## Eudoxus Planetary Model

It has been claimed (Landels (1983)) that Eudoxus' model of the Universe represents the first ever example of the technique of mathematical modelling, but while the most obvious mathematical phenomena are modelled quantitatively by Eudoxus' theory, the more subtle effects are only reproduced qualitatively, if at all. Simplicius on De caelo. Translation from Heath (1932). 5 See, for example, Goldstein (1980), (1983), Knorr (1990).

of the spheres performed exactly the same function: it rotated once every 24 h and accounted for the daily rotation of the heavens. Thus, the motions of the Sun, Moon and planets with respect to the fixed stars actually were modelled using only nineteen uniform circular motions. It is not clear whether Eudoxus thought of the spheres as being real physical entities, or whether he regarded them as purely mathematical constructions. Indeed, he may not have thought that the distinction was significant.6

In Eudoxus' theory for the Moon (see Figure 2.1, in which AB represents the celestial equator and CD the ecliptic), the outermost of the three spheres completes 1 revolution from east to west in 1 day, thus accounting for the diurnal rotation of the Moon. In order to account for the motion of the Moon around the ecliptic, Eudoxus introduced a second sphere attached to the first through an axis passing through the poles of the ecliptic and rotating from west to east. Eudoxus also knew that the Moon did not follow exactly the ecliptic, but deviated a little above and below it, and accounted for this by including a

Wright (1973) has argued that the very nature of the system makes it probable that Eudoxus did think of his spheres as having a physical reality.

third sphere, the axis of rotation of which makes a small angle with that of the second sphere.

It is not known what speeds Eudoxus chose for his second and third spheres, or what inclination he gave the third sphere relative to the second; Simplicius merely states that the rotation of the third sphere was slow. It seems likely that Eudoxus would have noted that the Moon's variation in latitude amounted to something like ± 5° and so would have used avalue close to 5° forthe inclination of the third sphere to the second. For the inclination of the second sphere to the first he would have used his value for the obliquity of the ecliptic and there is reason to believe that this was 1/15 of a circle, or 24°. This is the angle subtended at the centre by a regular fifteen-sided polygon, the construction of which is given in Euclid's Elements, Book IV, Proposition 16, and which, according to Proclus, was included because of its usefulness in astronomy.7

Eudoxus' theory for the Sun was very similar to his lunar theory. The axis of the innermost sphere was inclined at a very small angle to the second, thus giving the Sun a deviation from the ecliptic. The ecliptic seems to have been defined vaguely as some great circle passing through the zodiac rather than as the path of the Sun, the definition in terms of the motion of the Sun not being formulated until the second century BCby Hipparchus. The solar theory implied that the Sun moves with constant speed relative to the fixed stars, from which it follows that the seasons are all of equal length - a statement known to be false, since in 432 BC two astronomers from Athens - Meton and Euctemon -measured the times between summer solstice and autumn equinox, autumn equinox and winter solstice, winter solstice and spring equinox, and spring equinox and summer solstice, as 90,90, 92 and 93 days, respectively. Eudoxus' theory also assumed that the heavenly bodies were each at a fixed distance from the Earth, and so was incapable of modelling the observed changes in the apparent diameters of the Sun and Moon. This is most significant for the Moon,

7 As to the speeds of the second and third spheres, there is a certain amount of disagreement. According to Dicks (1970), the period of rotation of the second sphere would have been the synodic month and the slow rotation of the third sphere in the opposite direction would not have affected the speed of the Moon very much, but would have accounted for the Moon's movement in latitude and the fact that its greatest deviations occur at points which shift steadily westwards. However, many have taken the view that Simplicius got it wrong and that it was the second sphere that rotated slowly, with a period of over 18 years, and the innermost sphere rotated once in a period of a little over 27 days. This provides a much more accurate representation of the Moon's deviations in latitude and also models the so-called Saros period of 223 synodic months (^ 18 years), which was used by the Babylonians to investigate the periodic recurrence of various lunar phenomena, including eclipses. This view, which originated in the nineteenth century and has been repeated many times since, is, according to Dicks, an example of a misleading interpretation of early astronomical thought by attributing to it a sophistication inconsistent with contemporary knowledge. The fact that Eudoxus' system of concentric spheres is poor quantitatively in virtually all other respects lends credence to Dicks' opinion, but others (e.g. Thoren (1971)) do not share his view.

the greatest apparent diameter of which is 14 per cent larger than its smallest. Simplicius was well aware that, even judged by the astronomical knowledge of Eudoxus' day, the system of concentric spheres had serious deficiencies.

Nevertheless the theories of Eudoxus and his followers fail to save the phenomena, and not only those which were first noticed at a later date, but even those which were before known and actually accepted by the authors themselves.

Perhaps the main fascination with Eudoxus' theory is with his method for accounting for the retrograde motion of the planets, which he did by adding an extra sphere (see Figure 2.2).9 Just as in the case of the Sun and the Moon, the first and second sphere together account for the diurnal rotation and the regular motion around the ecliptic in the appropriate zodiacal period. The axis of the third sphere was in the plane of the equator of the second sphere (i.e. in the ecliptic plane) and the axis of the fourth sphere was inclined at a small angle a to that of the third. Crucially, the third and fourth spheres rotated in

8 Simplicius onDe caelo. Translation from Heath (1932). Since Simplicius lived nearly 1000 years after Eudoxus, it is quite possible that the phenomena he attempted to describe using Eudoxus' scheme, such as retrograde motion, were not those that Eudoxus originally had in mind (see Goldstein (1997)). However, Yavetz (1998) claims that, given the nature of the Eudoxan model, the most likely explanation is that retrograde motion was the object of the design.

opposite directions but at the same rate of 1 revolution per synodic period of the planet.

