O Male Factum : Rectilinearity and Kepler's Discovery of the Ellipse

Kepler Johannes , Mysterium cosmographicum: The secret of the universe, transl. by Duncan A. M. (New York, 1981), 95–97. Kepler does admit finite straightness as the “distinguishing features” — I.e., the geometrical boundaries — Of solid bodies. His argument is that the universe as a whole is inherently ordered by God and, therefore, laid out spherically about a single centre.

Ibid., 102–3.

Jammer Max , Concepts of space (Cambridge, 1954); and Koyré Alexandre , From the closed world to the infinite universe (Baltimore, 1957).

DiSalle Robert , “Spacetime theory as physical geometry”, Erkenntnis, xlii (1995), 317–37; and DiSalle Robert , Understanding space-time (Cambridge, 2006). See also Earman John , World enough and space-time (Cambridge, 1989).

Ruffner J. A. , “The curved and the straight: Cometary theory from Kepler to Hevelius”, Journal for the history of astronomy, ii (1971), 178–94.

Goldstein Bernard R. Hon Giora , “Kepler's move from orbs to orbits: Documenting a revolutionary scientific concept”, Perspectives on science, xiii (2005), 74–111.

Kepler Johannes , Johannes Kepler gesammelte Werke , ed. by von Dyck Walther Caspar Max (Munich, 1937 — Hereafter KGW), xv, 240–80.

See, for examples, Dreyer J. L. E. , A history of astronomy from Thales to Kepler (New York, 1953), 402; Koyré Alexandre , The astronomical revolution: Copernicus, Kepler, Borelli, transl. by Maddison R. E. W. (Ithaca, 1973), 259–61; Stephenson Bruce , Kepler's physical astronomy (Princeton, 1994), 107; Voelkel James R. , The composition of Kepler's Astronomia nova (Princeton, NJ, 2001), chap. 8; and Wilson Curtis , “Kepler's derivation of the elliptical path”, Isis, lix (1968), 1968–25, pp. 13–14.

Kepler , Mysterium (ref. 1), 93. Kepler is here referring to Nicholas of Cusa. See Aiton E. J. , “Celestial spheres and circles”, History of science, xix (1981), 75–114, p. 91. For more regarding Kepler's epistemology and theology, see Barker Peter Goldstein Bernard R. , “Theological foundations of Kepler's astronomy”, Osiris, xvi (2001), 2001–113; Field J. V. , Kepler's geometrical cosmology (Chicago, 1988); Jardine Nicholas , The birth of history and philosophy of science: Kepler's A defence of Tycho against Ursus (Cambridge, 1984), chap. 7; Kozhamthadam Job , The discovery of Kepler's laws (Notre Dame, 1994); Lindberg David C. , “The genesis of Kepler's theory of light: Light metaphysics from Plotinus to Kepler”, Osiris, 2nd ser., ii (1986), 5–42; and Martens Rhonda , Kepler's philosophy and the new astronomy (Princeton, 2000).

The full title of the Astronomia nova is New astronomy based upon causes or celestial physics.

We say rather “plausible”, rather than “possible” to emphasize the epistemic constraint on hypotheses. All hypotheses are “possible”, but only those that admit reasonable explanation are “plausible”. Note that Kepler's criteria for admissible hypothesis is not so different from Copernicus's reason for rejecting the Ptolemaic system. Copernicus argued that Ptolemy's use of an equant did not admit of explanation on the basis of accepted physical principles, and was therefore implausible.

Kepler wrote that all physical sciences include “a certain amount of conjecture”. Kepler Johannes , New astronomy , transl. by Donahue William H. (Cambridge, 1992; hereafter Astronomia nova), 47.

E.g., in ibid., chap. 57; Kepler Johannes , Epitome of Copernican astronomy & Harmonies of the world, transl. by Wallis Charles Glenn (Amherst, 1995), 52ff.

Stephenson , Kepler's physical astronomy (ref. 8), 3. Kepler's method here is akin to Descartes's subsequent method of radical doubt in his Meditations on first philosophy. Just as Kepler assumes planetary minds as a limiting case of comprehensibility, Descartes assumes a deceiving Demon as a limiting case of incomprehensibility. See Descartes René , Meditations on first philosophy, transl. by Cottingham John (Cambridge, 1996), 12–15.

“Computaui inde aequationes Eccentri in situbus acronychiis, officium faciunt ad unguem, de distantiis quominus idem dicam fecit earum inquirendarum Methodus paulò laxior, quae semper me circa 100 particulas in dubio relinquit, etiam cum optimae sunt obseruationes. Nosti enim optimas obseruationes uno minuto peccare posse. At unum minutum vitiat distantiam immaniter, si Planeta propè ⊙ vel ⊙ fuerit. Hoc tamen certum habeas; quam proximè verum venire.” Kepler , KGW, xv, 249–50.

