Astronomia nova

  • Bruce Stephenson
Part of the Studies in the History of Mathematics and Physical Sciences book series (HISTORY, volume 13)

Abstract

In 1609 Kepler published the Astronomia nova, the record of a decade’s intense labor. The full title of the work1 proclaims that his new astronomy is causal, that it is a physics of the heavens based upon an examination of the motions of the planet Mars. This book is the pinnacle of pre-Newtonian astronomy, and points the way toward modern astronomy as established by Newton. Yet the material treated is scarcely that of a definitive treatise. No trace does one find, in this book, of the assurance with which Ptolemy had explained his predecessors’ models, and his own, and no trace of the orderly and comprehensive arrangement of Ptolemy’s Almagest. The Astronomia nova is instead the account of a trip into unknown territory. We enjoy today a comfortable familiarity with the end results, the area law and the ellipse, so that the journey narrated by Kepler seems even more circuitous than it really is.

Keywords

Mercury Attenuation Resis Sine Clarification 

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References

  1. 1.
    Astronómia nova AITIOAOГHTOΣ, sev physica coelestis, tradita commentariis de motibus stellae Martis, ex observationibus G. V. Tychonis Brahe.Google Scholar
  2. 2.
    G.W., 3: 89: 7–11.Google Scholar
  3. 3.
    M. Maestlin, in G. W., 1: 138; E. S. Kennedy and V. Roberts, “The Planetary Theory of Ibn al-Shatir,” Isis 50 (1959): 227–235;MATHCrossRefGoogle Scholar
  4. 3a.
    O. Neugebauer, “On the Planetary Theory of Copernicus,” Vistas in Astronomy 10 (1968): 89–103.MathSciNetCrossRefGoogle Scholar
  5. 4.
    G. W.: 68: 40–41.Google Scholar
  6. 6.
    G. W., 3: 69: 1–3.Google Scholar
  7. 7.
    G. W, 3: 69: 4.Google Scholar
  8. 10.
    G. W, 3: 74: 10–11.Google Scholar
  9. 11.
    Proemium to Book Five of the Harmonice Mundi, in G. W., 6: 289–290.Google Scholar
  10. 19.
    G. W., 3: 84: 12–16.Google Scholar
  11. 20.
    Secundum praeconceptas et in Mysterio meo Cosmographico expressas opiniones. G. W., 3: 109: 40–41.Google Scholar
  12. 21.
    Kepler called them tabulae conditores, and did not remark on whether he thought them to have acted on Tycho’s instructions or their own initiative. G. W., 3: 115: 17–21.Google Scholar
  13. 22.
    For Ptolemaic latitude theory see O. Pedersen, A Survey of the Almagest (N.p.: Odense University Press, 1974), pp. 355–386;MATHGoogle Scholar
  14. 22a.
    O. Neugebauer, A History of Ancient Mathematical Astronomy (New York: Springer-Verlag, 1975), 1: 206–230;MATHGoogle Scholar
  15. 22b.
    R. C. Riddell, “The Latitudes of Mercury and Venus in the Almagest,” Archive for History of Exact Sciences 19 (1978): 95–111.MathSciNetCrossRefGoogle Scholar
  16. 23.
    N. Swerdlow, “The Derivation and First Draft of Copernicus’s Planetary Theory: A Translation of the Commentariolus with Commentary,” Proceedings of the American Philosophical Society 117 (1973): 484–489.Google Scholar
  17. 24.
    G. W., 3: 116–117.Google Scholar
  18. 25.
    G. W, 3: 130: 5–8.Google Scholar
  19. 26.
    See R. Small, An Account of the Astronomical Discoveries of Kepler (originally published London, 1804; reprint Madison: University of Wisconsin Press, 1963), pp. 163–177, 324–330.Google Scholar
  20. 27.
    G. W., 3: 141: 3–16.Google Scholar
  21. 28.
    G. W., 3: 131: 1–6.Google Scholar
  22. 30.
    G. W., 3: 155: 13–15.Google Scholar
  23. 31.
    For detailed exposition of the procedure, see Delambre, 1: 409–417; Small, pp. 180–185. O. Gingerich has shown that Kepler made so many calculations primarily because he started over repeatedly, with different sets of observations, between 1600 and 1604. The nested iterations were not as laborious as they first appear, because previous theory provided good initial values for the apsidal line and mean anomaly, and because the results of each adjustment usually indicated the direction and approximate size of the next adjustment. The calculations must still have been very tedious. “Kepler’s Treatment of Redundant Observations,” Internationales Kepler-Symposium, Weil-der-Stadt 1971, ed. F. Krafft, K. Meyer, B. Stickler (Hildesheim: Verlag Dr. H. A. Gerstenberg, 1973), pp. 307–314.Google Scholar
  24. 32.
