in many aspects of his work …”, and that he was considered an ally in Leibniz's battles with Newton.

Koyré , La révolution astronomique (ref. 3), 457, n. 4. He thought it unlikely, however.

For Kepler's use of the term ‘vortex’, see, for example, the Introduction and chapter summaries in the Astronomia nova, Kepler , GW (ref. 14), iii, 34, 44.

For Descartes's use of the Keplerian terms in correspondence, see Aiton , Vortex theory (ref. 6), 43; Descartes, Principles of philosophy , transl. by Miller Valentine R. Miller Reese P. (Dordrecht, 1983), Part iii, Art. 36. In the first edition of the Mysterium cosmographicum only the word aphelion (in Greek) appears (Duncan transl. (ref. 12), 161, 163, 183). The second edition (1621), annotated by Kepler, uses both terms, noting that he invented them. Both terms also appear in a marginal note on p. 32 of the Astronomia nova: “Aphelium et perihelium quid?”, GW (ref. 14), iii, 93; in the chapter summaries of chaps. 28 and 50, ibid., 42, 46, 49; and in Kepler's Index of Terms, in the front matter of the work. Kepler gave a new connotation to the Latin term inertia, applying it to the tendency of planets to remain at rest unless put into motion by a mover. The term appears in his Epitome, and the second edition of the Mysterium cosmographicum, GW, vii, 94, 296, 330; Cosmographic mystery (ref. 12), 171. See also Cohen , “Newton and Keplerian inertia” (ref. 6), 209, n. 11, and De Edward , “Kepler's harmonics and his concept of inertia”, American journal of physics, xxxiv (1966), 610–13.

Principles (ref. 20), iii, Art. 153; Kepler , GW, iii, chapter summary for chap. 37. Descartes may, however, have encountered the idea in Galileo or in another author. See, for example, Galileo , Dialogue concerning the two chief world systems — Ptolemaic and Copernican, transl. by De Stillman (Berkeley and Los Angeles, 1962), 453.

Principles (ref. 20), Part iii, Art. 35. Descartes likewise may have acquired this from secondary sources. The opinion of Miller and Miller (Principles of philosophy (ref. 20), 99, n. 31) that “Descartes seems acquainted only with Kepler's work in optics” would therefore seem to require further analysis.

Pelseneer , op. cit. (ref. 2), 181–2; McGuire J. E. De Martin , Certain philosophicall questions: Newton's Trinity notebook (Cambridge, 1983), 169; Aiton , Vortex theory (ref. 6), 43, 62, n. 60, 72; De Daniel , Descartes' metaphysical physics (Chicago, 1992), 349, n. 32; Shea William R. , Magic of numbers and motion: The scientific career of René Descartes (Canton, Mass., 1991), 285.

See, for example, De Pierre , “Réception de la cosmologie nouvelle à la fin du xvii^e siècle”, Avant, avec, après Copernic (ref. 1), 261–6, p. 262. On Huygens and Kepler, see also De Ernst , Die Reformation der Sternkunde (Jena, 1852), 245 and Whiteside Derek T. , “Newton's early thoughts on planetary motion: A fresh look”, The British journal for the history of science, ii (1964), 117–37, p. 121, n. 16.

De Christiaan , Oeuvres complètes , ed. by Société Hollandaise des Sciences (The Hague, 1888–1950), i, 463–4; iii, 438; viii, 376. Costabel's comment (ref. 24, loc. cit.) that Huygens's notes of 1682 made no allusion to Kepler's laws is misleading.

Huygens , Oeuvres (ref. 25), xxi, 124–32, 349–50.

Ibid. , xxi, 272 ff; Costabel , op. cit. (ref. 24), 263.

Alexandre Koyré , “Attitude esthétique et pensée scientifique”, Critique, ix (1955), 835–47, p. 840. He was seconded by Giorgio de Santillana, who called it “one of the strangest mysteries of the history of natural philosophy”, op. cit. (ref. 4), 36, n. 8, 169. Expressions of similar sentiment could doubtless be traced back to the eighteenth century.

Kepler , GW , iii, 26; Galileo , Dialogue (ref. 21), 462; De Stillman , “Galileo's theory of the tides”, Galileo studies: Personality, tradition, and revolution (Ann Arbor, 1970), 200–13, pp. 209, 213, n. 24 (original version in Physis, iii (1961), 185–94); idem, “Galileo and the concept of inertia”, ibid. , 240–56, p. 254 (originally in “An unpublished letter of Galileo to Peiresc”, Isis, liii (1952), 201–11).

De Erwin , Galileo as a critic of the arts (The Hague, 1951), 23; Koyré , “Attitude esthétique” (ref. 28), 840.

I am preparing a detailed analysis of their relationship.

For the Bologna chair, see Kepler to Roffenius, 17 Apr. 1617, Kepler , GW , xvii, 222–4; for Wotton's letter, ibid. , xviii, 42. Kepler had shortly before been visited by John Donne, who had referred anonymously to Kepler in his early works ( De Wilbur , “Donne's meeting with Kepler: A previously unknown episode”, Philological quarterly, 1 (1971), 132–4).

Russell John L. , “Kepler's laws of planetary motion: 1609–1666”, The British journal for the history of science, ii (1964), 1–24, p. 20.

Russell ascribes the less than enthusiastic initial reception of the Epitome “to the influence of Tycho Brahe, and the learned world was not much disposed to listen to” the defence of Copernicanism (ibid., 7). “No other work is mentioned so frequently or, for the most part, with so much respect where planetary theory is concerned” (p. 20). The First Part, containing Books i-iii, was printed in 1617 ( Caspar , Kepler (ref. 2), 293). The entire work was reprinted at Frankfurt in 1635. The least read of Kepler's works was the Harmonice mundi ( Russell , op. cit. (ref. 33), 6). According to Kepler, the Astronomia nova had been issued in few copies and at a steep price; the exact number is unknown (Max Caspar, Bibliographia Kepleriana: Ein Führer durch das gedruckte Schrifttum von Johannes Kepler (Munich, 1936), 55).

Russell , op. cit. (ref. 33), 6–9. Russell notes that in 1615 Magini “used Kepler's laws in calculating ephemerides for Mars”, but besides the acknowledgement, he gave no details.

De Wilbur , “Kepler in England: The reception of Keplerian astronomy in England, 1599–1687”, unpubl. Ph.D. diss., State University of New York at Buffalo, 1969, chap. ii; Apt Adam J. , “The reception of Kepler's astronomy in England: 1596–1650”, unpubl. D.Phil. diss., Oxford University, 1983; De Wilbur , “Wilhelm Schickard”, DSB (ref. 11), xii, 162–3, p. 163. Schickard's little treatise on the transit of Mercury of 1631, published the following year, mentions the ellipse and the inverse-distance rule, but he had been a friend and correspondent of Kepler's and had known about the ellipses during Kepler's lifetime.

One was Christopher Heydon, as shown in his correspondence with Henry Briggs in 1610. Bodleian Library: MS Ashmole 242, ff. 168b-170b.

The second law was mentioned by Pierre Hérigone (1642), Riccioli (1651) and John Wallis (1659), with Noël Durret presenting a geometrical construction equivalent to it. Russell , op. cit. (ref. 33), 20–21.

Ibid., 1, 10–19; Wilson , “From Kepler's laws” (ref. 6), 106–27. During the decade and a half from 1630 to 1645 most French astronomers accepted the idea of elliptical orbits; most English by 1655.

The almanac was written by Nathaniel Chauncy, son of Harvard's President ( Morison Samuel E. , Harvard College in the seventeenth century (Cambridge, Mass., 1936), 217). Several of Vincent Wing's astronomical textbooks published in the 1650s and 1660s were owned by Harvard students ( ibid. , 216). Donald K. Yeomans's claim (“The origins of North American astronomy — Seventeenth century”, Isis, lxviii (1977), 414–25, p. 417, n. 16), therefore, that in 1665 “Johannes Kepler's ideas were not well known in colonial America” requires some modification. He notes that in 1674, however, an almanac states that “astronomers are of the opinion (received from Kepler) that planets move in ellipses not circles”.

