Barker & Goldstein’s article “Theological Foundations of Kepler’s Astronomy” (Osiris, 16, 2001, pp. 88–113) claims that theological factors were crucial in Kepler’s derivation of the ellipse in the Astronomia Nova (AN). For this discussion we need to be aware of two laws announced earlier in the same work. The distance–velocity law says that the velocity of a planet is inversely proportional to its distance from the sun. The reciprocation law says, in brief, that “the planet’s body reciprocates [i.e., deviates from a circle] according to the measure of the versed sine of the eccentric anomaly” (AN, Donahue trans., p. 558), the meaning of which I shall explicate in detail below. These two laws are discovered essentially empirically, though of course by means of intermediate models, since the data is not available directly (AN, chs. 32 and 56 resp.). It is then shown that they can be physically realised by taking the sun and the planets to be magnetic (AN, chs. 33 and 57 resp.).

Now let us look at Barker & Goldstein’s reconstruction of Kepler’s derivation of the ellipse. They claim that “religious ideas contribute directly” to this derivation (p. 89) in the form of a theologically entrenched argument type called exemplum (§VII). A precise definition of exemplum appears elusive, but we need only be concerned with its alleged rôle in the Astronomia Nova. The crucial point according to Barker & Goldstein is that “the distance–velocity law and the reciprocation law are individually defensible by exemplum arguments” (p. 111), as follows:

“[O]n the grounds that the physical influence responsible for [these laws] is a species of an established genus [i.e., magnetism], this influence, whatever it is, can be recognizes as part of God’s governance of his creation and hence a law of nature.” (pp. 109–110)

I am not denying that these considerations may be part of the explanation for Kepler’s fondness for analogies. The textual evidence for this is very weak, however, as Barker & Goldstein are well aware. So to support their claim of direct religious influence, the exemplum theory needs a lot more to show for it. For this purpose they reconstruct Kepler’s derivation of the ellipse as follows.

“[The influence of the exemplum] is especially clear in the case of the last alternative to the ellipse, eliminated by Kepler in chapter 58, the via buccosa. Here the alternative curve is eliminated, not because it fails to fit the observations but because the ellipse—and only the ellipse—follows from the combination of the distance–velocity law and the reciprocation law. And what makes that a good basis for selecting between otherwise equally successful curves is that the two laws invoked here have already been shown to be parts of the providential plan, by means of exemplum inferences.” (p. 110)

Almost everything in this quotation is wrong. First of all we must note a mistake which does not seriously affect their main point: Kepler’s proof that the orbit is an ellipse (AN, ch. 59, thm. XI) uses only the reciprocation law, not the distance–velocity law, while Barker & Goldstein state repeatedly that both laws are needed for this proof (pp. 110, 111). The Newtonian idea of velocity and gravity interacting to generate the shape of the orbit is not present in Kepler. (The distance–velocity law is of course needed to compute the planet’s position on the ellipse at any given time, but this is not what Barker & Goldstein are referring to, for they specify explicitly that the theorem concerns “its two-dimensional track” only [p. 110].)

As for the rest of the above quotation, its many errors can only be sorted out after first describing what Kepler actually said. My account agrees with the excellent studies by Aiton (Mathematical Gazette, 59(410), 1975, pp. 250–260), Wilson (Isis, 59(1), 1968, pp. 4–25), and Stephenson (Kepler’s Physical Astronomy, Springer–Verlag, 1987, ch. 3). Barker & Goldstein refer to the first two of these “for different interpretations” (p. 111). I shall show that this is not a matter of “interpretations,” since in fact every step of Barker & Goldstein’s account is flatly contradicted by Kepler’s own words.

Consider first the reciprocation law, which says, in the notation of figure 1, that the Mars–sun distance is r = 1 + e cos x. The way in which Kepler discovered this law is important. He discovered “quite by chance” that “if the radius is substituted for the secant at the middle longitude [i.e., at x = 90°], this accomplishes what the observations suggest” (AN, p. 543). He then conjectured the reciprocation law as a generalisation of this: “the effect [of the reciprocation law] on all the eccentric positions will be the same as what was done here at the middle longitudes” (AN, p. 543). He then immediately (ch. 57) proceeded to satisfy himself that this law can be accounted for magnetically.

Now comes the problematic part: finding the orbit determined by this law. The problem is that the generalisation is ambiguous: it can generate either the via buccosa or the ellipse depending on how one interprets the equation. These two possible interpretations are shown in figures 2 and 3 respectively. Note that both interpretations agree when x = 90°, which, as we saw, was the empirical case that the law was induced from. Thus they are both permissible interpretations of the law. Indeed, Kepler makes this perfectly clear: on the basis of the reciprocation law “an orbit can be made for the planet … in a ‘puff-cheeked’ [i.e., via buccosa] form as well” (AN, p. 104). However, “this orbit is convicted of error by the equations” (AN, p. 104), i.e. empirically. Having “convicted” the via buccosa, Kepler “began … recalling the ellipses, quite convinced that I was following an hypothesis far, far different from the reciprocation hypothesis” (AN, p. 575). He then realised that the second interpretation of the reciprocation law is also possible, i.e., “that the distances constructed by the reciprocation and supported by the observations are contained in the perfect ellipse no less than in the puff-cheeked orbit” (AN, p. 104, italics added).

The truth is thus the exact opposite of Barker & Goldstein’s account above: the via buccosa is rejected solely for empirical reasons; despite the fact that it follows from the reciprocation law; which law does not uniquely determine the ellipse.