From the observed periods of fluctuation of their Cepheid variables, in combination with the other methods just explained, Dr Hubble of Mount Wilson Observatory has recently found that even the nearest of these nebulae, namely the nebula M 33 shewn in Plate XX (p. 213), is so remote that light takes some 850,000 years to travel from it to us. The Great Nebula M 31 in Andromeda (p. 30) is at the slightly greater distance of about 900,000 light-years. This abundantly proves that these nebulae lie right outside the galactic system, justifying the term “extra-galactic” nebulae.

One might attempt to estimate the total number of stars in these nebulae by counting those visible in a selected average small area, but more precise methods are available. Just as we have supposed that the outermost stars in the galactic system are describing orbits under the gravitational attraction of the galaxy as a whole, so we must suppose that the outermost stars in a nebula are describing orbits under the gravitational attraction of the main mass of the nebula; the forces which keep them from running away from the nebula are similar to those which keep the earth moving in its orbit round the sun. If so, we can weigh the nebulae, precisely in the same way as we weigh the sun (p. 44) or the galactic system (p. 69). Dr Hubble in this way estimates that the weight of the Great Nebula M 31 in Andromeda, shewn in Plate IV, must be about 3500 million times that of the sun, while the nebula N.G.C. 4594 in Virgo, shewn in Plate XV (p. 204), must have about 2000 million times the weight of the sun. In general it seems likely that each of the extra-galactic nebulae contains about enough matter to make some 2000 million stars.

This is not the same thing as saying that each nebula already contains 2000 million stars. While many of these nebulae appear to consist largely of clouds of stars, yet most of them contain also a large central region which no telescopic power has so far succeeded in resolving into distinct points. For instance, Plate XII shews the central region of the Great Nebula in Andromeda magnified to the same degree as the left-hand top corner shewn in Plate XI, and this is clearly not resolved into stars in the same way as the outer regions shewn in Plate XI. The whole of the nebula N.G.C. 4594 in Virgo, shewn in Plate XV, also refuses to be resolved into separate stars. We shall find reasons later (Chapter IV) for interpreting these central regions as masses of gas which are destined in time to form stars, but have not yet done so. We shall in fact find that the nebulae are the birthplaces of the stars, so that each nebula consists of stars born and stars not yet born. It is the total weight of stars already born and of matter which is destined to form stars that aggregates 2000 million suns.

About 2,000,000 of these extra-galactic nebulae are visible in the great 100-inch telescope. In general, they appear to be scattered with a tolerable approach to uniformity through space, their average distance apart being something of the order of 2,000,000 light-years, although here and there this uniformity is broken by clouds and clusters of nebulae. For instance, the sky is remarkably rich in nebulae in the constellations of Virgo and Coma Berenices. Here a cloud of about 300 nebulae is collected, according to Shapley, within a space having only from 5 to 10 times the dimensions of the galactic system, at a distance of some ten million light-years from the sun. The same region of the sky appears also to contain three other and more remote clouds. Shapley has suggested that our galactic system, the Andromeda nebula and other near nebulae may constitute a similar cloud.

THE REMOTEST DEPTHS OF SPACE. Hubble estimates that the most distant of the 2,000,000 nebulae revealed by the 100-inch telescope must be about 140 million light-years away from us. This is the greatest distance which the human eye has so far seen into space. The 220,000 light-years which formed the diameter of the galactic system seemed staggeringly large at first, but we are now speaking of distances some 600 times greater. For all but a 500th part of its long journey, the light by which we see this remotest of visible nebulae travelled towards an earth uninhabited by man. Just as it was about to arrive, man came into being on earth, and built telescopes to receive it. So at least it appears when viewed on the astronomical scale. Yet even this last 500th part of the journey covers the lives of 10,000 generations of men, through all of which, as well as through 500 times as great a span of time, the light has been travelling steadily onward at 186,000 miles a second.

