# Popular Science Monthly/Volume 58/January 1901/Chapters on the Stars VII

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* CHAPTERS ON THE STARS.
By Professor SIMON NEWCOMB, U. S. N.
THE STRUCTURE OF THE HEAVENS.
THE problem of the structure and duration of the universe is the most far-reaching with which the mind has to deal. Its solution may be regarded as the ultimate object of stellar astronomy, the possibility of reaching which has occupied the minds of thinkers since the beginning of civilization. Before our time the problem could be considered only from the imaginative or the speculative point of view. Although we can to-day attack it by scientific methods, to a limited extent, it must be admitted that we have scarcely taken more than the first step toward the actual solution. We can do little more than state the questions involved, and show what light, if any, science is able to throw upon the possible answers.
Firstly, we may inquire as to the extent of the universe of stars. Are the latter scattered through infinite space, so that those we see are merely that portion of an infinite collection which happens to be within reach of our telescopes, or are all the stars contained within a certain limited space? In the latter case, have our telescopes yet penetrated to the boundary in any direction? In other words, as, by the aid of increasing telescopic power, we see fainter and fainter stars, are these fainter stars at greater distances than those before known, or are they smaller stars contained within the same limits as those we already know? Otherwise stated, do we see stars on the boundary of the universe?
Secondly, granting the universe to be finite, what is the arrangement of the stars in space? Especially, what is the relation of the galaxy to the other stars? In what sense, if any, can the stars be said to form a permanent system? Do the stars which form the Milky Way belong to a different system from the other stars, or are the latter a part of one universal system?
Thirdly, what is the duration of the universe in time? Is it fitted to last forever in its present form, or does it contain within itself the seeds of dissolution? Must it, in the course of time, in we know not how many millions of ages, be transformed into something very different from what it now is? This question is intimately associated with the question whether the stars form a system. If they do, we may suppose that system to be permanent in its general features; if not, we must look further for our conclusion.
​The first and third of these questions will be recognized by students of Kant as substantially those raised by the great philosopher in the form of antinomies. Kant attempted to show that both the propositions and their opposites could be proved or disproved by reasoning equally valid in either case. The doctrine that the universe is infinite in duration and that it is finite in duration are both, according to him, equally susceptible of disproof. To his reasoning on both points the scientific philosopher of to-day will object that it seeks to prove or disprove, à priori, propositions which are matters of fact, of which the truth can be therefore settled only by an appeal to observation. The more correct view is that afterward set forth by Sir William Hamilton, that it is equally impossible for us to conceive of infinite space (or time), or of space (or time) coming to an end. But this inability merely grows out of the limitations of our mental power, and gives us no clue to the actual universe. So far as the questions are concerned with the latter, no answer is valid unless based on careful observation. Our reasoning must have facts to go upon before a valid conclusion can be reached.
The first question we have to attack is that of the extent of the universe. In its immediate and practical form, it is whether the smallest stars that we see are at the boundary of a system, or whether more and more lie beyond, to an infinite extent. This question we are not yet ready to answer with any approach to certainty. Indeed, from the very nature of the case, the answer must remain somewhat indefinite. If the collection of stars which forms the Milky Way be really finite, we may not yet be able to see its limit. If we do see its limit, there may yet be, for aught we know, other systems and other galaxies, scattered through infinite space, which must forever elude our powers of vision. Quite likely the boundary of the system may be somewhat indefinite, the stars gradually thinning out as we go further and further, so that no definite limit can be assigned. If all stars are of the same average brightness as those we see, all that lie beyond a certain distance must evade observation, for the simple reason that they are too far off to be visible in our telescopes.
There is a law of optics which throws some light on the question. Suppose the stars to be scattered through infinite space in such a way that every great portion of space is, in the general average, about equally rich in stars.
