In the High Heavens/Chapter 10

by Robert Stawell Ball

CHAPTER X.

IS THE UNIVERSE INFINITE?

ISHOP BUTLER has well remarked that "probability is the guide of life," and, assuredly, if it be our guide in all practical concerns, in a still more significant sense may it be claimed as the source of the greater part of human knowledge. Indeed, in a rapid survey of the field of astronomy we are tempted to affirm, not merely that the theory of probability is of the utmost service to us, but that it is almost our sole method of discovering the truth. This will not seem a paradox to any one who will reflect that there is hardly an astronomical doctrine, even of the most elementary kind, of which it might not be said that our belief in it depends simply on the fact that its truth is, in a high degree, more probable than its falsehood. To those who are accustomed to apply the doctrine of probabilities habitually and, indeed, almost unconsciously, it affords the readiest touchstone by which many fallacious scientific notions can be dissipated. Let me give an illustration of what I mean. In the first book about astronomy which I read in my boyhood there was a glowing description of an investigation which at one time seemed to have attracted a great deal of attention. I allude to the discovery, or the alleged discovery I should rather say, of a certain "central sun," about which it was believed or stated that all the bodies in the universe revolved. This marvellous centre was becomingly located in the Pleiades indeed, if I remember aright, it was actually identified with the star Alcyone. The doctrine was certainly a splendid and captivating one, but it was too good to be true. No one ever hears anything about the central sun hypothesis nowadays, and that, perhaps, for the simple reason that it stood condemned on the face of it by the theory of probabilities. It is wholly unnecessary at this time of day to attempt to appraise the value of the observations by which an astronomer, justly esteemed for other labours, demonstrated, or thought he had demonstrated, the existence of a "central sun." Even if the apparent movements of certain stars offered quite unequivocal testimony (which, indeed, was by no means the case) to show that they were revolving around Alcyone, still it is obvious, on a little consideration, that even this famous star could not be regarded as the centre of the whole universe without doing unwarrantable violence to all notions of probability. For just look at the facts in their due proportion.

Alcyone, no doubt, is a star of magnificent dimensions. It may be a hundred or a thousand times more massive and more brilliant than our sun. Alcyone is so remote from the earth that the light which now arrives at our eyes, even though it speeds on its way at the rate of 186,000 miles a second, has not improbably taken a century, or more than a century, to reach us. The Pleiades form a cluster of bright stars almost unique in their interest; and these circumstances might certainly render the notion that there lay the centre of the universe highly attractive to the imagination and perhaps even quite plausible. But the theory of probabilities at once upsets the whole doctrine when the facts are viewed in their proper light.

No doubt the theory of probabilities has nothing to say against Alcyone in comparison with any other star visible in the heavens, but what it does say is that it would be utterly preposterous to imagine that any one of the stars in the visible firmament could be the central sun around which all the bodies in the universe revolved. For summon up to your imagination the most distant star that can be seen with the unaided eye. Then think of the minutest star that our most potent telescope can disclose. Think of the tiniest stellar point of light which could possibly be depicted on the most sensitive photographic plate after hours of exposure to the heavens. Think, indeed, of the very remotest star which, by any conceivable device, can be rendered perceptible to our consciousness. Doubtless that star is thousands of billions of miles from the earth; doubtless the light from it requires thousands of years, and some astronomers have said millions of years, to span the abyss which intervenes between our globe and those distant regions. But, nevertheless, there is a certain number of miles, even though we know it not, at which the remotest stars known to us must lie. I do not speak of the most distant star which the universe may possibly contain; I only refer to the most distant star that we can possibly bring within our ken.