There are so many faint nebulae at the very limit of vision of the 100-inch telescope, that it seems certain that a still larger telescope would reveal a great many more. The 200-inch telescope, which it is hoped will shortly be built, having twice the aperture of the present 100-inch, ought to probe twice as far into space, and so may perhaps be expected to shew about eight times as many, or 16 million, nebulae.

THE STRUCTURE OF THE UNIVERSE

So far every increase of telescopic power has carried us deeper and deeper into space, and space has seemed to expand at an ever-increasing rate. We may well ask whether this expansion is destined to go on for ever: are there any limits at all to the extent of space?

Even a generation ago, I think most scientists would have answered this last question in the negative. They would have argued that space could be limited only by the presence of something which is not space. We, or rather our imaginations, could only be prevented from journeying for ever through space by running up against a wall of something different from space. And, hard though it may be to imagine space extending for ever, it is far harder to imagine a barrier of something different from space which could prevent our imaginations from passing into further space beyond.

The argument is not a sound one. For instance, the earth’s surface is of limited extent, but there is no barrier which prevents us from travelling on and on as far as we please. A traveller who did not understand that the earth’s surface is spherical, would naturally expect that longer and longer journeys from home would for ever open up new tracts of country awaiting exploration. Yet, as we know, he would necessarily be reduced in time to repeating his own tracks. As a result of its curvature, the earth’s surface, although unlimited, is finite in extent.

THE THEORY OF RELATIVITY

Through his theory of relativity, Einstein claims to have established that space also, although unlimited, is finite in extent. The total volume of space in the universe is of finite amount just as the surface of the earth is of finite amount, and for the same reason; both bend back on themselves and close up. The analogy is valid and useful only so long as we are careful to compare the whole of space to the surface of the earth, and not to its volume. The volume of the earth is also finite in amount, but for quite different reasons. A mole which burrowed on and on through the earth in a straight line would come in time to something which is not earth—it would emerge into the open air; but we can go on and on over the surface of the earth without ever coming to anything which is not the surface of the earth. The properties of space are those of the surface, not of the volume, of the earth.

As a consequence of space bending back into itself, a projectile or a ray of light can travel on for ever without going outside space into something which is not space, and yet it cannot go on for ever without repeating its own tracks. For this reason it is probable that light can travel round the whole of space and return to its starting point, so that if we pointed a sufficiently powerful telescope in the right direction in the midnight sky, we should see the sun and its neighbours in space by light which had made the circuit of the universe. We should not see them as they now are, but as they were many millions of years ago. Light which had left the sun so long ago would have travelled round almost the whole of space and then, just as it was about to complete the circle, it would be caught in our telescope instead of being allowed to start on its second journey round space.

This curvature of space has other functions than that which it performs on the grand scale, of limiting the total volume of space. Before Einstein’s day the curvatures of the paths of planets, cricket balls and projectiles in general were all attributed to the pull of a “force” of gravitation. The theory of relativity dismisses this supposed force as a pure illusion, and attributes the curved paths of projectiles of all kinds to their efforts to keep a straight track through a curved space. This curved space is not, it is true, the ordinary space of the astronomer. It is a purely mathematical and probably wholly fictitious space, in which the astronomers’ space and the astronomers’ time are inextricably bound together and enter as equal partners. To be absolutely exact, there are four equal partners. The first three are the three dimensions of ordinary space—breadth, width, and height, or, if we prefer, north-south, east-west and up-down. The fourth is ordinary time measured in a way appropriate to the way in which we have measured our space (a year of time corresponding to a light-year of space, and so on), and then multiplied by the square-root of -1. This last multiplication by the square-root of -1 is of course the remarkable feature of the whole affair. For the square-root of -1 has no real existence; it is what the mathematician describes as an “imaginary” number. No real number can be multiplied by itself and give -1 as the product. Yet it is only when time is measured in terms of an imaginary unit of √-1 years that there is true equal partnership between space and time. This shews that the equal partnership is purely formal—it is nothing but a convenient fiction of the mathematician. Indeed had it been anything more, our intuitive conviction that time is something essentially different from space could have had no basis in experience and so would have vanished ere now.