De Angus , John Kepler (London, 1955), 179–80; Russell , op. cit. (ref. 33), 10; Schofield , op. cit. (ref. 12), 189.

For Christopher Wren, for example, see Hall A. Rupert , “Wren's problem”, Notes and records of the Royal Society of London, xx (1963), 140–4, p. 141. For De Robert , see Hooke to Newton , 17 Jan. 1679/80, The correspondence of Isaac Newton, ed. by Turnbull Herbert W. (Cambridge, 1959–77), ii, 309.

Kepler , GW, x, Precepts, chap. xx. Russell notes that “It is possible … that the exact form was in fact known to many who never actually stated it” (op. cit. (ref. 33), 5). Jeremiah Horrocks used his own method of approximation to it for his lunar theory: Wilson Curtis A. , “Predictive astronomy in the century after Kepler”, in De Michael (ed.), The general history of astronomy (Cambridge, 1989-), ii, ed. by René Taton De Curtis , Planetary astronomy from the Renaissance to the rise of astrophysics, Part A: Tycho Brahe to Newton, 161–206, p. 198a.

GW, vii, 376ff.

The manuscript was written in 1640, but not published until 1662: “Venus in sole visa, seu tractatus astronomicus” in De Johann , Mercurius in sole visus (Gdansk, 1662), 111–45. See Horrocks J. , The transit of Venus across the Sun, transl. by Whatton Arundell B. (London, 1859), 204.

Russell , op. cit. (ref. 33), 1, 14. Russell names Horrocks, Holwarda, Hérigone, Riccioli, and Streete.

Ibid. , 11–12, 15; Armitage , op. cit. (ref. 41), 181; Pierre Hérigone , Cursus mathematicus (Paris, 1634–42; reprinted 1644); De Giambattista , Almagestum novum (Bologna, 1651; 2nd edn, Frankfurt, 1653).

Harriot seems usually to have had quick access to Kepler's publications. Roche John J. , “Thomas Harriot's astronomy”, unpubl. Ph.D. diss., Oxford, 1977, 37, n. 1; idem, “Harriot, Galileo, and Jupiter's satellites”, Archives internationales d'histoire des sciences, xxxii (1982), 9–51, p. 19.

De Henry , Thomas Hariot: The mathematician, the philosopher, and the scholar (London, 1900), 122.

Ibid., 123–4.

Apt , op. cit. (ref. 36), 193.

Copernicus had insisted on circles, and Kepler himself did not easily leave the circle when initial evidence from his effort to find a mathematical relationship governing planetary speed and distance from the Sun presented itself ( Donahue , “Kepler's first thoughts” (ref. 7), 71, 75). Even after the ellipse, the circle retained its importance in Kepler's philosophical and theological outlook. See Brackenridge , op. cit. (ref. 7), 117.

De Nathanael , Philosophia libera (2nd edn, Oxford, 1622). On Longomontanus, see Russell , op. cit. (ref. 33), 7.

Fabricius to Kepler, 20 Jan. 1607, Kepler , GW , xv, 377. See also Tycho Brahe's letter to Kepler of 9 Dec. 1599, some years before Kepler's discovery of the elliptical orbit (ibid., xiv, 94). On Brush and Shakerley, see Apt , op. cit. (ref. 36), 76.

De Samuel , Miscellanies: Or mathematical lucubrations (London, 1659), 25.

The grip of the geoheliocentric theory in Tycho's native land was so strong that no one used the Rudolphine tables there until Römer moved to Copenhagen from Paris in 1681. Longomontanus's creation of tables based on a theory that was a compromise betweeen Tycho and Copernicus, and his having been a professor at Copenhagen for several decades, exerted a powerful influence. Moesgaard Kristian P. , “How Copernicanism took root in Denmark and Norway”, in De Jerzy (ed.), The reception of Copernicus' heliocentric theory (Dordrecht and Boston, 1972), 116–51, pp. 126–34, 141.

Several of their authors may have been dissembling, Scheiner and Riccioli among them. Additional Jesuits taking a nominally Tychonic position were Inchofer, Biancani, Kircher, Polaccus, Beati, Tacquet and de Chasles ( Schofield , op. cit. (ref. 12), 281–9).

Christine Schofield shows that Tycho Brahe made use of it (ibid., 64). Wilson conjectures that this may have been through Kepler's influence (“From Kepler's laws” (ref. 6), 93).

Schofield , op. cit. (ref. 12), 64.

See, for example, William Lower in Stevens , op. cit. (ref. 49), 122; De Jeremy , Anatomy of urania practica (London, 1649), 15–16; Wilson , “From Kepler's laws” (ref. 6), 94.

Wilson , “Predictive astronomy” (ref. 43), 166b.

Wilson , “Kepler's laws” (ref. 6), 105. The problem earlier had arisen from the use of very small apertures, which, with the resulting diffraction, yielded diameters that were too large. Riccioli now used Kepler's method, requiring a larger aperture whose diameter is subtracted from the diameter of the solar image. After many observations between 1661 and 1665, his figures yielded an eccentricity of 0.0169, almost one-half the eccentricity of the equant using Ptolemaic procedures. Similar confirmations were made by Grimaldi, Cassini and Flamsteed. Wilson , “Predictive astronomy” (ref. 43), 161b, 167, 185.

Letter of 30 Oct. 1607, Kepler , GW , xvi, 71. On resistance to Kepler's physical ideas, see De Fritz , “The new celestial physics of Johannes Kepler”, in De Sabetai (ed.), Physics, cosmology and astronomy, 1300–1700 (Dordrecht, 1991), 185–227.

Letter of 21 Dec. 1616, Kepler , GW, xvii, 187.

Crüger to Philipp Müller, 1 Jul. 1622, Kepler , GW, xviii, 92.

Hatch Robert A. , The collection Boulliau (BN, FF. 13019–13059): An inventory (Philadelphia, 1982), p. xxix.

De Fritz , “Sphaera activitatis — Orbis virtutis. Das Entstehen der Vorstellung von Zentralkraften”, Sudhoffs Archiv, liv (1970), 113–40, pp. 134–5; Baldwin Martha R. , “Magnetism and the anti-Copernican polemic”, Journal for the history of astronomy, xvi (1985), 155–74, pp. 159–60, 168–9.

Quoted in Kircher's Latin in De Fritz , “Keplers Beitrag zur Himmelsphysik”, in De Fritz (eds), Internationales Kepler-symposium Weil der Stadt 1971 (Hildesheim, 1973), 55–139, p. 134.

De Michael De Christine , “Problems in late Renaissance astronomy”, in La soleil à la Renaissance: Sciences et mythes (Brussels, 1965), 21–31, p. 26.

For Lower, see Stevens , Hariot (ref. 49), 121; for Bainbridge, see Apt , op. cit. (ref. 36), 195; for Holwarda, see Wilson , “Predictive astronomy” (ref. 43), 166b; for Horrocks Wilbur Applebaum , “Between Kepler and Newton: The celestial dynamics of Jeremiah Horrocks”, Actes du xiii^me Congrès International d'Histoire des Sciences 1971 (Moscow, 1974), iv, 292–9; Shakerley , op. cit. (ref. 60), 15–16. Kepler believed that the rate of the Earth's diurnal rotation and annual revolution fluctuated, which was denied by Horrocks and Holwarda.

De Otto , “Notes on Kepler”, in De Arthur De Peter (eds), Kepler: Four hundred years (Vistas in astronomy, xviii (1975)), 781–5, pp. 781–2. See also Donahue William H. , “Kepler's fabricated figures: Covering up the mess in the New Astronomy”, Journal for the history of astronomy, xix (1988), 217–37. Donahue's conclusions are challenged in De Volker , “Keplers komplizierter Weg zur Wahrheit: Von neuen Schwierigkeiten die ‘Astronomia nova’ zu lesen”, Berichte zur Wissenschaftsgeschichte, xiii (1990), 167–76.

Letter of 4 Feb. 1605, Kepler , GW, xv, 149.

Briggs to Archbishop Ussher, August 1610, De Richard , The life of the most reverend father in God, James Usher (London, 1686), 12.

Briggs to Kepler, 20 Feb. 1625, Kepler , GW, xviii, 225–5, 229.

