[49] Omnis enim quæ videtur secundum locum mutatio, aut est propter locum mutatio, aut est propter spectatæ rei motum, aut videntis, aut certe disparem utriusque mutationem. Nam inter mota æqualiter ad eadem non percipitur motus, inter rem visam dico, et videntem (De Rev., I. v.).
I have tried to remove some of the crabbedness of the original passage by translating freely.
[50] To Coppernicus, as to many of his contemporaries, as well as to the Greeks, the simplest form of a revolution of one body round another was a motion in which the revolving body moved as if rigidly attached to the central body. Thus in the case of the earth the second motion was such that the axis of the earth remained inclined at a constant angle to the line joining earth and sun, and therefore changed its direction in space. In order then to make the axis retain a (nearly) fixed direction in space, it was necessary to add a third motion.
[51] In this preliminary discussion, as in fig. 40, Coppernicus gives 80 days; but in the more detailed treatment given in Book V. he corrects this to 88 days.
[52] Fig. 42 has been slightly altered, so as to make it agree with fig. 41.
[53] Coppernicus, instead of giving longitudes as measured from the first point of Aries (or vernal equinoctial point, chapter I., [§§ 11], 13), which moves on account of precession, measured the longitudes from a standard fixed star (α Arietis) not far from this point.
[54] According to the theory of Coppernicus, the diameter of the moon when greatest was about 1∕8 greater than its average amount; modern observations make this fraction about 1∕13. Or, to put it otherwise, the diameter of the moon when greatest ought to exceed its value when least by about 8′ according to Coppernicus, and by about 5′ according to modern observations.
[55] Euclid, I. 33.
[56] If P be the synodic period of a planet (in years), and S the sidereal period, then we evidently have (1∕P) + 1 = 1∕S for an inferior planet, and 1 - (1∕P) = 1∕S for a superior planet.
[57] Recent biographers have called attention to a cancelled passage in the manuscript of the De Revolutionibus in which Coppernicus shews that an ellipse can be generated by a combination of circular motions. The proposition is, however, only a piece of pure mathematics, and has no relation to the motions of the planets round the sun. It cannot, therefore, fairly be regarded as in any way an anticipation of the ideas of Kepler (chapter VII.).