    G. W., 3: 174:9–12.Google Scholar
  25. 34.
    Kepler expressed this caution clearly, using the same series of astronomical examples as in this passage, in the long letter he finally sent to Fabricius in July of 1603, letter #262 in G. W., 14: 425–426.Google Scholar
  26. 35.
    G.W., 3: 183: 4–6.Google Scholar
  27. 36.
    Furthermore, the stars themselves were mapped with the aid of lunar theory, which in turn depended on a rough solar theory. See the discussion in Neugebauer, 1: 53ff.Google Scholar
  28. 37.
    For example, Copernicus, Book three, Chap. 16.Google Scholar
  29. 38.
    G. W.:191: 26–32; quoting from letter No. 92, G. W., 13:198: 51–55. In the following discussion we shall continue to neglect the Ptolemaic and Tychonic interpretations of the second anomaly. Kepler, however, thought the analysis of Chaps. 22–26 sufficiently important that he presented it in all three interpretations.Google Scholar
  30. 39.
    G. W., 3: 192: 19–21.Google Scholar
  31. 40.
    Actually, this fact and the earth’s periodic time give only the angles between the points E. For actual longitudes Kepler had to assume an epoch, and hence an apsidal line. Again we see how important it was that Kepler was willing (happy!) to suppose that the earth’s apsidal line passed through the sun, and thus in the direction where Tycho had put it, mutatis mutandis. Google Scholar
  32. 41.
    G. W., 3: 203: 29–33.Google Scholar
  33. 42.
    G. W., 3:213: 19–21.Google Scholar
  34. 44.
    G. W., 3: 229: 36–38.Google Scholar
  35. 46.
    Noted by Caspar, G. W., 3: 467.Google Scholar
  36. 47.
    E. J. Aiton, “Kepler’s Second Law of Planetary Motion,” Isis 60 (1969): 75–90.MATHCrossRefGoogle Scholar
  37. 48.
    The classical theory of ratios is in Book five of Euclid. Concerning the importance of proper ratios to Galileo, see S. Drake in his edition of Two New Sciences (Madison: University of Wisconsin Press, 1974), pp. xxii-xxv.Google Scholar
  38. 49.
    G. W.,3:95.Google Scholar
  39. 52.
    G. W., 3: 233: 36–234: 2. Note that Kepler does not object to speaking, picturesquely, of the planet’s celeritatem in perihelio; but this phrase is not invested with any precise meaning. The geometrical proof which follows, and which is set off in italics, contains no mention of celeritas or anything similar.Google Scholar
  40. 53.
    There is a textual error in the statement of (18). In G. W., 3: 234: 27, read “Sed γδ est ad γυ…Google Scholar
  41. 54.
    Aiton, “Kepler’s Second Law,” p. 78, has appreciated the fact that in this chapter it was the equant, and not the distance law, which Kepler thought to be an approximation; also “Infinitesimals and the Area Law,” in Internationales Kepler-Symposium, Weil-der-Stadt 1971, p. 293.Google Scholar
  42. 55.
    G. W., 3: 236: 12–13.Google Scholar
  43. 56.
    The circumferences Kepler spoke of in this argument (G. W.: 239: 9–16) were not of any real orbits, but rather of imaginary paths centered on the sun, which a planet would follow if no other force affected its distance from the sun. Notice that for Kepler no centripetal force was needed to bend the orbit into a circle, since he did not think that the planet’s inertia kept it moving on the tangent.Google Scholar
  44. 57.
    G. W.: 240: 19–22. “Image” is our rendering of Kepler’s species, which has for the most part been left untranslated in other accounts. As Kepler used it the word seems to mean the appearance or visible manifestation of the sun; we shall present evidence for such an interpretation. Meanwhile we will use the English word “image” for concreteness. There should be no confusion with imago, an optical image, a word which Kepler used in his optics but not in his astronomy.Google Scholar
  45. 58.
    G. W., 3: 240: 28–29.Google Scholar
  46. 59.
    For example, J. L. E. Dreyer, A History of Astronomy from Thaïes to Kepler (New York: Dover, 1953), pp. 387–388;Google Scholar
  47. 59a.
