The apparent distance of two points on the celestial sphere is measured numerically by the angle between the lines joining the eye to the two points, expressed in =degrees=, =minutes=, and =seconds=.[3] We might of course agree to regard the celestial sphere as of a particular size, and then express the distance between two points on it in miles, feet, or inches; but it is practically very inconvenient to do so. To say, as some people occasionally do, that the distance between two stars is so many feet is meaningless, unless the supposed size of the celestial sphere is given at the same time. It has already been pointed out that the observer is always at the centre of the celestial sphere; this remains true even if he moves to another place. A sphere has, however, only one centre, and therefore if the sphere remains fixed the observer cannot move about and yet always remain at the centre. The old astronomers met this difficulty by supposing that the celestial sphere was so large that any possible motion of the observer would be insignificant in comparison with the radius of the sphere and could be neglected. It is often more convenient—when we are using the sphere as a mere geometrical device for representing the position of the stars—to regard the sphere as moving with the observer, so that he always remains at the centre. 8. Although the stars all appear to move across the sky (§ 5), and their rates of motion differ, yet the distance between any two stars remains unchanged, and they were consequently regarded as being attached to the celestial sphere. Moreover a little careful observation would have shown that the motions of the stars in different parts of the sky, though at first sight very different, were just such as would have been produced by the celestial sphere—with the stars attached to it—turning about an axis passing through the centre and through a point in the northern sky close to the familiar pole-star. This point is called the =pole=. As, however, a straight line drawn through the centre of a sphere meets it in two points, the axis of the celestial sphere meets it again in a second point, opposite the first, lying in a part of the celestial sphere which is permanently below the horizon. This second point is also called a pole; and if the two poles have to be distinguished, the one mentioned first is called the =north pole=, and the other the =south pole=. The direction of the rotation of the celestial sphere about its axis is such that stars near the north pole are seen to move round it in circles in the direction opposite to that in which the hands of a clock move; the motion is uniform, and a complete revolution is performed in four minutes less than twenty-four hours; so that the position of any star in the sky at twelve o’clock to-night is the same as its position at four minutes to twelve to-morrow night.