When Aristotle had arrived at this result, he further investigated the subject, and found, by proof, that there were many spheres, and that all moved in circles, but each with its peculiar motion as regards velocity and direction. He naturally argued that the ideal comprehended by the one sphere, which completes its circuit in one day, is different from that of another sphere which completes its circuit in thirty years; he thus arrived at the conclusion that there were as many ideals as there were spheres; each sphere has a desire for that ideal which is the source of its existence, and that desire is the cause of its individual motion, so that in fact the ideal sets the sphere in motion. Aristotle does not say, nor does any other authority, that there are ten or a hundred ideals; he simply states that their number agrees with that of the spheres. When, therefore, some of his contemporaries held that the number of spheres was fifty, he said, if that was true, the number of ideals must likewise be fifty. For the scholars in his time were few and possessed but imperfect learning; they thought that there must be a separate sphere for each movement, because they did not know that what appear to be several distinct movements can be explained as resulting from the inclination of one sphere as is, e.g., the case with the change in the longitude of a star, its declination and the places of its rising and setting noticed in the circle of the horizon. This point, however, does not concern us at present; let us therefore return to our subject.