Then imagine that, at some great distance, say that of the average stars of the sixth magnitude, we describe a sphere having its center in our system. Outside this sphere, describe another one, having a radius greater by a certain quantity, which we may call S. Outside that let there be another of a radius yet greater, and so on indefinitely. Thus we shall have an endless succession of concentric spherical shells, each of the same thickness, S. The volume of each of these regions will be ​proportional to the square of the diameters of the spheres which bound it. Hence, supposing an equal distribution of the stars, each of these regions will contain a number of stars increasing as the square of the radius of the region. Since the amount of light which we receive from each individual star is as the inverse square of its distance, it follows that the sum total of the light received from each of these spherical shells will be equal. Thus, as we include sphere after sphere, we add equal amount of light without limit. The result of the successive addition of these equal quantities, increasing without limit, would be that if the system of stars extended out indefinitely the whole heavens would be filled with a blaze of light as bright as the sun.
Now, as a matter of fact, such is very far from being the case. It follows that infinite space is not occupied by the stars. At best there can only be collections of stars at great distances apart.
The nearest approximation to such an appearance as that described is the faint, diffused light of the Milky Way. But so large a fraction of this illumination comes from the stars which we actually see in the telescope that it is impossible to say whether any visible illumination results from masses of stars too faint to be individually seen. Whether the cloud-like impressions, which Barnard has found in long-exposed photographs of the Milky Way, are produced by countless distant stars, too faint to impress themselves even upon the most sensitive photographic plate, is a question of extreme interest which cannot be answered. But even if we should answer it in the affirmative, the extreme faintness of light shows that the stars which produce it are not scattered through infinite space; but that, although they may extend much beyond the limits of the visible stars, they thin out very rapidly. The evidence, therefore, seems to be against the hypothesis that the stars we see form part of an infinitely extended universe.
But there are two limitations to this conclusion. It rests upon the hypothesis that light is never lost in its passage to any distance, however great. This hypothesis is in accordance with our modern theories of physics, yet it cannot be regarded as an established fact for all space, even if true for the distances of the visible stars. About half a century ago Struve propounded the contrary hypothesis that the light of the more distant stars suffers an extinction in its passage to us. But this had no other basis than the hypothesis that the stars were equally thick out to the farthest limits at which we could see them.
It might be said that he assumed the hypothesis of an infinite universe, and from the fact that he did not see the evidence of infinity, concluded that light was lost. The hypothesis of a limited universe and no extinction of light, while not absolutely proved, must be regarded ​as the one to be accepted until further investigation shall prove its unsoundness.
The second limitation has been the possible structure of an infinite universe. The mathematical reader will easily see that the conclusion that an infinite universe of stars would fill the heavens with a blaze of light, rests upon the hypothesis that every region of space of some great but finite extent is, on the average, occupied by at least one star. In other words, the hypothesis is that if we divide the total number of the stars by the number of cubic miles of space, we shall have a finite quotient. But an infinite universe can be imagined which does not fill this condition. Such will be the case with one constructed on the celebrated hypothesis of Lambert, propounded in the latter part of the last century. This author was an eminent mathematician, who seems to have been nearly unique in combining the mathematical and the speculative sides of astronomy. He assumed a universe constructed on an extension of the plan of the Solar System. The smallest system of bodies is composed of a planet with its satellites. We see a number of such systems, designated as the Terrestrial, the Martian (Mars and its satellites), the Jovian (Jupiter and its satellites), etc., all revolving round the Sun, and thus forming one greater system, the Solar System. Lambert extended the idea by supposing that a number of solar systems, each formed of a star with its revolving planets and satellites, were grouped into a yet greater system. A number of such groups form the great system which we call the galaxy, and which comprises all the stars we can see with the telescope. The more distant clusters may be other galaxies. All these systems again may revolve around some distant center, and so on to an indefinite extent. Such a universe, how far so ever it might extend, would fill the heavens with a blaze of light, and the more distant galaxies might remain forever invisible to us. But modern developments show that there is no scientific basis for this conception, attractive though it is by its grandeur.
So far as our present light goes, we must conclude that although we are unable to set absolute bounds to the universe, yet the great mass of stars is included within a limited space of whose extent we have as yet no evidence. Outside of this space there may be scattered stars or invisible systems. But if these systems exist, they are distinct from our own.
The second question, that of the arrangement of the stars in space, is one on which it is equally difficult to propound a definite general conclusion. So far, we have only a large mass of faint indications, based on researches which cannot be satisfactorily completed until great additions are made to our fund of knowledge.
