5 o8 THE POPULAR SCIENCE MONTHLY. 



tered throughout space, or that portion of space in which visible stars 

 are situated. In this case, the number of them out to any distance 

 from our solar system must vary as the cube of that distance, while 

 their light, supposing no important variations in real size and bright- 

 ness among them, is inversely proportionate, in the mean, to the square 

 of their distance. And since we have a constant ratio of light between 

 each magnitude and the next, we must accordingly have a constant 

 ratio of mean distance, equal to the square root of this ratio inverted, 

 and a constant ratio of number, equal to the cube of the ratio of dis- 

 tance. Mr. Peirce adopted the ratio 3f . While introducing no per- 

 ceptible change in the traditional magnitude-scale, except to rid it of 

 irregularities, this number has the convenience of being exactly the 

 cube of 1|-. Considering differences in brightness as due exclusively 

 to differences in distance, we may conclude that a star of the second 

 magnitude, for instance, is just half as far again from us as one of the 

 first, and two thirds as far as one of the third. The magnitude of any 

 star, then, is to be regarded as a logarithm of the number expressing 

 its ordinal rank, 3 being the base of the system. We may thus find 

 to what magnitudes the ordinal numbers, 200 and 225 in the example 

 given, correspond, and take these as the superior and inferior limits of 

 our observer's magnitude 4. The probable corrected magnitude may 

 be considered as half way between these limits, and we can not be 

 more exact than this in our reduction, because his discrimination has 

 not been close enough to admit of it. 



There are, it will thus be seen, three ways of stating the rank 

 of the stars: by magnitudes or other devices to express differences 

 of visual sensibility, by quantities of light, and by positions on a list 

 arranged in order of decreasing luster. These three are reduced to 

 one, through Fechner's law connecting the first two, and the hy- 

 pothesis of equable distribution connecting the second and third. 



But before accepting this hypothesis of equable distribution as part 

 of our knowledge, we must see how well it agrees with the facts. Ob- 

 servation must determine if the "ratio of light" and the "ratio of 

 number " have actually the mathematical relation given above. On 

 the scale adopted by Mr. Peirce, as we have seen, the distance of a 

 star should be two thirds that of one one magnitude fainter, and its 

 light, by the law of the inverse square, 2 times as great. But the 

 actual ratio of light between successive magnitudes is found by photo- 

 metric measurement to be not far from 2^; different observers vary- 

 ing from 2-3 to 2'8, but all giving values greater than the theory. By 

 the fact, however, that the ratio thus found is constant or very nearly 

 so for all grades of brightness, we are yet justified, notwithstanding 

 the objection from its too high value, in determining magnitudes by 

 counting, and so clearing individual estimates of much of their uncer- 

 tainty and irregularity. 



The conclusion seems unavoidable that a uniform distribution of 