These complications with respect to time need not concern us here; the essential point is that Einstein’s theory of relativity teaches that space ultimately bends back on itself like the earth’s surface, so that the total amount of space is finite.

THE COSMOLOGY OF EINSTEIN. According to Einstein’s original theory, the dimensions of space are determined by the amount of matter it contains. The more matter there is, the smaller space must be, and conversely; space could only be of literally infinite extent if it contained no matter at all. The problem of determining the extent of space accordingly reduces to that of determining how much matter it contains. We have no means of estimating how much matter may exist outside those regions of space which are within the reach of our telescopes, but within these regions matter seems to be fairly uniformly distributed in the form of extra-galactic nebulae.

From the known weights of these, Hubble estimates that the mean density of matter in space must be about 1·5 × 10⁻³¹ times that of water. On the assumption that matter is distributed with this density through the whole of space, including those parts which our telescopes have not yet penetrated, we can calculate quite definitely that the radius of space is 84,000 million light-years, or 600 times the distance of the furthest visible nebula. The journey round space would take 500,000 million light-years, and if ever our telescopes shew us the solar system from behind, we shall see it as it was 500,000 million years ago.

Thus, according to Einstein’s original theory, even the 140 million light-years through which we can range with our telescopes form only a small fraction of the whole of space—something like one part in a thousand million. There is plenty of space still awaiting exploration. It is perhaps not surprising. Mankind, who has been possessed of telescopes for only 300 years out of the 300,000 of his residence on earth, could hardly hope to discover the whole of space in so short a time. Our astronomer-explorers are moving from island to island in the small archipelago which surrounds their home in space, but they are still far from circumnavigating the globe. And, just as the earliest geographers tried to estimate the size of the earth, long before they thought of circumnavigating it, from the curvature of a small part of its surface, so astronomers are now trying to form estimates, although necessarily vague, of the size of the whole universe from the curvature of that part of it with which they are already acquainted.

The general theory of relativity has long passed the stage of being regarded as an interesting speculation. It not only accounts for phenomena of planetary motion before which Newton’s law of gravitation failed, but it has predicted other phenomena—the apparent displacements of stars near the sun at an eclipse, resulting from the light by which we see them being bent as it passes through the sun’s gravitational field, and a certain displacement of stellar spectra towards the red end—which were entirely unsuspected when the predictions were first made, but have subsequently been fully confirmed by observation. Indeed the theory has by now qualified as one of the ordinary working tools of astronomy. It has been used to measure the diameter of the small faint star Sirius B, the companion to Sirius (p. 262), as well as to discuss the nature of the stars at the centres of the “planetary nebulae” (p. 323).

Nevertheless, the general theory of relativity does not lead up to Einstein’s cosmology in a unique way. It is perfectly possible for the former to be true and the latter false. The general theory of relativity fixes the attributes of any small fraction of the universe quite definitely, but leaves open several alternative ways in which these small fractions can be pieced together to form a whole. Einstein’s particular view of the cosmos cannot therefore claim the prestige which attaches to the general theory of relativity as a whole. And indeed for some years it fell somewhat into disfavour, and appeared likely to be superseded by an alternative cosmology which de Sitter of Leiden propounded and developed in some detail in 1917.

THE COSMOLOGY OF DE SITTER. Let us first try to understand the essential differences between these two cosmologies.

Einstein’s cosmology supposes that the size of the cosmos is determined by the amount of matter it contains. If it was decided, at the creation, to create a universe containing a certain amount of matter which was to obey certain natural laws, then space must at once have adjusted itself to the size suited for containing just this amount of matter and no more. Or, if the size of the universe and the natural laws were decided upon, the creation of a certain definite amount of matter became an inevitable necessity. De Sitter’s universe is less simple, or, if we prefer so to put it, allowed more freedom of choice in its creation. After the laws of nature had been fixed, it was still possible to make a universe of any size, and to put any amount of matter, within limits, into it. Looked at from the strictly scientific point of view, Einstein’s universe has one element of arbitrariness fewer than de Sitter’s universe, and to this extent it has the advantage of simplicity.