A century ago Sir William Herschel reached the conclusion that our universe was composed of a comparatively thin but widely ​extended stratum of stars. To introduce a familiar object, its figure was that of a large thin grindstone, our Solar System being near the center. Considering only the general aspect of the heavens, this conclusion was plausible. Suppose a mass of a million of stars scattered through a space of this form. It is evident that an observer in the center, when he looks through the side of the stratum, would see few stars. The latter would become more and more numerous as he directed his vision toward the circumference of the stratum. In other words, assuming the universe to have this form, we should see a uniform, cloud-like arch spanning the heavens—a galaxy in fact.
This view of the figure of the universe was also adopted by Struve, who was, the writer believes, the first astronomer after Herschel to make investigations which can be regarded as constituting an important addition to thought on the subject. To a certain extent we may regard the hypothesis as incontestable. The great mass of the visible stars is undoubtedly contained within such a figure as is here supposed.
To show this let Fig. 1 represent a cross section of the heavens at
right angles to the Milky Way, the Solar System being at S. It is an observed fact that the stars are vastly more numerous in the galactic regions G G than in the regions P P. Hence, if we suppose the stars equally scattered, they must extend much farther out in G G than in P P. If they extend as far in the one direction as in the other, then they must be more crowded in the galactic belt. It will still remain true that the greater number of the stars are included in the flat region A B C D, those outside this stratum being comparatively few in number.
But we cannot assume that this hypothesis of the form of the universe affords the basis for a satisfactory conception of its arrangement. Were it the whole truth, the stars would be uniformly dense along the whole length of the Milky Way. Now, it is a familiar fact that this is not the case. The Milky Way is not a uniformly illuminated belt, but a chain of irregular, cloud-like aggregations of stars. Starting from this fact as ​a basis, our best course is to examine the most plausible hypotheses we can make as to the distribution of the stars which do not belong to the galaxy, and see which agrees best with observation.
Let Fig. 2 represent a section of the galactic ring or belt in its own plane, with the sun near the center S. To an observer at a vast distance in the direction of either pole of the galaxy, the latter would appear of this form. Let Fig. 3 represent a cross section as viewed by an observer in the plane of the galaxy at a great distance outside of it. How would the stars that do not belong to the galaxy be situated? We may make three hypotheses:
1. That they are situated in a sphere (A B) as large as the galaxy itself. Then the whole universe of stars would be spherical in outline, and the galaxy would be a dense belt of stars girdling the sphere.
2. The remaining stars may still be contained in a spherical space
(K L), of which the diameter is much less than that of the galactic girdle. In this case our Sun would be one of a central agglomeration of stars, lying in or near the plane of the galaxy.
3. The non-galactic stars may be equally scattered throughout a flat region (M N P Q), of the grindstone form. This would correspond to the hypothesis of Herschel and Struve.
There is no likelihood that either of these hypotheses is true in all the geometric simplicity with which I have expressed them. Stars are doubtless scattered to some extent through the whole region M N P Q, and it is not likely that they are confined within limits defined by any geometrical figure. The most that can be done is to determine to which of the three figures the mutual arrangement most nearly corresponds.
The simplest test is that of the third hypothesis as compared with the other two. If the third hypothesis be true, then we should see the ​fewest stars in the direction of the poles of the galaxy; and the number in any given portion of the celestial sphere, say one square degree, should continually increase, slowly at first, more rapidly afterwards, as we went from the poles toward the circumference of the galaxy. At a distance of 60° from the poles and 30° from the central line or circumference we should see more than twice as many stars per square degree as near the poles.
The general question of determining the precise position of the galaxy naturally enters into our problem. There is no difficulty in mapping out its general course by unaided eye observations of the heavens or a study of maps of the stars. Looking at the heavens, we shall readily see that it crosses the equator at two opposite points; the one east of the constellation Orion, between 6h. and 7h. of right ascension; the other at the opposite point, in Aquila, between 18h. and 19h. It makes a considerable angle with the equator, somewhat more than 60°. Consequently it passes within 30° of either pole. The point nearest of approach to the north pole is in the constellation Cassiopeia. In consequence of this obliquity to the equator, its apparent position on the celestial sphere, as seen in our latitude, goes through a daily change with the diurnal rotation of the earth. In the language of technical astronomy, every day at 12h. of sidereal time, it makes so small an angle with the horizon as to be scarcely visible. If the air is very clear, we might see a portion of it skirting the northern horizon. This position occurs during the evenings of early summer. At 0h. of sidereal time, which during autumn and early winter fall in the evening, it passes nearly through our zenith, from east to west, and can, therefore, then best be seen.