Imagine a great sphere to be described with its centre at our earth, and with a radius extending all the way from the earth to this last star knowable by man. Every star that we can see, every star whose existence becomes disclosed to us on our photographs, lies inside this sphere; as to the orbs which may lie outside that sphere we can know nothing by direct observation. The imagination doubtless suggests with irresistible emphasis, that this outer region is also occupied by stars and nebulae, suns and worlds, in the same manner as the interior of that mighty sphere whose contents are more or less accessible to our scrutiny. It would do utter violence to our notions of the law of continuity to assume that all the existent matter in the universe happened to lie inside this sphere; we need only mention such a supposition to dismiss it as wholly indefensible. I do not now make any attempt to express the number of miles in the diameter of the sphere which limits the extent of space known directly to man. What that number may be is quite immaterial for our present purpose. But the point that I specially want to bring out is that the volume occupied by this stupendous globe, which includes within it all possible visible material, must be but a speck when compared with the space which contains it. Think of the water in the Atlantic Ocean, and think of the water in a single drop. As the drop is to the Atlantic Ocean so is the sphere which we have been trying to conceive to the boundless extent of space. As far as we know it would seem that there could be quite as many of such spheres in space as there are drops of water in the Atlantic Ocean. And, in all probability, these other spheres throughout space are tenanted by stars, systems, and galaxies just as grand in themselves, just as imposing in their collocations, and just as overwhelming in their myriads as are those which lie within that one particular sphere of which alone we know anything with certainty.

Provided with this conception, we see at once that the doctrine of a visible central sun is an absurdity. As to whether there may be some central sun somewhere or other I can express no opinion, save that I do not see any reason whatever to think that such a body should exist. But we may feel practically certain, according to all rational grounds of probability, that even if there were a central sun in the universe it would not lie within our ken. Suppose that in the wide extent of the Atlantic Ocean there was one individual diatom of a specially interesting character; I do not mean one species with its myriad individuals, but one solitary specimen of a particular microscopic organism, which happened to flourish somewhere or other in the North or South Atlantic Ocean at some depth or other from the surface. Supposing that absolutely nothing further was known as to the whereabouts of this individual object; it might, for anything we could tell, lie beneath a mighty ice-floe in the Arctic regions; it might be miles deep in the Caribbean Sea; it might be basking on the surface in the Equatorial calms; it might be tossed in the surf on the shores of St. Helena; it might be floating at the mouth of the Amazons; it might be off the Cape of Good Hope, or amid the Antarctic icebergs. Would any reasonable man who desired to obtain that unique and extraordinary specimen for his collection imagine that if he went down to the coast of Cornwall and lifted a single drop from the Atlantic he would have such inconceivably good fortune as to find in it this rare diatom of which but a single individual existed throughout the millions of cubic miles of water which compose that mighty ocean? Of course, the mere statement of such a case is sufficient to show its absurdity. But the improbability that the ardent naturalist would secure the prize in the way I have described is not one whit greater than the improbability that, even if there were a central sun, it should lie within the domain of our scrutiny.

There is another line of reasoning by which the theory of probability will often give us invaluable information, which is not this time merely of a negative kind. There are many instances which might be taken of the principles now to be employed. I shall, however, adopt that particular one which presents, perhaps, the greatest interest to astronomers. The question often arises as to whether two objects which appear to us to lie near each other on the surface of the heavens are really neighbours in space, or whether their contiguity is only apparent. It often happens, for instance, that two stars appear very close together through the telescope, and we desire to know whether the two bodies are indeed allied by any bond of physical association, or whether the appearance may not be a mere accidental coincidence. The latter would be the case if the line joining the two stars happened to be so nearly directed towards the earth that, though in reality one of the stars is so much more remote than the other, yet that from our point of view the two happen to be projected on the same part of the sky. We are generally at fault in determining this question by direct observation, because it is usually impossible to find the actual distances by which the earth is separated from the objects, and, therefore, we are deprived of any direct assurance that those distances are so far equal as to enable us to assert that the apparent contiguity is indeed a real contiguity. Here the theory of probabilities will come to our aid and supply reliable information of the most convincing character.

The illustration I shall take is one connected with a famous object. The Great Nebula in Orion is known to be the most glorious body of its class that the heavens display. Seen through a powerful telescope, like that of Lord Rosse at Parsonstown, the appearance of this grand "light-stain" is of indescribable glory to one whose previous acquaintance with practical astronomy enables him to inform the picture before him with the knowledge necessary for its comprehension. It is a vast volume of bluish gaseous material with hues of infinite softness and delicacy. Here it presents luminous tracts which glow with an exquisite blue light; there it graduates off until it is impossible to say where the nebula ceases and the black sky begins. But from our present point of view I am only thinking of the nebula as the nimbus of glory which surrounds the marvellous multiple star known to astronomers as Theta Orionis.

