Kepler's Fabricated Figures: Covering up the Mess in the New Astronomy

For example, Koestler Arthur , The sleepwalkers (New York, 1963), 331–3.

Wilson Curtis , “Kepler's derivation of the elliptical path”, Isis, lix (1968), 75–90.

Johannes Kepler Gesammelte Werke (hereafter cited as KGW) (Munich, 1937–), iii, 341.

Wilson correctly points out that the agreement was not remarkably good. In fact, the difference in one place amounts to nearly six minutes of arc. This discrepancy appears to have resulted from Kepler's use of an incorrect value for the eccentricity of the Earth's orbit (for which see Wilson's article, “The error in Kepler's acronychal data for Mars”, Centaurus, xiii (1969), 263–8). However, Kepler would correctly respond that his discrepancies are far less than those of previous theories. Indeed, the correction in Mars's position resulting from the new solar theory alone can amount to more than a degree: See the New astronomy, Chapter 6, in KGW, iii, 96.

“Taediosum esset”, KGW, iii, 342.

P. 342, lines 27–28.

In KGW, iii, 337–8.

The two parts of this table appear in KGW, iii, 339 and 340, respectively.

KGW, iii, 340: “Debuit esse ex praecognitione mediocri hypotheseos, major differentia, scilicet 1275.” The word ‘hypothesis’ is used only in connection with the lower part of the table; however, Kepler says that “we have previous knowledge” (“praecognoscimus”) of the correct differences for the upper table, and it seems reasonable to presume that the source of this prior knowledge is the same for both sets of data.

The standard for these units is the semimajor axis of the Earth's orbit, which is taken to be 100,000.

KGW, iii, 339–40.

KGW, iii, 328, line 4, and 330, lines 11–15.

P. 332, line 27.

P. 325, lines 14–15.

P. 330, lines 24–29.

P. 340, lines 22–23.

P. 339, lines 31–33.

P. 342, lines 5–6, where he presents a revised aphelion and mean longitude without pointing out the change.

For example, the epoch given at the end of Chapter 53 is “anno MDC completo”. The convention at that time was to measure time from noon, and Kepler's figures show that it was noon on 1 January that he used (and not 31 December).

The position of the aphelion is determined as follows. On KGW, iii, 156, line 20, Kepler chooses as a provisional epoch the time of the first of the four oppositions upon which he is building the vicarious hypothesis. It is clear from the table on p. 150 and the positions on p. 153, lines 28–29 that the first date is 6 March 1587. From p. 166, lines 28–29, the aphelion at that time was at 28°48′55″ Leo, a position also shown in the table on p. 172. From the table on p. 170, the forward motion of the aphelion is 1′4″ per tropical year, measured with respect to the vernal equinox. From 6 March 1587 to 1 January 1601 is 5050 days, during which the aphelion progresses 14′45″, making its new epochal position 29°3′40″ Leo.

I have computed ‘Keplerian’ mean longitudes as follows. Mars's sidereal period is 686 days 23h 31m (KGW, iii, 198), which corresponds to a tropical period of 686 days 22h 17m 47s, or a mean diurnal motion of 0°.5240716149 measured with respect to the vernal equinox. From the table on p. 150, the mean longitude at the epoch (6 March 1587 at 7h 23m from noon) was 6 signs 0°47′40″. However, Kepler adjusted this Tychonic mean longitude by adding 3′55″ (p. 166, lines 27–28). So the Keplerian mean longitude at the provisional epoch was 6 signs 0°51′35″ (this number appears in the table on p. 172). To this one adds the product of the diurnal motion and the number of days. In this instance, there are 7h 23m less than 5050 days, so the mean motion is 126°24′2″ beyond full circles, making the mean longitude at the new epoch 10 signs 7°15′37″. This was then compared with the mean longitudes deduced from Kepler's other data to find the stated differences.

KGW, iii, 342, line 6.

Kepler's letter to Herwart von Hohenburg of 10 February 1605 (Letter no. 325, in KGW, xv, 145–7) states that he had actually sent the manuscript to the Emperor, although “a few chapters [were] still missing”. For the contents of this “proto-New astronomy” see Kepler's letter to Michael Maestlin, 5 March 1605 (Letter no. 335 in KGW, xv, 170–6). An account of Kepler's early work on Mars is given by Gingerich Owen , “Kepler's treatment of redundant observations; or, the computer versus Kepler revisited”, in Kraft F. Meyer K. , and Sticker B. (eds), Internationales Kepler-Symposium Weil der Stadt 1971 (Hildesheim, 1973), 307–14, and especially p. 313.

KGW, iii, 457–60. This is dated much earlier than the January 1605 version by Gingerich , “Kepler's treatment of redundant observations”, 313, note 20.

KGW, iii, 459, final paragraph: “Hic ratio reddenda, quare physica cum experientia non consentiat, et quatenus consentiat, et quomodo ex physica vere computare possimus, quomodoque ex vicaria veram eliciamus.”

P. 342, lines 27–28.

From KGW, iii, 341.

KGW, iii, 347, lines 27–34.

P. 342, lines 5–7.

The ellipse computations were carried out using the eccentricity, aphelion, and mean longitude given by Kepler (KGW, iii, 342).

Kepler himself points this out in Chapter 30 (KGW, iii, 229, lines 15–16).

The assumptions upon which these computations were based are these: The semimajor axis of the ellipse, or radius of the eccentric, is 152,342; the distance of the focus from the centre, or eccentricity of the eccentric, is 14,113; and the aphelion and mean longitude at the epoch are as given by Kepler on p. 342. Note that the circular hypothesis is not the same as the vicarious hypothesis: This is the circle whose diameter coincides with the major axis of the ellipse. For the sake of consistency, one must avoid Kepler's mean anomalies, since (as has been shown) they are derived from different versions of the vicarious hypothesis. Instead, the three curves were compared using the same mean longitude (that is, at the same moment). It would also be possible to compare them at the same true anomaly (that is, at the same heliocentric position), or at the same position with respect to the centre of the eccentric or ellipse. As it turns out, however, that makes little difference in the results.

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), 27.

KGW, xiv, 281.

“… nempe id, quod in hisce falsum, speciale est et abesse potest; quod vero necessitatem affert veritati, sub generali ratione verum omnino et ipsum est”, KGW, iii, 186.

This account closely follows Nicholas Jardine's shrewd exegesis of Kepler's Defence of Tycho. See Jardine , Birth (ref. 33), 216–19.