Its position can readily be determined by noting the general course of its brighter portions on a map of the stars, and then determining by inspection, or otherwise, the circle which will run most nearly through those portions. It is thus found that the position is nearly always near a great circle of the sphere. From the very nature of the case the position of this circle will be a little indefinite, and probably the estimates made of it have been based more on inspection than on computation. The following numerical positions have been assigned to the pole of the galaxy:
| Gould, | R. A. = | 12h. | 41m. | Dec. = | +27° | 21' |
| Herschel, W | Herschel, W | 12h. | 29m. | 31° | 31° | 30' |
| Seeliger | Seeliger | 12h. | 49m. | 27° | 27° | 30' |
| Argelander | Argelander | 12h. | 40m. | 28° | 28° | 5' |
Were it possible to determine the distance of a star as readily as we do its direction, the problem of the distribution of the stars in space would be at once solved. This not being the case, we must first study the apparent arrangement of the stars with respect to the galaxy, with a ​view of afterward drawing such conclusions as we can in regard to their distance.
APPARENT DISTRIBUTION OF THE STARS IN THE SKY.
Distribution of the Lucid Stars: Our question now is how are the stars, as we see them, distributed over the sky? We know in a general way that there are vastly more stars round the belt of the Milky Way than in the remainder of the heavens. But we wish to know in detail what the law of increase is from the poles of the galaxy to the belt itself.
In considering any question of the number of stars in a particular region of the heavens, we are met by a fundamental difficulty. We can set no limit to the minuteness of stars, and the number will depend upon the magnitude of those which we include in our account. As already remarked, there are, at least up to a certain limit, three or four times as many stars of each magnitude as of the magnitude next brighter. Now, the smallest stars that can be seen, or that may be included in any count, vary greatly with the power of the instrument used in making the count. If we had any one catalogue, extending over the whole celestial sphere, and made on an absolutely uniform plan, so that we knew it included all the stars down to some given magnitude, and no others, it would answer our immediate purpose. If, however, one catalogue should extend only to the ninth magnitude, while another should extend to the tenth, we should be led quite astray in assuming that the number of stars in the two catalogues expressed the star density in the regions which they covered. The one would show three or four times as many stars as the other, even though the actual density in the two cases were the same.
If we could be certain, in any one case, just what the limit of magnitude was for any catalogue, or if the magnitudes in different catalogues always corresponded to absolutely the same brightness of the star, this difficulty would be obviated. But this is the case only with that limited number of stars whose brightness has been photometrically measured. In all other cases our count must be more or less uncertain. One illustration of this will suffice:
I have already remarked that in making the photographic census of the southern heavens, Gill and Kapteyn did not assume that stars of which the images were equally intense on different plates were actually of the same magnitude. Each plate was assumed to have a scale of its own, which was fixed by comparing the intensity of the photographic impressions of those stars whose magnitudes had been previously determined with these determinations, and thus forming as it were a separate scale for each plate. But, in forming the catalogue from the international photographic chart of the heavens, it is assumed that the photographs taken with telescopes of the same aperture, in which ​the plates are exposed for five minutes, will all correspond, and that the smallest stars found on the plates will be of the eleventh magnitude.