Henry Bourchier to Archbishop Usher, 26 Mar. 1629, Parr , op. cit. (ref. 73), 404.

Ismailis Bullialdi astronomia Philolaica (Paris, 1645).

Ismaël Boulliau , Philolai sive dissertationis de vero systemate mvndi, libri iv (Amsterdam, 1639).

Hatch , Collection Boulliau (ref. 66), pp. xxviii–xxix.

Boulliau reaffirmed his position in a third work, Astronomia philolaica fundamenta clarius explicata (Paris, 1657), 5.

Aiton characterizes Boulliau's approach as Platonic as does Russell; Wilson as Aristotelian. Both characterizations are just, as they refer in the one case to the role of geometry, in the other, to the separation of disciplines. Aiton , Vortex theory (ref. 6), 91; Russell , op. cit. (ref. 33), 16; Wilson , “From Kepler's laws” (ref. 6), 109, n. 74.

Astronomia Philolaica (ref. 76), 3–7, 21–24.

On Boulliau's system, see Hatch , op. cit. (ref. 66), Introduction; Wilson , “Predictive astronomy” (ref. 43), 172–3; Boyer Carl B. , “Ismael Boulliau”, DSB (ref. 11), ii, 348–9, p. 349; idem, “Notes on the epicycle and the ellipse from Copernicus to Lahire”, Isis , xxxviii (1947), 55–56; Delambre , Histoire (ref. 3), ii, 146–50; Dreyer John L. E. , History of astronomy from Thales to Kepler , rev. by Stahl William H. (2nd edn, New York, 1953), 420. Delambre mistakenly says Boulliau gave no reason for his rejection of Kepler's second law (op. cit. (ref. 3), ii, 147).

Russell , op. cit. (ref. 33), 19; Wilson , “Predictive astronomy” (ref. 43), 176.

Russell , op. cit. (ref. 33), 18. See also Aiton , Vortex theory (ref. 6), 91. This seems an oversimplification, as some employed Boulliau's modification of the equant, but were “physicists” as well.

“Horrocks was a genius of the same stamp as Kepler. He appeared to have the same imagination and … he joined to it the same perseverance in calculation” ( Delambre , Histoire (ref. 3), ii, 499). See De Wilbur , “Jeremiah Horrocks”, DSB (ref. 11), vi, 514–16.

Horrocks , Opera posthuma (ref. 13), 8, 181–2.

Cambridge University Library: Royal Greenwich Observatory MSS, Flamsteed papers, lxviii, Horrocks , “Philosophicall exercises”, 1. The manuscript appears to have been begun about mid-1637.

Ibid., 23.

Letter of 24 Apr. 1637, Horrocks , Opera posthuma (ref. 13), 276.

Ibid. , 35, 60. Horrocks , Transit of Venus (ref. 45), 204. Koyré is therefore mistaken in asserting (Revolution astronomique (ref. 3), 458, n. 8) that Horrocks defended Kepler only in generalities and failed to mention the second and third laws.

He was introduced to Kepler's tables by his friend and correspondent William Crabtree, who lived near Manchester ( De Wilbur , “William Crabtree”, DSB , iii, 547–8). For Horrocks's conviction of the superiority of Kepler's tables, see his letter of 3 Jun. 1637, Opera posthuma (ref. 13), 287.

De Curtis , “On the origin of Horrocks's lunar theory”, Journal for the history of astronomy, xviii (1987), 77–94.

A few astronomers had learned of it earlier. See De Wilbur Hatch Robert A. , “Boulliau, Mercator and Horrocks's Venus in sole visa: Three unpublished letters”, Journal for the history of astronomy, xiv (1983), 166–79.

De Robert , “Cometa or, remarks about comets”, in The Cutler lectures of Robert Hooke , ed. by Gunther Robert T. (Early science in Oxford, viii; Oxford, 1931), 217–71, p. 252.

De Olaf , “Some early European observatories”, in De Arthur De Peter (eds), The origins, achievement and influence of the Royal Observatory, Greenwich: 1675–1975 (Vistas in astronomy, xx (1976)), 17–28, pp. 24–25; Wilson , “Predictive astronomy” (ref. 43), 168a. Jeremiah Horrocks paid considerable attention to the accuracy of his angle-measuring devices, observing conditions, sources of observational error, and the need to compensate for atmospheric refraction and ocular parallax (Opera posthuma (ref. 13), passim).

Among the the most important was the correspondence maintained by Boulliau with astronomers in Germany, Poland, Italy and England, as well as in France. Boulliau and Hevelius wrote to one another over several decades during the middle of the century ( Hatch , Collection Boulliau (ref. 66), pp. xvii, xxxii, n. 44; xlix).

De Volker , “Ephemerides in the early 17th century”, Vistas in astronomy, xxii (1978), 21–26, p. 25, n. 6.

Russell , op. cit. (ref. 33), 7–8.

It was the only “published and usable observation” ( Van Helden Albert , “The importance of the transit of Mercury of 1631”, Journal for the history of astronomy , vii (1976), 1–10, p. 3). See also De Bernard , “Pierre Gassendi”, DSB, v, 284–90, p. 285a.

Russell , “Kepler's laws” (ref. 33), 10–11. De Wilbur , “Wilhelm Schickard”, DSB, xii, 162–3.

Wilson , “Predictive astronomy” (ref. 43), 165a. See also idem, “Inner planets” (ref. 15), 242–4. All other tables were off by several degrees of longitude (idem, “From Kepler's laws” (ref. 6), 100).

Noël Durret , Nouvelle théorie des planètes (Paris, 1635). His accompanying Supplementi tabularum Richelianarum pars prima were for the most part translations of Lansberge (Owen Gingerich, “Kepler's place in astronomy”, in Beer Beer (eds), Kepler (ref. 71), 261–78, p. 272, n. 8).

Hatch , Collection Boulliau (ref. 66), p. xxxvi, n. 62; Wilson , “Inner planets” (ref. 15), 243. The quotation is from the Astronomia Philolaica, 355 as transl. by Wilson , “From Kepler's laws” (ref. 6), 100.

Apt , op. cit. (ref. 36), 86–87.

Horrocks , Opera posthuma (ref. 13), 306; De Mordechai , The mathematician's apprenticeship (Cambridge, Mass., 1984), 156–7; Wilson , “From Kepler's laws” (ref. 6), 101.

John Digby wrote from Paris in 1656 that he could not obtain a copy of Kepler's ephemerides; six months later he was successful in borrowing a set, which he thought might be the only one in Paris (Historical Manuscripts Commission, Eighth report (London, 1881), part i, vol. vii (append.), 219b). Digby may have meant the ephemerides of Andreas Argoli or Lorenz Eichstadt, which were based on the Rudolphine tables ( Wilson , “Predictive astronomy” (ref. 43), 187–9). The last year for which Kepler calculated ephemerides was 1636; two versions of Kepler's tables were published in 1650, one in 1657 and an English version in 1676 ( Caspar , Bibliographia Kepleriana (ref. 34)).

[ De Henry ], “Observations made in several places …”, Philosophical transactions of the Royal Society , ii (1667), 295–7. “Kepler's planetary positions were generally about thirty times better than any of his predecessors'…” ( De Owen , “Ptolemy, Copernicus, Kepler” in Adler Mortimer J. Van Doren J. (eds), The great ideas today (Chicago, 1983), 137–80, p. 179).

Small , op. cit. (ref. 4), 297; Dreyer , op. cit. (ref. 82), 393.

De Joseph , A tutor to astronomy and geography (London, 1659), 268.

Armitage , Kepler (ref. 41), 166.

See De Yasukatsu , “On the order of accuracy of Kepler's solar theory”, in Beer Beer (eds), Kepler (ref. 71), 769–80, p. 780.

On Durret, see Wilson , “Predictive astronomy” (ref. 43), 165b. Examining 26 calculations for Saturn from the Rudolphine tables and comparing them with a modern ephemeris, Wilson finds an average error of 3′21′ with some amounting to 12′ or 13′ (“From Kepler's laws” (ref. 6), 102, n. 40). De James , Ephemerides of the celestial motions (London, 1652), sig. b5b; he reiterated his complaint in his ephemerides published in 1672. See also Delambre , Histoire moderne (ref. 3), ii, 456; Van Helden Albert , “Huygens and the astronomers”, in Bos Henk J. M. (eds), Studies on Christiaan Huygens (Lisse, 1980), 147–65, p. 165, n. 76.