    M. Caspar, Kepler 1571–1630 (New York: Collier, 1962), p. 143;Google Scholar
  48. 59b.
    A. Koyré, The Astronomical Revolution (Ithaca: Cornell University Press, 1973), p. 409; Aiton, “Kepler’s Second Law,” p. 78, and “Infinitesimals,” p. 294.Google Scholar
  49. 60.
    G. W., 2: 21–22.Google Scholar
  50. 62.
    G.W., 3: 240: 8–10.Google Scholar
  51. 64.
    G. W., 3: 243: 9–15. In view of some statements in Chapter 36, it may be that the visual terminology in this metaphor is not accidental. The word “aspect,” used both here and in that chapter, is interesting because of its prominence in Kepler’s astrology, and because it did not commit him to saying whether the metaphor was about the sun seeing the planets, or the planets seeing the sun.Google Scholar
  52. 65.
    Delambre (1:439) objected to this statement, pointing out that Kepler had no justification for identifying the solar equator with the orbital plane of the earth. Kepler obviously knew this; indeed he was the first astronomer to consistently treat the earth as a planet like the others. Delambre’s oversight indicates only that he failed to study Part V of the Astronomia nova. In fairness, I know of no one else who has studied those final chapters.Google Scholar
  53. 66.
    G. W., 3: 244: 19–21.Google Scholar
  54. 67.
    Probably following Gilbert, De Magnete, Book 2, Chap. 7.Google Scholar
  55. 68.
    G. W., 3: 246: 1–3 (my emphasis). Note that the virtue is explicitly not confined to the ecliptic.Google Scholar
  56. 69.
    For example, Astronomia Pars Optica, 1, prop. 9, in G. W., 2: 22.Google Scholar
  57. 70.
    G. W., 3: 250:19–27.Google Scholar
  58. 71.
    G. W., 3:251: 3–30. On 251: 22 read Planeta, with the 1609 text, rather than Planetae as printed in G. W. Google Scholar
  59. 72.
    G. W.: 251–252.Google Scholar
  60. 76.
    In penuria melioris sententiae, G. W., 3: 260: 5. By the time he published the book, he was able to add a promise here of observational evidence that the path was in fact not an eccentric circle.Google Scholar
  61. 77.
    G.W., 3: 262: 31–36.Google Scholar
  62. 79.
    G. W., 3: 265: 1–2.Google Scholar
  63. 80.
    Much confusion has surrounded this point. The clearest and most reliable discussions are those of Aiton, particularly in “Infinitesimals,” pp. 295–6 and n. 38.Google Scholar
  64. 81.
    G. W., 3:268:20–21.Google Scholar
  65. 83.
    Kepler drew this conchoid, and drew from it the conclusion that the area law was an inexact measure of the distance law, in his letter to Fabricius dated February, 1605, #281 in G. W., 15: 30–31.Google Scholar
  66. 84.
    G. W., 3: 267: 6–9; discussed above.Google Scholar
  67. 86.
    G.W., 3: 269: 24–25.Google Scholar
  68. 87.
    G. W., 3: 268: 36–269: 2.Google Scholar
  69. 88.
    Caspar noticed the figures, and remarked that Kepler “braucht diese Überlegungen im folgenden nicht mehr, hat aber seine Freude an ihnen.” Neue Astronomie (Munich: R. Oldenbourg, 1929), p. 406; also G. W.: 470.Google Scholar
  70. 89.
    G. W., 3: 46: 36–37.Google Scholar
  71. 90.
    He supposed this width to vary about as sin2 β; see Caspar’s discussion in G. W., 3: 471, or in Neue Astronomie, p. 407.Google Scholar
  72. 91.
    G. W., 3:283:38–284:3.Google Scholar
  73. 92.
    Kepler reached this conclusion in 1603, while performing the analysis of the earth’s orbit reported in Chapter 26: see the letter to Fabricius, #262, in G. W., 14: 410–411. Contrary to one’s first impression, the Astronomia nova is arranged not chronologically, but rather for didactic reasons.Google Scholar
  74. 93.
    He was using the uniformly-rotating epicycle as a model for the oval by October of 1602, as shown by the letter to Fabricius, #226 in G. W., 14: 277–278.Google Scholar
  75. 94.
