Not unfrequently it is the lot of the philosophers to contend against the doctrines of the vulgar, but when it happens, as in the case of Ptolemy's researches, that the doctrines of the vulgar are corroborated by philosophical investigation which bear the stamp of the highest authority, it is not to be wondered at that such doctrines should be deemed well-nigh impregnable. In this way we may, perhaps, account for the remarkable fact that the theories of Ptolemy held unchallenged sway over the human intellect for the vast period already mentioned. Up to the present we have been speaking only of those primary motions of the heavens, by which the whole sphere appeared to revolve once every twenty-four hours. We have now to discuss the remarkable theories by which Ptolemy endeavoured to account for the monthly movement of the moon, for the annual movement of the sun, and for the periodic movements of the planets which had gained for them the titles of the wandering stars. Possessed with the idea that these movements must be circular, or must be capable, directly or indirectly, of being explained by circular movements, it seemed obvious to Ptolemy, as indeed it had done to previous astronomers, that the track of the moon through the stars was a circle of which the earth is the centre. A similar movement with a yearly period must also be attributed to the sun, for the changes in the positions of the constellations in accordance with the progress of the seasons, placed it beyond doubt that the sun made a circuit of the celestial sphere, even though the bright light of the sun prevented the stars in its vicinity, from being seen in daylight. Thus the movements both of the sun and the moon, as well as the diurnal rotation of the celestial sphere, seemed to justify the notion that all celestial movements must be "perfect," that is to say, described uniformly in those circles which were the only perfect curves. The simplest observations, however, show that the movements of the planets cannot be explained in this simple fashion. Here the geometrical genius of Ptolemy shone forth, and he devised a scheme by which the apparent wanderings of the planets could be accounted for without the introduction of aught save "perfect" movements.