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TOR more than two centuries and a half astronomers have T studied the depths of heaven with the telescope, piercing farther and yet farther into wondrous abysms of space, gathering clearer and yet clearer information as to the structure of celestial objects, and accumulating an untold wealth of knowledge respecting the habitudes of the great system whereof our sun is a constituent orb. During all this process of research, the great end and aim of astronomers has been to extend the range of their instrumental appliances, in order to analyse

depths. Now and then it has occurred to some among their number to endeavour to combine the results which have been gathered together with so much pains; but these attempts have been almost lost sight of amidst the continual accumulation of fresh facts. The efforts made to arrange and systematise our knowledge have been altogether out of proportion with its extent.

And, very strangely, when any attempts are made to educe from the labours of observers their proper significance, to reap the harvest which is already ripe, or rather to grind the corn which is already in our garners, the cry is raised that such attempts are fit only for the theorists, that they argue a want of appreciation of the labours of observers, and that we have more to hope from fresh observations than from any process of mere reasoning. Surprising, indeed, that those who say “ Let us use the observations already made," should be accused of undervaluing observation; and that those who can find no value or significance in past observations, should call so eagerly for fresh ones!

I make these remarks because I am about to exhibit certain

views respecting the habitudes of interstellar space, which have been formed from the study of the past labours of astronomers. I am fully sensible of the fact that to many I should seem better worthy of a hearing, if I nightly timed my watch by the stars, if I had spent a few years of labour in attempting to divide well-known double stars with inadequate telescopic power, or if I had in some other equally convincing manner exhibited my title to be regarded as a member of the now large array of amateur telescopists who work so hard and effect so little, and suppose themselves to be practical astronomers. Let me not be misunderstood, however. It is only because I wish to see amateur telescopists engaged on more useful researches, because I wish to see them devote a little more consideration than they do now to the thought of advancing astronomy, that I speak slightingly of the modes in which at present they are for the most part wasting time. We want all their help, and more, to advance the interest of our well-loved science; all their telescopic appliances are too few for the work astronomers would like to see them doing.

In studying the heavens, we have always this great difficulty, that we are looking at objects which are in reality at very different distances, but which appear to lie on the concave surface of a vast spherical enclosure. It seems almost hopeless to attempt by any processes of observation to obtain reliable estimates of the distances of all, save a very few, of the fixed stars. It is not going too far to say that we are tolerably certain of the distance of only one star in the heavens—the star, Alpha Centauri. This being the case, and the heavens spangled with millions of objects at altogether unknown distances, we must look carefully round us for evidence of another kind than that derived from actual measurement—we must look for signs of association, for definite laws of aggregationif any such exist—and, if possible, we must apply that mode of inquiry from analogy which Sir William Herschel found in many instances so effective.

And here, as I have mentioned the name of this great astronomer, to whom we owe the first systematic survey of the heavens, and the first attempt to reduce the results of observation into law and order, I wish with, extreme diffidence, to point to what I cannot but consider an error of judgment in his selection of the principles which were to guide his survey of the heavens. It appears to me, that it would have been in all respects better had his first processes of stellar observation been directed to gauge the probability that this or that law of distribution prevails in the heavens, rather than to the application of a system of star-gauging, which, if founded on a mistaken assumption, was necessarily but a waste of labour. It would have been a misfortune if the unequalled observing qualities of either the elder or the younger Herschel had been misapplied for a single hour; but the possibility that the labours of both these astronomers should have been devoted year after year to a process which (if my views are just) was practically useless, is painful indeed to reflect upon. It is true that the labours of the Herschels have been so numerous and so widely extended, that even the recognition of their star-gaugings as of little real utility would leave the great mass of useful results credited to them almost unaffected; but it would remain none the less a misfortune that labours, which in the case of other men would have worthily filled a lifetime, should have been misdirected.

And yet, when one considers the matter apart from preconceived notions, how inconceivably small the chance appears that these laws of distribution believed in by Sir William Herschel actually prevail within the sidereal depths. How amazing that to his clear perceptions the idea should ever have seemed probable that the celestial spaces are occupied only by orbs resembling our sun. For, be it distinctly noted, that his belief in the existence of gaseous nebulæ, and orbs in various stages of development, belongs to the later part of his career as an observer. Undoubtedly the whole system of star-gauging was founded upon the belief that the sidereal system consists of stars, varying greatly perhaps in size, but still not so greatly but that the least of them would be visible in Herschel's great telescope, as far as the very limits of the sidereal system, and that these stars are distributed with a certain general uniformity throughout space.

