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Annals of Science Volume 39, 1982 - Issue 3
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Original Articles

Kepler, elliptical orbits, and celestial circularity: A study in the persistece of metaphysical commitment

Part II

J. Bruce Brackenridge Lawrence University, Appleton, Wisconsin, 54911, USA
Pages 265-295 | Received 23 Aug 1981, Published online: 23 Aug 2006

  • The Latin edition: Kepler Johannes Gesammelte Werke von Dyck Walther Caspar Max Hammer F. List Martha Munich 1937–75 VI 287 368 19 vols (to date) this edition is cited hereafter as KGW. The German edition: translated by Max Caspar, Harmonice Mundi, Weltharmonik (Munich, 1939). The English edition: translated by Charles Glen Wallis, Great Books of the Western World, 54 vols (Chicago, 1952), XVI, 1005–85; this edition is cited hereafter as GBWW.
  • Caspar , Max . 1959 . Kepler New York translated by C. Doris Hellman Also in paperback (New York, 1962), with slightly different pagination. First German edition in 1948.
  • Bruce Brackenridge , J. 1982 . Kepler, Elliptical Orbits, and Celestial Circularity: A Study in the Persistence of Metaphysical Commitment. Part I . Annals of Science , 39 : 117 – 143 . this work is cited hereafter as Part I.
  • Russell Hanson , Norwood . 1961 . The Copernican Disturbance and the Keplerian Revolution . Journal of the History of Ideas , 22 : 169 – 184 . (pp. 182–3). See also Part I (footnote 3), 120, for a discussion of this point.
  • Holton , Gerald . 1955 . Johannes Kepler's Universe: Its Physics and Metaphysics . American Journal of Physics , 22 : 340 – 351 . (p. 347). See also Part I (footnote 3), 120, for a discussion of this point.
  • Caspar . 1959 . Kepler 287 – 287 . New York translated by C. Doris Hellman
  • Physical Cause: arises from considerations within astronomy itself. It has a physical portion that Kepler identifies with Aristotelian ‘efficient cause of motion or movers’ and a metaphysical portion that he identifies with Aristotelian ‘formal causes or figures which the movers follow (in their paths)’ and ‘Material causes or orbs and the physical and intension and remission of motions’ KGW Gesammelte Werke von Dyck Walther Caspar Max Hammer F. List Martha Munich 1937–75 II 14 14 Astronomiae Pars Opticae Cited in Robert S. Westman, ‘Kepler's Theory of Hypothesis and the Realist Dilemma’, Studies in the History and Philosophy of Science, 8 (1972), 233–64. Archetypal Cause: arises from considerations without astronomy. It is clearly metaphysical and is identified by Kepler with Aristotelian ‘formal and final cause’. This type of cause can be seen most clearly in geometry and musical harmony as made manifest in the structure of the world. For a more extended discussion of these two causes, see Part I (footnote 3), 118–19.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 9 – 9 . Munich Cited in Alexander Koyré, translated by R. E. W. Maddison, The Astronomical Revolution (London, Paris, Ithica, 1973), p. 128.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 12 – 12 . Munich Cited in Koyré (footnote 8), 141–2.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 12 – 12 . Munich Cited in Koyré (footnote 8), 141.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 13 – 13 . Munich Cited in Koyré (footnote 8), 142.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , Munich Translated into German with an introduction by Max Caspar in 1923, with a new edition in 1936. A complete English translation, Johannes Kepler, Mysterium Cosmographicum. The Secret of the Universe, translated by A.M. Duncan (New York, 1981), has just been published by Abaris Books, with introduction and commentary by Eric Aiton and preface by I. Bernard Cohen. Sections of it are also translated in Koyré (footnote 8). There is a detailed discussion in Aiton (footnote 17), and a more general analysis appears in Caspar (footnote 2).
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VII , Munich An English translation by C. G. Wallis of books IV and V appears in GBWW (footnote 1), XVI, 845–1004.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VIII , 7 – 128 . Munich It is this second edition that is translated by Duncan (footnote 12).
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 29 – 30 . Munich Cited and discussed in Eric Aiton, ‘Johannes Kepler and the Astronomy Without Hypothesis’, Japanese Studies in the History of Science, 14 (1975), 49–71 (pp. 54–5).
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VII , 267 – 270 . Munich and GBWW (footnote 13), XVI, 864–7.
  • Aiton , Eric . 1977 . Johannes Kepler and the “Mysterium Cosmographicum” . Sudhoffs A`rchiv , 61 : 185 – 185 .