This complex sidereal system consists of four bright stars quite close together, with at least two smaller ones which evidently belong to the same scheme. The whole sextuple group makes a spectacle unique in the heavens. Wherever Theta Orionis happened to be in the sky it must necessarily be known as the most elaborately composed of all multiple stars. But, as a matter of fact, we find the wonderful star apparently occupying the most imposing site in the Great Nebula, so that the latter serves as a splendid setting to the complex star. The appearance presented would, of course, be explained if it should happen that the wondrous multiple did actually lie inside the nebula wherein it was seen gleaming. But it is, no doubt, conceivable that the effect actually witnessed might be accounted for if it should happen that the multiple star were billions of miles in the foreground, only so placed that from our point of view we beheld it projected with the brightest part of the nebula as a background. Fig. 31.—The Multiple Star (Theta Orionis) in the great nebula of Orion.Such, too, is the translucency of nebulous material that it is at least a conceivable hypothesis that the nebula might be the object which lay in the foreground and that the star occupied a position billions of miles in the rear, but that from where we were situated our line of sight towards the star conducted our vision directly through the centre of the nebula.

We have really no means of certainly knowing which of these notions is the correct one. At least, I should say, direct observation cannot be held to have shown conclusively that one of these doctrines is true and that the other two are false. It could only have done so when we had measured the distances of both the nebula and the multiple star from the earth. As a matter of fact we have not measured the distance of either the one or the other. This is eminently a case in which the theory of probabilities can be suitably applied, and the result to which it leads is of no uncertain kind. It demonstrates, by a line of reasoning the cogency of which cannot be impugned, that the famous stars are not standing out in front of the nebula, that they are not sunk far behind, but that they do veritably lie at the heart of the nebula itself, the combined object forming one glorious organization. To simplify the application of the argument, let us assume that the visible heavens are constituted, not of hosts of stars and nebulae, but of one single star and one single nebula. Let us suppose that the nebula occupies an area of about one square degree, that is, about five times the area of the full moon, and let us suppose that from our point of view the star appears to lie within the confines of the nebula. Would it be more reasonable to believe that the presence of the star in that particular locality of the heavens was only an accidental circumstance due to the line of vision from the nebula to the star passing through the eye, or that it was due to the fact that there was some physical connection between the two bodies, in which case, of course, the star would lie within the confines of the nebula, and the contiguity would be real as well as apparent?

Suppose that the star and the nebula were both planted down absolutely at random on the surface of the heavens; then, as the nebula occupies a space of one square degree, and as there are forty thousand square degrees on the surface of the sphere, there are obviously forty thousand chances to one against the star happening to lie within the confines of the nebula, if the connection between the two bodies were merely casual and apparent. For the ordinary purposes of life, when we find that there are forty thousand chances to one against a particular phenomenon occurring, we generally exclude from the realm of practical duty the supposition that the unlikely event will occur. If a sum of £150 is to be raffled by the sale of enough tickets at a penny a-piece to leave a reasonable profit on the undertaking, the purchaser of a ticket builds but little hope on his chances of success. He knows that the chances against him are about forty thousand to one. We are entitled to say that there must be forty thousand chances to one against the star lying in the nebula, unless it should happen that there was some physical connection between the two. We see, however, that the star does lie in the nebula; therefore, for all practical purposes, we conclude that there must be some physical reason for this coincidence, but we can see no physical reason whatever why the line joining the star and the nebula should pass near the earth if the two objects were totally distinct. We are, therefore, forced to the conclusion that the star must be directly associated with the nebula. There are forty thousand chances to one that this is the case, and, as rational people, we adopt this conclusion as the basis of our belief.

