I should like to express my gratitude to the National Science Foundation for support through much of the long period over which this paper was written.

¹ Russell, J. L., ‘Kepler's laws of planetary motion: 1609–1666’, The British journal for the history of science, ii (1964), 1–24.CrossRefGoogle Scholar Derek T. Whiteside, ‘Newton's early thought on planetary motion: a fresh look’, ibid., pp. 117–37. Delambre, J-B. J., Histoire de l'astronomie moderne (Paris, 1821)Google Scholar, ii. passim.

² Russell, , op. cit. (1), pp. 1, 20.Google Scholar

³ Whiteside, , op. cit. (1), p. 121.Google Scholar

⁴ Descriptions of the empty-focus equant and its various modifications are given in Wilson, Curtis, ‘From Kepler's laws, so-called, to universal gravitation: empirical factors’, Archive for history of exact sciences, vi (1970), 89–170CrossRefGoogle Scholar; see especially pp. 117–19, 124, 132.

⁵ For a survey of the surprising extent to which various of Kepler's works were read in England, see Applebaum, Wilbur, ‘Kepler in England: the reception of Keplerian astronomy in England, 1599–1687’ (State University of New York, Buffalo, Ph.D. dissertation, 1969).Google Scholar

⁶ Russell, , op. cit. (1), p. 20Google Scholar, concluded that ‘Kepler's laws and theories attracted little attention until the publication of the Rudolphine tables in 1627’.Google Scholar

⁷ Draft of ‘Philosophicall exercises’ extant at Greenwich Observatory; quoted in Gaythorpe, S. B., ‘On Horrocks' treatment of the evection and the equation of the centre, with a note on the elliptic hypothesis of Albert Curtz and its correction by Boulliau and Newton’, Monthly notices of the Royal Astronomical Society, lxxxv (1925), 858–65.CrossRefGoogle Scholar

⁸ While there has not been a systematic documentation of the subject, the fact of a midseventeenth-century burgeoning of confidence in technology, and a concomitant material expectation from scientific studies, can scarcely be disputed. It constitutes the prima facie foundation of the so-called Merton thesis and is not challenged even by those who doubt the validity of the putative causal connexion between the spread of Puritanism and the rise of modern science. That astronomy had already enjoyed a privileged position from Elizabethan days up to the interregnum is argued by Johnson, F. R. in his Astronomical thought in renaissance England (Baltimore, 1937).Google Scholar No bibliographical study of the later technical literature has yet been carried out. But C. Blagden has traced records of ‘The distribution of Almanacks in the second half of the seventeenth century’ (Papers of the Bibliographical Society of the Univ. of Virginia, [1958], 107–16)Google Scholar, which show an astounding circulation of 350,000 annually, in England alone, from 1664 to 1687. Further discussion of this intense astrological interest during the period is in Thomas, Keith, Religion and the decline of magic: studies in popular belief in sixteenth- and seventeenth-century England (London, 1971)Google Scholar, especially pp. 283–322.

⁹ See the title pages to the third and fourth books, pp. 39, 103.

¹⁰ See Rigaud, S. P. (ed.), Correspondence of scientific men of the seventeenth century (2 vols., London, 1841), ii. 528, 531Google Scholar, for references to the preservation and publication of Horrox's papers. For a judgment that ‘From the late 1650's until their eventual publication, Horrocks' manuscripts were widely circulated’, see Applebaum, Wilbur, ‘Horrocks, Jeremiah’, Dictionary of scientific biography (New York: Charles Scribner's Sons, 1970), vi. 514–16.Google Scholar

¹¹ See especially pp. 12–13. Shakerley had looked rather carefully at Horrox's papers (p. 18), and had even taken the trouble to translate the precepts for his lunar theory, which he did with such understanding that Flamsteed later ‘thought them clearer expressed than the English ones in Crabtree's letter’; see Rigaud, , op. cit. (10), ii. 157.Google Scholar

¹² This is Russell's assessment (p. 18) of Wing's rare Speculum uranicum of 1649.Google Scholar He does not say whether Wing literally used the Rudolphine table to generate the ephemeris (which is, presumably, what it is), or whether his references were of the more usual ‘inspirational’ nature.

