THE EXTENT OF THE SIDEREAL HEAVENS. 257 



globes, one of which has a diameter a thousand times as 

 great as the other. The volumes of two such spheres have 

 a ratio so great that perhaps we do not readily compre- 

 hend it. The point I now wish to illustrate is one which 

 very frequently arises in the consideration of problems in 

 astronomy, and, indeed, in other subjects as well. 



Let us take a very simple example. Suppose a cow is 

 tethered by chain in a field ; she will, of course, be only 

 able to graze over the grass which grows within the circle 

 to which her movements are confined. If we desire to 

 give the cow double as much grass as she had before, 

 would that end be accomplished by doubling the length of 

 the chain ? It would be more than accomplished ; in fact, 

 she would then have four times as much pasture as she 

 had before. The circle over which she can graze has 

 double the diameter, no doubt, but then its area is four 

 times as great. Thus we see that the area increases in a 

 far more rapid ratio than that in which the radius 

 increases. Think next of two bird cages. "We may sup- 

 pose them to be both shaped like globes, the diameter of 

 the larger being double that of the smaller. What will 

 be the freedom enjoyed by the bird in the larger cage as 

 compared with that of the less fortunate bird in the other ? 

 The volume of space over which the bird in the big globe 

 can fly is not twice nor even four times as great as that 

 allotted to the other bird ; it is no less than eight times as 

 great. 



The bulk of a grain of sand as compared with the bulk 

 of a football may illustrate the space accessible to our eyes 

 when compared with the space accessible to one of the 

 great telescopes. The larger of these spaces has a thousand 