This will illustrate the argument used in the actual case of the Great Nebula in Orion and the multiple star in the same constellation. It is true that there are thousands of stars and thousands of nebulae, but there is only one star so marvellously complex in its character as Theta Orionis, and there is only one nebula so ample in its magnificence as that in the sword-handle of Orion. But we find the unique multiple star apparently located in the richest part of the unique nebula. If, therefore, we remember that the region of the nebula referred to is perhaps about a square degree in extent, we are entitled to affirm that there must be forty thousand chances to one that Theta Orionis, the star, is veritably immersed in the glorious nebulosity of Orion. The theory of probabilities allows reasonable beings to draw no other conclusion.

The theory of probabilities is also very instructive in the information which it gives us with reference to the existence of an invisible myriad of bodies through space which can never be discerned by any means at our disposal. It is, of course, well known that the stars, properly so-called, are each of them brilliant suns, intrinsically of majestic proportions, but dwarfed to comparative insignificance by the tremendous distance in space at which they are placed. These bodies are all self-luminous, and it may no doubt happen that there are dark bodies in the vicinity of some of the bright stars to which these stars act as illuminants, just in the same way as the sun dispenses light to the planets. But it is utterly impossible for us to discern objects illuminated in this fashion, for the light which they receive from suns that lie in their neighbourhood would be altogether insufficient to render them visible to us across the vast abyss of space by which they are separated from the earth. There are, no doubt, certain indirect processes of reasoning by which astronomers have learned, with more or less accuracy, something with regard to these dark stars. Thus, for instance, it has been shown that the extraordinary fluctuations in the light of Algol must be attributed to the eclipses of a brilliant star by the interposition at regular intervals of a dark body revolving around it. There are also cases in which it has happened that two dark stars have come so near each other that they may be almost said to have collided, and the sudden transformation of energy of motion into energy of light and heat has been sufficient to announce far and wide through the universe the character of the event which has taken place.

But such instances are few and far between, and we should receive a very erroneous impression as to the population of the celestial regions by bodies devoid of light if we thought that the few whose presence has been occasionally disclosed in some very indirect and casual manner bore anything like a considerable proportion to the total number which actually exist. It is just at this point that the theory of probabilities comes to supplement our knowledge, and the results to which it conducts us are of a most startling description. By this theory we are assured, with a logic which cannot be controverted, that the invisible bodies must be vastly more numerous than the visible stars, so that even the millions of bright stars which we see afford only an utterly inadequate conception of the full extent of the material universe. Remember, I am not now referring to objects beyond our ken merely because they lie so far off. What I mean is that even within the sphere which contains the visible stars that we know, there is such a stupendous quantity of matter of a dark character, that the visible part bears an almost imperceptible proportion to it. It may well be asked how we know that there is this exuberant abundance of invisible matter. Let the theory of probabilities answer the question.

I shall suppose that we have to deal with a lapse of time, which for our present argument may be regarded as indefinitely long. It can be demonstrated that the conditions under which a mass of matter becomes so highly heated as to shine with star-like radiance are wholly exceptional in their character. So far as our present knowledge goes it would seem that the brightness of any sun-like body is to be attributed solely to the transformation in some fashion of mechanical power into heat. To take our own sun as an example, it is now an assured doctrine that the heat so necessary for our welfare is sustained by the gradual contraction of the solar volume. The energy available for transformation into heat in this process seems sufficient to supply the radiation of the sun, not only for ages such as those which we reckon in the human period, but even throughout a lapse of time so vast as that which geology demands for the formation of the earth's crust. But it is certain that the quantity of possible light and heat to be dispensed by the sun is limited in amount. The sun cannot shine on for ever. A time must assuredly come when the mighty orb at present so brilliant will have no more potency for the radiation of light than is at present possessed by the earth or the moon.

