P

ecliptic

Fig. 6.7. Geocentric latitudes.

Kepler's procedure was nothing like the above, however. He used a complicated iterative geometrical procedure to work out from Tycho's observations the best possible value for f /g. This clearly was incredibly tedious, since he wrote:

If this wearisome method has filled you with loathing, it should more properly fill you with compassion for me, as I have gone through it at least 70 times at the expense of a great deal of time ...

Kepler's calculations led him to conclude that, based on a radius of 100 000, he should take f = 7232 and g = 11 332, for which f /g « 0.64, which is very close to the optimal value. The maximum deviation from the ideal elliptical value is about 2' of arc, which represents a huge improvement over Copernicus and is less than the errors introduced from other sources in Kepler's calculations. As far as longitudes were concerned, Kepler's vicarious hypothesis succeeded brilliantly, and he continued to use it as a method for determining longitudes long after he had discarded it as a physical theory.

Next, Kepler turned his attention to latitudes. In Kepler's physically motivated astronomy there was no place for Copernicus' cumbersome theory in which the planetary latitudes were linked to the motion of the Earth. As far as Kepler was concerned, Mars was orbiting the Sun, so it followed that its motion lay in a plane through the Sun, and he determined that this plane had an inclination of 1° 50' to the ecliptic.63 In a heliocentric system, the latitude as viewed from the Earth can be considerably larger than this orbital inclination, as the Earth can be much nearer the planet than the Sun. Thus, when a planet P is at opposition, we have the situation shown in Figure 6.7, in which a is the latitude as viewed from the Sun, which, in the case of Mars, has a maximum value of 1° 50', and 0 is the geocentric latitude. By measuring 0 and determining a from

61 Kepler New Astronomy, Chapter 16. The reason Kepler carried out so many iterations was due largely to the lack of any theory for dealing with redundant observations (see Gingerich (1973b)). Kepler's iterative scheme is described succinctly in Kozhamthadam (1994), Chapter 6.

62 See Whiteside (1974).

63 Kepler actually used three different methods for computing the inclination, each yielding the same answer. This was confirmation that the plane of the orbit did pass through the Sun (see Jacobsen (1999)).

his simple latitude theory, Kepler then could solve the triangle SEP and, hence, determine the distance to the planet in terms of the Earth-Sun distance.

Before Kepler, astronomers had been satisfied with separate theories for latitude and longitude, but to Kepler, who for the first time was basing his astronomy on physical principles, both phenomena should have the same cause. Thus, he checked to see whether his hypothesis matched latitude observations. The angles involved are small but, nevertheless, the accuracy of Tycho's observations was sufficient to show Kepler that his vicarious hypothesis was wrong. For an ellipse with major axis of length 2R and with the Sun at one focus, the Sun-Planet distance is given in powers of the eccentricity by r/R = 1 + e cos a — e2 sin2 a + O(e3), but the vicarious hypothesis has (from the cosine rule applied to the triangle OSM in Figure 6.6: R2 = r2 + g2R2 — 2rgR cos a):

Thus, to obtain agreement to first order, we need to take g = e, and Kepler found (using completely different methods, of course) that to construct a model based on the equant that predicted accurately planetary distances, he needed to place the equant and the Sun equidistant from the centre of the orbit, exactly as Ptolemy had done in his geocentric scheme. The longitudes are then given, from Eqn (6.2), by a = a — 2e sin a + e2 sin 2a + O(e3), which now differs from the true relation at second order, the error being (e2/4) sin 2a. The maximum error of about 8' occurs when a = ± 45°, i.e. near the octants. There is no way that anybody working with observations made prior to Tycho's could have detected such an error, but Kepler was well aware of the accuracy of the data he was working with and realized there was a problem:

Since the divine benevolence has vouchsafed us Tycho Brahe, a most diligent observer, from whose observations the 8' error in this Ptolemaic computation is shown, it is fitting that we with thankful mind both acknowledge and honour this benefit of God. For it is in this that we shall carry on, to find at length the true form of the celestial motions ...

Kepler now had perfectly good theories for both longitude and latitude, but they involved different geometrical constructions. His refusal to accept an error of 8' when he tried to explain everything from a single model led

Kepler New Astronomy, Chapter 19.

him onto a tortuous, but ultimately rewarding, path of discovery. Errors that previously would have been considered acceptable became the driving force behind the development of a radical, new astronomy. He turned his attention to the theory of the Earth's motion which he recognized as the key to a deeper understanding of planetary orbits. Observations necessarily are made from the Earth and then calculations performed to infer things about the orbit of a planet round the Sun. Because of the high degree of precision to which Kepler was working, this procedure could only work if the position of the Earth was known accurately. Now, in Kepler's physical conception of the Universe, there should be no difference between the theory for the Earth and for the other planets. But in On the Revolutions, all the planets except the Earth had an epicycle that played the role of a Ptolemaic equant. In essence, neglecting the variations in eccentricity that took place over a long period of time, Copernicus' theory for the orbit of the Earth was simply an eccentric circle, exactly like Hipparchus' solar theory devised over 1500 years previously. The reason that solar theory had lagged behind theories for the planets was that the methods developed for making accurate observations of planetary positions were no use for the Sun as it is never seen against the backdrop of fixed stars, and the one technique for obtaining very accurate solar positions was based around eclipse observations, which in turn depend on the highly complex motions of the Moon.