Kepler , Mysterium , 62–65; Kepler , Astronomia nova, 372–5. By 1605, Kepler only possessed a preliminary version of the Area Law for which he is famous. It was not worked out in full generality until the Epitome of Copernican astronomy. See Aiton E. J. , “Kepler's second law of planetary motion”, Isis, lx (1969), 75–90; Aiton E. J. , “Infinitesimals and the area law”, in Internationales Kepler-symposium, Weil der Stadt 1971, ed. by Krafft Fritz Meyer Karl Sticker Bernhard (Hildesheim, 1973), 285–305; Barker Peter Goldstein Bernard R. , “Distance and velocity in Kepler's astronomy”, Annals of science, li (1994), 1994–73; Davis A. E. L. , “The mathematics of the area law: Kepler's successful proof in Epitome astronomiae Copernicanae (1621)”, Archive for history of exact sciences, lvii (2003), 2003–93; and Stephenson , Kepler's physical astronomy (ref. 8), 161ff.

Kepler , Astronomia nova, 404ff.

See Ibid., chap. 60.

“Sed et aliud est quod desidero in hac hypothesi: Nempe quod ad insaniam usque contendens causam naturalem confingere non possum, cur Mars cui tanta cum probabilitate libratio in diametro tribuebatur (res enim nobis ad virtutes magneticas pulchrè admodum recidebat) potius velit ire Ellipsin vel ei proximam uiam. Fortasse tamen puto uirtutes magneticas non omnino respicere sinus sed aliud aliquid.” Kepler , KGW , xv, 251. See the corresponding passage in Astronomia nova, chap. 58: Kepler , Astronomia nova, 576.

In fact, it generates the infamous via buccosa. The via buccosa is especially promising because the libration along the diameter of an epicycle is “measured” by the increase of the eccentric anomaly caused by the anima motrix. In other words, the “magnetic” vis insita moving the body toward and away from the Sun is straightforwardly related to the “magnetic” anima motrix and everything “reduces so beautifully to magnetic virtues”. Declaring himself a “wretch [miser]”, Kepler reports the failure of this model to Fabricius. See Kepler , KGW , xv, 249. See also Kepler , Astronomia nova, chap. 58.

Though it was initially troubling, Kepler eventually dispensed with this second problem by redefining astronomical terms. In the elliptical hypothesis, the eccentric angle does not measure the anomaly to the planet, but to an imaginary point on the circumscribing eccentric circle, while the planet is found on a perpendicular dropped from that point. At first, Kepler worried that this meant a descent along a radius to the Sun could not generate an ellipse, since he believed the anima motrix would carry the planet according to the eccentric anomaly, and the planet's radial descent according to the versed sine would place it on the via buccosa. To handle this particular difficulty, Kepler redefines his astronomical terms. He uses angular measures on the circumscribing eccentric to designate points on the ellipse. Thus, in Figure 1, arc DE is “named” by arc DF. The “eccentric anomaly”, now really arc DE, is specified by angle DBF, where F is a point on the circle. From the properties of an ellipse, Kepler knows that area DEA is proportional to area DFA. This allows him to employ his provisional area law, since he knows that area DFA (not area DEA, as in the later formulation) is proportional to the mean anomaly. Kepler can say, therefore, that the anima motrix, which is governed by the area law, causes the forward motion of point E, not point F. So the anima motrix causes the increase of the eccentric anomaly as it has been redefined, and allows a descent along a radius to mimic the “incursion” along a perpendicular required by the ellipse. Without this remarkable redefinition, Kepler could not have dealt with the elliptical geometry he needed. Since this solution does not require the planet itself to recognize or measure the perpendicular directly, we set it aside, though the same problem arises again in relation to the vis insita, as seen below. Note, however, the crucial importance of expressing the action of the anima motrix by an area law, as opposed to the earlier distance law. Kepler , KGW , xv, 250–1. See the corresponding discussion in Kepler , Astronomia nova, chaps. 59–60, esp. p. 593. See also Stephenson, Kepler's physical astronomy (ref. 8), 126–30.

Specifically, the axis of the Earth rotates around the normal to the ecliptic, making roughly one revolution a year in the sense opposite to the Earth's annual orbit around the Sun. If the Earth were not to “move” this way, by contrast, the axis would always point in the “same direction” — Which would require it to tilt toward or away from the Sun at all times.

“Omnino sapit magneticam vim Eccentricitas, vt est in meis Commentariis: Ut si globus Martis haberet axem magneticum, vno polo Solis appetentem, altero fugientem, eoque axe porrigeretur in longitudines medias, tunc quamdiu versatur in descendente semicirculo, maximè in longitudine media, porrigit polum appetentem versus Solem, itaque semper ad Solem accedit, sed maxime in longitudine media, nihil in apsidibus.” Kepler , KGW, xv, 251.

