ARE THE STARS INFINITE IN NUMBER?
Most of the critics of my first short discussion of this subject laid great
stress upon the impossibility of proving that the universe, a part of which
we see, is not infinite; and a well-known astronomer declared that unless
it can be demonstrated that our universe is finite the entire argument
founded upon our position within it fall to the ground. I had laid myself
open to this objection by rather incautiously admitting that if the
preponderance of evidence pointed in this direction any inquiry as to our
place in the universe would be useless, because as regards infinity there
can be no difference of position. But this statement is by no means exact,
and even in an infinite universe of matter containing an infinite number of
stars, such as those we see, there might well be such infinite diversities
of distribution and arrangement as would give to certain positions all the
advantages which I submit we actually possess. Supposing, for example, that
beyond the vast ring of the Milky Way the stars rapidly decrease in number
in all directions for a distance of a hundred or a thousand times the
diameter of that ring, and that then for an equal distance they slowly
increase again and become aggregated into systems or universes totally
distinct from ours in form and structure, and so remote that they can
influence us in no way whatever. Then, I maintain, our position within our
own stellar universe might have exactly the same importance, and be equally
suggestive, as if ours were the only material universe in existence--as if
the apparent diminution in the number of stars (which is an observed fact)
indicated a continuous diminution, leading at some unknown distance to
entire absence of luminous--that is, of active, energy-emitting
aggregations of matter.[1] As to whether there are such other material
universes or not I offer no opinion, and have no belief one way or the
other. I consider all speculations as to what may or may not exist in
infinite space to be utterly valueless. I have limited my inquiries
strictly to the evidence accumulated by modern astronomers, and to direct
inferences and logical deductions from that evidence. Yet, to my great
surprise, my chief critic declares that 'Dr. Wallace's underlying error is,
indeed, that he has reasoned from the area which we can embrace with our
limited perceptions to the infinite beyond our mental or intellectual
grasp.' I have distinctly _not_ done this, but many astronomers have done
so. The late Richard Proctor not only continually discussed the question of
infinite matter as well as infinite space, but also argued, from the
supposed attributes of the Deity, for the necessity of holding this
material universe to be infinite, and the last chapter of his _Other Worlds
than Ours_ is mainly devoted to such speculations. In a later work, _Our
Place among Infinities_, he says that 'the teachings of science bring us
into the presence of the unquestionable infinities of time and of space,
and the presumable infinities of matter and of operation--hence therefore
into the presence of infinity of energy. But science teaches us nothing
about these infinities as such. They remain none the less inconceivable,
however clearly we may be taught to recognise their reality.' All this is
very reasonable, and the last sentence is particularly important.
Nevertheless, many writers allow their reasonings from facts to be
influenced by these ideas of infinity. In Proctor's posthumous work, _Old
and New Astronomy_, the late Mr. Ranyard, who edited it, writes: 'If we
reject as abhorrent to our minds the supposition that the universe is not
infinite, we are thrown back on one of two alternatives--either the ether
which transmits the light of the stars to us is not perfectly elastic, or a
large proportion of the light of the stars is obliterated by dark bodies.'
Here we have a well-informed astronomer allowing his abhorrence of the idea
of a finite universe to affect his reasoning on the actual phenomena we can
observe--doing in fact exactly what my critic erroneously accuses me of
doing. But setting aside all ideas and prepossessions of the kind here
indicated, let us see what are the actual facts revealed by the best
instruments of modern astronomy, and what are the natural and logical
inferences from those facts.
ARE THE STARS INFINITE IN NUMBER?
The views of those astronomers who have paid attention to this subject are,
on the whole, in favour of the view that the stellar universe is limited in
extent and the stars therefore limited in number. A few quotations will
best exhibit their opinions on this question, with some of the facts and
observations on which they are founded.
Miss A.M. Clerke, in her admirable volume, _The System of the Stars_, says:
'The sidereal world presents us, to all appearance, with a finite
system.... The probability amounts almost to certainty that star-strewn
space is of measurable dimensions. For from innumerable stars a limitless
sum-total of radiations should be derived, by which darkness would be
banished from our skies; and the "intense inane," glowing with the mingled
beams of suns individually indistinguishable, would bewilder our feeble
senses with its monotonous splendour.... Unless, that is to say, light
suffer some degree of enfeeblement in space.... But there is not a particle
of evidence that any such toll is exacted; contrary indications are strong;
and the assertion that its payment is inevitable depends upon analogies
which may be wholly visionary. We are then, for the present, entitled to
disregard the problematical effect of a more than dubious cause.'
