Kepler’s Physical Astronomy pp 21-137 | Cite as

# Astronomia nova

## Abstract

In 1609 Kepler published the *Astronomia nova*, the record of a decade’s intense labor. The full title of the work^{1} 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

Optical Equation Equant Center Eccentric Anomaly Distance Theory Planetary Theory## Preview

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## References

- 1.Astronómia nova
*AITIOAO*Г*HTO*Σ, sev physica coelestis, tradita commentariis de motibus stellae Martis, ex observationibus G. V. Tychonis Brahe.Google Scholar - 2.
*G.W.*, 3: 89: 7–11.Google Scholar - 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 - 3a.O. Neugebauer, “On the Planetary Theory of Copernicus,”
*Vistas in Astronomy*10 (1968): 89–103.MathSciNetCrossRefGoogle Scholar - 4.
*G. W.*: 68: 40–41.Google Scholar - 6.
*G. W.*, 3: 69: 1–3.Google Scholar - 7.
*G. W*, 3: 69: 4.Google Scholar - 10.
*G. W*, 3: 74: 10–11.Google Scholar - 11.
- 19.
*G. W.*, 3: 84: 12–16.Google Scholar - 20.
*Secundum praeconceptas et in Mysterio meo Cosmographico expressas opiniones. G. W.*, 3: 109: 40–41.Google Scholar - 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 - 22.For Ptolemaic latitude theory see O. Pedersen,
*A Survey of the Almagest*(N.p.: Odense University Press, 1974), pp. 355–386;MATHGoogle Scholar - 22a.O. Neugebauer,
*A History of Ancient Mathematical Astronomy*(New York: Springer-Verlag, 1975), 1: 206–230;MATHGoogle Scholar - 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 - 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 - 24.
*G. W.*, 3: 116–117.Google Scholar - 25.
*G. W*, 3: 130: 5–8.Google Scholar - 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 - 27.
*G. W.*, 3: 141: 3–16.Google Scholar - 28.
*G. W.*, 3: 131: 1–6.Google Scholar - 30.
*G. W.*, 3: 155: 13–15.Google Scholar - 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 - 32.
*G. W.*, 3: 174:9–12.Google Scholar - 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 - 35.
*G.W*., 3: 183: 4–6.Google Scholar - 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
- 37.For example, Copernicus, Book three, Chap. 16.Google Scholar
- 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 - 39.
*G. W.*, 3: 192: 19–21.Google Scholar - 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 - 41.
*G. W.*, 3: 203: 29–33.Google Scholar - 42.
*G. W.*, 3:213: 19–21.Google Scholar - 44.
*G. W.*, 3: 229: 36–38.Google Scholar - 46.Noted by Caspar,
*G. W.*, 3: 467.Google Scholar - 47.E. J. Aiton, “Kepler’s Second Law of Planetary Motion,”
*Isis*60 (1969): 75–90.MATHCrossRefGoogle Scholar - 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 - 49.
*G. W.*,3:95.Google Scholar - 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 - 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 - 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 - 55.
*G. W.*, 3: 236: 12–13.Google Scholar - 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 - 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 - 58.
*G. W.*, 3: 240: 28–29.Google Scholar - 59.For example, J. L. E. Dreyer,
*A History of Astronomy from Thaïes to Kepler*(New York: Dover, 1953), pp. 387–388;Google Scholar - 59a.M. Caspar,
*Kepler*1571–1630 (New York: Collier, 1962), p. 143;Google Scholar - 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 - 60.
*G. W.*, 2: 21–22.Google Scholar - 62.
*G.W.*, 3: 240: 8–10.Google Scholar - 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 - 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
- 66.
*G. W.*, 3: 244: 19–21.Google Scholar - 67.Probably following Gilbert,
*De Magnete*, Book 2, Chap. 7.Google Scholar - 68.
*G. W.*, 3: 246: 1–3 (my emphasis). Note that the virtue is explicitly not confined to the ecliptic.Google Scholar - 69.
- 70.
*G. W.*, 3: 250:19–27.Google Scholar - 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 - 72.
*G. W.*: 251–252.Google Scholar - 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 - 77.
*G.W.*, 3: 262: 31–36.Google Scholar - 79.
- 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
- 81.
*G. W.*, 3:268:20–21.Google Scholar - 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 - 84.
*G. W.*, 3: 267: 6–9; discussed above.Google Scholar - 86.
*G.W.*, 3: 269: 24–25.Google Scholar - 87.
*G. W.*, 3: 268: 36–269: 2.Google Scholar - 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 - 89.
*G. W.*, 3: 46: 36–37.Google Scholar - 90.He supposed this width to vary about as sin
^{2}β; see Caspar’s discussion in*G. W.*, 3: 471, or in*Neue Astronomie*, p. 407.Google Scholar - 91.
*G. W.*, 3:283:38–284:3.Google Scholar - 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 - 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 - 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 - 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 - 96.
*G.W.*, 14:279:628–645.Google Scholar - 97.
*G. W.*, 14: 277: 572–575.Google Scholar - 98.
*G. W.*, 3: 292: 17–19.Google Scholar - 99.
*G. W.*, 3: 292: 27–293: 20.Google Scholar - 103.
*G. W.*, 3:310:23–31.Google Scholar - 104.
*G*. W., 3: 312:15–16.Google Scholar - 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 - 106.
*G. W.*, 3: 310: 31–34.Google Scholar - 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 - 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 - 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
- 111.
*G. W.*, 3: 329: 2–3.Google Scholar - 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 - 114.
- 115.
*G. W.*, 3: 346: 2–4.Google Scholar - 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
- 117.
- 118.
*G. W.*,3:349:1–26.Google Scholar - 119.
*G. W.*, 3: 350: 4–5.Google Scholar - 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 - 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 - 122.
*G. W.*, 3: 354: 33–355: 2.Google Scholar - 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 - 124.
*G. W.*, 3: 355: 3–6.Google Scholar - 125.
*G. W.*, 3: 355: 14.Google Scholar - 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
- 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 - 131.
*G. W.*, 3: 359: 3–7.Google Scholar - 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 - 133.
*G. W.*, 3: 362:14–16.Google Scholar - 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 - 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 - 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 - 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 - 138.See the discussion in Whiteside, “Keplerian Planetary Eggs,” especially pp. 13–15.Google Scholar
- 139.
*G. W.*, 3: 365: 38.Google Scholar - 141.
*G. W.*, 3: 299, marginal note.Google Scholar - 144.
*G. W.*: 392–393.Google Scholar - 145.
*G. W.*, 3: 403: 10–17; 13: 292, 307.Google Scholar - 146.
*G. W.*: 404: 20–39.Google Scholar - 147.
*G. W.*, 3: 404: 40–405: 15.Google Scholar - 148.
*G. W.*, 3: 405: 16–22.Google Scholar - 149.
*G. W.*, 3: 405: 23–27, 4–6.Google Scholar