    G. W., 14:263–280. The section of interest here is from 277: 563–279: 615. The figure on p. 278 is not consistent with the text, which sometimes (but not always) reads as if the labels should be reversed for points C and E. The diagram in Frisch’s edition has the two labels reversed, and his text is always consistent with his diagram. Joannis Kepleri Opera Omnia (Frankfurt: Heyder & Zimmer, 1858–1891), 3: 68 (hereafter cited as O.O). Google Scholar
  76. 95.
    “Is motus Martis consistit in nitendo contra virtutem Solis. Nam in principio temporis restitutorii seu anomaliae nititur directè retro….” G. W., 14: 277: 567–568.Google Scholar
  77. 96.
    G.W., 14:279:628–645.Google Scholar
  78. 97.
    G. W., 14: 277: 572–575.Google Scholar
  79. 98.
    G. W., 3: 292: 17–19.Google Scholar
  80. 99.
    G. W., 3: 292: 27–293: 20.Google Scholar
  81. 103.
    G. W., 3:310:23–31.Google Scholar
  82. 104.
    G. W., 3: 312:15–16.Google Scholar
  83. 105.
    G. W., 3: 311:40–312:1. For Kepler, remember, forces caused motion rather than acceleration. The planet’s velocity from its own moving virtue could be simply added to its circumsolar velocity (even though each of these motions was curvilinear), just as a Newtonian physicist would add accelerations.Google Scholar
  84. 106.
    G. W., 3: 310: 31–34.Google Scholar
  85. 108.
    “… It is a disgrace for an astronomer to try out something with numbers whose foundation he has not already seen in geometry…,” G. W., 3: 336: 32–33. In his letter to Maestlin dated 5 March, 1605, Kepler referred to Chapter 51 as Chapter 51, indicating that by that date he had assembled that much of the Astronomia nova: letter #335, in G. W., 15: 171: 62–65. At the beginning of 1605 Kepler wrote to Longomontanus that he had completed (probably these) 51 chapters: letter #323, in G. W., 15: 141: 255–256. This is one of the few indications we have of a completion date for a portion of the book as he finally published it.Google Scholar
  86. 109.
    Near the apsides the distances changed very little in any plausible theory, so that one could not be badly mistaken about the increments. (No comparison of different increments was needed.) Kepler made the curious statement that the increments were also known, reasonably well, in the mean distances. G. W., 3: 337: 33–338: 1.Google Scholar
  87. 110.
    W. Donahue has shown that the numbers Kepler published in Chapter 53 were not all derived from any one set of assumptions, and that the table published in that chapter probably represents Kepler’s attempt, after discovering his final distance law, to smooth the exposition leading up to it. “The Peccadillo of Johannes Kepler,” read at the meetings of the History of Science Society, Bloomington, Indiana, 2 November 1985.Google Scholar
  88. 111.
    G. W., 3: 329: 2–3.Google Scholar
  89. 113.
    C. Wilson has correctly emphasized both the impossibility of determining the true orbit from distance calculations alone, and the role of the equations of center alongside the rough distance calculations in suggesting the kind of improvements needed: “Kepler’s Derivation of the Elliptical Path,” Isis 59 (Spring 1968): 4–25.Google Scholar
  90. 114.
    Kepler to Fabricius, letter #308, in G. W., 15: 79: 71–80: 1.Google Scholar
  91. 115.
    G. W., 3: 346: 2–4.Google Scholar
  92. 116.
    C. Wilson (“Kepler’s Derivation,” p. 12, n. 32) appears to believe that since the libration theory supplied the distances in the table of Chapter 53, these distances, or at least the twenty of them not calculated explicitly in the text, were not observationally determined. On the contrary, the technique of that chapter was a way of using observations to confirm or refine a postulated distance. Although obtained initially from the libration theory, the distances in Chapter 53 really had been confirmed as (approximately) correct by the observations.Google Scholar
  93. 117.
    Letter #335, in G. W., 15,170–176.Google Scholar
  94. 118.
    G. W.,3:349:1–26.Google Scholar
  95. 119.
    G. W., 3: 350: 4–5.Google Scholar
  96. 120.
    Copernicus had hypothesized an annual rotation of the earth to account for its axis being always pointed in the same (tropical) direction. Kepler, whose frame of reference for the earth’s motion around the sun was no longer that of a rotating shell, asserted that mere constancy of direction of the axis did not require a special cause, although the very slow precession of the equinoxes perhaps did. G. W., 3: 350: 30–41.Google Scholar
  97. 121.