Kepler's method for obtaining accurate positions of the Earth was ingenious. He could determine the position of the Earth from observations of Mars, provided he could place Mars accurately. Martian longitudes were fine, as we have seen, but not distances. Kepler got round this by using observations of Mars separated by its zodiacal period (687 days) so that, however far Mars was from the Sun, it was the same each time. With this device he managed to show that the Earth's orbit was better represented by a Ptolemaic equant-eccentric mechanism, exactly like that of the other planets.65

Kepler appreciated the equivalence between Copernican and Ptolemaic planetary theory, and he knew that, in Ptolemy's scheme, the epicycle represented the orbit of the Earth round the Sun. In order to modify Ptolemy's astronomy so as to take advantage of the refined Earth orbit, the whole of Kepler's theory, complete with equant and bisected eccentricity, had to be attached to the planet's deferent. With his improvement in the theory of the Earth's motion, Kepler

65 Koyre (1973) suggested that Kepler's calculations demonstrated that the Earth's eccentricity should be bisected as in Ptolemaic planetary theory, but Kepler's method was not accurate enough to show that the centre of the orbit of the Earth should be placed exactly half way between the Sun and the equant (Wilson (1968)). He assumed simply that this was the case because it fitted-in best with his physical theory in which speeds vary inversely with the distance from the Sun.

demonstrated conclusively the greater simplicity of the heliocentric structure over the geocentric one, and realized he had hammered another nail in the coffin of Ptolemaic astronomy:

And finally ... the sun itself... will melt all this Ptolemaic apparatus like butter, and will disperse the followers of Ptolemy, some to Copernicus' camp, and some to Brahe's.

Kepler had suggested in the Secret of the Universe that the speed of an individual planet was inversely proportional to its distance from the Sun.67 In the New Astronomy, he provides a geometrical demonstration of the fact that, at least at perihelion and aphelion, this is precisely what Ptolemy's equant mechanism achieved. He also appreciated that, at other points on the orbit, this distance law was only satisfied approximately, but he put this down to the equant mechanism being a geometrical hypothesis that gave only approximate agreement with the physical law. The latter he assumed to be true throughout the orbit.

What was the 'force' that made the planets move in this way? Kepler believed that it was similar to magnetism, which had been the subject of a very influential book, On the Magnet,68 by the respected Englishman (and royal physician) William Gilbert in 1600. Gilbert concluded that the Earth was a giant magnet and Kepler reasoned that the 'motive virtue' present in the Sun which drove the planets on their orbits was also present, to a lesser extent, in the Earth, and that this was responsible for the Moon's motion. Kepler's force was not the same as magnetism, as it did not attract the planets to the Sun; instead, he envisioned a rotating Sun with rotating filaments emanating from it that drive the planets round. Kepler's erroneous conception of inertia meant that in order for the planets to be moving continually, they had to be subject constantly to a force in the direction of motion. This motive virtue spread out like light as one moved away from the Sun and, hence, its effect diminished with distance. But, unlike light - which spreads out spherically and, hence, varies inversely as the square of the distance - Kepler wanted to produce a motion that decayed in direct proportion to the distance, and so he believed that the effect of the

66 Kepler New Astronomy, Chapter 26.

Kepler did not express the law in terms of the instantaneous speed of the planet, a concept with which he was not entirely at ease, but instead said that the time taken to traverse equal arcs was proportional to the distance from the Sun. This still leaves the concept of instantaneous distance, but Kepler seems to have been more comfortable with this, even if, understandably, he did not know how to treat it properly in his mathematics.

A translation of De magnete was produced in the late nineteenth century and reprinted in the 1950s (Gilbert (1958)). Gilbert argued strongly in favour of the daily rotation of the Earth, but chose to sit on the fence regarding heliocentrism (Russell (1973)). Margolis (2002) argues that his outlook was clearly Copernican even if he never stated the fact explicitly.

filaments was in some way concentrated near the ecliptic plane in which the planets move.

This could not be the whole story, though, because otherwise the planets simply would orbit on circular paths centred on the Sun. So Kepler thought that each planet had to possess its own innate ability to steer itself along its orbit. He tried to work out what sort of additional motion a planet would have to have in order to turn a circular orbit centred on the Sun into an eccentric circular orbit, and he tried various devices including epicycles and librations (oscillations toward and away from the Sun). He concluded that the necessary variation in solar distance was extremely complicated and, thus, that in a physical astronomy, eccentric circular orbits were unnatural.

Nevertheless, he persevered with them and attempted to work out how a planet would move around such an orbit under the influence of his distance law. Since the distance was varying continuously, this was no easy task without

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