On the other hand this simplicity is acquired at a price. The fundamental corner-stone of the whole theory of relativity is the equal partnership of space and time in the sense already explained. Einstein’s cosmology gains its simplicity only at the expense of supposing that this equality of partnership disappears when we view the cosmos as a whole. It supposes that space and time are indistinguishable (in the purely formal sense already indicated) only to a being whose experience is limited to a small fraction of the universe; they become utterly distinct for a being who can range through the whole of space and time. It is not altogether clear how much weight ought to be attached to this objection, if objection it is. Real space and real time undoubtedly are distinct. Even if we deny the reality of both, they still remain distinguishable as modes of perception. What reproach, then, can it be to a cosmology that it admits that, in the last resort, when the universe is contemplated on the grand scale, space and time resolve themselves into distinct types of entity? Somehow we knew it already, before ever we began to contemplate the universe on the grand scale.

Whatever the answer to this last question may be, de Sitter’s cosmology avoids all possible reproach by maintaining the equal partnership of space and time, not only in individual fractions of the cosmos, but throughout the cosmos as a whole. It will of course be understood that we are still speaking of equal partnership in the purely formal sense already explained, a light-year entering the cosmology on the same footing as the square-root of -1 years. Even de Sitter’s cosmology does not pretend that a light-year (9·46 million million kilometres) is the same thing as twelve months.

Although Einstein’s main theory of relativity has been amply confirmed by observation, the cosmological part of it did not predict any special features such as permitted of a direct observational test. De Sitter’s cosmology, on the other hand, predicted that the spectra of all distant objects must shew a displacement towards the red, of amount depending on the distance of the object. The equal partnership of space and time results in the vibrations of the light-waves emitted by any specified source being slower in distant than in near parts of the universe; the stream of time rolls more rapidly just where we happen to be than anywhere else. This sounds paradoxical at first, but examination shews that it is not; de Sitter is not asking us to return to a geocentric universe, because he shews that the inhabitant of a distant star would also find that terrestrial atoms were keeping slower time than his own. The paradox is completely resolved by the concept of the relativity of all measures of space and time.

This displacement to the red as a result of mere distance is peculiar to de Sitter’s cosmology. It is additional to the displacement which, as all cosmologies agree, the spectrum of a moving body must shew as the result of its motion, this latter being towards the red only if the body is receding from the earth (p. 51). On de Sitter’s cosmology, the two displacements are not entirely independent, for it is an essential feature of this cosmology that near bodies should tend to move further apart from one another. Just as bits of straw thrown together into a stream tend to get separated as they float down the stream, so objects in de Sitter’s universe move further apart as they float down the stream of time.

Thus on de Sitter’s theory a displacement of spectral lines to the red cannot be interpreted as evidence either of motion or of distance; it is a mixture of both. This does not mean that we have been altogether wrong in deducing the velocities of stars in the galactic system from the observed displacements of their spectral lines. No appreciable displacement is produced by distance alone, unless this distance forms an appreciable fraction of the radius of the universe. Systematic displacements to the red are, it is true, observed in the spectra of the most distant stars, but they are of very small amount. It is only when we look to the remote extra-galactic nebulae that we can expect to observe the effect in appreciable strength.

Now it has long been one of the outstanding puzzles of astronomy that the spectra of the distant nebulae are uniformly displaced towards the red. The observed displacements are not small. Interpreted as velocities, many of them would represent speeds of over 1000 miles a second. Humason has found that two faint nebulae N.G.C. 4860 and 4853 shew apparent speeds of recession of 4900 and 4600 miles a second respectively. They are both members of a cloud of nebulae in Coma Berenices (p. 72), which is probably at a distance of about 50 million light-years. Quite recently the spectrum of another very faint nebula in Ursa Major has been found to indicate an apparent motion of recession with a speed of 7200 miles a second. If de Sitter’s theory is rejected, almost all the extra-galactic nebulae must be running away from us with terrific, almost unimaginable, speeds. And the further away they are the more precipitately they are increasing their distance. Yet we can hardly reintroduce simplicity by adopting de Sitter’s theory, and treating the whole apparent stampede of nebulae as spurious, since this theory involves that the nebulae may well, in actual fact, be running away from us, scattering being an inherent property of objects in a de Sitter universe.