¹³ Wing, Vincent, Harmonicon coeleste (London, 1651), p. 44.Google Scholar

¹⁴ Nor is he even alluding to planetary theory when on p. 26 he says: ‘Some things we have set down according to the opinion of Bullialdus, but in most things we have credited Kepler’. Russell, (p. 19)Google Scholar appears to have taken the statement as a general credo rather than as the specific reference to Shakerley's list of geographical coordinates that it is.

¹⁵ Miscellanies of Mr. Samuel Foster (London, 1659)Google Scholar: ‘Of the Planetary Instruments’, p. 25. Despite the date of publication, Foster died in 1652.

¹⁶ For the issues involved in this dispute, see Allen, Phyllis, ‘Scientific studies in the English universities of the seventeenth century’, Journal of the history of ideas, (1949), 219–53CrossRefGoogle Scholar, and Curtis, Mark H., Oxford and Cambridge in transition, 1558–1642: an essay on the changing relation between the English universities and English society (Oxford, 1959).Google Scholar The most important pamphlets in the exchange have been reprinted in Debus, Allen (ed.), Science and education in the seventeenth century (London, 1970).Google Scholar

¹⁷ All except Ward were among the small group of founder-members of the Royal Society. Biographical sketches of them are included in the commemorative issue (xv, 1960) of Notes and records of the Royal Society of London. It is possible that Hooke should be numbered in this group, since he was at Oxford in 1653 and displayed a life-long interest in astronomy. Unfortunately, he provides no independent information on the question at hand. His celebrated letter to Newton of 6 January 1679/80 (The correspondence of Isaac Newton [London, 1959], ii. 309)Google Scholar shows that he was sufficiently familiar with Kepler to cite his alternative formulation of the second law (namely, that the velocity of a planet in its orbit is inversely proportional to its distance from the sun). It does not rule out the possibility that he knew trie proper form, since the alternative version was crucial to his supposed derivation of the inverse square law; see Westfall, R. S., ‘Hooke and the law of universal gravitation’, The British journal for the history of science, iii (1967), 245–61.CrossRefGoogle Scholar In any case, Hooke's well-documented weakness in mathematics renders him almost useless as an indicator of contemporary mathematical knowledge.

¹⁸ The only thorough account of Boulliau's work and Ward's critique of it is in Wilson, , op. cit. (4), pp. 106–22.Google Scholar But Wilson does not comment on the fact that Wing had recognized Boulliau's mechanism as an empty-focus equant some years before Ward's publication.

¹⁹ Russell cites discussions of Kepler's physical theories and/or his law of areas in Cunitia, Maria, Urania propitia (Oels, 1650)Google Scholar; Holwarda, John Phocylides, Philosophia naturalis (Franeker, 1651)Google Scholar; and Riccioli, G. B., Almagestum novum (Bologna, 1651 and 1653).Google Scholar

²⁰ ‘Qua vero ratione Geometrice querenda sit prima inaequalitas, quomodo ex dato medio planetae alicujus motu in Ellipsi haberi, directe posset Angulus ad solem, nescivit & se nescire professus est Magnus Keplerus; Interea Falsae positionis repetita operatione Apheliorum loca satis accurate determinavit, Calculumque (in Tabulis Rudophinis) satis felicter absolvit’; in Ismaelis Bullialdi astronomiae philolaicae fundamenta, inquisitio brevis (Oxford, 1653)Google Scholar, preface.

²¹ Quanquam Bullialdus Calculum primae inaequalitatis legitime atq; ex Geometriae norma nusquam absolverit, Atq; Keplerus Impossibile existimaverit ex dato medio Planetae alicujus motus, verum atq; apparentem accurate quasi a priori invenire. Non desunt tamen Methodi, eaeq: plures, hanc rem praestandi, ego duas proponam quarum alteram D.P.N. [Sir Paul Neile] Alteram nos ipsi invenimus’; ibid., p. 26.

