The Story of the Heavens

by Robert S. Ball

THE ASTRONOMICAL SIGNIFICANCE OF HEAT.

Heat and Astronomy—Distribution of Heat—The Presence of Heat in the Earth—Heat in other Celestial Bodies—Varieties of Temperature—The Law of Cooling—The Heat of the Sun—Can its Temperature be Measured?—Radiation connected with the Sun's Bulk—Can the Sun be Exhausting his Resources?—No marked Change has occurred—Geological Evidence as to the Changes of the Sun's Heat Doubtful—The Cooling of the Sun—The Sun cannot be merely an Incandescent Solid Cooling—Combustion will not Explain the Matter—Some Heat is obtained from Meteoric Matter, but this is not Adequate to the Maintenance of the Sun's Heat—The Contraction of a Heated Globe of Gas—An Apparent Paradox—The Doctrine of Energy—The Nebular Theory—Evidence in Support of this Theory—Sidereal Evidence of the Nebular Theory—Herschel's View of Sidereal Aggregation—The Nebulæ do not Exhibit Changes within the Limits of our Observation.

That a portion of a work on astronomy should bear the title placed at the head of this chapter will perhaps strike some of our readers as unusual, if not actually inappropriate. Is not heat, it may be said, a question merely of experimental physics? and how can it be legitimately introduced into a treatise upon the heavenly bodies and their movements? Whatever weight such objections might have once had need not now be considered. The recent researches on heat have shown not only that heat has important bearings on astronomy, but that it has really been one of the chief agents by which the universe has been moulded into its actual form. At the present time no work on astronomy could be complete without some account of the remarkable connection between the laws of heat and the astronomical consequences which follow from those laws.

In discussing the planetary motions and the laws of Kepler, or in discussing the movements of the moon, the proper motions of the stars, or the revolutions of the binary stars, we proceed on the supposition that the bodies we are dealing with are rigid particles, and the question as to whether these particles are hot or cold does not seem to have any especial bearing. No doubt the ordinary periodic phenomena of our system, such as the revolution of the planets in conformity with Kepler's laws, will be observed for countless ages, whether the planets be hot or cold, or whatever may be the heat of the sun. It must, however, be admitted that the laws of heat introduce certain modifications into the statement of these laws. The effects of heat may not be immediately perceptible, but they exist—they are constantly acting; and in the progress of time they are adequate to effecting the mightiest changes throughout the universe.

Let us briefly recapitulate the circumstances of our system which give to heat its potency. Look first at our earth, which at present seems—on its surface, at all events—to be a body devoid of internal heat; a closer examination will dispel this idea. Have we not the phenomena of volcanoes, of geysers, and of hot springs, which show that in the interior of the earth heat must exist in far greater intensity than we find on the surface? These phenomena are found in widely different regions of the earth. Their origin is, no doubt, involved in a good deal of obscurity, but yet no one can deny that they indicate vast reservoirs of heat. It would indeed seem that heat is to be found everywhere in the deep inner regions of the earth. If we take a thermometer down a deep mine, we find it records a temperature higher than at the surface. The deeper we descend the higher is the temperature; and if the same rate of progress should be maintained through those depths of the earth which we are not able to penetrate, it can be demonstrated that at twenty or thirty miles below the surface the temperature must be as great as that of red-hot iron.

We find in the other celestial bodies abundant evidence of the present or the past existence of heat. Our moon, as we have already mentioned, affords a very striking instance of a body which must once have been very highly heated. The extraordinary volcanoes on its surface place this beyond any doubt. It is equally true that those volcanoes have been silent for ages, so that, whatever may be the interior condition of the moon, the surface has now cooled down. Extending our view further, we see in the great planets Jupiter and Saturn evidence that they are still endowed with a temperature far in excess of that which the earth has retained; while, when we look at our sun, we see a body in a state of brilliant incandescence, and glowing with a fervour to which we cannot approximate in our mightiest furnaces. The various fixed stars are bodies which glow with heat, like our sun; while we have in the nebulæ objects the existence of which is hardly intelligible to us, unless we admit that they are possessed of heat.

From this rapid survey of the different bodies in our universe one conclusion is obvious. We may have great doubts as to the actual temperature of any individual body of the system; but it cannot be doubted that there is a wide range of temperature among the different bodies. Some are hotter than others. The stars and suns are perhaps the hottest of all; but it is not improbable that they may be immeasurably outnumbered by the cold and dark bodies of the universe, which are to us invisible, and only manifest their existence in an indirect and casual manner.

The law of cooling tells us that every body radiates heat, and that the quantity of heat which it radiates increases when the temperature of the body increases relatively to the surrounding medium. This law appears to be universal. It is obeyed on the earth, and it would seem that it must be equally obeyed by every other body in space. We thus see that each of the planets and each of the stars is continuously pouring forth in all directions a never-ceasing stream of heat. This radiation of heat is productive of very momentous consequences. Let us study them, for instance, in the case of the sun.