Flamsteed to the Royal Society, 24 Nov. 1669, Rigaud Stephen P. Rigaud Stephen J. (eds), Correspondence of scientific men of the seventeenth century (Oxford, 1841), ii, 89.

Letter to Collins, 5 May 1673, ibid., ii, 163.

Flamsteed to Seth Ward, 31 Jan. 1679/80, in De Francis , An account of the rev'd. John Flamsteed… (London, 1835), 121–2.

Wilson , “Predictive astronomy” (ref. 43), 171b. Using observations of parallaxes of Mars in opposition and of Venus near inferior conjunction and conjectures about the relative sizes of the planets in sequence from the Sun as well as the actual size of the Sun compared to the planets, Horrocks claimed that a reduction in solar parallax gives better results for lunar and solar eclipses and yields better elements for the planets, particularly Venus (ibid., 167b-169a).

De Thomas , Astronomia Carolina (London, 1661), 12. See also De Wilbur , “Thomas Streete”, DSB , xiii, 96. Huygens and Picard at the Académie were also familiar with the Venus in sole visa and accepted the necessity of a reduction in the solar parallax. In the 1670s Cassini and Flamsteed had, through determinations of the Martian parallax, reduced the solar parallax to less than 10”. Flamsteed even thought it likely that it could be as small as 7”. Flamsteed to Collins, 20 Feb. 1672/73, Rigaud , Correspondence (ref. 113), ii, 160; Wilson , “Predictive astronomy” (ref. 43), 176a, 189a; De Eric , “Early researches of John Flamsteed”, Journal for the history of astronomy, vii (1976), 124–38, p. 129; Maeyama , “Kepler's hypothesis vicaria” (ref. 7), 87.

Wilson , “Predictive astronomy” (ref. 43), 171a.

Among the published tables based on the Rudolphine were those of Noël Durret (Paris, 1639), De Vincent (Florence, 1639), De Maria (Oels, 1650), Morin J. B. (Paris, 1650; London, 1675, 1676), Streete T. (London, 1661), Coley H. (London, 1675), Mercator N. (London, 1676) ( De Owen , “Kepler”, DSB , vi, 289–312, p. 308a). For English almanacs and ephemerides based on Kepler, see Applebaum , “Kepler in England” (ref. 36), 130–1.

Hatch , Collection Boulliau (ref. 66), p. xxvii.

Wilson , “Kepler's laws” (ref. 6), 100, citing Astronomia Philolaica, 354–92. Boulliau adopted Kepler's fixed inclinations of planetary orbits as simplifying the problem of the latitudes (ibid., 110).

Boulliau to Huygens, in Huygens , Oeuvres (ref. 25), ii, 492. At the end of the Philolaic tables, Boulliau, possibly having had second thoughts about the accuracy of his own figures for Mars compared to Kepler's, inserted a table from Kepler behind his own. Modifying Kepler's eccentricity for Mars in 1657, Boulliau obtained a slight improvement, making his tables slightly better than Kepler's. Hatch , Collection Boulliau (ref. 66), p. xlvii.

Ibid. , note 141; Boulliau's calculating procedures were used by Streete and Wing, and Mercator adopted Boulliau's before developing his own hypothesis. Shakerley's Tabulae Britannicae were essentially the Philolaic tables calculated for London and the Julian calendar, as were the tables of John Newton's Astronomia Britannica ( Wilson , “Predictive astronomy” (ref. 43), 176b).

Flamsteed to the Royal Society, 24 Nov. 1669, in Rigaud , op. cit. (ref. 113), ii, 88. See also De John , Cometographia, or a view of the celestial and terrestrial globes (London, 1679).

Testing those corrected positions against Tuckerman's ephemeris indicates their accuracy to “within 2′— And frequently to within less than 1′ — Of arc” ( Wilson , “Predictive astronomy” (ref. 43), 168b).

Mercator's Institutionum astronomicarum of 1676 was Englished and included in William Leybourn's Cursus mathematicus … (London, 1690). The passage is cited from the latter work, p. 803.

The subject of the non-uniformity of planetary motion before Kepler could benefit from close examination and clarification. It is not always clear whether a particular astronomer is violating Aristotelian precepts about uniform motion, or referring to apparent motion, or to a description of a geometrical model employing epicycles or equants which was not meant to reflect physical reality. At any event, Kepler's eventual insistence on non-uniform motion in ‘simple’ orbits was unique.

The problem was first posed in the Astronomia nova (GW, iii, 381).

The general form of ‘Kepler's Problem’ was presented by Christopher Wren in a broadside printed in 1659 and is reproduced in Hall , “Wren's problem” (ref. 42), 142–3. Other mathematicians offering solutions, either geometrical or analytical, were Boulliau Seth Ward Wallis John Gregory James Newton Keill John Machin John Euler . See Russell , “Kepler's laws” (ref. 33), 3; De John , Lives of the professors of Gresham College (London, 1740), 97; Turnbull Herbert W. (ed.), James Gregory tercentenary volume (London, 1939), 220, n. 4; Newton , Correspondence (ref. 42), i, 149, n. 4.

Stevens , op. cit. (ref. 49), 123. ‘Atechnies’ is a term used by Kepler (in Greek) in chap. xlviii of the Astronomia nova (GW, iii, 310) to characterize the complex methods he had employed in an early version of the area rule. He was not enamoured of them either.

Wilson , “Predictive astronomy” (ref. 43), 174b–175a. Wilson also notes that the demand by Boulliau and others for a direct method of deriving true anomaly from mean anomaly “seems associated with a kind of neo-classic purism; astronomy is taken to be both a mathematical art and an esoteric science of quasi-divine things, and the astronomer becomes a supreme artifex, following strict rules that are imposed both by the nature of the art and by the supposedly sublime nature of the celestial objects” (“From Kepler's laws” (ref. 6), 115).

De John , Astronomia Britannica (London, 1657), sig. A₂6. For Flamsteed, see De Edward , The sphere of Marcus Manilius (London, 1675), 84–85.

Koyré , Révolution astronomique (ref. 3), 130, 444 n. 103, 495; Whiteside , “Newton's early thoughts” (ref. 24), 124. For Hooke, see Hooke to Newton , 6 Jan. 1680, Correspondence (ref. 42), ii, 309. Thoren avers that the area rule was “both known and appreciated by most of the astronomers of the period” and concludes that in the 1660s and 1670s there existed a “reluctant” belief in the area rule, but that its difficulty of application led to its neglect in practice, but also to its omission from the works of the period ( De Victor , “Kepler's second law in England”, The British journal for the history of science, vii (1974), 243–56, pp. 243–4, 255). This appears to ignore the persistent confusion between the area and inverse-distance rules until the early 1670s.

Ptolemy had not applied the equant to the solar orbit. The strategy, tactics and reasoning employed by Kepler in his struggles with the Martian orbit have been elaborated in great detail in the several works cited in ref. 7 above.

Kepler , GW , x, 172. Bailly has Curtz as the originator of the empty focus theory; Delambre has Boulliau in that role ( Bailly , op. cit. (ref. 3), ii, 144, 211–12; Delambre , Astronomie moderne (ref. 3), ii, 161). See Gaythorpe Sidney B. , “Horrocks's treatment of evection and the equation of the centre, with a note on the elliptic hypothesis of Albert Curtz …”, Monthly notices of the Royal Astronomical Society, lxxxv (1925), 858–65, pp. 861–2. The first to suggest the use of circles to generate the ellipse or to employ the empty focus model, however, appears to have been David Fabricius in a letter to Kepler, 20 Jan. 1607, to which Kepler replied on 1 Aug. of that year ( Kepler , GW , xv, 376–86; xvi, 14–30). See also Herz , op. cit. (ref. 4), ii, 219–20.