Professor Simon Newcomb, one of the first of American mathematicians and
astronomers, arrives at a similar conclusion in his most recent volume,
_The Stars_ (1902). He says, in his conclusions at the end of the work:
'That collection of stars which we call the universe is limited in extent.
The smallest stars that we see with the most powerful telescopes are not,
for the most part, more distant than those a grade brighter, but are mostly
stars of less luminosity situate in the same regions' (p. 319). And on page
229 of the same work he gives reasons for this conclusion, as follows:
'There is a law of optics which throws some light on the question. Suppose
the stars to be scattered through infinite space so that every great
portion of space is, in the general average, equally rich in stars. Then at
some great distance we describe a sphere having its centre in our sun.
Outside this sphere describe another one of a greater radius, and beyond
this other spheres at equal distances apart indefinitely. Thus we shall
have an endless succession of spherical shells, each of the same thickness.
The volume of each of these shells will be nearly proportional to the
squares of the diameters of the spheres which bound it. Hence each of the
regions will contain a number of stars increasing as the square of the
radius of the region. Since the amount of light we receive from each 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 add sphere after sphere we add equal amounts of light without limit.
The result 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.'
But the whole light given us by the stars is variously estimated at from
one-fortieth to one-twentieth or, as an extreme limit, to one-tenth of
moonlight, while the sun gives as much light as 300,000 full moons, so
that starlight is only equivalent at a fair estimate to the six-millionth
part of sunlight. Keeping this in mind, the possible causes of the
extinction of almost the whole of the light of the stars (if they are
infinite in number and distributed, on the average, as thickly beyond the
Milky Way as they are up to its outer boundary) are absurdly inadequate.
These causes are (1) the loss of light in passing through the ether, and
(2) the stoppage of light by dark stars or diffused meteoritic dust. As to
the first, it is generally admitted that there is not a particle of
evidence of its existence. There is, however, some distinct evidence that,
if it exists, it is so very small in amount that it would not produce a
perceptible effect for any distances less remote than hundreds or perhaps
thousands of times as far as the farthest limits of the Milky Way are from
us. This is indicated by the fact that the brightest stars are _not_
always, or even generally, the nearest to us, as is shown both by their
small proper motions and the absence of measurable parallax. Mr. Gore
states that out of twenty-five stars, with proper motions of more than two
seconds annually, only two are above the third magnitude. Many first
magnitude stars, including Canopus, the second brightest star in the
heavens, are so remote that no parallax can be found, notwithstanding
repeated efforts. They must therefore be much farther off than many small
and telescopic stars, and perhaps as far as the Milky Way, in which so many
brilliant stars are found; whereas if any considerable amount of light were
lost in passing that distance we should find but few stars of the first
two or three magnitudes that were very remote from us. Of the twenty-three
stars of the first magnitude, only ten have been found to have parallaxes
of more than one-twentieth of a second, while five range from that small
amount down to one or two hundredths of a second, and there are two with no
ascertainable parallax. Again, there are 309 stars brighter than magnitude
3.5, yet only thirty-one of these have proper motions of more than 100" a
century, and of these only eighteen have parallaxes of more than
one-twentieth of a second. These figures are from tables given in Professor
Newcomb's book, and they have very great significance, since they indicate
that the brightest stars are _not_ the nearest to us. More than this, they
show that out of the seventy-two stars whose distance has been measured
with some approach to certainty, only twenty-three (having a parallax of
more than one-fiftieth of a second) are of greater magnitudes than 3.5,
while no less than forty-nine are smaller stars down to the eighth or ninth
magnitude, and these are on the average much nearer to us than the brighter
stars!
Taking the whole of the stars whose parallaxes are given by Professor
Newcomb, we find that the average parallax of the thirty-one bright stars
(from 3.5 magnitude up to Sirius) is 0.11 seconds; while that of the
forty-one stars below 3.5 magnitude down to about 9.5, is 0.21 seconds,
showing that they are, on the average, only half as far from us as the
brighter stars. The same conclusion was reached by Mr. Thomas Lewis of the
Greenwich Observatory in 1895, namely, that the stars from 2.70 magnitude
down to about 8.40 magnitude have, on the average, double the parallaxes
of the brighter stars. This very curious and unexpected fact, however it
may be accounted for, is directly opposed to the idea of there being any
loss of light by the more distant as compared with the nearer stars; for if
there should be such a loss it would render the above phenomenon still more
difficult of explanation, because it would tend to exaggerate it. The
bright stars being on the whole farther away from us than the less bright
down to the eighth and ninth magnitudes, it follows, if there is any loss
of light, that the bright stars are really brighter than they appear to us,
because, owing to their enormous distance some of their light has been lost
before it reached us. Of course it may be said that this does not
_demonstrate_ that no light is lost in passing through space; but, on the
other hand, it is exactly the opposite of what we should expect if the more
distant stars were perceptibly dimmed by this cause, and it may be
considered to prove that if there is any loss it is exceedingly small, and
will not affect the question of the limits of our stellar system, which is
all that we are dealing with.