Our great luminary emits an incessant flood of radiant heat in all directions. A minute fraction of that heat is intercepted by our earth, and is, directly or indirectly, the source of all life, and of nearly all movement, on our earth. To pour forth heat as the sun does, it is necessary that his temperature be enormously high. And there are some facts which permit us to form an estimate of what that temperature must actually be.

It is difficult to form any numerical statement of the actual temperature of the sun. The intensity of that temperature vastly transcends the greatest artificial heat, and any attempt to clothe such estimates in figures is necessarily very precarious. But assuming the greatest artificial temperature to be about 4,000° Fahr., we shall probably be well within the truth if we state the effective temperature of the sun to be about 14,000° Fahr. This is the result of a recent investigation by Messrs. Wilson and Gray, which seems to be entitled to considerable weight.

The copious outflow of heat from the sun corresponds with its enormous temperature. We can express the amount of heat in various ways, but it must be remembered that considerable uncertainty still attaches to such measurements. The old method of measuring heat by the quantity of ice melted may be used as an illustration. It is computed that a shell of ice 43-1⁄2 feet thick surrounding the whole sun would in one minute be melted by the sun's heat underneath. A somewhat more elegant illustration was also given by Sir John Herschel, who showed that if a cylindrical glacier 45 miles in diameter were to be continually flowing into the sun with the velocity of light, the end of that glacier would be melted as quickly as it advanced. From each square foot in the surface of the sun emerges a quantity of heat as great as could be produced by the daily combustion of sixteen tons of coal. This is, indeed, an amount of heat which, properly transformed into work, would keep an engine of many hundreds of horse-power running from one year's end to the other. The heat radiated from a few acres on the sun would be adequate to drive all the steam engines in the world. When we reflect on the vast intensity of the radiation from each square foot of the sun's surface, and when we combine with this the stupendous dimensions of the sun, imagination fails to realise how vast must be the actual expenditure of heat.

In presence of the prodigal expenditure of the sun's heat, we are tempted to ask a question which has the most vital interest for the earth and its inhabitants. We live from hour to hour by the sun's splendid generosity; and, therefore, it is important for us to know what security we possess for the continuance of his favours. When we witness the terrific disbursement of the sun's heat each hour, we are compelled to ask whether our great luminary may not be exhausting its resources; and if so, what are the prospects of the future? This question we can partly answer. The whole subject is indeed of surpassing interest, and redolent with the spirit of modern scientific thought.

Our first attempt to examine this question must lie in an appeal to the facts which are attainable. We want to know whether the sun is showing any symptoms of decay. Are the days as warm and as bright now as they were last year, ten years ago, one hundred years ago? We can find no evidence of any change since the beginning of authentic records. If the sun's heat had perceptibly changed within the last two thousand years, we should expect to find corresponding changes in the distribution of plants and of animals; but no such changes have been detected. There is no reason to think that the climate of ancient Greece or of ancient Rome was appreciably different from the climates of the Greece and the Rome that we know at this day. The vine and the olive grow now where they grew two thousand years ago.

We must not, however, lay too much stress on this argument; for the effects of slight changes in the sun's heat may have been neutralised by corresponding adaptations in the pliable organisms of cultivated plants. All we can certainly conclude is that no marked change has taken place in the heat of the sun during historical time. But when we come to look back into much earlier ages, we find copious evidence that the earth has undergone great changes in climate. Geological records can on this question hardly be misinterpreted. Yet it is curious to note that these changes are hardly such as could arise from the gradual exhaustion of the sun's radiation. No doubt, in very early times we have evidence that the earth's climate must have been much warmer than at present. We had the great carboniferous period, when the temperature must almost have been tropical in Arctic latitudes. Yet it is hardly possible to cite this as evidence that the sun was then much more powerful; for we are immediately reminded of the glacial period, when our temperate zones were overlaid by sheets of solid ice, as Northern Greenland is at present. If we suppose the sun to have been hotter than it is at present to account for the vegetation which produced coal, then we ought to assume the sun to be colder than it is now to account for the glacial period. It is not reasonable to attribute such phenomena to fluctuations in the radiation from the sun. The glacial periods prove that we cannot appeal to geology in aid of the doctrine that a secular cooling of the sun is now in progress. The geological variations of climate may have been caused by changes in the earth itself, or by changes in its actual orbit; but however they have been caused, they hardly tell us much with regard to the past history of our sun.

The heat of the sun has lasted countless ages; yet we cannot credit the sun with the power of actually creating heat. We must apply to the tremendous mass of the sun the same laws which we have found by our experiments on the earth. We must ask, whence comes the heat sufficient to supply this lavish outgoing? Let us briefly recount the various suppositions that have been made.