In like manner it can be shown that the materials constituting the sun have not always been luminous. We cannot indeed say with certainty by what influence their brightness was originally kindled. It probably arose from a collision, or an approach to a collision, between two dark masses which happened to come to an encounter with enormous velocities in their progress through space. It is, however, plain that the ages during which the sun has been brilliant form only an incident, so to speak, in the infinite history of that quantity of matter of which the solar system is constituted. Notwithstanding the millions, or thousands of millions, of years for which that matter has existed, it has perhaps only once become so heated, owing to circumstances which we may describe as accidental or casual, as to have acquired the ample light-dispensing power of a sun. It is, however, possible that such periods of light-radiating capacity should have occurred more than once; they may possibly have occurred several times throughout the ages of time past. Nor is it likely that the last phenomena of this kind have yet arrived.

The sun, after the lapse of countless years, will lose all its heat and pass into a dark mass. In that form it may endure for a period so protracted that the spell during which it has acted as the luminary to our system will appear but a moment in comparison with the dark ages which succeed the solar splendour. But we can conceive that the darkness, which is the doom of our system, need not necessarily be eternal so far as its materials are concerned; it may be that again in the course of its wanderings through space, the tide of chance may at length bring the dark and tremendous globe so near some other orb that another collision should take place with appalling vehemence. The solid materials may again become transformed into a stupendous glowing nebula, and then, in the course of the tedious contraction of this nebula, another protracted period of brilliance will diversify the career of this great body, and may last long enough for the evolution of planets and of whole races of highly organized creatures.

The essential point for our present consideration must not be misunderstood. A little reflection will show that any periods of brilliance must be regarded as exceptional periods in the history of each body. Think, for instance, of all the iron on the surface of the earth. There is the iron in the ore; there are the great stores of pig-iron lying ready for use; there are the vast bridges which span our rivers and straits; there are the thousands of miles of railway lines; there are the countless wheels and pieces of machinery; there are iron vessels on every ocean, and objects of every size made of iron, from the smallest nails up to hundred-ton guns. There is also at this moment, and every moment, a good deal of hot iron on the earth. While I write, iron is doubtless flowing from blast furnaces in England, Wales, and Scotland; while I write, ingots of white-hot Bessemer steel are being dealt with under the steam-hammer or in the rolling-mills; while I write, horse-shoes are being forged, and, at each moment, in one way or another, pieces of iron of every temperature could be found, from those which are as cold as the iron apparatus used by Sir James Dewar in his experiments in the liquefaction of air, up to the glittering melted steel which is poured from the tilted converter. But it must be admitted that the highly-heated pieces of iron bear a very small proportion indeed to the total mass of iron in the world at any moment. No doubt there are many tons of iron now white-hot, but there are many millions of tons of iron once white-hot, but now no warmer than the air around. At certain phases in its history every piece of iron has to undergo the operation of being raised to incandescence, or even of being transformed into a liquid. But the laws of cooling are such that, as soon as the opportunity is afforded, the iron parts with its redundant heat and returns to a stable condition, in which it is at the temperature of the air.

Suppose that some percipient being, who was viewing this earth from above, could only recognise iron when it was red-hot or white-hot, but that he had every facility for perceiving such iron as happened to be in this condition. With such faculties, he would, no doubt, be able to discern here and there a stream of molten iron issuing from a blast furnace, or perhaps to witness the operation of the forging of an anchor under the steam-hammer, to watch the rolling of the plates for an armour-clad, or to see the more humble operations of the blacksmith or nail-maker. But he would surely form an entirely erroneous impression as to the quantity of iron on this earth, or as to the extent in which it was employed in the varied purposes of the arts, if he concluded that there was no iron on our globe at all except that which happened at the moment to be in that particular incandescent state in which alone it was visible to him. If he were gifted with reasoning powers he would say, "It is quite true that I can only see the iron while it is red-hot, but I know that for iron to be red-hot on the earth's surface is an exceptional and abnormal condition of a very temporary or intermittent character. No doubt, every piece of iron may have to be red-hot once, or more than once, but the total duration of such phases of incandescence is quite insignificant under ordinary circumstances when compared with the periods in which the iron is cold and invisible. I, therefore, cannot refuse to believe that there must be an amount of iron on the earth which I do not see, but which bears a proportion to that which I do see in the ratio of thousands or millions to one."