In addition to those mentioned above, the most prominent examples during the century were Cavalieri in 1632, Boulliau in 1645 and 1657, Seth Ward in 1654 and 1656, Pagan and John Newton in 1657, Streete in 1661, Mercator in 1664, 1670 and 1676, Wing and Cassini 1669, Isaac Newton in 1670 and 1679, Halley in 1676 and Huygens in 1681. Cavalieri and Horrocks independently found means of handling ‘Kepler's Problem’ by a method of approximation. Bailly , op. cit. (ref. 3), ii, 209–14; Delambre , Astronomie moderne (ref. 3), ii, passim; Whiteside , “Newton's early thoughts” (ref. 24), 122, n. 18; idem, “Before the Principia: The maturing of Newton's thoughts on dynamical astronomy”, Journal for the history of astronomy , i (1970), 5–19, p. 9; Wilson , “From Kepler's laws” (ref. 6), 117–33; idem, “Predictive astronomy” (ref. 43), 169b–70b; De Yasukatsu , Hypothesen zur planetentheorie des 17. jahrhunderts (Frankfurt am Main, 1971).

Wilson , “Predictive astronomy” (ref. 43), 174b.

De Seth , In Ismailis Bullialdi astronomiae Philolaica fundamenta inquisitio brevis (Oxford, 1653). That its publication actually took place in 1654 was pointed out by Hatch Robert A. , Collection Boulliau (ref. 66), p. xlvi, n. 132.

Ward , Inquisitio brevis , 3; [Boulliau], Astronomia Philolaica (ref. 76), 286. See Wilson , “From Kepler's laws” (ref. 6), 117–18.

Wilson , ibid., 121.

Ismailis Bullialdi astronomiae Philolaicae (ref. 79).

Wilson , “From Kepler's laws” (ref. 6), 140. With a proper eccentricity, using a librating equant point along the major axis of the ellipse, an accuracy of 20” is possible ( Wilson , “Predictive astronomy” (ref. 43), 178a).

De Blaise , La théorie des planètes… (Paris, 1657). See Wilson , “From Kepler's laws” (ref. 6), 122–3.

De John , op. cit. (ref. 132), 66. The statement is confused, as the motion of the radius vector at the empty focus is “equal”, i.e. uniform, but the motion of the planet is not.

Whiteside , “Before the Principia” (ref. 136), 9. A similar division between theory and practice seems to have occurred among seventeenth-century mathematicians regarding the use of indivisibles. Quite a few sought pragmatic solutions for quadratic equations despite what they knew as violations of mathematical rigour. De Douglas , “Philosophical theory and mathematical practice in the seventeenth century”, Studies in the history and philosophy of science, xx (1989), 215–44.

Kepler , GW, vii, 380.

De Vincent , An ephemerides of the celestial motions for xiii years (London, 1658), 140. John Collins wrote to James Gregory the following year in words that almost repeat those used by Wing earlier (Gregory tercentenary volume (ref. 129), 202).

Mercator set his equant on the line of apsides somewhat closer to the Sun than the empty focus. It yielded a maximum error for Mars of less than 2' ( Hypothesis astronomia nova (London, 1664), Sig. 3a). See also Whiteside Derek T. , “Mercator”, DSB , ix, 310–12, p. 310. Maeyama points out that Mercator could have obtained better results utilizing circles and the vicarious hypothesis had he used his own more precise data rather than Tycho's, since he had an improved figure for solar parallax through his familiarity with Horrocks's modification of it (“Kepler's hypothesis vicaria” (ref. 7), 89–90).

[Mercator], “Some considerations of Mr. Nic. Mercator …”, Philosophical transactions of the Royal Society, v (1670), 1168–75, p. 1174.

Nicolai Mercatoris … institutionum astronomicorum libri duo (London, 1676), 162–73. Kepler of course had made it clear that the equant was inadequate.

Victor Thoren asserts that “virtually all the English text-writers—the very people who adopted, adapted, and disseminated the empty-focus equant theories — Held decidedly relaxed views on the subject of astronomical exactitude” (op. cit. (ref. 133), 244. This seems to go too far, since such theories were also employed by the best astronomers for whom both convenience as well as mathematical equivalence or near-equivalence in saving the appearances served as it had for two thousand years.

Hall , “Wren's problem” (ref. 42), 141; Bennett J. A. , “Hooke and Wren and the system of the world: Some points toward an historical account”, The British journal for the history of science, viii (1975), 32–61, pp. 35, 37.

Streete , op. cit. (ref. 117), 342; De Curtis , “Predictive astronomy” (ref. 43), 179.

Attempts were made by Cavalieri Horrocks Boulliau Wing Ward Mercator Cassini Flamsteed Halley (Wilson, ibid. ; see also Maeyama , Hypothesen zur Planetentheorie (ref. 136) and De Owen De Barbara , appendix to Thoren , op. cit. (ref. 133), 257–8).

De Thomas (ed.), The history of the Royal Society … (London, 1756–57), ii, 417. Oldenburg read Cassini's paper from the Journal des Savants for 2 Sept. 1669 to the Society.

Philosophical transactions of the Royal Society, v (1670), 1169–75.

Ibid. , 1174–5. Thoren's assertion (op. cit. (ref. 133), 255) that there must have been extra-empirical grounds for acceptance of the second law since its empirical aspects “had already been essentially duplicated by the refined equant theories” fails to recognize Mercator's contribution. Nor, despite Alexandre Koyré (Newtonian studies (London, 1965), 130), was Newton the first to recognize that inverse-distance and area rules were not equivalent. Brian S. Baigrie mistakenly claims that the “area rule is absent in the scientific literature prior to Newton”, and that the ellipse was treated by astronomers as a “mere computational device” (“The justification of Kepler's ellipse”, Studies in the history and philosophy of science, xxi (1990), 633–64, pp. 652–3).

Possibly three tangents or positions (see De Angus , Edmond Halley (London and Edinburgh, 1966), 15–16; Wilson , “From Kepler's laws” (ref. 6), 158). Halley's method is described in “Methodus directa et geometrica …”, Philosophical transactions , xi (1676), 683–6. His English draft title was “A direct geometrical process to find the aphelion, eccentricities, and proportions of the orbs of the primary planets, without the supposition, hitherto employed, of the equality of motion at the other focus of the ellipsis”, in Rigaud , op. cit. (ref. 113), ii, 237. The complete draft is on pp. 237–41.

See Thoren , “Kepler's second law” (ref. 133), 254 and the appendix by Gingerich and Welther, pp. 257–8. Here “best” must be understood in the context of Boulliau's and Mercator's equant theories, which were accurate to within one minute of arc.

Wilson , “Predictive astronomy” (ref. 43), 161b; idem, “Kepler's derivation of the elliptical path”, Isis , lix (1968), 5–25, p. 21; idem, “From Kepler's laws” (ref. 6), 101.

Wilson Curtis A. , “Newton and some philosophers on Kepler's ‘laws’”, Journal of the history of ideas, xxxv (1974), 231–58, p. 257.

Hobbes , On body, in his English works (ref. 10), i, 435.

[ Boulliau ], Astronomia Philolaica (ref. 76), 25.

Wilson , “From Kepler's laws” (ref. 6), 104.

Whiteside , “Newton's early thoughts” (ref. 24), 121, n. 16; Cohen I. Bernard , Newtonian revolution (ref. 7), 225; Russell , “Kepler's laws” (ref. 33), 14.

De Robert , Micrographia (London, 1665), 238.

See De Wilbur , “Horrocks”, DSB, 514–16, p. 516.

De John , “… Hypothesis about the flux and reflux of the sea”, Philosophical transactions, i (1666), 263–89, pp. 272, 280–1. Wallis surely knew that if the orbit was not elliptical, it was certainly not circular.

De Thomas , in Gunther Robert T. (ed.), The life and work of Robert Hooke (Early science in Oxford, vi; Oxford, 1930), 265. A possible source for the statements by Hooke and Wallis on whether orbits are circular or elliptical may have been Boulliau's assertion in the Astronomia Philolaica, 25, that the eccentricities of Earth and Venus were too small to detect a difference between a circle and an ellipse.

Newton , Correspondence (ref. 42), ii, 305.