The fact that the spectra of the most distant nebulae shew these large displacements provides a certain presumption in favour of the truth of de Sitter’s cosmology; this at least explains them twice over, while no other cosmology explains them at all. If we tentatively accept this cosmology, then each observed spectral shift must be regarded as the sum of two parts, one arising in the ordinary way from a recession of the nebula, and the other arising merely from its distance.

A preliminary study by Dr Hubble has shewn that on the whole the spectral displacements are largest for the most distant nebulae, and that their amounts are roughly proportional to the distances of the nebulae from us. If we interpret the whole of the observed displacements purely as evidence of recession, we can calculate that the radius of the universe is about 2000 million light-years, or some fourteen times the distance of the furthest visible nebula. With so large a radius of the universe, the further displacement resulting from the mere distance of even remote nebulae is negligible, so that our assumption that the displacements arise almost entirely from velocities of recession receives à posteriori vindication. If the observed displacements of the nebular spectra had been strictly proportional to their distances from us, we should have obtained a consistent explanation of the observed facts by assuming that we lived in a de Sitter universe having a radius of about 2000 million light-years.

THE EXPANDING UNIVERSE. It used to be thought that the cosmologies of Einstein and de Sitter were antagonistic to one another, since obviously no one universe could be an Einstein universe and a de Sitter universe at the same time. A recent investigation by a Belgian mathematician, the Abbé G. Lemaître, has put a different complexion on the problem. In brief Lemaître has shewn that no universe could stay permanently in the state considered by Einstein. A universe in this state is an unstable structure; immediately it came into being it would start to expand, and would not cease from expanding until it had become a de Sitter universe. Even after this the expansion would continue, but it would now become merged in the normal expansion of the de Sitter universe, such as we have already considered.

In the light of these results, the question at issue is not whether our universe is an Einstein universe or a de Sitter universe, but rather how far it has travelled along the road which begins with an Einstein universe and ends with a de Sitter universe. Whatever the answer may be, we are led to suppose that at the beginning of time the nebulae were much nearer to one another than they now are, that ever since then they have obeyed their inherent tendency to scatter, or rather the tendency of the flowing stream of time to scatter them, and that they are now moving away from us, and from one another, with speeds which are proportional to their distances. This is in accordance with Hubble’s conclusion, that the apparent speeds of recession of the nebulae are roughly proportional to their distances. From Hubble’s data Eddington has calculated that the original Einstein universe must have had a radius of about 1200 million light-years. If the apparent speeds of recession of the nebulae had been found to be strictly proportional to their distances, we could have explained everything by supposing that we lived in an expanding universe, which had started as an Einstein universe of 1200 million light-years’ radius, had now expanded to something of the order of 2000 million light-years’ radius, and was destined to go on expanding to all eternity.

This provides a simple and rather fascinating picture of the universe, but there are many reasons against supposing that it is a true one. In the first place, if we interpret the spectral displacements as evidence of velocity alone, the speeds of the nebulae are probably very far from being (as the foregoing picture would require) accurately proportional to their distances from us. A group of three nebulae, all believed to be at the same distance of about 50 million light-years, differ by nearly 2000 miles a second in their speeds, which average about 5000 miles a second. Oort has found a general tendency for the speeds of very distant nebulae not to be strictly proportional to their distances. At from 20 to 40 million light-years the apparent divergences average 750 miles a second, but it is not clear how far these result merely from inaccurate estimates of the distances of these remote nebulae.