In the case of the lucid stars this difficulty does not arise, because the photometric estimates are on a sufficiently exact and uniform scale to enable us to make a count, which shall be nearly correct, of all the stars down to, say, magnitude 6.0 or some limit not differing greatly from this. Several studies of the distribution of these stars have been made; one by Gould in the Uranometria Argentina, one by Schiaparelli, and another by Pickering. The counts of Gould and Schiaparelli, having special reference to the Milky Way, are best adapted to our purpose. The most striking result of these studies is that the condensation in the Milky Way seems to commence with the brightest stars. A little consideration will show that we cannot, with any probability, look for such a condensation in the case of stars near to us. Whatever form we assign to the stellar universe, we shall expect the stars immediately around us to be equally distributed in every direction. Not until we approach the boundary of the universe in one direction, or some great masses like those of the galaxy in another direction, should we expect marked condensation round the galactic belt. Of course we might imagine that even the nearest stars are most numerous in the direction round the galactic circle. But this would imply an extremely unlikely arrangement, our system being as it were at the point of a cone. It is clear that if such were the case for one point, it could not be true if our Sun were placed anywhere except at this particular point. Such an arrangement of the stars round us is outside of all reasonable probability. Independent evidence of the equal distribution of the stars will hereafter be found in the proper motions. If then, the nearer stars are equally distributed round us, and only distant ones can show a condensation toward the Milky Way, it follows that among the distant stars are some of the brightest in the heavens, a fact which we have already shown to follow from other considerations.
Very remarkable is the fact, pointed out first by Sir J. Herschel and heavens very nearly in a great circle, but not exactly in the Milky Way. heavens very nearly in a great circle, but not exactly in the Milky Way. In the northern heavens the brightest stars in Orion, Taurus, Cassiopeia, being near the Southern Cross and the other in Cassiopeia. This belt includes the brightest stars in a number of constellations, from Canis Major through the southern region of the heavens and back to Scorpius. In the northern heavens the brightest stars in Orion, Taurus, Cassiopeia, Cygnus and Lyra belong to this belt. It would not be safe, however, to assume that the existence of this belt results from anything but the chance distribution of the few bright stars which form it. In order to reach a definite conclusion bearing on the structure of the heavens, ​it is advisable to consider the distribution of the lucid stars as a whole.
Dr. Gould finds that the stars brighter than the fourth magnitude are arranged more symmetrically relatively to the bright stars we have just described than to the galactic circle. This and other facts suggested to him the existence of a small cluster within which our sun is eccentrically situated and which is itself not far from the middle plane of the galaxy. This cluster appears to be of a flattened shape and to consist of somewhat more than 400 stars of magnitudes ranging from
the first to the seventh. Since Gould wrote, the extreme inequality in the intrinsic brightness of the stars has been brought to light and seems to weaken the basis of his conclusion on this particular point.
A very thorough study of the subject, but without considering the galaxy, has also been made by Schiaparelli. The work is based on the photometric measures of Pickering and the Uranometria Argentina of Gould. One of its valuable features is a series of planispheres, showing in a visible form the star density in every region of the heavens for stars of various magnitudes. We reproduce in a condensed form two of ​these planispheres. They were constructed by Schiaparelli in the following way: The entire sky was divided into 36 zones by parallels of declination 5° apart. Each zone was divided into spherical trapezia by hour-circles taken at intervals of 5° from the equator up to 50° of north or south declination; of 10° from 50 to 60; of 15° from 60 to 80; of 45° from 80 to 85, while the circle within 5° of the pole was taken as a single region. In this way 1,800 areas, not excessively different from each other, were formed.
The star density, as it actually is, might be indicated by the number
of stars of these regions. As a matter of fact, however, the density obtained in this way would vary too rapidly from one area to the adjoining one, owing to the accidental irregularities of distribution of the stars. An adjustment was, therefore, made by finding in the case of each area the number of stars contained in 1/200 of the entire sphere, including the region itself and those immediately round it. The number thus obtained was considered as giving the density for the central region. The total number of stars being 4,303, the mean number in 1/200 of the whole sphere is 21.5, and the mean in each area is 10.4.
​The numbers on the planisphere given in each area thus express the star density of the region, or the number of stars per 100 square degrees, expressed generally to the nearest unit, the half unit being sometimes added.
A study of the reproduction which we give will show how fairly well the Milky Way may be traced out round the sky by the tendency of those stars visible to the naked eye to agglomerate near its course. In other words, were the cloud forms which make up the Milky Way invisible to us, we should still be able to mark out its course by the condensation of the stars. As a matter of interest, I have traced out the central line of the shaded portions of the planispheres as if they were the galaxy itself. The nearest great circle to the course of this line was then found to have its pole in the following position:
R.A.; 12h. 18m.