It is well to observe how fatally any error in this fundamental hypothesis affects the significance of any system of stargauging. We turn a telescope in a given direction, and we see, perhaps, but four or five faint stars. According to the Herschelian hypothesis, the limits of the sidereal system are near to us in that direction, because the stars seen are so few, and those stars being necessarily within those limits, and faint, belong probably to the lower orders of real magnitude. But what if that hypothesis be erroneous—if there may exist in this or that direction vast blank spaces a thousandfold larger, perhaps, than the whole sphere of the visible stars in extent? Then, perchance, these four or five faint stars may lie farther from us than the farthest belonging to some of the richer star-fields; they may form a group of orbs which individually surpass Sirius or Canopus in magnificence, and are separated from each other by distances exceeding many thousandfold those which separate our sun from neighbouring luminaries. But, yet again, suppose that in any direction our telescope reveal crowded star-fields, orbs of all orders of apparent brightness, “ strewn as


by handfuls, and both hands full," and each increase of power adding fresh riches to the display. According to the Herschelian hypothesis, there is but one explanation of these wonders; we are looking into a widely extended part of the sidereal system, and those different orders of stars lie at different orders of distance—the farthest at distances so enormous that we cannot attain to them. But, in what a different light we must regard the scene if we remember the possibility that that wondrous wealth of stellar display need by no means argue enormous extension. All these sparkling orbs may be gathered into one region of space, their various orders of apparent lustre arguing various orders of real magnitude. Instead of looking into star-lit depths, which extend linearly from the eye, far out into space beyond the ordinary limits of distance separating from us the outer bounds of the sidereal system, we may in fact be contemplating a wondrously variegated star-group.

But the conclusions we are to form must be founded not on the consideration of what may be, but on our observation of what is. There is abundant evidence for forming probable views respecting the general laws prevailing within the sidereal system; at any rate, for deciding whether it is more probable that there is or not any general uniformity of distribution within its limits.

One direct consequence of the laws of probability has been very much lost sight of in dealing with the subject we are now engaged upon. It has been urged that where so many stars are spread over the heavens, at so many various distances, we ought not to be surprised if very great varieties of distribution should be observed, nor conclude, therefore, that the general uniformity predicated by Sir William Herschel may not prevail as respects distribution in space. It has been forgotten that the vastness of the numbers in question should tend to a uniformity of apparent distribution, instead of the reverse.

I had been led myself to overlook this consideration, obvious as it is, until it was impressed upon me in a very striking manner during a somewhat novel process of research.

I wished to determine what peculiarities of distribution might be expected to appear among a number of points spread over a plane surface perfectly at random. It is clear that this is a preliminary consideration very necessary for the purpose of determining whether the laws of distribution seen among the stars are accidental or not. Now, the problem of determining by purely mathematical considerations what peculiarities would probably appear in a chance distribution of any given number of points, is one which may be regarded as altogether too difficult for solution. Very simple problems of probability have been found perplexing, insomuch that two eminent ma

thematicians of the last century are said to have disputed over the question whether the chance of tossing one head and one tail in two throws of a coin were one-half or one-third.* But problems concerning the chance distribution of points are specially difficult, as any one will find who tries a few apparently simple ones.f Therefore, I sought to solve this particularly complex problem in a practical manner, by simply spreading a number of points at random, and examining the result. But how to distribute points perfectly at random? It seems very easy, but is not so by any means. Suppose we take a handful of grains, and throw them upon a table. Will they then be strewn without law or order ? Very far from it. The fact that they have all come from the same hand will lead to very obvious effects, taking away altogether from the desired random character of the distribution. Then, again, suppose we were to distribute grains over a table from a sieve as large in extent as the table, and uniformly filled. In this case the grains would be distributed with a uniformity not appertaining to chance distribution. And so of a number of other contrivances which may be thought of; in every case of mechanical distribution, we always find either an enforced inequality or an enforced equality of distribution, not that really random distribution which we require.

The plan I actually adopted, if laborious, was at least satisfactory in this respect. I took a table of logarithms (any other book full of tabulated figures would have done equally well), and opening the book at random, brought down the point of a pencil upon the page of figures. The numeral on which, or nearest to which, the point fell, I entered in a book. In this way I took out several thousand figures, following each other in altogether random sequence. Then, having divided two adjacent sides of a square into 100 equal parts, I drew parallels to the sides, through the points of division, thus dividing the square into 10,000 small squares. Now, suppose the first four figures in my list to have been 7324. I took the seventy-third

• The erroneous reasoning by which the answer is made to be one-third seldom fails to puzzle the uninitiated. “There are,” said D'Alembert, “ three possible events : either two heads must be thrown, or two tails, or head and tail; of these three possible events, only one is favourable. The chance of that event is, therefore, precisely the same as the chance of drawing one particular ball out of a bag containing three, that is, it is one-third.'

† For instance, here are two: (1.) On a square surface of given size (say one square foot) two points are marked in at random; what is the chance that they will be within a given distance (say one inch) of each other? (2.) Three bullets strike a circular target three feet in diameter; what is the chance that the lines joining the three points where the target is struck will include a triangle less than one square foot in area ?

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