This remarkable fact of the enormous remoteness of the majority of the
brighter stars is equally effective as an argument against the loss of
light by dark stars or cosmic dust, because, if the light is not
appreciably diminished for stars which have less than the fiftieth of a
second of parallax, it cannot greatly interfere with our estimates of the
limits of our universe.
Both Mr. E.W. Maunder of the Greenwich Observatory and Professor W.W.
Turner of Oxford lay great stress on these dark bodies, and the former
quotes Sir Robert Ball as saying, 'the dark stars are incomparably more
numerous than those that we can see ... and to attempt to number the stars
of our universe by those whose transitory brightness we can perceive would
be like estimating the number of horseshoes in England by those which are
red-hot.' But the proportion of dark stars (or nebulæ) to bright ones
cannot be determined _a priori_, since it must depend upon the causes that
heat the stars, and how frequently those causes come into action as
compared with the life of a bright star. We do know, both from the
stability of the light of the stars during the historic period and much
more precisely by the enormous epochs during which our sun has supported
life upon this earth--yet which must have been 'incomparably' less than its
whole existence as a light-giver--that the life of most stars must be
counted by hundreds or perhaps by thousands of millions of years. But we
have no knowledge whatever of the rate at which true stars are born. The
so-called 'new stars' which occasionally appear evidently belong to a
different category. They blaze out suddenly and almost as suddenly fade
away into obscurity or total invisibility. But the true stars probably go
through their stages of origin, growth, maturity, and decay, with extreme
slowness, so that it is not as yet possible for us to determine by
observation when they are born or when they die. In this respect they
correspond to species in the organic world. They would probably first be
known to us as stars or minute nebulæ: at the extreme limit of telescopic
vision or of photographic sensitiveness, and the growth of their
luminosity might be so gradual as to require hundreds, perhaps thousands of
years to be distinctly recognisable. Hence the argument derived from the
fact that we have never witnessed the birth of a true permanent star, and
that, therefore, such occurrences are very rare, is valueless. New stars
may arise every year or every day without our recognising them; and if this
is the case, the reservoir of dark bodies, whether in the form of large
masses or of clouds of cosmic dust, so far from being incomparably greater
than the whole of the visible stars and nebulæ, may quite possibly be only
equal to it, or at most a few times greater; and in that case, considering
the enormous distances that separate the stars (or star-systems) from each
other, they would have no appreciable effect in shutting out from our view
any considerable proportion of the luminous bodies constituting our stellar
universe. It follows, that Professor Newcomb's argument as to the very
small total light given by the stars has not been even weakened by any of
the facts or arguments adduced against it.
Mr. W.H.S. Monck, in a letter to _Knowledge_ (May 1903), puts the case
very strongly so as to support my view. He says:--'The highest estimate
that I have seen of the total light of the full moon is 1/300000 of that of
the sun. Suppose that the dark bodies were a hundred and fifty thousand
times as numerous as the bright ones. Then the whole sky ought to be as
bright as the illuminated portion of the moon. Every one knows that this is
not so. But it is said that the stars, though infinite, may only extend to
infinity in particular directions, _e.g._ in that of the Galaxy. Be it so.
Where, in the very brightest portion of the Galaxy, will we find a part
equal in angular magnitude to the moon which affords us the same quantity
of light? In the very brightest spot, the light probably does not amount to
one hundredth part that of the full moon.' It follows that, even if dark
stars were fifteen million times as numerous as the bright ones, Professor
Newcomb's argument would still apply against an infinite universe of stars
of the same average density as the portion we see.
