Empirical Equivalence and Approximative Methods in the New Astronomy : A Defence of Kepler against the Charge of Fraud

“Planet fakery exposed”, The Times (London), 25 January 1990, 31a. (This article is mostly an excerpt from: Broad William J. , “After 400 years, a challenge to Kepler: He fabricated his data, scholar says”, New York Times, 23 January 1990, C1, 6).

Donahue William , “Kepler's fabricated figures: Covering up the mess in the New astronomy”, Journal for the history of astronomy, xix (1988), 217–37.

Kepler Johannes , New astronomy, translated with introduction and notes by Donahue William (Cambridge, 1992).

Bialas Volker (“Keplers komplizierter Weg zur Wahrheit: Von neuen Schwierigkeiten, die Astronomia nova zu lesen”, Berichte zur Wissenschaftsgeschichte, xiii (1990), 167–76) takes exception to Donahue's claim that the area law in the ellipse is the only method by which Kepler could have calculated the table at the end of Chapter 53, and suggests another. Bialas's suggestion, and an argument about why it cannot be correct, will be discussed in ref. 19 below.

Donahue , op. cit. (ref. 2), 217, 232, and 234.

See Jardine Nicholas , The birth of history and philosophy of science: Kepler's “A defence of Tycho against Ursus”, with essays on its provenance and significance (Cambridge, 1984).

Donahue , op. cit. (ref. 2), 221.

Kepler argues for this view in his Defence of Tycho against Ursus (see Jardine , op. cit. (ref. 6).

Wilson Curtis , “Kepler's derivation of the elliptical path”, Isis, lix (1968), 4–25, reprinted in Astronomy from Kepler to Newton: Historical studies (London, 1989).

We find here a “real life example” of the Duhem-Quine thesis of the underdetermination of theory by evidence, familiar to many in the form of the “curve-fitting” problem: No matter how many data points you have, there are always many possible curves that will fit the data. (This problem is of course compounded when observational error is taken into account — even more curves will fit the data in the sense of passing through the error bars.) It follows that it is necessary to use extra-empirical considerations, such as simplicity (or, as in Kepler's case, physical arguments), to choose which line to draw through the data points.

See for example Wilson , op. cit. (ref. 9), 14 and 19.

Wilson , op. cit. (ref. 9), 21.

An excellent, more detailed, summary of the argument of Kepler's New astronomy, one not focused on contextualizing Chapter 53 as my summary is, can be found in Gingerich's Owen “Johannes Kepler”, in The general history of astronomy, ii: Planetary astronomy from the Renaissance to the rise of astrophysics. Part A: Tycho Brahe to Newton, ed. by Taton René and Wilson Curtis (Cambridge, 1989), 54–78.

Donahue , op. cit. (ref. 2), 228–9 and 233.

The overall point would not be affected were these two dates included, but since there are grounds for excluding them, they are so excluded, since that allows the point to be made more forcefully.

Donahue , op. cit. (ref. 2), 229, italics added.

Ibid.

It is interesting, given that Donahue is worried about Kepler's integrity, that Donahue accepts Kepler's assertion in Chapter 56 without re-computing the distances using the elliptical hypothesis. It is also interesting to note (and I thank an anonymous referee for pointing this out to me) that the method of reciprocation is not uniquely in agreement with the ellipse. Whiteside (Journal for the history of astronomy, v (1974), 1–21; see especially p. 14) has shown that the method of reciprocation, also known as “libration”, does not provide empirical justification for preferring the final ellipse over the via buccosa, the “puffy cheek” path. This is as we should expect, since we know (through Wilson's argument) that the margin of error in Kepler's data was too large for the choice of the ellipse to be a purely empirical matter. Whiteside notes (as Kepler did not) that the maximum optical difference between the via buccosa and the final ellipse is just one arcminute, i.e., less than the margin of error in the observations, and it follows that the two orbits are not empirically distinguishable. But Kepler, in his argument against the “puffy cheek” orbit (Chapter 58), seems to be in too much of hurry, and leaves the impression that the method of reciprocation, since it is successful when used in the ellipse, does give sufficient empirical reason to reject the via buccosa, and he then “leaps” to the ellipse. What he fails to note is that the distances and equations in the via buccosa differ from those in the ellipse by less than the margin of error in the data, which means that in fact no empirical argument will be able to provide grounds for preferring the ellipse. This is another respect in which Kepler's argument to the ellipse is imperfect, and more evidence that it is really the physical argument (especially Chapter 57) that determines the choice of the ellipse over its (empirically indistinguishable) competitors.

Bialas (op. cit. (ref. 4)) suggests that Kepler could have calculated the distances in the table of Chapter 53 by taking the mean of the distances in the circle of Chapter 44 and in the oval of Chapter 45. For the results Bialas reports (pp. 173–4), this method yields good agreement with Kepler's table of Chapter 53. (Unfortunately, among the dates Bialas considers are the ones of December 1582 — observations which, Donahue argues, we have reason to think are corrupt.) If correct, Bialas's suggestion would absolve Kepler of fraudulently using the elliptical hypothesis as support for itself. But there are two problems. First, Donahue's hypothesis, that Kepler calculated the table of Chapter 53 using the ellipse, gains additional support from consideration of the longitudes in that table as well (see the fifth section of this paper); Bialas's suggestion can tell us nothing about the longitudes. Second, Bialas's explanation cannot make sense of Kepler's remarks in Chapter 56, telling us that he calculated the table of Chapter 53 using the elliptical hypothesis. It would seem, then, that Bialas's suggestion makes Kepler a dissimulator too, albeit in Chapter 56 instead of 53. Had Kepler used Bialas's method to calculate the table, and then in Chapter 56 claimed to have computed the table using the ellipse, intending to make the elliptical hypothesis seem to have greater empirical support, his fraud would be worse than the one with which Donahue charges him. (Whether or not Kepler used Bialas's method, that method will automatically yield results that are within the margin of error of the distances in the table of Chapter 53: We know by Chapter 51 that the true distances fall about halfway between the circle and the first oval!).

