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 brightness 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 distance. Mr. Peirce adopted the ratio 338. While introducing no perceptible change in the traditional magnitude-scale, except to rid it of irregularities, this number has the convenience of being exactly the cube of 112. 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, 338 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 413. 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 hypothesis 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. Observation 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, 214 times as great. But the actual ratio of light between successive magnitudes is found by photometric measurement to be not far from 212; different observers varying 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 uncertainty and irregularity.
The conclusion seems unavoidable that a uniform distribution of
