of the resulting cosmologies may be, they must, nevertheless, be regarded as little more than guesses, since the scientific data necessary to a sound conception of the problem are still lacking. The chief elements required are the determination of the number, distribution, constitution, brightness, distance and motions of the stars. All these determinations are now possible to science, at least in part, although only a few decades ago several of them would have seemed impossible.
A recent contribution of great importance in this line is a study of the 'Distribution of Stars,' by Professor Edward C. Pickering, director of the Harvard Observatory. The memoir forms Part V., Vol. XLVIII., of the Annals of the Observatory. This research was rendered possible only as a result of the extended work in stellar photometry which has been carried on by the author during the last quarter of a century. Nearly two million observations of the brightness of the stars have been made by him and his assistants at the Cambridge and Arequipa stations of the observatory. Upon these results especially, and also upon the various Durchmusterungs, and the work of Father Hagen, the investigation is based.
Two very important facts are brought out by this research. Hitherto it has been supposed that the proportion of faint stars to the whole number is much greater in the milky way than in other parts of the sky. This may still be true for very faint stars, but Professor Pickering shows conclusively that, for all stars whose magnitudes have been determined, the proportion of bright and faint stars is practically the same in the milky way as elsewhere in the sky, and that the stellar density is only about twice as great.
From theoretical considerations Professor Pickering derives the formula, log in which N is the number of stars brighter than a given magnitude M. A comparison of the results of observation, however, shows that the coefficient of M is never greater than 0.52, and that this value, which is fairly uniform for the brighter stars, grows rapidly less for the faintest stars whose magnitudes have been determined. An inspection of Table XXI. of the memoir shows that for the stars visible to the naked eye, the whole number brighter than a given magnitude is from three to four times as great as that of the next lower magnitude; that is, for example, there are 3.3 times as many stars of the fifth magnitude and brighter, as of the fourth magnitude and brighter. For fainter stars the ratio steadily decreases, until for the twelfth magnitude the number is little more than twice that for the eleventh magnitude. This curve of distribution is remarkable. If the rate of decrease continues with equal rapidity for successive magnitudes, it would lead apparently toward the ratio unity at the eighteenth or twentieth magnitude, which would imply the limit—possibly very ill-defined—of our universe. This conclusion is, however, still very uncertain, since reliable observations of brightness are available only to the twelfth magnitude. Professor Pickering is careful to draw few conclusions beyond the reach of actual observation. He says, however, 'As estimates are given which are still more uncertain than these, it may be stated that the number of stars corresponding to the magnitude 15, or which would be visible in a telescope of 15 inches aperture, would be about eighteen million, and the increase for larger apertures would be surprisingly small.' With the mounting of the great five-foot reflector at Cambridge, however, there seems to be little doubt but that the determinations of brightness will be extended to the faintest stars which can be reached at the present day. Before many years we shall perhaps know whether our universe is simply a limited region in the infinite.