There are two ways to calculate this: (i) take the average of the differences between the two tables, or (ii) take the difference between the average distances. This second way yields a difference between Donahue's and Kepler's distances for these six dates (that is, leaving the signs of the differences intact) of a tiny 4 parts, where the distance being measured is about 164 500 parts! The first way, taking the average of the absolute value of each of the distance differences (as given in the text) is a fairer comparison, however, for the simple reason that the method of Chapter 53 is not a theory of Mars's orbit; it is, rather, designed to give the Mars-Sun distance very accurately given a very particular observing situation. Therefore, the distance results obtained by the two methods should be compared pair-wise, and not as an ensemble.

Compare Wilson , op. cit. (ref. 9), 13 et passim.

For example, in his discussion of the longitudes in the table (there will be more on longitudes in Section 5 of this paper) Donahue writes, “Although the agreement … is within the limits of observational precision, we are concerned here with agreement of theories” (p. 229). The point is well taken: It is the lack of precise agreement that allows us to know that Kepler did not use the method of Chapter 53 to calculate the table. But to go on from there to a charge of fraud is to forget the fact that the results are empirically equivalent.

34.7 ÷ 164 490 = 0.000 210 955…. Donahue's average Mars-Sun distance is 164 489, with a maximum distance of 166 412; corresponding numbers for Kepler are 164 493.3, and 166 400.

Kepler is not innocent of this sort of over-precision himself, insofar as he reports orbital distances as if all six digits were empirically meaningful. He at least demonstrates elsewhere that he is aware that the distances are accurate only to within a few hundred parts.

This assumes that Donahue is right that Kepler calculated the table with the ellipse — but whatever method Kepler used to calculate the table, Donahue's calculations show that its results were empirically indistinguishable from the results of the method of Chapter 53. Kepler is therefore justified in using one method in place of the other. To be clear about the kind of interchangeability meant here, we should not ignore the fact that Kepler was not an instrumentalist about astronomical hypotheses, as is clear from his insistence that any theory of Mars's orbit must capture the planet's true motions through space, and not just its geocentric positions. That is why Kepler is interested in finding Mars's distance from the Sun in the first place. But the method described in Chapter 53 (which is not a theory of Mars's orbit) is empirically equivalent to the corresponding ellipse calculation of the Mars-Sun distance, and therefore the two are interchangeable as devices for computing the table.

The second sentence in this quotation, and Kepler's remarks that follow about the details of how the constants of the calculations were set, do indeed give the impression that the table was calculated using the method of Chapter 53. Donahue's hypothesis, that the chapter was written before the discovery of the ellipse and afterwards poorly revised, accounts for this discrepancy.

Donahue , op. cit. (ref. 2), 232ff.

An anonymous referee has posed a plausible objection: One could rightly object that “using the final theory — ellipse plus area rule — is inappropriate, when Chapter 53 is proposing to give ‘another way of exploring Mars-Sun distances’. The table would not be giving the result of an empirical exploration at all. I would not cry ‘fraud’ here, but I would be unhappy, because the empirical justification for the ellipse and area rule have to be in question.” I have several comments in response to this sort of objection, with whose spirit I must say I am sympathetic. I agree that we shouldn't be perfectly happy with what Kepler has done in Chapter 53, because he simply has not been clear enough about what is going on. The table indeed does not give the results of the advertised “other way” of exploring the distances, and this is clearly inappropriate given the lack of explanation. The results reported are, however, empirically indistinguishable from the promised results, and this seems to lessen the inappropriateness. It is important to notice, too, that Kepler makes no claim that the table provides empirical reason to prefer the elliptical hypothesis. The justification for choosing the ellipse is, in the end, not an empirical one at all. Merely to count as a competitor, any theory of Mars's orbit must give distances within the margin of error of those determined by the method of Chapter 53 (empirical adequacy is a minimum standard for theories), and in this respect any number of theories are equally empirically justified. (Empirical considerations have, therefore, a mainly negative force: They tell us which theories are non-starters.) Since empirical considerations are insufficient, we need extra-empirical reasons (Kepler uses physical arguments invoking the cause of planetary motion) to prefer one of these empirically adequate theories (the ellipse, say) over the others (like the via buccosa). This is what I meant above when I said that there is nothing that the supposed fraud could actually achieve.

Donahue , op. cit. (ref. 2), 233.

Ibid., 229.

In the notes to his translation of the New astronomy, Donahue does not explicitly mention his own (1988) article or the accusation of fraud. He does, however, discuss the fact that the table of Chapter 53 was not calculated according to the method described in that chapter, and there are more (and longer) translator's notes to Chapter 53 than for almost any other part of the New astronomy. In addition, in the “Translator's Introduction” to the New astronomy, Donahue writes, “[A]lthough Kepler often seems to have been chronicling his researches, the New astronomy is actually a carefully constructed argument that skillfully interweaves elements of history and (it should be added) of fiction” (New astronomy, 3). The footnote to the passage just quoted remarks: “The entire table at the end of Chapter 53, for example, is based upon computed longitudes presented as observations” (New astronomy, 3). This seems to indicate that Donahue didn't change his mind about what Kepler had done in Chapter 53 between writing his (1988) article and publishing the (1992) translation of the New astronomy.