  • Neugebauer , Otto . 1968 . On the Planetary Theory of Copernicus . Vistas in Astronomy , 10 : 89 – 103 .
  • See Heath Thomas L. The Thirteen Books of Euclid's Elements New York 1956 III 467 503 3 vols Euclid's propostions 13–17 of Book XIII. KGW (footnote 1), I, 46, for Kepler's numerical ratios. Also GBWW (footnote 1), XVI, 1012 for these values and for comments upon rationality.
  • For Maestlin's computed Copernican data, polyhedral postulate values, see KGW Gesammelte Werke von Dyck Walther Caspar Max Hammer F. List Martha Munich 1937–75 I 54 54 19 vols (to date) columns 2 and 3. The data appears there as an angle, and may be converted to the distance scale in which the Earth's mean radius is unity, by expressing the angle in degrees; i.e., 9° 59′ 15″ = 9·9875°. The Copernican data also appears in KGW (footnote 1), I, 74–5.
  • See KGW Gesammelte Werke von Dyck Walther Caspar Max Hammer F. List Martha Munich 1937–75 I 426 429 for Caspar's analysis of these errors.
  • Aiton . 1977 . Johannes Kepler and the “Mysterium Cosmographicum” . Sudhoffs A`rchiv , 61 : 177 – 177 .
  • See KGW Gesammelte Werke von Dyck Walther Caspar Max Hammer F. List Martha Munich 1937–75 I 429 429 for Caspar's table of corrected values, given both as distances and as angles. Also Aiton (footnote 17), 189.
  • Tycho's data was obtained from KGW Gesammelte Werke von Dyck Walther Caspar Max Hammer F. List Martha Munich 1937–75 VI 309 309 or GBWW (footnote 1), XVI, 1026. The values for the polyhedral postulate in column 3 were calculated following the procedure outlined for the calculations in Tables 1 and 2. Also see KGW (footnote 1), VI, 298–9, and GBWW (footnote 1), XVI, 1017 for Kepler's comments on using ‘Brahe's observations' in an analysis in which the tetrahedron produces an exact fit for the spheres of Mars and Jupiter.
  • Aiton . 1977 . Johannes Kepler and the “Mysterium Cosmographicum” . Sudhoffs A`rchiv , 61 : 187 – 188 .
  • See Brouwer Dirk Clemence G.M. The Solar System Kuiper G. Middlehurst B. Chicago 1961 III 31 94 5 vols (p. 47). Also Curtis Wilson, ‘Horrocks, Harmonies, and the Exactitude of Kepler's Third Law’, Studia Copernicana, XVI (Warsaw, 1978), 235–59 (p. 240).
  • The data for 1970 was obtained from Allen C.W. Astrophysical Quantities London 1973 140 141 The data for 1913 was obtained from F. Warrain, Essai Sur l'Harmonices Mundi (Paris, 1942), I, 68. The mean radius appears as the modern semi-major axis and the eccentricity, as given by Kepler, is the product of the modern eccentricity and the semi-major axis.
  • See Koyre Gesammelte Werke von Dyck Walther Caspar Max Hammer F. List Martha Munich 1937–75 I 150 155 19 vols (to date) and Aiton (footnote 17), 190.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. XIII , 32 – 33 . Munich 19 vols (to date) Cited in Aiton (footnote 17), 174.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 74 – 75 . Munich 19 vols (to date) Also see Aiton (footnote 17), 191, for a revised version of this table that is designed ‘to clarify Kepler's intention to interpolate the polyhedra so as to obtain the best fit’.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. XIV , 130 – 130 . Munich 19 vols (to date) Cited in Koyré (footnote 8), 396.
  • See the following: Koyro Gesammelte Werke von Dyck Walther Caspar Max Hammer F. List Martha Munich 1937–75 I 172 279 19 vols (to date) (By contrast to the 102 pages devoted to the Astronomia Nova, Koyré spends only 28 pages on the Mysterium Cosmographicum and 17 pages on the Harmonice Mundi.) D. T. Whiteside, ‘Keplerian Planetary Eggs, Laid and Unlaid, 1600–1605’, Journal for the History of Astronomy, 5(1974), 1–21. See note 1 for a review of earlier surveys of the Martian battles. Also Aiton (footnote 15) 57–69, Eric Aiton, ‘Kepler's Path to the Construction and Rejection of His First Oval Orbit for Mars’, Annals of Science, 35 (1978), 173–90, and Curtis Wilson, ‘Kepler's Derivation of the Elliptical Path’, Isis, 59 (1968), 5–25.
