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SCIENCE PROGRESS 



calculation shows that the resulting increase is as the cube 

 of the radius, so that if there were i,ooo (or lo^) stars on the 

 first shell of radius lo, we should find 4,096 (or 16^) on the 

 next shell of radius 16, which is about 4 times as many : on 

 the next shell of radius 25 we should find 15,625 (or 25^), which 

 is again about 4 times the number. Had we taken more 

 accurate figures for the successive distances, instead of only 

 approximate values, we should have found a constant ratio, 

 slightly less than 4, for the numbers on successive surfaces : 

 that is to say, that on this erroneous hypothesis of stellar 

 brightness being merely an effect of distance, we should expect 

 to find 4 times as many stars of the 3rd magnitude as of the 

 2nd : 4 times as many of the 4th as of the 3rd : 4 times as many 

 again of the 5th : and so on continually. Now this expectation 

 is not fulfilled : the ratio is nearer 3 than 4, as the following 

 figures (taken from Newcomb's The Stars : a Study of the 

 Universe^ p. 54) show : 



[We begin with the second magnitude because stars brighter 

 than this are so few that the numbers have an accidental 

 character.] 



What reason can be assigned for this discrepancy between 

 expectation and observation ? The one first to be suspected 

 is that the considerable assumption just made, that the stars 

 are all of the same intrinsic brightness, is not correct and is 

 answerable for the discrepancy. But on examination we very 

 soon find that error in this assumption can only increase the 

 discrepancy and is without effect in diminishing it. Suppose for 

 simplicity there were two kinds of stars, one much brighter 

 intrinsically than the other. It will remind us that the 

 difference is in the stars themselves, not an effect of distance, 

 if we use two special words such as "brilliant" and "glowing" 

 to distinguish them. Then in the first shell there will be say 

 50 brilliant and 50 glowing stars (the numbers are only 



