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ratio 3| instead of 4 we should have to be 20 per cent, wrong 

 in the measure of magnitude, so that we should be estimating 

 erroneously as a difference of 6 magnitudes what was really 

 only 5. No human measures are perfect but, for reasons 

 which it would take too long to give here, it is practically 

 certain that our estimate is not so wrong as this. 



We must go back to an earlier assumption that the stars 

 are scattered impartially through space ; this cannot be the 

 case, at any rate in the neighbourhood of our sun. We have 

 so far been considering only the brighter stars (roughly 

 speaking those visible to the naked eye) and these must be 

 nearer to us (other things being equal) than the fainter. It 

 is after all not unnatural that in the neighbourhood of our 

 sun the stars should not be scattered at random : for we see 

 in the sky many " clusters " of stars and it is not unreasonable 

 to suppose that our sun may belong to such a cluster or 

 cloud of stars. The result would be an excess of stars near 

 us and therefore bright : and to see that this will explain the 

 observed facts we have only to turn our argument round. 

 Hitherto we have argued from the number of bright stars 

 how many faint ones there ought to be and found the estimate 

 deficient : if we start with the observed number of faint stars 

 and calculate how many bright ones there should be we shall 

 find them in excess and the excess is due to the solar cluster. 

 As an illustration, suppose we start with the number of stars 

 of the 6th magnitude in the table given above and divide 

 continually by 4, we get : 



and we have accordingly assigned 748 stars to the solar cluster. 

 We have not much guidance as to the accuracy of this crude 

 supposition but it is certainly well within the limits suggested 

 by other clusters. On a photograph taken at the Yerkes 

 Observatory of the great cluster in Hercules, Mr. W. E. Plummer 



