6 A Short History cf Astronomy [Cn. I. 



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/, ^, 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 / 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 / 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 im- 

 portant 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. 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.* 



We might of course agree to regard the celestial sphere 

 as of a particular size, and then express the distance be- 

 tween 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 



* A right angle is divided into ninety degrees (90), a degree into 

 sixty minutes (60'), and a minute into sixty seconds (60"). 