Herschel John F. W. , A preliminary discourse on the study of natural philosophy (London, 1830), 178; Koestler , op. cit. (ref. 3), 328.

De David , early in the eighteenth century, was unusual in recognizing the importance of physical theory in Kepler's discoveries ( Aiton , “Kepler in recent research” (ref. 7), 78).

Stephenson , Kepler's physical astronomy (ref. 7), 2–3, 22.

See ref. 163 above. For Hooke , Wallis and Newton, see below.

Apt , op. cit. (ref. 36), 183–5. Briggs's model appeared to reject even the unequal motion of the vicarious theory. Christopher Heydon was aware of the importance of realism with respect to unequal motion for Kepler's “hypothesis wc he cals genuine” in contrast to the fictive model of Briggs (Heydon to Briggs, c. 1610, Bodleian Library: MS Ashmole 242, f. 168b).

All of Kepler's major works make this clear, as reflected in the title-pages of the Astronomia nova, and of Book iv of the Epitome, which Kepler calls a supplement to Aristotle's De caelo. For pre-Keplerian efforts in the wake of the dissolution of the celestial spheres, see Kelly Mary S. , “Celestial motors: 1543–1632”, unpubl. Ph.D. diss., University of Oklahoma, 1964. The importance of Kepler's physical theories in the construction of his rules for planetary motion has been investigated in concrete detail. See Hanson Norwood R. , Patterns of discovery: An inquiry into the conceptual foundations of science (Cambridge, 1958), 73–85; Wilson , “Kepler's derivation” (ref. 160); and Stephenson , op. cit. (ref. 7).

Jardine , op. cit. (ref. 6), 144.

Ibid., 154, 156.

Kepler , Astronomia nova, GW, iii, 142. See also Jardine , op. cit. (ref. 6), 143 for Kepler on the nature of hypotheses.

Stephenson , op. cit. (ref. 7), 116. Hanson remarks that “his discovery of Mars' orbit is physical thinking at its best” (op. cit. (ref. 176), 72–73).

GW, xv, 72.

Kepler , Epitome, GW, vii, 257.

This is not to say that physical precepts are not embodied in the Almagest, nor that Ptolemy was unconcerned about them.

See Aristotle on the difficulty of gaining knowledge of the heavens, De partibus animalium, 644b 25; on the supremacy of fact over theory, De generatione animalium, 760b. To say as Drake does that Ptolemy in the Almagest “explicitly excluded physics and metaphysics from its purview as a treatise on mathematical astronomy” (Almagest, I, 1, preface), ignores the fact that some of the mathematical hypotheses of Ptolemy's models are based on physical and metaphysical assumptions ( De Stillman , “Galileo's steps to full Copernicanism and back”, Studies in the history and philosophy of science, xviii (1987), 93–105, p. 95).

Westman Robert S. , “Three responses to the Copernican theory” in Westman Robert S. (ed.), The Copernican achievement (Berkeley, 1975), 285–345, p. 303; idem, “The astronomer's role in the sixteenth century: A preliminary study”, History of science, xxii (1980), 105–47, p. 107; idem, “Magical reform and astronomical reform: The Yates thesis reconsidered”, in McGuire J. E. Westman Robert S. (eds), Hermeticism and the scientific revolution (Los Angeles, 1977), 3–91, pp. 68–69.

Donahue , Dissolution (ref. 11), 66–69.

Ibid., 71. This does not seem to be true at least of Clavius, who seems to have held that epicycles and eccentric circles exist in nature, on the grounds that true conclusions could not come from false premises ( Wallace William A. , discussion comments in De Owen (ed.), The nature of scientific discovery (Washington, D.C., 1975), 382–7, p. 383). Ironically, Kepler, arguing from the same principle, would conclude that epicycles and eccentric circles do not exist in the natural realm.

“It can therefore be said that by the beginning of the seventeenth century there was no longer any clear-cut distinction between traditional and non-traditional theories [about the composition of the heavens], nor between astronomy and physics” ( Donahue , Dissolution (ref. 11), 71).

The problem is complex and has by no means been adequately addressed. Even Donahue hedges (ibid., preface and 219). Field points out that Tycho rejected solid celestial spheres by 1588, but did not cite the comet observations of 1577 as the reason. Moreover, Mästlin still accepted them, while Kepler didn't. See Field Judith V. , “Kepler's rejection of solid celestial spheres”, Vistas in astronomy , xxiii (1978), 207–11. See also the criticism of John Heilbron, that Donahue fails to distinguish between the opinions of the natural philosophers and the astronomers on the relation of physics to mathematical astronomy and has ignored the opinion of medieval speculative natural philosophers, where he would have found many of the ideas he attributes as novel to the minor thinkers of the later sixteenth century (“Commentary: Duhem and Donahue” in Westman (ed.), The Copernican achievement (ref. 185), 276–84, pp. 277–8).

Among them were Tycho, Mästlin, Fabricius, Longomontanus, Boulliau and Pagan ( Russell John L. , “Kepler and scientific method”, in Beer Beer (eds), Kepler (ref. 71), 733–45, pp. 741–2). To Russell's list may be added Brengger, Briggs and Riccioli among others (for Brengger, see Kepler , GW , xvi, 71; for Briggs , GW , xviii, 225). Rejecting both Kepler's physical causes and Boulliau's geometrical necessities, Riccioli held that God's ultimate means were unknown and the planetary motions are governed by intelligences, following a divine harmony and Divine Providence ( Wilson , “From Kepler's laws” (ref. 6), 103–4).

Horrocks , Opera posthuma (ref. 13), 179.

Kepler , Epitome, Book iv, pref., GW . vii, 249; Kepler to Bianchi V. , 17 Feb. 1619, ibid., xvii, 321–8.

Wilson , “Predictive astronomy” (ref. 43), 49–50, citing the preface to the Rudolphine tables. See also Kepler to Bernegger , 25 Jun. 1625, GW, xviii, 237.

Wilson , “Inner planets” (ref. 15), 244.

Donahue , Dissolution (ref. 11), 191–2.

The idea was adopted by several, including Wallis (“Hypothesis about the flux and reflux of the sea” (ref. 168), 270).

Discussions of this speculation are found in Stephenson , op. cit. (ref. 7), 143–4; Gingerich , “Kepler” (ref. 119), 303b; idem, “Ptolemy, Copernicus, Galileo”, in Adler Van Doren (eds), The great ideas today (ref. 107), 137–80, p. 170. At first Kepler believed that the surface areas of the planets were proportional to their distances from the Sun. In the Epitome, however, citing telescopic observations, but not without a dash of speculation concerning archetypes, he concluded that volumes were proportional to distance (GW, iii, 281–2). A number of astronomers, including Horrocks, Streete, Wendelin, Remus Quietanus and “possibly Huygens” were attracted to these ideas. For Huygens, see Van Helden Albert , Measuring the universe: Cosmic dimensions from Aristarchus to Halley (Chicago, 1985), 122–4; idem, “Halley and the dimensions of the solar system”, in Thrower N. J. W. (ed.), Standing on the shoulders of giants: A longer view of Newton and Halley (Berkeley and Los Angeles, 1990), 143–56.

Donahue , Dissolution (ref. 11), 162–3, 251, 272–4, 291, 293–4; Schofield (ref. 12), 189, 243; Horrocks , Transit of Venus (ref. 45), 181; Mercator , Institutionum astronomicarum (ref. 150), 145. Boulliau, while rejecting Kepler's physical speculations on the cause of planetary motion, accepted his conjecture concerning an annual variation in the Earth's rate of diurnal rotation, as did Wing and Streete ( Wilson , “Predictive astronomy” (ref. 43), 196–7).

De Walter , Physiologia Epicuro-Gassendo-Charletoniana (London, 1654), 277.

Birch , in Günther (ed.), op. cit. (ref. 169), 256. Hooke continued to use the analogy of magnetism for gravity as late as 1678 ( Gunther (ed.), op. cit. (ref. 94), 228–9.

Applebaum , “Kepler in England” (ref. 36), 110–14.

See Stevens , op. cit. (ref. 49), 121; Hobbes , op. cit. (ref. 10), i, 434 and vii, 102; Schofield , op. cit. (ref. 12), 243; Koyré , Newtonian studies (ref. 157), 117.