²² In his Astronomia geometrica (Oxford, 1656)Google Scholar, for example, Ward devotes a serious portion of his effort to the adaptation of his scheme for use with circular orbits.

²³ Newton, John, Astronomia britannica (London, 1657)Google Scholar, preface.

²⁴ See Wilson, , op. cit. (4), pp. 119–20Google Scholar, and Delambre, , op. cit. (1), pp. 168–72.Google Scholar

²⁵ Wing, Vincent, An ephemerides of the coelestial motions for XIII years (London, 1658), 140.Google Scholar

²⁶ ibid., pp. 143–4.

²⁷ Whiteside, Derek T., ‘Before the Principia: the maturing of Newton's thoughts on dynamical astronomy, 1664–84’, Journal for the history of astronomy, i (1970), 8.Google ScholarWilson, , op. cit. (4), p. 128.Google Scholar saw Wren's statement as ‘a clear echo of the Astronomia nova’.

²⁸ Russell, , op. cit. (1), p. 19.Google Scholar

²⁹ Lilly, William, Merlini anglici ephemeris (London, 1663)Google Scholar; see ‘October’ for the notation of a conjunction of Jupiter and Saturn ‘By the Rudolphin or Kepler's Tables’.

³⁰ Wing, Vincent, Astronomia britannica (London, 1669).Google Scholar

³¹ For a discussion of Streete's ideas, see Wilson, , op. cit. (4), p. 126.Google Scholar

³² ‘Some considerations of Mr. Nic. Mercator, concerning the geometrick and direct method of signor Cassini for finding the apogees, excentricities, and anomalies of the planets, as that was printed in the Journal des Scavans of Septemb. 2, 1669 …’, Philosophical transactions of the Royal Society, 25 03 1670, pp. 1168–75.Google Scholar Wilson, who regards the article as ‘the coup de mort to the simple elliptic theory’, provides a more extended discussion; see Wilson, , op. cit. (4), pp. 129–33.Google Scholar In particular, he shows that the issue was, for a short time, a cause célèbre that made the areal law notorious.

³³ ‘Neque inventus fuit hactenus, qui areas Kepleri phaenomenis satisfacere posse negaret; sed, cum eas Calculo directo exhibere nec ipse nec post eum quisquam potuerit … Quamvis autem religio fuerit Keplero, ab Hypothesi, quam Naturalem esse plane persuasum habebat, recedere; quidni liberum foret aliis periculum facere, num via quaevis alia detur, inaequalitatem Planetarum primam directo Calculo investigandi?’; ibid., p. 1174.

³⁴ See especially pp. 162–73.

³⁵ Mercator, however, died in 1687, so that the appearance of the cited translation as the astronomical section of the Cursus mathematicus three years after the Principia reflects the judgment of Leybourne rather than that of Mercator.

³⁶ The public airing of astronomical disputes was quite common at the time. Reference has already been made to the exchange between Shakerley and Wing in 1649. In 1658 (op. cit. [25], post-script) Wing complained that an author ‘lately publishing something of Astronomy, is pleased to carp at me, publiquely affirming, the Eccentrick Inequality found according to our method, is not so legitimate as that he delivers’. Streete, in particular, was willing to quarrel on any occasion. Already, in 1664, he had attacked Mercator in ‘An Advertisement concerning a new hypothesis’; see Streete, An appendix to astronomia carolina (London, 1664), p. 28.Google Scholar In the same publication (pp. 25–8) he issued ‘A Monitum to Mr. Vincent Wing’, which was duly followed by a response from Wing (Examen astronomiae carolinae ‘London, 1665’)Google Scholar and a rejoinder by Streete (Examen examinatum; or Wing's examination of astronomia carolina examined [London, 1666]Google Scholar). In 1675 he challenged Flamsteed; see Streete, , The description & use of the planetary systeme (London, 1674)Google Scholar, ‘post-script’, p. 4.