Precisely similar is the way in which the astronomer who is properly familiar with the theory of probabilities will approach the study of the stars. He will reflect that each mass of matter must be cold and invisible for by far the greater part of the period of its existence; he will reflect that on rare occasions, separated by intervals of appalling length, certain exceptional conditions arise by which this dark piece of matter may be so kindled that, for an epoch, long it may be in years but brief indeed when compared with the span of its total existence, the body would glow as a star.

Provided with this conception let us look on the universe with its millions of orbs. These orbs will be found in every state possible to such bodies; but the enormous majority of them must, in accordance with the principles just laid down, be in the dark and invisible state. Out of some millions it may perhaps be concluded that, at any particular moment, a dozen or so might, by accidental circumstances, be in those phases of their several careers in which luminosity is a characteristic. In such cases only will the orbs be visible. The instructed astronomer will, therefore, believe that the non- visible orbs must be hundreds, thousands, or perhaps millions of times more numerous than those which he can see. When we remember that, by our telescopes and on our photographs, we can discern something like one hundred million luminous stars in the sky; when we remember that every one of these is the indication of a wholly exceptional incident in the career of the body from which the light emanates; and when we further believe, as believe we must, that for each one star which we can thus see there must be a stupendous number of invisible masses, then, indeed, we begin to get some notion of the extraordinary multitude in which material orbs are strewn through space. The theory of probabilities declares to us with a certainty, hardly, in my opinion, inferior to that of optical demonstration, that even within the distance which can be penetrated by our telescopes the visible stars cannot form the hundredth, probably not the thousandth, perhaps not the millionth part of the total quantity of matter.

On the question as to whether space is finite, our observations with the telescope have but little information to give. The question here involved is rather of a metaphysical complexion. The extent of space depends more upon the facts of consciousness than upon those of astronomical observation. It may, perhaps, simplify the discussion of the subject if we first of all consider the question as to whether the quantity of matter in the universe may be presumed to be infinite or not. We can put the question into a perfectly concise form by reflecting that every particle of matter, whether solid, liquid, or gaseous, is composed of molecules. No doubt these molecules are so numerous that even in the air we breathe the capacity of a lady's thimble would contain a multitude of molecules so great that it has to be enumerated by billions. But we are not at present merely concerned with the actual number of molecules that may exist in the atmosphere, even in its whole extent, or in the whole earth, or in the whole sun. Let us try to conceive the number of molecules that are present in all the stars, bright and dark, which exist not only within those regions of space accessible to our telescopes, but elsewhere as well. In short, let us try to conjure up in our imagination the kind of figures which are to express the total number of molecules in the universe. Is that number finite, or is it not?

This is, perhaps, one of the most fundamental questions in nature which could possibly be proposed. Let us consider the consequences which would follow from adopting a negative answer to this question. If we suppose that the number of molecules is indeed infinite, then we are necessarily forced to admit at once that space must be infinite too; for had space any boundary, then, since the molecules do not admit of being crowded together beyond a certain extent, it would be impossible that they could exist in infinite abundance. Adopting the sound principle that we need not assume more than is necessary to explain the phenomena actually presented by our consciousness, it seems to me to be clear that the number of molecules of matter in the universe must be finite. The row of figures which would express that number, whatever it may be, is the most remarkable descriptive constant which the universe possesses. It matters not for our present argument what may be the range of figures by which this number can be expressed. It may not be too large to be written even on the thumb-nail by the compendious method of notation now in general use.

Let us next see whether we can learn anything as to the extent of space itself. It is apparent that we seem to be in the presence of about equal difficulties whether we attempt to think of space as finite or as infinite; for, imagine that you go up in a straight line into the sky, and on, and on, and on, in thought for millions of miles, it would seem that the journey ought to be endless; for, supposing that you try to conceive a boundary, the imagination will equally suggest that there is something on the other side of that boundary from which you can commence again. It appears almost equally impossible to suppose that the journey could be carried on for ever as to suppose that it could ever be brought to an end. It was, however, long ago shown by Kant that space was rather to be regarded as a form in which the human mind was compelled to regard objects than as a self-existing fact of external nature. We have no power in our own consciousness to surmount the difficulties of conception to which I have referred. They arise from the conditions of our mental constitution, and reasoning about space will do no more to remove its mysteries than it will suffice to give to the man born blind a notion of the colour scarlet. But mathematicians, while fully aware of the imperfection of their powers of conception as regards the facts of space, are still enabled to frame a perfectly consistent theory according to which the observed phenomena of nature can be presented within a space which is finite in dimensions. They are even able, as it were, to lay their finger upon the exact point in which the subjective difficulty has arisen.