Hevelius moved from a Keplerian to a Cartesian mechanism ( Donahue , Dissolution (ref. 11), 293–4). For the rejection of magnetism on empirical grounds, see Newton , Correspondence (ref. 42), ii, 341–2.

Kepler's role in the history of astronomy extends beyond his planet laws and for too long has been restricted to them. For a brief discussion of Kepler's physical ideas, see Hall , Revolution in science (ref. 4), 144–5. It misses the mark to say that because his Aristotelian “dynamics was already outmoded, Kepler's physical explanations could exert little influence” ( Aiton , Vortex theory (ref. 6), 2).

We see this in Borelli, who insists on ellipses as the true planetary path ( Koyré , Revolution astronomique (ref. 3), 468). The Jesuit defenders of Tycho, however, insisting on the preservation of circular motion, argued that the planets actually move in “spirals” ( Schofield , op. cit. (ref. 12), 227–30).

Applebaum , “Between Kepler and Newton” (ref. 70); Donahue , Dissolution (ref. 11), 192. For White , see Russell John L. , “The Copernican system system in Great Britain”, in De Jerzy (ed.), The reception of Copernicus' heliocentric theory (Dordrecht and Boston, 1972), 189–239, p. 223.

Among them were Roberval, Holwarda, Hobbes, Streete, Wing and the early Newton. See Aiton , Vortex theory (ref. 6), 90–91; Donahue , Dissolution (ref. 11), 249–50; Wilson , “Kepler's laws” (ref. 6), 125; Applebaum , “Kepler in England” (ref. 36), 158–60; Bennett J. A. , “Cosmology and the magnetic philosophy, 1640–1680”, Journal for the history of astronomy, xii (1981), 165–77, p. 175, where a magnetic cosmological tradition in England is ascribed to the influence of Gilbert.

Descartes , Principles , Part iii, sec. xxx; Hall A. Rupert , “Sir Isaac Newton's notebook, 1661–65”, Cambridge historical journal, ix (1948), 239–50, p. 244. Wilson surmises that Newton's query derives from his reading of Wing's Astronomia Britannica of 1669 (“From Kepler's laws” (ref. 6), 142).

Sir Isaac Newton's mathematical principles of natural philosophy and his system of the world, transl. by De Andrew , rev. and ed. by De Florian (Berkeley, 1960), 550.

The chief discussions of Borelli worth noting are De Angus , “Borell's hypothesis and the rise of celestial mechanics”, Annals of science , vi (1950), 268–82; Koyré , Revolution astronomique (ref. 3), part iii; Westfall Richard S. , Force in Newton's physics: The science of dynamics in the seventeenth century (London, 1971), 213–30.

The Keplerian influence may be seen in the very title of his work on celestial dynamics, which includes the expression “ex causis physicae deductae”, corresponding to the Greek aitiologetos in the full title of Kepler's Astronomia nova ( Borelli G. A. , Theoricae Mediciorum planetarum ex causis physicis deductae (Florence, 1666)). See Koyré , Revolution astronomique (ref. 3), 510, n. 2.

Koyré , ibid., 462.

Ibid., 466. For Kepler this occurs only for the Earth's satellite.

“Impact was a dynamic action, and to deal with it he grasped blindly at available dynamic concepts whatever their import for his concept of motion” (Westfall, Force in Newton's physics (ref. 210), 216).

De Stephen , “Magnetical philosophy and astronomy”, in Taton Wilson (eds), op. cit. (ref. 43), 45–53, p. 53a; Bennett , “Magnetic philosophy” (ref. 207), 172; Westfall Richard S. , “Hooke and the law of universal gravitation: A reappraisal of a reappraisal”, The British journal for the history of science, iii (1967), 245–61, pp. 249–50; idem, Force in Newton's physics (ref. 210), 268–72. In the 1660s Hooke and Newton, among others, had not yet abandoned Cartesian vortices while considering central attractive forces.

See, for example, Hooke's review of the fate of Kepler's hypothesis and its variants, concluding that “they are fain to be most thrown aside when they come to calculation” ( De Richard (ed.), The posthumous works of Robert Hooke S.R.S . (London, 1705), 179). See also Mercator , Hypothesis astronomia nova (ref. 148), sig. B2r.

For Kepler , see, for example, Rudolphine tables, GW, x, 42–43.

For a partial list see Schofield , op. cit. (ref. 12), 226–7.

De Vincent , Ephemerides …for… 1659… 1671 (London, 1657), sig. Ar. Similar language may be found in New England almanacs shortly afterwards ( Morison , op. cit. (ref. 40), 11).

Hooke , Posthumous works (ref. 216), 167. See also Centore F. F. , “The philosophy of heliocentrism in pre-Newtonian English science”, Organon, x (1974), 75–85. The same point was made by Huygens (see text and ref. 25 above).

Cohen , “Newton and Kepler's inertia” (ref. 6), 201. The works through which Newton learned of Kepler's ideas are discussed in Whiteside , “Newton's early thoughts” (ref. 24), 131, n. 48; idem, “Sources and strengths of Newton's early mathematical thought”, Texas quarterly, x (1967), 69–85, pp. 72–75; McGuire J. E. De Martin , “Newton's mathematical apprenticeship: Notes of 1664/5”, Isis, lxxvi (1985), 349–65, p. 352; idem, Certain philosophical questions (ref. 23), 300; Westfall Richard S. , Never at rest: A biography of Isaac Newton (Cambridge, 1980), 94; Cohen I. B. , Newtonian revolution (ref. 7), 345, n. 12; idem, “Newtonian astronomy: The steps toward universal gravitation”, Vistas in astronomy, xx (1976), 85–98, pp. 89–96.

Herivel John W. , The background to Newton's Principia: A study of Newton's dynamical researches in the years 1664–84 (Oxford, 1965), 121; Hall, however, dates Newton's notes as having been made in 1661 or 1662 ( Hall A. Rupert , Isaac Newton: Adventurer in thought (Oxford, 1992), 62). See Whiteside , “Newton's early thoughts” (ref. 24), 124, where Whiteside points out that Newton's citation of the Kepler's third law is from Streete's Astronomia Carolina. Streete learned of it from the papers of Horrocks, who encountered it in Kepler's Harmonice mundi.

Newton to Halley, 14 Jul. 1686, Correspondence (ref. 42), ii, 445, where Newton writes: “for ye duplicate proportion I can affirm yt I gathered it from Kepler's theorem about 20 yeares ago”. He repeated the assertion in the draft of a letter penned in 1718 ( Whiteside , “Newton's early thoughts” (ref. 24), 117).

Newton expressed his uncertainty in the endpapers of his copy of Wing's book ( Wilson , “Newton and philosophers” (ref. 161), 238; Cohen , Newtonian revolution (ref. 7), 345, n. 12).

Newton , “System of the world” in Newton's mathematical principles (ref. 209), 549–626, p. 559; Wilson , “Kepler's laws” (ref. 33), 91. After discovering the law of gravitation, Newton could no longer assume the exactitude of the third law. Flamsteed was no help one way or another in 1684–85 when queried by Newton on the matter with respect to the satellites of Jupiter ( Wilson , “Horrocks, harmonies” (ref. 15), 258).

Wilson , “Kepler's laws” (ref. 6), 90, n. 4. I. B. Cohen says Newton learned of the second law in 1678 after seeing it in Mercator's Institutionum astronomicarum (Introduction to Newton's Principia (Cambridge, Mass., 1971), 52, n. 17; idem, Newtonian revolution (ref. 7), 250–1, where Cohen adds that Newton never thought of it in astronomical terms until 1679). See also Whiteside Derek T. , “Newton and Kepler”, Nature, ccxlviii (1974), 634, where Cohen's dating is questioned. John Herivel also questions Cohen's dating in the latter's Introduction to the Principia (review of Cohen I. B. Alexandre Koyré (eds), Philosophia naturalis principia mathematica, Nature, ccxlvii (1974), 163–4, p. 164b). Newton could well have encountered the second law in 1670 from seeing Mercator's article in the Philosophical transactions of that year (Curtis Wilson, private communication).