³⁷ Since Mercator was born and educated in Germany, he presumably had more exposure to the Keplerian tradition than his English contemporaries.

³⁸ For a general discussion of the teaching of science in the universities, see Allen, op. cit. (16). For a quotation of John Wallis's notorious lament about the state of mathematical education at Cambridge in the 1630s, and a partial rebuttal of it, see Curtis, , op. cit. (16), pp. 244–7.Google Scholar Everything known about the requirements of the curriculum and the scheduling and attendance of mathematical lectures after the Restoration suggests that the level of formally taught mathematics continued to be very low.

³⁹ Flamsteed's progress from Sacrobosco to Streete (as described in his twenty-first year) is documented by him on pp. 9–11 of Francis Baily's An account of the Revd. John Flamsteed (London, 1835).Google Scholar

⁴⁰ However, Halton did have two rather obscure pieces printed as appendices to Foster, 's Miscellanies (see note 15, above) in 1659Google Scholar: a letter describing a procedure he termed ‘Reflexed dialling’, and one describing an apparatus he called ‘The parallactic instrument improved’.

⁴¹ ‘… he promised me a sight of the Richleian Tables … composed by Natalis Durret, a Frenchman, more laborious, in my opinion, than ingenious … But the prescript to the tables (which is full of various faults, not to be excused by the press) I suppose may be wholly his; for the ingenious Kepler could hardly be thought guilty of such oversight, or rather errors; see Baily, , op. cit. (39), p. 21.Google Scholar Flamsteed made a translation of the introduction, which is preserved in the Sloane Manuscripts (no. 533, ff. 33 et seq.) at the British Museum; ff. 42^r–43^v contain the discussion of motion in the orbit.

⁴² Letter to John Collins, in Rigaud, , op. cit. (10), ii. 139.Google Scholar Flamsteed's distrust of such speculations extended even to Newtonian physics, and lasted throughout his life.

⁴³ ibid., p. 77.

⁴⁴ Collins's first letter to Flamsteed (15 Jan. 1669/70) is not extant, but Flamsteed's long reply (24 January; ibid., p. 94) contains the following paragraph:

‘I thank you for the catalogue of mathematical pieces. I desire you inform me of Hecker's Ephemerides, on what tables they are framed, at what year they begin and how long continue, to what meridian they are fitted, where they may be had, and at what rates; and likewise where and at what rates have the Urania propitia Mariae Gunitiae.’ It can scarcely be sheer coincidence that Hecker (Motum caelestium ephemerides ex observationibus correctis nobilissim. TYCHONIS BRAHEI, & JOH. KEPLERI hypothesibus physicis, tabulisque rudolphinis [Danzig, 1662])Google Scholar and Cunitia (described in Delambre, , op. cit. [i], pp. 324–6)Google Scholar were the two working astronomical pieces done from Kepler during the third quarter of the seventeenth century. In 1672 Collins appears to have solicited solutions to Kepler's equation; see his responses from Gregory and Strode (Rigaud, , op. cit. [10], ii. 237, 438).Google Scholar

⁴⁵ Flamsteed spent over a year getting Hecker's Ephemerides (ibid., pp. 101, 106), but was using them by early 1672 (ibid., p. 127). Thirty-five years later, when writing his auto-biographical sketches, Flamsteed would still remember specifically his use of Hecker; see Bailey, , op. cit. (39), p. 32.Google Scholar

⁴⁶ Sherburne, , The sphere of Marcus Manilius made an english poem (London, 1675), pp. 84–5Google Scholar of the Appendix.

⁴⁷ Gaythorpe shows that Horrox's approximation is equivalent to Kepler as far as the terms in e³, where it differs by only 1/6 e³ sin³M; see Gaythorpe, , op. cit. (7), p. 861.Google Scholar Both Boulliau and Wing achieved the same order of accuracy with their corrections, but with constant terms four times as large; see Whiteside, op. cit. (1), notes 25 and 31. For details of the computation, see the Appendix to this article by Gingerich and Welther.

