Moreover modern astronomers, as well as ancient, find it convenient for very many purposes to make use of this sphere, though it has no material existence, as a means of representing the directions in which the heavenly bodies are seen and their motions. For all that direct observation can tell us about the position of such an object as a star is its _direction_; its distance can only be ascertained by indirect methods, if at all. If we draw a sphere, and suppose the observer’s eye placed at its centre O (fig. 1), and then draw a straight line from O to a star S, meeting the surface of the sphere in the point _s_; then the star appears exactly in the same position as if it were at _s_, nor would its apparent position be changed if it were placed at any other point, such as S′ or S″, on this same line. When we speak, therefore, of a star as being at a point _s_ on the celestial sphere, all that we mean is that it is in the same direction as the point _s_, or, in other words, that it is situated somewhere on the straight line through O and S. The advantages of this method of representing the position of a star become evident when we wish to compare the positions of several stars. The difference of direction of two stars is the angle between the lines drawn from the eye to the stars; _e.g._, if the stars are R, S, it is the angle R O S. Similarly the difference of direction of another pair of stars, P, Q, is the angle P O Q. The two stars P and Q appear nearer together than do R and S, or farther apart, according as the angle P O Q is less or greater than the angle R O S. But if we represent the stars by the corresponding points _p_, _q_, _r_, _s_ on the celestial sphere, then (by an obvious property of the sphere) the angle P O Q (which is the same as _p_ O _q_) is less or greater than the angle R O S (or _r_ O _s_) according as the arc joining _p q_ on the sphere is less or greater than the arc joining _r s_, and in the same proportion; if, for example, the angle R O S is twice as great as the angle P O Q, so also is the arc _p q_ twice as great as the arc _r s_. We may therefore, in all questions relating only to the directions of the stars, replace the angle between the directions of two stars by the arc joining the corresponding points on the celestial sphere, or, in other words, by the distance between these points on the celestial sphere. But such arcs on a sphere are easier both to estimate by eye and to treat geometrically than angles, and the use of the celestial sphere is therefore of great value, apart from its historical origin. It is important to note that this =apparent distance= of two stars, _i.e._ their distance from one another on the celestial sphere, is an entirely different thing from their actual distance from one another in space. In the figure, for example, Q is actually much nearer to S than it is to P, but the apparent distance measured by the arc _q s_ is several times greater than _q p_.