In the second place, if the observed spectral shifts represent mere scattering, we can calculate the time since this scattering began; it proves to be many thousands of millions of years. Enormous though such a length of time is, it does not appear to be enough. We shall see later (Chapter III) how time leaves its mark, its wrinkles and its grey hairs, on the stars, so that we can guess their ages tolerably well, and the evidence is all in favour of stellar lives, not of thousands of millions, but of millions of millions, of years. If the nebulae owe their present motions to mere scattering, then the stars must have lived the greater parts of their lives before this scattering began. Such a hypothesis seems too artificial for acceptance, at any rate so long as any alternative is open.

Of course we must frankly admit that our estimates of stellar ages may be found to need revision. Indeed they have been calculated on the supposition that no appreciable scattering of the type required by de Sitter’s cosmology has ever taken place. If not only the nebulae, but also the stars composing the galactic system, were huddled together at the beginning of time, our estimates of the lives of the stars would have to be substantially shortened, and it is conceivable, although I think very unlikely, that they could be reduced to lengths of the kind we have just considered; they certainly could not be so reduced if the original universe had a radius anywhere near to Eddington’s estimate of 1200 million light-years.

Nevertheless the cosmologies of de Sitter and Lemaître undoubtedly require that the present-day universe must be expanding, that the nebulae must be retreating both from one another, and from us, and that their spectra must, as a consequence, shew displacements to the red. But there is nothing to compel us to identify these displacements with those actually observed. Many causes, other than motions of recession, are capable of reddening spectral lines, and only after we have deducted all the reddening due to these various other causes shall we be in a position to say that the residue represents a true motion of recession.

Quite recently Dr Zwicky of Pasadena has suggested that gravitating matter diffused through space may redden all light which passes through the space. He gives dynamical reasons for his suggestion, calculates the amount of spectral shift to be anticipated in the light reaching us from the nebulae, and finds that it would account for practically the whole of the observed shift. It is possible that the greater part of the observed reddening of nebular light may be due to this or some similar cause, only a small fraction representing true motions of recession. If so, we can extend the age of the universe indefinitely, and are free to assign to the stars the very great ages which the evidence of general astronomy seems to demand. This we shall discuss fully in Chapter III.

In de Sitter’s original form of this cosmology, light would take an infinite time to travel round the universe, and this would prevent any object being seen by light which had travelled the long way round. This results from de Sitter having considered only the ideal case of a universe entirely empty of all matter. With even a little matter in the universe, the path of a ray of light would presumably bend back on itself and return to its starting point after a finite time. It has been quite seriously suggested that two faint nebulae (h 3433 and M 83) may actually be our two nearest neighbours in the sky, M 33 and M 31, seen the long way round space. If so, we see the fronts of two objects when we look at M 33 and M 31, and the backs of the same two objects when we point our telescopes in exactly the opposite directions and look at h 3433 and M 83. No doubt this is only a conjecture, and perhaps rather a wild one, but many more startling conjectures have been made in astronomy, and subsequently proved to be true.

All these discussions of the structure of space are of course highly speculative, but they agree in suggesting the general conclusion that, if we cannot yet see the whole of space, we can at least survey an appreciable fraction of it. Our astronomer-explorers may not as yet have circumnavigated the globe, but they are perhaps discovering America, and we can well imagine that even the next generation will have completed the circumnavigation of space, and will think of a finite but unbounded space in the same way, and with the same ease, as we think of the finite but unbounded surface of the earth.

A MODEL OF THE UNIVERSE

We found it difficult enough to visualise the 4¼ light-years which constitute the distance to the nearest star, so we may be well advised not even to attempt to visualise this last distance of thousands of millions of light-years, the conjectured circumference of the universe. Yet we may try to see all these distances in proper proportion relative to one another by the help of a model drawn to scale. We can escape the effort of trying to imagine unimaginably great distances by keeping the scale very small.