TELESCOPIC EVIDENCE AS TO THE LIMITS OF THE STAR SYSTEM
Throughout the earlier portion of the nineteenth century every increase of
power and of light-giving qualities of telescopes added so greatly to the
number of the stars which became visible, that it was generally assumed
that this increase would go on indefinitely, and that the stars were really
infinite in number and could not be exhausted. But of late years it has
been found that the increase in the number of stars visible in the larger
telescopes was not so great as might be expected, while in many parts of
the heavens a longer exposure of the photographic plate adds comparatively
little to the number of stars obtained by a shorter exposure with the same
instrument.
Mr. J.E. Gore's testimony on this point is very clear. He says:--'Those who
do not give the subject sufficient consideration, seem to think that the
number of the stars is practically infinite, or at least, that the number
is so great that it cannot be estimated. But this idea is totally
incorrect, and due to complete ignorance of telescopic revelations. It is
certainly true that, to a certain extent, the larger the telescope used in
the examination of the heavens, the more the number of the stars seems to
increase; but we now know that there is a limit to this increase of
telescopic vision. And the evidence clearly shows that we are rapidly
approaching this limit. Although the number of stars visible in the
Pleiades rapidly increases at first with increase in the size of the
telescope used, and although photography has still further increased the
number of stars in this remarkable cluster, it has recently been found that
an increased length of exposure--beyond three hours--adds very few stars to
the number visible on the photograph taken at the Paris Observatory in
1885, on which over two thousand stars can be counted. Even with this great
number on so small an area of the heavens, comparatively large vacant
places are visible between the stars, and a glance at the original
photograph is sufficient to show that there would be ample room for many
times the number actually visible. I find that if the whole heavens were as
rich in stars as the Pleiades, there would be only thirty-three millions in
both hemispheres.'
Again, referring to the fact that Celoria, with a telescope showing stars
down to the eleventh magnitude, could see almost exactly the same number of
stars near the north pole of the Galaxy as Sir William Herschel found with
his much larger and more powerful telescope, he remarks: 'Their absence,
therefore, seems certain proof that very faint stars do _not_ exist in that
direction, and that here, at least, the sidereal universe is limited in
extent.'
Sir John Herschel notes the same phenomena, stating that even in the Milky
Way there are found 'spaces absolutely dark _and completely void of any
star_, even of the smallest telescopic magnitude'; while in other parts
'extremely minute stars, though never altogether wanting, occur in numbers
so moderate as to lead us irresistibly to the conclusion that in these
regions we see _fairly through_ the starry stratum, since it is impossible
otherwise (supposing their light not intercepted) that the numbers of the
smaller magnitudes should not go on continually increasing ad infinitum. In
such cases, moreover, the ground of the heavens, as seen between the stars,
is for the most part perfectly dark, which again would not be the case if
innumerable multitudes of stars, too minute to be individually discernible,
existed beyond.' And again he sums up as follows:--'Throughout by far the
larger portion of the extent of the Milky Way in both hemispheres, the
general blackness of the ground of the heavens on which its stars are
projected, and the absence of that innumerable multitude and excessive
crowding of the smallest visible magnitudes, and of glare produced by the
aggregate light of multitudes too small to affect the eye singly, which the
contrary supposition would appear to necessitate, must, we think, be
considered unequivocal indications that its dimensions _in directions where
these conditions obtain_, are not only not infinite, but that the
space-penetrating power of our telescopes suffices fairly to pierce
through and beyond it.'[2]
This expression of opinion by the astronomer who, probably beyond any now
living, was the most competent authority on this question, to which he
devoted a long life of observation and study extending over the whole
heavens, cannot be lightly set aside by the opinions or conjectures of
those who seem to assume that we must believe in an infinity of stars if
the contrary cannot be absolutely proved. But as not a particle of evidence
can be adduced to prove infinity, and as all the facts and indications
point, as here shown, in a directly opposite direction, we must, if we are
to trust to evidence at all in this matter, arrive at the conclusion that
the universe of stars is limited in extent.
Dr. Isaac Roberts gives similar evidence as regards the use of
photographic plates. He writes:--'Eleven years ago photographs of the
Great Nebula in _Andromeda_ were taken with the 20-inch reflector, and
exposures of the plates during intervals up to four hours; and upon some
of them were depicted stars to the faintness of 17th to 18th magnitude,
and nebulosity to an equal degree of faintness. The films of the plates
obtainable in those days were less sensitive than those which have been
available during the past five years, and during this period photographs
of the nebula with exposures up to four hours have been taken with the
20-inch reflector. No extensions of the nebulosity, however, nor
increase in the number of the stars can be seen on the later rapid
plates than were depicted upon the earlier slower ones, though the
star-images and the nebulosity have greater density on the later
plates.'