Dec. $+$ 27°.
This estimate was made without having at the time any recollection of the position of the galaxy given by other authorities. Compared with the positions given in the last chapter by Gould and Seeliger, it will be seen that the deviation is only 5° in right ascension, while the declinations are almost exactly similar. We infer that the circle of condensation found in this way makes an angle with the galaxy of less than 5°.
DISTRIBUTION OF THE FAINTER STAES.
The most thorough study of the distribution of the great mass of stars relative to the galactic plane has been made by Seeliger in a series of papers presented to the Munich Academy from 1884 to 1898. The data on which they are based are the following:
1. The Bonner Durchmusterung of Argelander and Schonfeld, described in our third chapter. These two works included under this title are supposed to include all the stars to the ninth magnitude, from the north pole to 24° of south declination. But there are some inconsistencies in the limit of magnitude which we shall hereafter mention.
2. The 'star gauges' of the two Herschels. These consisted simply in counts of the number of stars visible in the field of view of the telescope when the latter was directed toward various regions of the sky. Sir William Herschel's gauges were partly published in the 'Philosophical Transactions.' A number of unpublished ones were found among his papers by Holden and printed in the publications of the Washburn Observatory, Vol. II. The younger Herschel, during his expedition to the Cape of Good Hope, continued the work in those southern regions of the sky which could not be seen in England.
3. A count of the stars by Celoria, of Milan, in a zone from the equator to 6° Dec, extended round the heavens.
​From what has been said the question which will first occupy our attention is that of the distribution of the stars with reference to the galactic plane, or rather, the great circle forming the central line of the Milky Way.
The whole sky is divided by Seeliger into nine zones or regions, each 20° in breadth, by small circles parallel to the galactic circle. Region I. is a circle of 20° radius, whose center is the galactic pole. Round this central circle is a zone 20° in breadth, called Zone II. Continuing the division, it will be seen that Zone V. is the central one of the Milky Way, extending 10° on each side of the galactic circle.
The condensed result of the work is shown in the following table:
Column 'Area' shows the number of square degrees in each region, so far as included in the survey. It will be remarked that the catalogues in question do not include the whole sky, as they stop at 24° S. Dec.
Column 'Stars' shows the number of stars to magnitude 9.0 found in each area.
Column 'Density' is the quotient of the number of stars by the area, and is, therefore, the mean number of stars per square degree in each region. In column 'D' these numbers are corrected, for certain anomalies in the magnitudes given by the catalogues, so as to reduce them to a common standard.
| Region. | Area. Degrees. | Stars. | Density. | D. |
| I | 1,398.7 | 4,277 | 3.06 | 2.78 |
| II | 3,146.9 | 10,185 | 3.24 | 3.03 |
| III | 5,126.6 | 19,488 | 3.80 | 3.54 |
| IV | 4,589.8 | 24,492 | 5.34 | 5.32 |
| V | 4,519.5 | 33,267 | 7.36 | 8.17 |
| VI | 3,971.5 | 23,580 | 5.94 | 6.07 |
| VII | 2,954.4 | 11,790 | 3.99 | 3.71 |
| VIII | 1,790.6 | 6,375 | 3.56 | 3.21 |
| IX | 468.2 | 1,644 | 3.51 | 3.14 |
A study of the last two columns is decisive of one of the fundamental questions already raised. The star density in the several regions increases continuously from each pole (regions I. and V.) to the galaxy itself. If the latter were a simple ring of stars surrounding a spherical system of stars, the star density would be about the same in regions I., II. and III., and also in VII., VIII. and IX., but would suddenly increase in IV. and VI. as the boundary of the ring was approached. Instead of such being the case, the numbers 2.78, 3.03 and 3.54 in the north, and 3.14, 3.21 and 3.71 in the south, show a progressive increase from the galactic pole to the galaxy itself.
The conclusion to be drawn is a fundamental one. The universe, or, at least, the denser portions of it, is really flattened between the ​galactic poles, as supposed by Herschel and Struve. In the language of Seeliger: "The Milky Way is no merely local phenomenon, but is closely connected with the entire constitution of our stellar system."