The earth, travelling 1200 times faster than an express train, makes a journey of 600 million miles around the sun every year. Let us represent this journey by a pin-head ¹/₁₆ of an inch in diameter. This fixes the scale of our model; the sun has shrunk to a minute speck of dust ¹/₃₄₀₀ of an inch in diameter, while the earth is a still more minute speck which is too small to be seen at all even in the most powerful of microscopes. On this scale the nearest star in the sky, Proxima Centauri, must be placed about 225 yards away, and to contain even the hundred stars nearest to our sun in space, the model must be a mile high, a mile long and a mile wide.

Let us go on building the model. We may think of stars indiscriminately as specks of dust, because their sizes vary about as much as the sizes of specks of dust. In the vicinity of the sun we must place specks of dust at average distances of about a quarter of a mile apart. In other regions of space they are generally even farther apart, for, owing to the presence of the “local cluster,” the immediate neighbourhood of the sun happens to be a rather crowded part of the sky. We go on building the model for hundreds of miles in every direction, and then, if we are building in a direction well away from the galactic plane, the specks of dust begin to thin out; we are approaching the confines of the galaxy. In the galactic plane itself we build out for about 7000 miles before we come to the farthest globular cluster, and still we are inside the galactic system. With our earth’s long yearly journey round the sun as a pin-head the whole galactic system is about the size of the American continent. It may be well to pause and try to visualise the relative sizes of a pin-head and of the American continent, before we go on with our mental model-building.

After we have finished the galactic system, we must travel about 30,000 miles before we begin to set up the next bit of our model, at any rate if we are keeping it to scale. At this distance we place the next family of stars, a family which is probably substantially smaller and more compact than our own galactic family, but is comparable with it both in size and in numbers. So we go on building our model—a family of thousands of millions of stars every 30,000 miles or so—until we have two million such families. The model now stretches for about four million miles in every direction. This represents as far as we can see into space with a telescope; we can imagine the model going on, although we know not how nor where—all we know is that the part so far built represents only a fraction of the universe.

Every galactic system or island universe or extra-galactic nebula contains thousands of millions of stars, or gaseous matter destined ultimately to form thousands of millions of stars, and we know of two million such systems. There are, then, thousands of millions of millions of stars within the range covered by the 100-inch telescope, and this number must be further multiplied to allow for the parts of the universe which are still unexplored. At a moderate computation, the total number of stars in the universe must be something like the total number of specks of dust in London. Think of the sun as something less than a single speck of dust in a vast city, of the earth as less than a millionth part of such a speck of dust, and we have perhaps as vivid a picture as the mind can really grasp of the relation of our home in space to the rest of the universe.

An alternative procedure would have been to construct our scale-model by taking all the specks of dust in London and spreading them out to the right distances to represent the various stars in space. The average actual distances between specks of dust in London is a quite small fraction of an inch; to get our model to correct scale, this distance must be increased to about a quarter of a mile, even when we are building the part which represents the crowded part of space round the sun. If we build our model in this way, we obtain a vivid picture of the emptiness of space. Empty Waterloo Station of everything except six specks of dust, and it is still far more crowded with dust than space is with stars. This is true even of the comparatively crowded region inside the galactic system; it takes no account of the immense empty stretches between one system of stars and the next. On averaging throughout the whole of the model, the mean distance of a speck of dust from its nearest neighbour proves to be something like 80 miles. The universe consists in the main not of stars but of desolate emptiness—inconceivably vast stretches of desert space in which the presence of a star is a rare and exceptional event.

Let us in imagination take up a position in space somewhere near the sun, and watch the stars moving past with speeds about 1000 times that of an express train. If space were really crowded with stars our position would be as unenviable as if we sat down in the middle of Regent Street to watch the traffic go by—our life, though thrilling, would be brief. Yet, as exact calculation shews, the stellar traffic is so little crowded that we would have to wait about a million million million years before a star ran into us. Put in another form, the calculation shews that any one star may expect to move for something of the order of a million million million years before colliding with a second star. The stars move blindly through space, and the players in the stellar blind-man’s-buff are so few and far between that the chance of encountering another star is almost negligible. We shall see later that this concept is of the profoundest significance in our interpretation of the universe.