I must here be permitted to refer to a point in connection with the elements of Euclid. The beginner who studies that work commences, of course, by learning the axioms, and reads without any feeling of discontent or querulousness such venerable truths as that "the whole is greater than its part." But, after a number of propositions of this eminently unquestionable but somewhat puerile kind, he is suddenly brought up by the famous twelfth axiom in which Euclid lays down the theory of parallel lines. Here is a statement of a radically different kind from such assertions as that "if equals be added to equals the wholes are equal." In fact, Euclid's notion of parallel lines is so far from being an axiom of the same character as those other propositions that it is quite possible to doubt its truth without doing any violence to our consciousness.

The principle assumed in the twelfth axiom cannot be proved, and it has been well remarked, that it indicates the supreme genius of Euclid to have expressed this particular axiom in such language as challenges at once the attention and the caution of the student. It may, however, be said that nearly all our difficulties in connection with the conceptions of space take their origin in the ambiguities which arise from the assumption which the twelfth axiom implies. Some modern mathematicians have gone so far as to deny the existence of this axiom altogether as a truth of nature, and it is most important to notice that when free from the embarrassment which the assumption of Euclid involves, a geometry emerges which removes our difficulties. It seems to show that space is finite rather than infinite, so far as we can assign definite meaning to the words, but it would lead me into matters somewhat inconvenient for these pages if I were to pursue the matter with any further detail. I may, however, say that it can be demonstrated that all known facts about space are reconcilable with the supposition that if we follow a straight line through space using for the word straight the definition which science has shown properly to belong to it then, after a journey which is not infinite in its length, we shall find ourselves back at the point from which we started. If anyone should think this a difficulty, I would recommend him to try to affix a legitimate definition to the word straight. He will find that the strictly definable attributes of straightness are quite compatible with the fact that a particle moving along a straight line will ultimately be restored to the point from which it departed.

It is quite true that this seems to be a paradox, but it will not be so considered by the geometer. The truth it implies is indeed quite a familiar doctrine in modern geometry. But what is not so familiar to mathematicians is that the restoration of the travelling particle to the point from which it originally started need not involve a journey of infinite length. If we assume Euclid's twelfth axiom to be true, then no doubt the traveller can only get back to the point from which he started as the result of a journey of infinite length which will have occupied an infinite time. But now suppose that Euclid's twelfth axiom be not true, or suppose that, what comes to the same thing, the three angles of a triangle are not indeed equal to two right angles, then the journey may require neither an infinite lapse of time nor an infinitely great speed before the traveller regains his original position, even though he be moving in a straight line all the time. Space is thus clearly finite; for a particle travelling in a straight line with uniform speed in the same direction is never able to get beyond a certain limited distance from the original position, to which it will every now and then return.

Those who remember their Euclid may be horror-struck at the heresy which suggests any doubt as to the sanctions by which they believe in the equality of the three angles of a triangle to two right angles. Let them know now that this proposition has never been proved, and never can be proved, except by the somewhat illogical process of first assuming what is equivalent to the same thing, as Euclid does in assuming the twelfth axiom. Let it be granted that this proposition is to some very minute extent an untrue one—there is nothing we know of which shows that such a supposition is unwarrantable—no measurements that we can make with our instruments, no observations that we can make with our telescopes, no reasonings that we can make with our intellect, can ever demonstrate that the three angles of a triangle may not as a matter of fact actually differ from two right angles by some such amount as, let us say, the millionth part of a second. This does no violence to our consciousness, while it provides the needed loophole for escape from the illogicalities and the contradictions into which our attempted conceptions of space otherwise and us.