Exactly similar facts are recorded in the cases of the Great Nebula in
_Orion_, and the group of the Pleiades. In the case of the Milky Way in
_Cygnus_ photographs have been taken with the same instrument, but with
exposures varying from one hour to two hours and a half, but no fainter
stars could be found on one than on the other; and this fact has been
confirmed by similar photographs of other areas in the sky.
THE LAW OF DIMINISHING NUMBERS OF STARS
We will now consider another kind of evidence equally weighty with the two
already adduced. This is what may be termed the law of diminishing numbers
beyond a certain magnitude, as observed by larger and larger telescopes.
For some years past star-magnitudes have been determined very accurately by
means of careful photometric comparisons. Down to the sixth magnitude stars
are visible to the naked eye, and are hence termed lucid stars. All fainter
stars are telescopic, and continuing the magnitudes in a series in which
the difference in luminosity between each successive magnitude is equal,
the seventeenth magnitude is reached and indicates the range of visibility
in the largest telescopes now in existence. By the scale now used a star of
any magnitude gives nearly two and a half times as much light as one of the
next lower magnitude, and for accurate comparison the apparent brightness
of each star is given to the tenth of a magnitude which can easily be
observed. Of course, owing to differences in the colour of stars, these
determinations cannot be made with perfect accuracy, but no important error
is due to this cause. According to this scale a sixth magnitude star gives
about one-hundredth part of the light of an average first magnitude star.
Sirius is so exceptionally bright that it gives nine times as much light as
a standard or average first magnitude star.
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.
It seems to me that here we have a fairly direct proof that the stars of
our universe are really limited in number.
There are thus four distinct lines of argument all pointing with more or
less force to the conclusion that the stellar universe we see around us, so
far from being infinite, is strictly limited in extent and of a definite
form and constitution. They may be briefly summarised as follows:--
(1) Professor Newcomb shows that, if the stars were infinite in number, and
if those we see were approximately a fair sample of the whole, and further,
if there were not sufficient dark bodies to shut out almost the whole of
their light, then we should receive from them an amount of light
theoretically greater than that of sunlight. I have shown, at some length,
that neither of these causes of loss of light will account for the enormous
disproportion between the theoretical and the actual light received from
the stars; and therefore Professor Newcomb's argument must be held to be a
valid one against the infinite extent of our universe. Of course, this does
not imply that there may not be any number of other universes in space, but
as we know absolutely nothing of them--even whether they are material or
non-material--all speculation as to their existence is worse than useless.
(2) The next argument depends on the fact that all over the heavens, even
in the Milky Way itself, there are areas of considerable extent, besides
rifts, lanes, and circular patches, where stars are either quite absent or
very faint and few in number. In many of these areas the largest telescopes
show no more stars than those of moderate size, while the few stars seen
are projected on an intensely dark background. Sir William Herschel,
Humboldt, Sir John Herschel, R.A. Proctor, and many living astronomers hold
that, in these dark areas, rifts, and patches, we see completely through
our stellar universe into the starless depths of space beyond.
(3) Then we have the remarkable fact that the steady increase in the number
of stars, down to the ninth or tenth magnitudes, following one constant
ratio either gradually or suddenly changes, so that the total number from
the tenth down to the seventeenth magnitudes is only about one-tenth of
what it would have been had the same ratio of increase continued. The
conclusion to be drawn from this fact clearly is, that these faint stars
are becoming more and more thinly scattered in space, while the dark
background on which they are usually seen shows that, except in the region
of the Milky Way, there are _not_ multitudes of still smaller invisible
stars beyond them.
(4) The last indication of a limited stellar universe--the estimate of
numbers by the light-ratio of each successive magnitude--powerfully
supports the three preceding arguments.
The four distinct classes of evidence now adduced must be held to
constitute, as nearly as the circumstances permit, a satisfactory proof
that the stellar universe, of which our solar system forms a part, has
definite limits; and that a full knowledge of its form, structure, and
extent, is not beyond the possibility of attainment by the astronomers of
the future.
FOOTNOTES:
[1] In a letter to _Knowledge_, June 1903, Mr. W.H.T. Monck puts the same
point in a mathematical form.
[2] _Outlines of Astronomy_, pp. 578-9. In the passages quoted the italics
are Sor John Herschel's.
[3] Mr. J.E. Gore in _Concise Knowledge Astronomy_, pp. 541-2.