Kepler apparently acquired and read De magnete shortly after its appearance in 1600. He mentions it in some detail in his Apologia pro Tychone contra Ursum, which was composed shortly thereafter. Jardine , The birth of history and philosophy of science (ref. 9), 146.

“Sit nobis eadem figura coporis planetarii proposita quae supra. Dixi supra perinde esse, siue planeta consideretur vt globus, siue vt planum circuli; jam etiam hoc dico, perinde esse, siue vt planum circuli consideretur siue vt linea. Nam certam est ex Gilberto Anglo, et per se etiam sine eius authoritate, Virtutem magneticam porrigi in rectum. Quare vt globus fingitur constare ex infinitis circularibus planis, Eccentrico parallelis, quorum omnium eadem est ratio, ita circuli planum propter hanc virtutis rectitudinem, ex infinitis constat rectis, quarum rursum omnium eadem est ratio. Ergo planetae corpus ita considerari potest, vt quaelibet recta, cum nulla aliam impediat, vt supra falso confinxi.” Kepler , KGW, xv, 253.

“Sit ergo AD axis magneticus fugiens in A, appropinquans in D, repraesentans vnam ex infinitis rectis virtuosis corporis Martii. Sit autem B punctum medium inter AD, Sole in BI, dictum appropinquationem vt fugam fieri nullam, causa est, quia A et D sunt in opere aequali. Ergo hoc est quasi aequipondium. Vide me Optica cap. I.” Ibid., xv, 253–4.

Kepler Johannes , Optics: Paralipomena to Witelo & optical part of astronomy , transl. by Donahue William H. (Santa Fe, 2000; hereafter, Optics), 27.

In the Optics, Kepler defines “violent motion” or “impulse” as an attribute of light. Therefore, he conceives the action of light as something similar to the action of hard bodies colliding. Reflection, for example, is not merely a turning back of a light ray, but a “repercussion [repercussus]”. See ibid., 26, 34. The reference to optics is also significant because light, for Kepler, is both a geometrical and a natural phenomenon. Thus, if the Sun's action could be compared to light, it might similarly be brought under mathematical description, and thus made comprehensible. For recent work on this issue, see Chen-Morris Raz , “Optics, imagination, and the construction of scientific observation in Kepler's new science”, The monist , lxxxiv (2001), 453–86; Gal Ofer Chen-Morris Raz , “The archaeology of the inverse square law: (1) Metaphysical images and mathematical practices”, History of science, xliii (2005), 2005–414; and Gal Ofer Chen-Morris Raz , “Nature's drawing: Motion as mathematical order in Kepler and Galileo” (forthcoming).

Kepler , Optics, 33.

Ibid., 32.

Ibid.

In fact, this assertion is false.

Kepler , Optics, 32.

Ibid., 33.

Ibid., 33–34.

Ibid., 34.

This is apparently the source of the stream analogy that appears in chap. 57 of the Astronomia nova, although in that case, the oar is assumed to turn, rather than remain parallel.

“Sit iam Sol in BGK. Et centro B spacio BD circulus DG delineatur, et ex G, sectione circuli cum linea Solis perpendicularis in DA ducatur. Si igitur GB sit trutina, et AB, BD brachia librae, erit vt DC ad CA sic fortitudo anguli DBG ad fortitudinem ABG.“ Kepler , KGW, xv, 254.

“Itaque fuga hic tanta est, quanta DC appetentia tanta quanta AC. Aufer ab AC aequalem ipsi DC, quae sit AS. Ergo SC est hic modulus appetentiae, et AD mensura appententiae angulo nullo. Et vt AD ad SC, sic BD ad BC vel GH. Ergo sinus digressionis planetae ab apogaeo vel perigaeo metitur celeritatem accedendi.” Ibid., xv, 253–4.

Earlier, Kepler uses “digression of the planet from apogee” to signify the mean anomaly. Here, he means the eccentric anomaly.

Note that this only applies to the instantaneous motion caused by the vis insita. Kepler had yet to realize that the accumulation of such motions, each proportional to the sine of the anomaly, would be proportional to the versed sine (i.e., cosine), as required by the ellipse. In yet another remarkable flash of intuition, Kepler would come to this conclusion shortly hereafter, but that is a subject for elsewhere. See Kepler , KGW, xv, 255.