  • For a discussion of other sources for the concept of circularity, see note 118 in Aiton Johannes Kepler and the “Mysterium Cosmographicum” Sudhoffs A`rchiv 1977 61 193 193
  • Koyré . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 43 – 43 . Munich 19 vols (to date) claims ‘that by restricting himself to uniform circular motion, Copernicus, as he loudly proclaimed, got rid of the equant’. Neugebauer (footnote 18), 92–6, however, claims that buried in the actual calculations, there remains an essential equant. In any event, Kepler is to revive the equant in his first attack on Mars.
  • Koyré . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 173 – 175 . Munich 19 vols (to date) Whiteside (footnote 32), 7. Aiton (footnote 15), 58–9.
  • Koyré . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 178 – 178 . Munich 19 vols (to date) Whiteside (footnote 32), 8. Aiton (footnote 15), 60.
  • Koyré . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 178 – 178 . Munich 19 vols (to date)
  • Koyré . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 178 – 178 . Munich 19 vols (to date)
  • Aiton . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 61 – 61 . Munich 19 vols (to date)
  • Koyré . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 180 – 184 . Munich 19 vols (to date) Aiton (footnote 15), 61–4.
  • Koyré . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 182 – 182 . Munich 19 vols (to date)
  • Aiton . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 62 – 62 . Munich 19 vols (to date)
  • Aiton . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 62 – 63 . Munich 19 vols (to date) Koyré (footnote 8), 185–214.
  • See for example Symon Keith Mechanics , 3rd edn Reading, Maryland 1971 127 127
  • Koyré . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 185 – 185 . Munich 19 vols (to date)
  • See Aiton Eric Infinitesimals and the Area Law Internationales Kepler-Symposium, Weil-der-Stadt 1971 285 305 See also Eric Aiton, ‘Kepler's Second Law of Planetary Motion’, Isis, 60 (1969), 75–90.
  • Koyré . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. I , 232 – 232 . Munich 19 vols (to date)
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. III , 286 – 286 . Munich 19 vols (to date) Cited in Whiteside (footnote 32), 8.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. III , 263 – 263 . Munich 19 vols (to date) Cited in Koyré (footnote 8), 231.
  • Caspar . 1959 . Kepler 169 – 170 . New York translated by C. Doris Hellman
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. XVI , 201 – 201 . Munich 19 vols (to date) Cited in Caspar (footnote 2), 170.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. XVI , 205 – 206 . Munich 19 vols (to date) Cited in Caspar (footnote 2), 171.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 290 – 290 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1010.
  • Caspar . 1959 . Kepler New York translated by C. Doris Hellman
  • Caspar . 1959 . Kepler 86 – 86 . New York translated by C. Doris Hellman
  • Caspar . 1959 . Kepler 265 – 265 . New York translated by C. Doris Hellman
  • Westman . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. II , 252 – 252 . Munich S. K. Heninger, Touches of Sweet Harmony (San Marino, 1974).
  • Westman . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. II , 249 – 249 . Munich
  • Westman . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. II , 252 – 252 . Munich
  • Plato . 1971 . Timaeus 51 – 51 . London translated by H. D. P. Lee
  • Westman . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. II , 252 – 252 . Munich
  • Westman . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. II , 253 – 253 . Munich
  • Westman . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. II , 257 – 258 . Munich
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 302 – 302 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1020.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 306 – 306 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1023.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 313 – 313 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1031.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 313 – 313 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1032.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. XVI , 315 – 315 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1033.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 316 – 316 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1034.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 320 – 320 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1038.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 322 – 322 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1040.
  • See Caspar Kepler New York 1959 287 288 translated by C. Doris Hellman Koyre (footnote 8), 327 and 451, note 5; see also 333 and 452, note 16. Rudolf Haase, ‘Kepler's Harmonies, Between Pansophia and Mathesis Universalis’, Vistas in Astronomy, 18 (1975), 519–33, see p. 527: ‘This musical harmony could clearly not have been achieved with circular orbits, so the Creator had to sacrifice the perfect shape of the circle and instead use the ellipse’.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 323 – 323 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1040.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 324 – 324 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1041.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 328 – 328 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1048.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 328 – 328 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1048.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 321 – 321 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1039.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 330 – 330 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1050.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 330 – 330 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1050.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 331 – 331 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1051.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 332 – 332 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1052.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 339 – 339 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1057.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 356 – 356 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1073.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 358 – 358 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1074.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 359 – 359 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1076.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 357 – 359 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1074–6.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 348 – 348 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1066. See Proposition XXXVIII.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 357 – 357 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1074. The constant in the ‘distance law’ is proportional to the angular momentum of a given planet and is thus different for each planet. The constant in the ‘3/2 power law’ is proportional to the universal gravitational constant G and the mass of the Sun M, and thus is the same for each planet. In what is to follow, Kepler effectively sets this latter constant, i.e., 4π2 GM, equal to unity.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 359 – 359 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1076. The radius is given by Kepler as 9556 in the column headed ‘number of the ratio of the spheres found among the squares’.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 359 – 359 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1076.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VIII , 120 – 120 . Munich 19 vols (to date) note 10. Cited in Aiton (footnote 17), 192.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 359 – 359 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1076.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 290 – 290 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1010.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 360 – 360 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1077.