The motion of the third and fourth spheres together combine to generate a figure-of-eight-type curve, which Eudoxus termed a hippopede, named after the device used to tether a horse by its feet.

... the fourth sphere, which turns about the poles of the inclined circle carrying the planet and rotates in the opposite sense to the third, i.e. form east to west, but in the same period, will prevent any considerable divergence (on the part of the planet) from the zodiac circle, and will cause the planet to describe about this same zodiac circle the curve called by Eudoxus the hippopede ...

When superimposed on the regular motion induced by the outermost two spheres, the hippopede has the effect of producing small deviations in latitude and occasional periods of retrograde motion, as required. This curve was only of limited success in modelling the observed planetary motion, but is an early example of the study of curves in space and its construction demonstrates a great deal of mathematical skill.

With modern mathematical techniques, visualization of Eudoxus' hippopede is much simpler than it was for the ancients. If we introduce orthogonal unit vectors {i1, ji, k1}, where ki is along the axis of rotation of the fourth sphere (FG in Figure 2.2) and i1 and j1 lie in its equatorial plane and rotate with it, then the position of the planet is given, for all times, by the vector i1. Next, we introduce a second set of vectors {i2, j2, k2} which are the same as the first except that the vectors i2 and j2 do not rotate with the sphere. Hence, the vectors i2, j2 are related to i1, j1 via a 2 x 2 rotation matrix, and if the angular rotation rate of the fourth sphere is -Q we have:

for example. Finally, we introduce an orthogonal set of vectors fixed in the third sphere with k3 along its axis of rotation (CD in Figure 2.2) and, hence, in the ecliptic plane, which makes an angle a with k1 and k2. Since the angular speed

10 Simplicius on De caelo. Translation from Heath (1932).

11 The question as to how Eudoxus devised his construction ofthe hippopede has been the subject of much research and speculation (see, for example, Neugebauer (1953) and Riddell (1979)). Aaboe (1974) claims that Eudoxus' model could have been meant only as a qualitative description of the planets, because if one enters in the appropriate periods for Venus or Mars there is simply no way of producing retrograde motion. Yavetz (1998) disputes this for the case of Mars.

Fig. 2.3. An example of a planetary path in Eudoxus' theory.

Fig. 2.3. An example of a planetary path in Eudoxus' theory.

for example. It follows that the position of the planet in terms of {i3, j3, k3} is:

13 j3 k3

cos a for example. It follows that the position of the planet in terms of {i3, j3, k3} is:

This, then, is the parametric equation of the hippopede. For the purpose of illustrating the type of planetary paths that can result from Eudoxus' model, we can superimpose a uniform motion in the k3 direction, corresponding to the motion around the ecliptic, and obtain various curves like that shown in Figure 2.3, in which the angle a was taken as 5°. The curve shows how Eudoxus' construction leads to paths that deviate by a small amount above and below the ecliptic, and are predominantly in one direction, but with regular periods of retrograde motion.

As a quantitative predictive tool, Eudoxus' system of concentric spheres would not have been much use and, since it was developed before the Greeks came into contact with the accurate arithmetic astronomy of the Babylonians, it is reasonable to think that quantitative prediction was not its purpose. However, as the goals of Greek astronomy evolved, other astronomers attempted to modify the scheme so that it better represented the observed phenomena. Callippus of Cyzicus, who was a pupil of a pupil of Eudoxus, added seven more spheres to Eudoxus' scheme.

Callippus set down the same arrangement of spheres as did Eudoxus, and gave the same number as he did for Jupiter and Saturn, but for the sun and moon he thought there were two spheres still to be added if one were going to account for the appearances, and one for each of the remaining planets.

Aristotle Metaphysics, Book A8, 1073b. Translation from Aristotle (1999).

The two extra spheres for the Sun were introduced to accommodate the variation in the length of the seasons - summer, autumn, winter and spring - which were measured by Callippus, around 330 BC, as 92, 89, 90 and 94 days, respectively, each being accurate to within 1 day.

The details of Eudoxus' theory are not known with any certainty, but we do know that the scheme exerted a profound influence over the development of astronomical thought. It was not as good at predicting the future positions of heavenly bodies as the arithmetical schemes of the Babylonians, but it was far more influential. This was because it demonstrated the power of geometrical techniques, in that superpositions of simple uniform rotations could be used to model extremely complex behaviour, and because it (as modified by Callippus) was adopted by the giant of Greek scientific philosophy - Aristotle - whose teachings dominated intellectual thought for the next 2000 years. As a scientific theory, Eudoxus' system of concentric spheres is best described as ad hoc. It explains the phenomena only in as much as they are built into the model. It does not predict or explain any independent result, and is untestable because the model's intrinsic parameters simply can be modified whenever observations

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