Now it is found that from the first to the sixth magnitude the stars increase in number at the rate of about three and a half times those of the preceding magnitudes. The total number of stars down to the sixth magnitude is given by Professor Newcomb as 7647. For higher magnitudes the numbers are so great that precision and uniformity are more difficult of attainment; yet there is a wonderful continuance of the same law of increase down to the tenth magnitude, which is estimated to include 2,311,000 stars, thus conforming very nearly with the ratio of 3.5 as determined by the lucid stars.
But when we pass beyond the tenth magnitude to those vast numbers of faint stars only to be seen in the best or the largest telescopes, there appears to be a sudden change in the ratio of increased numbers per magnitude. The numbers of these stars are so great that it is impossible to count the whole as with the higher magnitude stars, but numerous counts have been made by many astronomers in small measured areas in different parts of the heavens, so that a fair average has been obtained, and it is possible to make a near approximation to the total number visible down to the seventeenth magnitude. The estimate of these by astronomers who have made a special study of this subject is, that the total number of visible stars does not exceed one hundred millions.[3]
But if we take the number of stars down to the ninth magnitude, which are known with considerable accuracy, and find the numbers in each succeeding magnitude down to the seventeenth, according to the same ratio of increase which has been found to correspond very nearly in the case of the higher magnitudes, Mr. J.E. Gore finds that the total number should be about 1400 millions. Of course neither of these estimates makes any pretence to exact accuracy, but they are founded on all the facts at present available, and are generally accepted by astronomers as being the nearest approach that can be made to the true numbers. The discrepancy is, however, so enormous that probably no careful observer of the heavens with very large telescopes doubts that there is a very real and very rapid diminution in the numbers of the fainter as compared with the brighter stars.
There is, however, yet one more indication of the decreasing numbers of the faint telescopic stars, which is almost conclusive on this question, and, so far as I am aware, has not yet been used in this relation. I will therefore briefly state it.
The Light Ratio as Indicating the Number
of Faint Stars
Professor Newcomb points out a remarkable result depending on the fact that, while the average light of successively lower magnitudes diminishes in a ratio of 2.5, their numbers increase at nearly a ratio of 3.5. From this it follows that, so long as this law of increase continues, the total of starlight goes on increasing by about forty per cent. for each successive magnitude, and he gives the following table to illustrate it:—
| Mag. | 1 | Total Light | =1 |
| " | 2 | " | = 1.4 |
| " | 3 | " | = 2.0 |
| " | 4 | " | = 2.8 |
| " | 5 | " | = 4.0 |
| " | 6 | " | = 5.7 |
| " | 7 | " | = 8.0 |
| " | 8 | " | = 11.3 |
| " | 9 | " | = 16.0 |
| " | 10 | " | = 22.6 |
| Total light to Mag. 10 = 74.8 |
Thus the total amount of the light given by all stars down to the tenth magnitude is seventy-four times as great as that from the few first magnitude stars. We also see that the light given by the stars of any magnitude is twice as much as that of the stars two magnitudes higher in the scale, so that we can easily calculate what additional light we ought to receive from each additional magnitude if they continue to increase in numbers below the tenth as they do above that magnitude. Now it has been calculated as the result of careful observations, that the total light given by stars down to nine and a half magnitude is one-eightieth of full moonlight, though some make it much more. But if we continue the table of light-ratios from this low starting-point down to magnitude seventeen and a half, we shall find, if the numbers of the stars go on increasing at the same rate as before, that the light of all combined should be at least seven times as great as moonlight; whereas the photometric measurements make it actually about one-twentieth. And as the calculation from light-ratios only includes stars just visible in the largest telescopes, and does not include all those proved to exist by photography, we have in this case a demonstration that the numbers of the stars below the tenth and down to the seventeenth magnitude diminish rapidly.
We must remember that the minuter telescopic stars preponderate enormously in and near the Milky Way. At a distance from it they diminish rapidly, till near its poles they are almost entirely absent. This is shown by the fact (already referred to at p. 146) that Professor Celoria of Milan, with a telescope of less than three inches aperture, counted almost as many stars in that region as did Herschel with his eighteen-inch reflector. But if the stellar universe extends without limit we can hardly suppose it to do so in one plane only; hence the absence of the minuter stars and of diffused milky light over the larger part of the heavens is now held to prove that the myriads of very minute stars in the Milky Way really belong to it, and not to the depths of space far beyond.