    Kepler was for a while concerned over the disagreement between his physics, which indicated a libratory force proportional to the sine of the eccentric anomaly, and his distance law, which indicated that the completed libration was in fact proportional to the versed sine of that angle, until he realized that the latter represented the cumulative effect of the former. G. W., 15:252–255.Google Scholar
  98. 122.
    G. W., 3: 354: 33–355: 2.Google Scholar
  99. 123.
    Aiton apparently missed this consideration, which is left largely implicit in Kepler’s very brief argument in G. W., 3: 354: 27–40. See “Kepler’s Second Law,” p. 83, n. 51, and “Infinitesimals,” p. 298.Google Scholar
  100. 124.
    G. W., 3: 355: 3–6.Google Scholar
  101. 125.
    G. W., 3: 355: 14.Google Scholar
  102. 126.
    Aiton believes that, at this time, Kepler conceived the sun’s rotating virtue to act along the oval path, “as if the magnetic force just steered [the planet] in the right direction” (“Infinitesimals,” pp. 298–299). Certainly Kepler realized later that the circumsolar force should be responsible only for the component of motion perpendicular to the radius. I believe that he had not worked out this distinction when he wrote Chapter 57, and that we should regard his later revision as mathematical rather than physical; but Aiton may be correct.Google Scholar
  103. 130.
    At this point Kepler used the eccentric anomaly as it had been defined on a circle, as the angle between aphelion and the planet, measured at the center of the circle. G. W., 3: 358: 38–40.Google Scholar
  104. 131.
    G. W., 3: 359: 3–7.Google Scholar
  105. 132.
    G. W., 3: 360: 8–32. Since the versed sine of the coequated anomaly was needed, we may presume that the anima would hold these fibers perpendicular to the apsidal line. The sine of the angle between the solar radius and the fibers would then be the cosine of the coequated anomaly; and changes in the cosine mirror the changes in the versed sine.Google Scholar
  106. 133.
    G. W., 3: 362:14–16.Google Scholar
  107. 134.
    The final stages in Kepler’s work on Mars are recounted in a long letter to Fabricius (#358 in G. W., 15: 240–280) dafed 11 October, 1605, which reveals little more than the account published in the Astronomia nova. Google Scholar
  108. 135.
    For example, Koyré, Astronomical Revolution, p. 259 (“a mistake”), p. 261 (“unfounded”); Aiton, “Second Law,” p. 83 (“gratuitous”). Aiton later corrected his opinion, recognizing that the placement was “natural”: “Infinitesimals,” p. 299.Google Scholar
  109. 136.
    D. Whiteside has calculated that the via buccosa was observationally indistinguishable from the final ellipse. At any given time the planet’s distance from the sun differs on the two orbits, but because the motion around the sun depends on the distance, the two positions lie within three quarters of a minute of arc in true anomaly. “Keplerian Planetary Eggs,” Journal for the History of Astronomy 5 (1974): p. 14, and “Newton’s Early Thoughts on Planetary Motion,” British Journal for the History of Science 2 (1964): p. 129 n. 42. Kepler himself never paused to make such calculations. Once he realized how to combine his physics with his distances—as discussed in the following pages—he knew that the combination was correct.Google Scholar
  110. 137.
    C. Wilson has quite properly emphasized that Kepler’s distance calculations did not nearly suffice to define the orbit, and that an essential element was his realization that the circle and the earlier ellipse gave opposite and roughly equal equal errors in the equations of center: “Kepler’s Derivation,” passim. I cannot agree, however, with his implication that this realization sufficiently determined the shape of the orbit (p. 18).Google Scholar
  111. 138.
    See the discussion in Whiteside, “Keplerian Planetary Eggs,” especially pp. 13–15.Google Scholar
  112. 139.
    G. W., 3: 365: 38.Google Scholar
  113. 141.
    G. W., 3: 299, marginal note.Google Scholar
  114. 144.
    G. W.: 392–393.Google Scholar
  115. 145.
    G. W., 3: 403: 10–17; 13: 292, 307.Google Scholar
  116. 146.
    G. W.: 404: 20–39.Google Scholar
  117. 147.
    G. W., 3: 404: 40–405: 15.Google Scholar
  118. 148.
    G. W., 3: 405: 16–22.Google Scholar
  119. 149.
    G. W., 3: 405: 23–27, 4–6.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Bruce Stephenson
    • 1
  1. 1.Department of Astronomy and AstrophysicsUniversity of ChicagoChicagoUSA

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