Gilbert does not seek to explain all Copernicus's earthly motions. He only aims to explain the motions Copernicus ascribes to the Earth in and of itself — I.e., the first and third motions. Gilbert does not, on the other hand, have anything to say about Copernicus's second motion, the annual orbit of the Earth around the Sun. In fact, Gilbert never explicitly affirms that the Earth moves through the cosmos. Nevertheless, Gilbert accepts, without comment, Copernicus's conclusion that the Earth is not the centre of planetary orbits. Far more telling, though, is the very fact that Gilbert sees the need to explain the third motion at all. Recall that Copernicus introduces the third motion to account for the apparent stability of the Earth's axis, given that the Earth is orbiting the Sun. If one assumes, conversely, that the Earth does not orbit, and remains in place, presumably one would also assume that its axis would remain in place, obviating any need to explain the appearance of stability. Hence, the very fact that Gilbert sees it necessary to explain the fixity of the Earth's axis implies that he accepts Copernicus's second motion. For further support of this position, see Freudenthal Gad , “Theory of matter and cosmology in William Gilbert's De magnete”, Isis, lxxiv (1983), 22–37, pp. 33ff.

Gilbert William , De magnete, transl. by Mottelay P. Fleury (New York, 1958), 180.

“Tertius his motus a Copernico inductus, non est motus omnino, sed telluris est directio stabilis, dum in circulo mango fertur, dum unam partem coeli constanter respicit.” Gilbert William , De mundo nostro sublunari philosophia nova (Amsterdam, 1651), 165.

This discussion ignores the very slow precession of the Earth's equinoxes, attributed by Copernicus to a small difference between the second and third motions and by Gilbert to a slow “wobble” of the “common mother” and, therefore, the Earth's axis.

Gilbert , De magnete (ref. 43), 66, 180.

Kepler explicitly credits Gilbert with the notion that magnetic axes remain parallel to themselves in the Astronomia nova, chap. 57, and in the Epitome, 4.3.1. Kepler , Astronomia nova , 550–1; Kepler , Epitome of copernican astronomy & Harmonies of the world (ref. 13), 95. In fact, Kepler was not alone in seeing the importance of Gilbert's thesis. There is a parallel discussion of Gilbert in Galileo Galilei, Dialogue concerning the two chief world systems, transl. by Drake Stillman (Berkeley, 1967), 345–55, 410. Galileo endorses Gilbert's conclusion that the “third motion” is not a motion, but fails to comprehend the conceptual shift underlying that conclusion. This confusion leads to difficulties surrounding Galileo's argument for the motion of the Earth based on the motion of sunspots. See Gingerich Owen , “The Galileo sunspot controversy: Proof and persuasion”, Journal for the history of astronomy, xxxiv (2003), 2003–78.

Kepler gives both explanations. See Kepler , Astronomia nova, 553.

Ibid., 567.

Ibid., 569.

Ibid., 570.

Ibid., 559. Kepler was disappointed to find, for instance, that the Earth's axis is roughly parallel to its apsidal line, not perpendicular, as needed in the case of Mars, and that its direction slowly changes. Nevertheless, Kepler decided to “relinquish” these objections, emphasizing, meanwhile, the conceptual importance of Gilbert's treatment of the third motion over its factual content. (“Nam in meis Commentariis relicta fuit haec obiectio: Si planetae per directionem axis in easdem mundi plagas virtute magnetica eccentricitates conficiunt, Terra idem faciet. At Terrae axis is solum directus est qui porrigitur à in …. Illam vero obiectionem de Telluris axe in apsidum lineam inconstanter tamen porrecto superis discutiendam relinquamus.” Kepler, KGW, xv, 254–6.) Kepler also ignored the very important point that the anomaly in question here is the eccentric anomaly, which violates his rejection of physical references to empty points, such as the eccentric centre. Thus, the anomaly should be the equated anomaly, which is measured to the body of the Sun. In the Astronomia nova, Kepler dismisses this difference as insignificant. Kepler, Astronomia nova, 558; Stephenson, Kepler's physical astronomy (ref. 8), 115–16. Later, in the Epitome of Copernican astronomy, Kepler introduces another libration of the magnetic axis precisely equal to the optical equation — The difference of the two anomalies. Thus, the magnetic axis “measures” the eccentric anomaly even though it is physically affected according to the equated anomaly. See Kepler , Epitome of Copernican astronomy & Harmonies of the world (ref. 13), 99–106; and Stephenson, Kepler's physical astronomy (ref. 8), 146–72.

In fact, Kepler emphasized the importance of the magnetic balance for planetary orbits throughout his career. He repeats the full explanation, including his “law of the balance”, in the Epitome, Kepler Book V. , Epitome of Copernican astronomy & Harmonies of the world (ref. 13), 128–33. One of the goddesses atop the Temple of Urania on Kepler's frontispiece of the Rudolphine Tables (1627) holds an unequally weighted balance with the Sun at its fulcrum, which may also be a reference to this mechanism. The goddess is flanked by images of Magnetica, holding a compass and lodestone, and Geometria, displaying the elliptical orbit with its perpendiculars to the apsidal line. Though he has previously claimed otherwise, Owen Gingerich has told me he concurs in this speculation. See Gingerich Owen , “Johannes Kepler and the Rudolphine Tables” in The great Copernicus chase (Cambridge, 1992), 123–31.