Fig. 5.4. Copernicus' theory for the orbit of the Earth.

The Earth E orbits uniformly about O, completing 1 revolution in 1 sidereal year, which Copernicus computes to be 365 days 6 h 9 min 40 s. The centre of the orbit of the Earth is no longer fixed but rotates around C with |CO | = r2 say, completing 1 revolution in 3434 years, and C rotates around the Sun S with |SC| = ri, completing 1 revolution in just over 53 000 years. The motion around the Sun is linked to the precession/obliquity mechanism shown in Figure 5.3 by assuming that SCO is a straight line when 0 = 0 and that this was the case in 65 BC. Copernicus took the radius of the Earth's orbit |OE| to be 10 000 units and determined from the observed values of the eccentricity at various times that r1 = 369, r2 = 48. Thus, the eccentricity varies between 0.0369 + 0.0048 = 0.0417 « 1/24 and 0.0369 - 0.0048 = 0.0321. The aphelion A oscillates about the mean aphelion A with a period of 3434 years, while A takes 53 000 years to complete 1 revolution of the Sun.

In the First Report, Rheticus added his own astrological interpretation to the motion of the centre of the orbit of the Earth:

I shall add a prediction Thus, when the eccentricity of the sun was at its maximum [i.e. when SCO was a straight line], the Roman government became a monarchy; as the eccentricity decreased, Rome too declined, as though aging, and then fell. When the eccentricity reached the boundary and quadrant of mean value [i.e. when SCO was a right angle], the Mohammedan faith was established; another great empire came into being and increased very rapidly, like the change in eccentricity. A hundred years hence, when the eccentricity will be at its minimum, this empire will complete its period. In our time it is at its pinnacle from which equally swiftly, God willing, it will fall with a mighty crash. We look forward to the coming of our Lord Jesus Christ when the centre of the eccentricity reaches the other boundary of mean value, for it was in that position at the creation of the world Thus it appears that this small circle is in very truth the Wheel of Fortune, by whose turning the kingdoms of the world have their beginnings and vicissitudes.

In describing the model illustrated in Figure 5.4 we have assumed tacitly that the Sun is at rest with the points C and O moving around small circles. However, it is possible to achieve precisely the same geometrical relationship between S and E by assuming that the centre of the orbit of the Earth O is motionless with S and C revolving around it on small circles. Copernicus realized that there was no way of saying which of O and S actually was at rest at the centre of the Universe, but it is quite clear that he believed this honour belonged to the Sun.

One of Copernicus' reasons for assuming a moving Earth is greatly to simplify the theories of the planets. But that would be of little use if it were not possible to devise a theory of the motion of the Earth that represented accurately the observed phenomena. This Copernicus achieved with a considerable degree of success, but the resulting scheme could hardly make any claim to simplicity. The theory is also full of holes. The values of many of the numerical parameters simply are assumed by Copernicus, particularly when the necessary mathematics would otherwise have been beyond him, and there are numerous arithmetical and historical inaccuracies.

The motion of the Moon

The lunar theory is much more straightforward than that for the Earth, since the observed motion of the Moon does not possess the same gross irregularities as the orbits of the planets. Copernicus agreed with the ancients that its motion takes place round the Earth; in fact, he went further. Since the Moon was now the only heavenly body orbiting the Earth, Copernicus attributed a certain similarity, or kinship, to the two bodies. This represents a sharp contrast with Aristotelian cosmology in which all celestial bodies were intrinsically different from the Earth. Copernicus objected to Ptolemy's theory of the Moon on precisely the same grounds as Ibn al-Shatir had done 200 years before him, i.e. that it violated the principle of uniform circular motion and it predicted a manifestly incorrect change in the apparent diameter of the Moon during the course of a month.

The model that Copernicus used for the longitude of the Moon is, in fact, identical to that of Ibn al-Shatir and the same as the one he had used previously

Rheticus First Report. Quoted from Rosen (1959).

in the Commentariolus, and the latitude theory is Ptolemy's. In his earlier work, Copernicus simply had adapted numerical parameters from the Alfonsine Tables, but in Book IV of On the Revolutions he set about deriving a new set of parameters for the lunar theory. He also discussed the parallax of the Moon, the distances and sizes of the Sun and Moon, and his theory of eclipses.

Whereas Ibn al-Shatir had used ri = 6; 35 and r2 = 1; 25 for the radii of the two lunar epicycles, based on a deferent radius of 60, Copernicus computes r1 = 1097 and r2 = 237 based on a deferent radius of 10 000. These parameters are almost identical, which is not surprising, since both men were attempting to reproduce essentially the same phenomena and the lunar parameters are not subject to the same long-term variations as those of the Sun. Copernicus worked out the greatest and least distances of the Moon as 68 20 and 5210 Earth radii, respectively (Ibn al-Shatir had 68; 0 and 52; 0), from which the Moon's apparent diameter should vary between 37' 34'' and 28' 45''. This, as Copernicus pointed out, is a significant improvement over Ptolemy's theory.

Copernicus' planetary theory

Unlike solar and lunar theories, planetary theory in the sixteenth century had changed little since the time of Ptolemy. The motion of the Sun and Moon was of practical importance for the construction of calendars and the determination of religious festivals, and these theories were fairly easy to test through eclipse observations. On the other hand, the only use for planetary theory was in the casting of horoscopes, and here great accuracy was not particularly relevant. It was also a very difficult matter to carry out the observations required to verify such a theory. The only significant contribution since AD 200 was Ibn al-Shatir's geometrical construction, which removed the equant from Ptolemy's theory.

What Copernicus realized - indeed, it was the raison d'etre of his heliocentric theory - was that the so-called second anomaly in the motion of all the planets (i.e. retrograde motion) that Ptolemy modelled using an epicycle for each planet, could be reduced to a single cause. Thus, with all the planets (including the Earth) orbiting the Sun, the occurrence of retrograde motion was simply a consequence of viewing the planet from a moving observatory (see Figure 5.1). This also explained the fact that retrograde motion happened always at opposition for the superior planets and at conjunction for Mercury and Venus, whereas Ptolemy had had to build these features into his model. Copernicus' theory has a further advantage, as we shall see, since actually it determines the distances of all the planets from the Sun in terms of the distance of the Earth, and so it is no longer necessary to invoke any extra metaphysical

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