This conclusion is strengthened by a study of the data given by Celoria. It will be remarked that the zone counted by this astronomer cuts the Milky Way diagonally at an angle of about 62°, and, therefore, does not take in either of its poles. Consequently, regions I. and IX. are both left out. For the remaining seven regions the results are shown as follows: We show first the area, in square degrees, of each of the regions, II. to VII., included in Celoria's zone. Then follows in the next column the number of stars counted by Celoria, and, in the third, the number enumerated in the Durchmusterung in these portions of each region. The quotients show the star-density, or the mean number of stars per square degree, recorded by each authority:
| | | Area | Number of stars | Number of stars | Star-Density | Star-Density |
| Region | Region | Degrees | Cel. | D. M. | Cel. | D. M. |
| II | | 404.4 | 27,352 | 1,230 | 67.6 | 3.04 |
| III | | 284.6 | 22,551 | 932 | 79.3 | 3.28 |
| IV | | 254.6 | 29,469 | 1,488 | 115.7 | 5.83 |
| V | | 284.6 | 41,820 | 1,833 | 146.9 | 6.44 |
| VI | | 284.6 | 31,706 | 1,472 | 111.4 | 5.22 |
| VII | | 329.5 | 25,618 | 1,342 | 77.7 | 4.07 |
| VIII | | 314.5 | 22,264 | 1,184 | 70.8 | 3.77 |
It will be seen that the law of increasing star-density from near the galactic pole to the galaxy itself is of the same general character in the two cases. The number of stars counted by Celoria is generally between 18 and 25 times the number in the Durchmusterung.
An important point to be attended to hereafter is that the star-density of the Milky Way itself, as derived from each authority, is between two and three times that near the galactic poles. Very different is the result derived from the Herschelian gauges, which is this:
| Region | I. | II. | III. | IV. | V. | VI. | VII. | VIII. | IX. |
| Density | 107 | 154 | 281 | 560 | 2019 | 672 | 261 | 154 | 111 |
From the gauges of the Herschels it follows that the galactic star density is nearly 20 times that of the galactic poles. At these poles the Herschels counted about 50 per cent, more stars than Celoria. In the galaxy itself they counted 14 for every one by Celoria. The principal cause of this discrepancy is the want of uniformity of the magnitudes.
The recent comparisons of the Durchmusterung with the heavens, mostly made since Seeliger worked out the results we have given, show that the limit of magnitude to which this list extends is far from uniform, and varies with the star-density. In regions poor in stars, all of the latter to the tenth magnitude are listed; in the richer regions of the galaxy the list stops, we may suppose, with the ninth magnitude, or ​even brighter. Yet, in all cases, the faintest stars listed are classed as of magnitude 9.5. Thus a ninth magnitude star in the galaxy, according to the Durchmusterung, is very different from one of this magnitude elsewhere.
DISTRIBUTION OF THE STARS HAVING A PROPER MOTION.
Having found that the stars of every magnitude show a tendency to crowd toward the region of the Milky Way, the question arises whether this is true of those stars which have a sensible proper motion. Kapteyn has examined this question in the case of the Bradley stars. His conclusion is that those having a considerable proper motion, say more than 10" per century, are nearly equally distributed over the sky, but that when we include those having a small proper motion, we see a continually increasing tendency to crowd toward the galactic plane.
But the irregularity in the distribution of the stars observed by Bradley seems to me to render this result quite unreliable. For every such star Auwers derived a proper motion. And, if these proper motions are considered, their distribution will be the same as that of the stars. To reach a more definite conclusion, we must base our work on lists of proper motions, which are as nearly complete within their limits as it is possible to make them. Such lists have been made by Auwers and Boss, their work being based on their observations of zones of stars for the catalogue of the Astronomische Gesellschaft. The zone observed by Auwers was that between 15° and 20° in N. Dec; while Boss's was between 1° and 5°. To speak more exactly, the limits were from 14° 50' to 20° 10' and 0° 50' to 5° 10', each zone of observation overlapping 10' on the adjoining one. Thus the actual breadths were 5° 20' and 4° 20'. Within these respective limits, Auwers, by a comparison with previous observations, found 1,300 stars having an appreciable proper motion, and Boss 295. But Boss's list is confined to stars having a motion of at least 10"; of such the list of Auwers contains 431. The number of square degrees in the two zones is 1,556 and 1,830, respectively. The corresponding number of stars with proper motions extending 10" is, for each 100 square degrees:
In Boss's zone, 18.9. In Auwers's zone, 23.9.