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VI , 360 – 361 . Munich 19 vols (to date) and GBWW (footnote 1), XVI, 1078-9.
  • Caspar . 1959 . Kepler 287 – 287 . New York translated by C. Doris Hellman
  • Holton . 1955 . Johannes Kepler's Universe: Its Physics and Metaphysics . American Journal of Physics , 22 : 349 – 349 . quoting H. Zaiser, Kepler als Philosoph (Stuttgart, 1932), p. 47.
  • Holton . 1955 . Johannes Kepler's Universe: Its Physics and Metaphysics . American Journal of Physics , 22 : 349 – 349 .
  • Mittlestrass , Jürgen . 1972 . Methodological Elements of Keplerian Astronomy . Studies in the History and Philosophy of Science , 8 : 203 – 232 . (translated by James Fearns) (p. 214), and p. 210, note 8.
  • Wilson . 1961 . The Solar System Edited by: Kuiper , G. and Middlehurst , B. Vol. III , 238 – 238 . Chicago 5 vols
  • Wilson . 1961 . The Solar System Edited by: Kuiper , G. and Middlehurst , B. Vol. III , 241 – 241 . Chicago 5 vols
  • KGW . 1937–75 . Gesammelte Werke Edited by: von Dyck , Walther , Caspar , Max , Hammer , F. and List , Martha . Vol. VII , 273 – 273 . Munich 19 vols (to date) Cited in Koyré (footnote 8), 453.
  • See Whiteside D.T. Newton's Early Thoughts of Planetary Motion: A Fresh Look British Journal for the History of Science 1965 2 122 122 See also Wilson (footnote 26), 235.
  • Wilson . 1961 . The Solar System Edited by: Kuiper , G. and Middlehurst , B. Vol. III , 253 – 253 . Chicago 5 vols
  • Wilson . 1961 . The Solar System Edited by: Kuiper , G. and Middlehurst , B. Vol. III , 259 – 259 . Chicago 5 vols
  • See Newton Isaac The Correspondence of Isaac Newton Turnbull H.W. Cambridge 1959 I 388 388 note 14, and A. E. Shapiro, ‘Newton's “Achromatic” Dispersion Law’, Achives for the History of Science, 21 (1979), 91–128 (pp. 109–13). Also Susi Jeans in The New Grove Dictionary of Music and Musicians, edited by Stanley Sadie, 20 vols (London, 1980), XIII, 170–1, who has a longer work in preparation entitled ‘Sir Isaac Newton and the Science of Music’. Penelope Gouk has also written on music and science; see P. Gouk, ‘The Role of Acoustics and Music Theory in the Scientific Work of Robert Hooke’, Annals of Science, 37 (1980), 573–605, and ‘The Music Theory Manuscripts of Isaac Newton’, forthcoming.
  • Jeans . 1959 . The Correspondence of Isaac Newton Edited by: Turnbull , H.W. Vol. XIII , 170 – 170 . Cambridge The basic unit or equally tempered semitone is for Newton 12√2, and he thus anticipates the later system of cents (100 cents = 12√2).
  • Newton . 1959 . The Correspondence of Isaac Newton Edited by: Turnbull , H.W. Vol. 1 , 376 – 376 . Cambridge
  • See Part I Kepler, Elliptical Orbits, and Celestial Circularity: A Study in the Persistence of Metaphysical Commitment. Part I Annals of Science 1982 39 141 141 Figure 13.
  • Shapiro . 1959 . The Correspondence of Isaac Newton Edited by: Turnbull , H.W. 109 – 109 . Cambridge note 38.
  • Gouk . 1959 . The Correspondence of Isaac Newton Edited by: Turnbull , H.W. 91 – 128 . Cambridge forthcoming.

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