Order of Distance.Number of Stars there is Room for.Magnitude.Number of Stars of that Magnitude.
1. . . . . . . . 261 17
2. . . . . . . . 982 57
3. . . . . . . . 2183 206
4. . . . . . . . 3864 454
5. . . . . . . . 60251,161
6. . . . . . . . 86666,103
7. . . . . . . .1,17876,146
8. . . . . . . .1,538

The result of this comparison is, that if the order of magnitudes could indicate the distance of the stars, it would denote at first a gradual and afterward a very abrupt condensation of them, at and beyond the region of the sixth-magnitude stars.

If we assume the brightness of any star to be inversely proportional to the square of its distance, it leads to a scale of distance different from that adopted by Herschel, so that a sixth-magnitude star on the common scale would be about of the eighth order of distance according to this scheme—that is, we must remove a star of the first magnitude to eight times its actual distance to make it shine like a star of the sixth magnitude.

On the scheme here laid down, Herschel subsequently assigned the order of distance of various objects, mostly star-clusters, and his estimates of these distances are still quoted. They rest on the fundamental hypothesis which has been explained, and the error in the assumption of equal intrinsic brilliancy for all stars affects these estimates. It is perhaps probable that the hypothesis of equal brilliancy for all stars is still more erroneous than the hypothesis of equal distribution, and it may well be that there is a very large range indeed in the actual dimensions and in the intrinsic brilliancy of stars at the same order of distance from us, so that the tenth-magnitude stars, for example, may be scattered throughout the spheres which Herschel would assign to the seventh, eighth, ninth, tenth, eleventh, twelfth, and thirteenth magnitudes. However this may be, the fact remains that it is from Herschel's groundwork that future investigators must build. He found the whole subject in utter confusion. By his observations, data for the solution of some of the most general questions were accumulated, and in his memoirs, which Struve well calls "immortal," he brought the scattered facts into order and gave the first bold outlines of a reasonable theory. He is the founder of a new branch of astronomy.

Researches for a Scale of Celestial Measures. Distances of the Stars.

If the stars are supposed all of the same absolute brightness, their brightness to the eye will depend only upon their distance from us. If we call the brightness of one of the fixed stars at the distance of Sirius, which may be used as the unity of distance, 1, then if it is moved to the distance 2, its apparent brightness will be one-fourth; if to the distance 3, one-ninth; if to the distance 4, one-sixteenth, and so on, the apparent brightness diminishing as the square of the distance increases. The distance may be taken as an order of magnitude. Stars at the distances two, three, four, etc., Herschel called of the second, third, and fourth magnitudes.

By a series of experiments, the details of which cannot be given here, Herschel determined the space-penetrating power of each of his telescopes. The twenty-foot would penetrate into space seventy-five times farther than the naked eye; the twenty-five foot, ninety-six times; and the forty-foot, one hundred and ninety-two times. If the seventh-magnitude stars are those just visible to the naked eye, and if we still suppose all stars to be of equal intrinsic brightness, such seventh-magnitude stars would remain visible in the forty-foot, even if removed to 1,344 times the distance of Sirius (1,344 = 7 × 192). If, further, we suppose that the visibility of a star is strictly proportional to the total intensity of the light from it which strikes the eye, then a condensed cluster of 25,000 stars of the 1,344th magnitude could still be seen in the forty-foot at a distance where each star would have become 25,000 times fainter, that is, at about 158 times the distance of Sirius (158 × 158 = 24,964). The light from the nearest star requires some three years to reach the earth. From a star 1,344 times farther it would require about 4,000 years, and for such a cluster as we have imagined no less than 600,000 years are needed. That is, the light by which we see such a group has not just now left it. On the contrary, it has been travelling through space for centuries and centuries since it first darted forth. It is the ancient history of such groups that we are studying now, and it was thus that Herschel declared that telescopes penetrated into time as well as into space.

Other more exact researches on the relative light of stars were made by Herschel. These were only one more attempt to obtain a scale of celestial distances, according to which some notion of the limits and of the interior dimensions of the universe could be gained. Two telescopes, exactly equal in every respect, were chosen and placed side by side. Pairs of stars which were exactly equal, were selected by means of them. By diminishing the aperture of one telescope directed to a bright star, and keeping the other telescope unchanged and directed to a fainter star, the two stars could be equalized in light, and, from the relative size of the apertures, the relative light of this pair of stars could be accurately computed, and so on for other pairs. This was the first use of the method of limiting apertures. His general results were that the stars of the first magnitude would still remain visible to the naked eye, even if they were at a distance from us twelve times their actual distance.

This method received a still further development at his hands. He did not leave it until he had gained all the information it was capable of giving. He prepared a set of telescopes collecting 4, 9, 16, etc. (2 × 2, 3 × 3, 4 × 4, etc.), times as much light as the naked eye. These were to extend the determinations of distance to the telescopic stars. For example, a certain portion of the heavens which he examined contained no star visible to the naked eye, but many telescopic stars. We cannot say that no one of these is as bright in itself as some of our first-magnitude stars. The smallest telescope of the set showed a large number of stars; these must, then, be twice as far from us, on the average, as the stars just visible to the naked eye. But first-magnitude stars, like Sirius, Procyon, Arcturus, etc., become just visible to the eye if removed to twelve times their present distance. Hence the stars seen in this first telescope of the set were between twelve and twenty-four times as far from us as Arcturus, for example.

"At least," as Herschel says, "we are certain that if stars of the size and lustre of Sirius, Arcturus, etc., were removed into the profundity of space I have mentioned, they would then appear like the stars which I saw." With the next telescope, which collected nine times more light than the eye, and brought into view objects three times more distant, other and new stars appeared, which were then (3 × 12) thirty-six times farther from us than Arcturus. In the same way, the seven-foot reflector showed stars 204 times, the ten-foot 344 times, the twenty-foot 900 times farther from us than the average first-magnitude star. As the light from such a star requires three years to reach us, the light from the faintest stars seen by the twenty-foot would require 2,700 years (3 × 900).