The question whether the greater richness of nearly 25 per cent, in Auwers's zone is real is one on which it is not easy to give a conclusive answer. The probability, however, seems to be that it is mainly due to the greater richness of the material on which Auwers's proper motions are based. The question is not, however, essential in the present discussion.
We now examine the question of the respective richness of proper motion stars in this way:
​Each of these zones cuts the galaxy at a considerable angle. Each zone, as a matter of course, has a far larger richness of stars per unit of surface in the galactic region than in the remaining region. We, therefore, divide each zone in four strips, two including the galactic regions and two the intermediate region. The boundaries are somewhat indefinite: we have fixed them by the richness of the total number of stars. For the galactic strips we take in Boss's zone the strip between 5h. and 8h. of B. A. and that between 17h. and 20h. Each of these strips being 3h. in length, the two together comprise one-quarter the total surface of the zone. If the proper motion stars crowd towards the galaxy like others do, then the numbers in the galactic region should be proportional to the total number observed in the region. But if they are equally distributed then there should be only one-quarter as many in the galactic region as in the other regions.
In the case of Boss's zone, the total number of stars observed and of those having a proper motion found in the four regions described are as follows:
| | Total Number | Proper Motions. | Proper Motions. |
| | Observed. | Actual. | Prop. |
| Galactic strip, 5h. to 8h | 1,614 | 24 | 37 |
| Galactic strip, 17h. to 20h | 1,340 | 36 | 37 |
| Intermediate strip, 8h. to 17h | 2,458 | 124 | 111 |
| Intermediate strip, 20h. to 5h | 2,831 | 111 | 111 |
The last column contains the number of proper motions we should find if the whole 295 were distributed proportionally to the areas of the several strips. There is evidently no excess of richness in the galactic strips, but rather a deficiency in the strip near 6h., which is accidental.
In the case of Auwers's zone, the galactic strips are those between 5h. and 8h., and again between 18h. and 21h. Here, as in the other case, the galactic strips include one-quarter of the whole area. But, owing to the greater richness of the sky, they include nearly 40 per cent, of the whole number of stars. Then, if the proper motion stars are equally distributed, one-quarter should be found in the region, and if they are proportional to the number of stars observed, 40 per cent, should be within this region. Grouping the regions outside the galaxy together, as we need not distinguish between them, the result is as follows:
| | Stars | Proper Motions. | Proper Motions. |
| | Observed. | Actual. | Prop. |
| Galactic strip, 5" to 8" | 1,797 | 155 | 157 |
| Galactic strip, 18" to 21" | 1,984 | 202 | 157 |
| Outside the galaxy | 6,008 | 901 | 944 |
We see that in the strip from 5h. to 8h. there is contained almost exactly one-eighth the whole number of proper motion stars. That is, ​in this region the stars are no thicker than elsewhere. In the region from 18h. to 21h. there is an excess of 45 stars having proper motions. Whether this excess is real may well be doubted. It is scarcely, if at all, greater than might be the result of accidental inequalities of distribution. Were the proper motion stars proportional to the whole number, there ought to be 240 within the strip. The actual number is 38 less than this.
It is to be remembered that Auwers's proper motions are not limited to a definite magnitude, as were Boss's, but that he looked for all stars having a sensible proper motion. The question, what proper motion would be sensible, is a somewhat indefinite one, depending very largely on the data. It may, therefore, well be that the small excess of 45 found within this strip is due to the fact that more stars were observed and investigated, and, therefore, more proper motions found. Besides this, some uncertainty may exist as to the reality of the minuter proper motions.
The conclusion is interesting and important. If we should blot out from the sky all the stars having no proper motion large enough to be detected, we should find remaining stars of all magnitudes; but they would be scattered almost uniformly over the sky, and show no tendency toward the galaxy.
From this again it follows that the stars belonging to the galaxy lie farther away than those whose proper motions can be detected.