⁴⁸ Details and documentation of Flamsteed's work are given in the Appendix.

⁴⁹ The doctrine of the sphere is frequently cited as part of SirMoore, Jonas's A new systeme of the mathematicks (London, 1681).Google Scholar Flamsteed's Gresham College lectures are in volume xxxviii of the notebooks of his papers and observations preserved at the Royal Observatory, Hurstmonceaux. They have been published on microfilm by the British Public Records Office (London, 1969) as Observations of the Royal Astronomers.

⁵⁰ Philosophical transactions (1676), p. 611.Google Scholar

⁵¹ ibid., p. 684. Halley's original English rendition was ‘which to lay aside observation gives us good reason’; see Rigaud, , op. cit. (10), i. 229.Google Scholar

⁵² Whiteside, , op. cit. (1), p. 121Google Scholar, note 17, implicitly equating knowledge of the second law with textbook use or literal citations of it, assessed the situation as follows: ‘Newton's statement in his Principia (London, 1687, Book III, p. 404Google Scholar, and equivalently in all later editions) that the “Propositio est astronomis notissima” thus becomes one more item of damning evidence of his relative unfamiliarity with contemporary astronomical literature in 1685 (when he apparently first wrote it down)’.

⁵³ Wilson, , op. cit. (4), p. 921.Google Scholar

⁵⁴ It would by no means be hopeless to argue that even the invention of the other methods was accomplished with one eye on the law of areas: compare, particularly, Boulliau's modification of the simple elliptic theory with Kepler's designation of the eccentric anomaly. Ostensibly, of course, all the work was purely empirical reconsideration of the phenomena, but even in this enterprise everyone used Kepler's selected Tychonic data.

⁵⁵ By 1900 an admittedly incomplete inventory of the ‘Bibliographie du problème de Kepler’ by the editors of the Bulletin astronomique, xvii (1900), 37–47Google Scholar, contained 123 citations.

⁵⁶ Greenwood, Nicholas, Astronomia anglicana (London, 1689)Google Scholar; discussed in Delambre, , op. cit. (1), pp. 547–8.Google Scholar

⁵⁷ Leybourne, William, Cursus mathematicus (London, 1690)Google Scholar; see above, note 35.

⁵⁸ David Gregory seems to have been the one who set the pattern for the discussion of equant hypotheses as explicit numerical approximations to Kepler's second law in the Newtonian tradition. See his Elements of astronomy (London, 1715Google Scholar: original Latin edition, 1702), pp. 381–92.

⁵⁹ See the second and third editions of Thomas Streete's Astronomia carolina (London), published by Halley, in 1710 and 1716.Google Scholar

⁶⁰ Whiston, William, Astronomical principles of religion (London, 1717).Google Scholar After correctly stating the law of areas on p. 41, Whiston refers, on p. 48, to ‘the Point of even Motion in or very near the superior Focus’. In sections XXVII and XXVIII of his Astronomical lectures (read in 1703, and published at Cambridge in Latin and English editions in 1707 and 1716), Whiston parallels very closely Gregory's discussion of Ward's hypothesis (and corrections of it) as approximations to Kepler's second law.

⁶¹ Keill, John's Introduction to the true astronomy (first Latin edition, London, 1721)Google Scholar was still citing Ward's hypothesis as an approximation as late as the sixth edition of 1778; see pp. 303–11.

⁶² Leadbetter, Charles, in his Uranoscopia (London, 1735)Google Scholar, uses Boulliau's modification only, with no reference to the law of areas.

⁶³ Heath, Robert, Astronomia accurata (London, 1760)Google Scholar, offers ‘A limited correction of Ward's hypothesis’ (pp. 109 ff.) as the first of ‘different methods of solving the Keplerian Problem’.