Kepler's ‘Via Ovalis Composita’: Unity from Diversity

Donahue W. H. , “Kepler's first thoughts on oval orbits: Text, translation, and commentary”, Journal for the history of astronomy, xxiv (1993), 71–100.

Johannes Kepler Gesammelte Werke (Munich 1937–), the superb modern edition of Kepler's works, edited by Caspar : Hereafter cited as KGW.

Kepler Johannes , Astronomia nova AITIOλOΓHTOΣ seu physica coelestis, tradita commentariis de motibus stellae Martis (Heidelberg, 1609), used here in the modern edition, KGW, iii. Also available is the English version, Johannes Kepler, New astronomy, transl. by Donahue William H. (Cambridge, 1992).

Whiteside D. T. , “Keplerian planetary eggs, laid and unlaid, 1600–1605”, Journal for the history of astronomy, v (1974), 1–21.

Aiton E. J. , “Kepler's path to the construction of his first oval orbit for Mars”, Annals of science, xxxv (1978), 173–90.

Davis A. E. L. , A mathematical elucidation of the bases of Kepler's laws (1981, published on demand by University Microfilms International, Ann Arbor, MI, since 1989).

KGW, iii, Nachbericht, 471–4.

KGW, iii, 257.

KGW, iii, 365.

Kepler adjusted the observations as he went along (with the help of Astronomia nova, chap. 17) to compensate for the motion of the apsides and other secular and periodic changes, to ensure that he could look for a curve that would, in theory, be closed (repeating).

Kepler introduced the term ‘ovoid’ as an alternative description of a curve with a single axis of symmetry in chap. 46 (KGW, iii, 295), incidentally implying strong criticism of people who use the term ‘ellipse’ imprecisely.

In the mathematics of an ellipse, QBC is known as the auxiliary angle.

Whiteside, “Keplerian planetary eggs” (ref. 4), 15.

Any suggestion that Kepler might be accused of “violation of the tradition of circularity” could be rebutted on the grounds that the typical points of all his astronomical proposed orbits — Of course including the correct ellipse — Were constructed by means of circular arcs.

By setting β = 90° in Equation (1), we can find a formula for the quadrant-distance, and thence the corresponding ‘lesser’ semi-axis, that will enable us to evaluate the constant A associated with each grade.

KGW, iii, 310.

As explained in 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), 355–93.

KGW, iii, 291.

Stephenson Bruce , Kepler's physical astronomy (New York, 1987), 92, Fig. 18a.

KGW, iii, 288.

KGW , iii, 291 and subsequently; reproduced by Donahue, op. cit. (ref. 1), 94.

Kepler chose symmetry on that basis for his diagram of chap. 58.

The situation when two sides and a non-included angle are equal is nowadays known as ‘the ambiguous case’ of congruence of a pair of triangles. A modern proof is given in Euclid, The thirteen books of ‘The Elements’, i, transl. with an introduction and commentary by Heath T. L. (Cambridge, 1908), 305–7.

Donahue , op. cit. (ref. 1), 75, 92, 97, does not regard the geometrical identification as significant, because he believes the two ovals differ in ‘physical’ characteristics.

Whiteside , “Keplerian planetary eggs” (ref. 4), 9.

KGW, iii, 311.

We apply the standard series-expansion: Sin⁻¹(e sin β) = e sin β + 1/6 e³ sin³ β + ….

This is confirmed, for example, in Davis A. E. L. , “Kepler's road to Damascus”, Centaurus, xxxv (1992), 143–64, pp. 160–1.

The limits of observation-controlled accuracy within which Kepler worked have been conveniently set out in Davis A. E. L. , “Grading the eggs”, Centaurus, xxxv (1992), 121–42, p. 124. We shall continue to rely on these limits throughout this paper.

See, for instance, Whiteside , “Keplerian planetary eggs” (ref. 4), 14.

It has been established in Davis, Elucidation (ref. 6), Appendix I (2), that the value of the true anomaly θ (in terms of common β) for an ellipse of the second grade is:

θ = β — e sin β + O(e³).

Also, from the relations set out in the Appendix, after some manipulation, we derive precisely the same result for the oval.

KGW, iii, 297.

KGW, iii, 320, lines 38–39: “Morae igitur, quas nectit in illis arcubus, qui ex centro Solis apparent aequales, sunt in dupla proportione distantiarum”.

The lack of a precise specification of the independent variable was a major obstacle to assessing the Keplerian time-measures, which was overcome only in Kepler's mature work. See Davis , op. cit. (ref. 17), §6.

Whiteside , “Keplerian planetary eggs” (ref. 4), fn. 25, has cited Fladt K. , “Das Keplersche Ei”, Elemente der Mathematik, xvii (1962), 73–78, who first discussed this relationship. But we should note that it arises in the Appendix from first principles, as an inherent property of the oval.

All twelve are briefly examined individually in Davis, Elucidation (ref. 6), Appendix III (5). Ovals II and III of chap. 50 each turned back into a circle and were therefore immediately eliminated, while Kepler himself described oval IV of chap. 50 as a monstrosity (but it has been inadvertently missed out of Appendix III (5)). A synopsis of the remaining nine ovals is set out in chap. 3, §5, Tabulation (b).

Whiteside , “Keplerian planetary eggs” (ref. 4), 7–8, 10.

Euclid , op. cit. (ref. 23), i: Kepler realized that the application was not quite exact.

Wilson Curtis , “Kepler's derivation of the elliptical path”, Isis, lix (1968), 5–25, pp. 17–18.

Whiteside , “Keplerian planetary eggs” (ref. 4), fn 23, series (a), (b), (c).

That expansion was later agreed in private correspondence, and was subsequently set out in Davis, Elucidation (ref. 6). Its radius vector appears as expression (1) in the Keplerian approach adopted in this paper.

See Davis , “Kepler's road to Damascus” (ref. 28), tabulation p. 158. The expressions for the radius vectors can be derived alternatively from the values of the constant A found as explained in ref. 15.

Kepler commented in chap. 51 (KGW, iii, 326) that he would be extremely satisfied if he could keep the uncertainties in the distances below 100 units.

“veritatem esse in utriusque medio”, chap. 55, KGW, iii, 345.

See, for instance, Davis , Elucidation (ref. 6), chap. II, §4; Davis, op. cit. (ref. 17), §5; Davis, “Kepler's road to Damascus” (ref. 28), §7.