78. The application of the same ideas to an apparent rotation round the observer, as in the case of the apparent daily motion of the celestial sphere, is a little more difficult. It must be remembered that the eye has no means of judging the direction of an object taken by itself; it can only judge the difference between the direction of the object and some other direction, whether that of another object or a direction fixed in some way by the body of the observer. Thus when after looking at a star twice at an interval of time we decide that it has moved, this means that its direction has changed relatively to, say, some tree or house which we had noticed nearly in its direction, or that its direction has changed relatively to the direction in which we are directing our eyes or holding our bodies. Such a change can evidently be interpreted as a change of direction, either of the star or of the line from the eye to the tree which we used as a line of reference. To apply this to the case of the celestial sphere, let us suppose that S represents a star on the celestial sphere, which (for simplicity) is overhead to an observer on the earth at A, this being determined by comparison with a line A B drawn upright on the earth. Next, earth and celestial sphere being supposed to have a common centre at O, let us suppose _firstly_ that the celestial sphere turns round (in the direction of the hands of a clock) till S comes to S′, and that the observer now sees the star on his horizon or in a direction at right angles to the original direction A B, the angle turned through by the celestial sphere being S O S′; and _secondly_ that, the celestial sphere being unchanged, the earth turns round in the opposite direction, till A B comes to A′ B′, and the star is again seen by the observer on his horizon. Whichever of these motions has taken place, the observer sees exactly the same apparent motion in the sky; and the figure shews at once that the angle S O S′ through which the celestial sphere was supposed to turn in the first case is equal to the angle A O A′ through which the earth turns in the second case, but that the two rotations are in opposite directions. A similar explanation evidently applies to more complicated cases. Hence the apparent daily rotation of the celestial sphere about an axis through the poles would be produced equally well, either by an actual rotation of this character, or by a rotation of the earth about an axis also passing through the poles, and at the same rate, but in the opposite direction, _i.e._ from west to east. This is the first motion which Coppernicus assigns to the earth.