Wilson , “Inner planets” (ref. 15), 244.

Elliptical orbits and the area rule, however, are attributed to Kepler in the tract De motu. Koyré speculates that it may have been due to an aversion to Kepler's “continuous mixture of ‘metaphysical hypotheses’ with ‘natural philosophy’” ( Koyré , Newtonian studies (ref. 157), 101–2, n. 2; see also Cohen , Introduction (ref. 225), 130–1; idem, “Newton and Keplerian inertia” (ref. 6), 199, n. 2; idem, Revolution in science (ref. 4), 496; Aiton , Vortex theory (ref. 6), 101). Aiton also points out that Newton also did so in a manuscript c. 1700, as shown in Newton , Correspondence (ref. 42), iv, 1.

Cohen , Introduction to Principia (ref. 226), 31.

De Edmond , Philosophical transactions of the Royal Society, xvi (1687), 292. Perhaps Newton was making a distinction between hypothesis in its traditional mathematical sense as an initial assumption, not necessarily true, and his recognition that they represented approximations to real orbits.

Cohen I. B. , Introduction to Principia (ref. 226), 295–6; idem, “Newton's theory vs. Kepler's theory and Galileo's theory: An example of a difference between a philosophical and a historical analysis of science” in De Yehuda (ed.), The interaction between science and philosophy (Atlantic Highlands, N.J., 1974), 299–338, p. 312.

Wilson , “Newton and some philosophers” (ref. 161), 233–5; also idem, “Kepler's laws” (ref. 6), 89.

Newton , Correspondence (ref. 42), ii, 436.

Hall A. Rupert Hall Marie B. , Unpublished scientific papers of Isaac Newton (Cambridge, 1962), 277; Herivel , op. cit. (ref. 222), 282; Whiteside Derek T. (ed.), The mathematical papers of Isaac Newton (Cambridge, 1967–81), vi, 49. In the Principia Newton credited Kepler only for the third law, but omitted mention of his name in connection with the second and third laws ( Cohen , “Newton's theory vs Kepler's theory” (ref. 231), 313).

Hall Hall , op. cit. (ref. 234), 378. Curtis Wilson has pointed out to me that Newton's use of “prove” (probare) meant “test”. For Newton the word demonstratio stood for mathematical proof.

Cohen , Newtonian revolution (ref. 7), 229.

Cohen , Introduction to Principia (ref. 226), 136.

Cohen , “Newton's theory” (ref. 231), 311–12. Kepler , however, was aware that the orbital shape and the area rule were underdetermined and particularly in the case of the Moon was he aware of attractive forces from both Sun and Earth as acting to modify the ideal orbit. Cohen's statement that Kepler “really investigated the orbit only in the neighborhood of the apsides” is a puzzling one. Newton became convinced of universal gravitation only in 1684 after satisfying himself about the extent of the perturbations and that only gravitation was at work. Only at the end of that year did Flamsteed confirm to him that Jupiter's satellites obeyed Kepler's third rule. Newton then concluded that they obeyed the second law and that Jupiter and Saturn perturbed one another's orbits. He had yet to solve the problem of cometary orbits, how spheres attracted one another and if the inverse-square rule holds at their surfaces. See Wilson , “From Kepler's laws” (ref. 6), 164–7.

Cohen , Newtonian revolution (ref. 7), 43.

Cohen , “Newton's theory” (ref. 231), 313, 300.

GW, iii, 244.

GW, vii, 296–7.

Kepler , Somnium seu opus posthumum de astronomia lunaris … (Sagan and Frankfurt, 1634), GW , xi (2), 317–67. The expression inertia materiae appears in GW, vii, 94. It also there refers to the Astronomia nova and Book iv of the Epitome.

Moles was first used in his note 90 to his translation of Plutarch's The face in the Moon. See note 5 to the 2nd edition, Cosmographic mystery (ref. 12), 171. See also De Edward (ed. and transl.), Kepler's Somnium: The dream, or posthumous work on lunar astronomy (Madison, 1967), 69, n. 142, and Kepler's Introduction to the Astronomia nova, GW, iii, 25.

The figure is arrived at by examining the refractive indices of air compared to water (30' to 48°), assuming a like ratio of ether to air and cubing the number (Epitome, GW, vii, 261).

Aiton , Vortex theory (ref. 6), 260.

Newton, however, as shown by his arguments in Book III of the Principia, “depended on none of them as precise empirical laws” ( Wilson , “Newton and philosophers” (ref. 161), 234; idem, “From Kepler's laws” (ref. 6), 89).

De Colin , An account of Sir Isaac Newton's discoveries (London, 1748), 47.

Ibid., 47–54.

See, for example, Koestler , op. cit. (ref. 3), 396; Neugebauer , “Notes on Kepler” (ref. 71), 384–5; Cohen , Revolution in science (ref. 4), 127, 229.

Brian Baigrie sees this as a difference in approach between historians of science who see Kepler's laws as important in their own right and those who see them as gaining in significance only after the Principia. He uses this as a case study in the “transformation of scientific problems”, this being the sub-title of his article “Kepler's laws before and after the Principia”, Studies in history and philosophy of science, xvii (1987), 177–208, p. 179. See also Stephenson , op. cit. (ref. 7), 202.

Cohen , Revolution in science (ref. 4), 132–3. See also Heniger S. K. , “Pythagorean cosmology and the triumph of heliocentrism” in La soleil à la Renaissance (ref. 69), 35–53, p. 53. Wilson, on the other hand, characterizes Kepler's innovations and the resulting improvements in predictive accuracy as revolutionary (“From Kepler's laws” (ref. 6), 92, 122).

Halley in Wotton William , Reflections upon ancient and modem learning (London, 1694), 280.

Hall , Revolution in science (ref. 4), 139.

Hanson Norwood R. , “The Copernican disturbance and the Keplerian revolution”, Journal of the history of ideas, xxii (1961), 169–84, p. 169.

Koyré , Revolution astronomique (ref. 3), 120.

An excellent summary of the central issues in this point of view is provided by Wilson , “Predictive astronomy” (ref. 43), 205.

See Bennett , “Cosmology and magnetic philosophy” (ref. 207), 168 for Foster ; Donahue , Dissolution (ref. 11), 250 for Boulliau.

Today, for example, through socially legitimated norms and means, the ‘community’ passes judgement in order to determine membership, support research and determine publication; in the seventeenth century, of course, norms and means for these functions were quite different.

The Scottish universities were rather traditional in 1680, then moved rapidly to Newtonianism ( Russell John L. , “Cosmological teaching in the seventeenth-century Scottish universities, Part 1”, Journal for the history of astronomy, v (1974), 122–32; Part 2, 145–54).

Among those familiar with and to a certain extent adopters of a portion of Keplerian astronomy were Magini (1615), Cavalieri (1632), Riccioli (1651) and Cassini (1662) ( Russell , “Kepler's laws” (ref. 33), 15). A clue to the Bologna story may be found in Kepler's novel effort at collaboration with Magini, who used data provided by Kepler for the construction of his ephemerides ( Bialas , “Ephemerides” (ref. 97), 21–22).

This is suggested by Thoren , “Kepler's second law” (ref. 133), 251, n. 37 in connection with Nicolaus Mercator, who was early a Keplerian and played an important role in clarifying Kepler's second law for English astronomers. But the professor at his alma mater, the University of Rostock, was Fabricius, a pupil of Tycho's ( Hofmann Joseph E. , “Nicolaus Mercator (Kauffman), sein Leben und Wirken, Vorzugsweise als Mathematiker”, Abhandlungen mathematisch-naturwissenschaftliche Klasse d. Akademie d. Wissenschaft Mainz, iii (1950), 45–103, p. 49, n. 18).

Westman , “Astronomer's role” (ref. 185), 120.

“The new disciplinary norms define a widened domain of options but they do not determine which must be chosen” (ibid., 134). See also Westman's “Two cultures or one? A second look at Kuhn's The Copernican revolution”, Isis, lxxv (1994), 79–115, pp. 104–11, where he provides a brief analysis of the selective reception of Kepler's ideas and of the role of his Epitome in attempting to create a novel disciplinary structure for astronomy in the context of both court and academic cultures.