Place two red-hot spheres of iron side by side, a large one and a small one. They have been taken from the same fire; they were both equally hot; they are both cooling, but the small sphere cools more rapidly. It speedily becomes dark, while the large sphere is still glowing, and would continue to do so for some minutes. The larger the sphere, the longer it will take to cool; and hence it has been supposed that a mighty sphere of the prodigious dimensions of our sun would, if once heated, cool gradually, but the duration of the cooling would be so long that for thousands and for millions of years it could continue to be a source of light and heat to the revolving system of planets. This suggestion will not bear the test of arithmetic. If the sun had no source of heat beyond that indicated by its high temperature, we can show that radiation would cool the sun a few degrees every year. Two thousand years would then witness a very great decrease in the sun's heat. We are certain that no such decrease can have taken place. The source of the sun's radiation cannot be found in the mere cooling of an incandescent mass.

Can the fires in the sun be maintained by combustion, analogous to that which goes on in our furnaces? Here we would seem to have a source of gigantic heat; but arithmetic also disposes of this supposition. We know that if the sun were made of even solid coal itself, and if that coal were burning in pure oxygen, the heat that could be produced would only suffice for 6,000 years. If the sun which shone upon the builders of the great Pyramid had been solid coal from surface to centre, it must by this time have been in great part burned away in the attempt to maintain its present rate of expenditure. We are thus forced to look to other sources for the supply of the sun's heat, since neither the heat of incandescence nor the heat of combustion will suffice.

There is probably—indeed, we may say certainly—one external source from which the heat of the sun is recruited. It will be necessary for us to consider this source with some care, though I think we shall find it to be merely an auxiliary of comparatively trifling moment. According to this view, the solar heat receives occasional accessions from the fall upon the sun's surface of masses of meteoric matter. There can be hardly a doubt that such masses do fall upon the sun; there is certainly no doubt that if they do, the sun must gain some heat thereby. We have experience on the earth of a very interesting kind, which illustrates the development of heat by meteoric matter. There lies a world of philosophy in a shooting star. Some of these myriad objects rush into our atmosphere and are lost; others, no doubt, rush into the sun with the same result. We also admit that the descent of a shooting star into the atmosphere of the sun must be attended with a flash of light and of heat. The heat acquired by the earth from the flashing of the shooting stars through our air is quite insensible. It has been supposed, however, that the heat accruing to the sun from the same cause may be quite sensible—nay, it has been even supposed that the sun may be re-invigorated from this source.

Here, again, we must apply the cold principles of weights and measures to estimate the plausibility of this suggestion. We first calculate the actual weight of meteoric indraught to the sun which would be adequate to sustain the fires of the sun at their present vigour. The mass of matter that would be required is so enormous that we cannot usefully express it by imperial weights; we must deal with masses of imposing magnitude. It fortunately happens that the weight of our moon is a convenient unit. Conceive that our moon—a huge globe, 2,000 miles in diameter—were crushed into a myriad of fragments, and that these fragments were allowed to rain in on the sun; there can be no doubt that this tremendous meteoric shower would contribute to the sun rather more heat than would be required to supply his radiation for a whole year. If we take our earth itself, conceive it comminuted into dust, and allow that dust to fall on the sun as a mighty shower, each fragment would instantly give out a quantity of heat, and the whole would add to the sun a supply of heat adequate to sustain the present rate of radiation for nearly one hundred years. The mighty mass of Jupiter treated in the same way would generate a meteoric display greater in the ratio in which the mass of Jupiter exceeds the mass of earth. Were Jupiter to fall into the sun, enough heat would be thereby produced to scorch the whole solar system; while all the planets together would be capable of producing heat which, if properly economised, would supply the radiation of the sun for 45,000 years.

It must be remembered that though the moon could supply one year's heat, and Jupiter 30,000 years' heat, yet the practical question is not whether the solar system could supply the sun's heat, but whether it does. Is it likely that meteors equal in mass to the moon fall into the sun every year? This is the real question, and I think we are bound to reply to it in the negative. It can be shown that the quantity of meteors which could be caught by the sun in any one year can be only an excessively minute fraction of the total amount. If, therefore, a moon-weight of meteors were caught every year, there must be an incredible mass of meteoric matter roaming at large through the system. There must be so many meteors that the earth would be incessantly pelted with them, and heated to such a degree as to be rendered uninhabitable. There are also other reasons which preclude the supposition that a stupendous quantity of meteoric matter exists in the vicinity of the sun. Such matter would produce an appreciable effect on the movement of the planet Mercury. There are, no doubt, some irregularities in the movements of Mercury not yet fully explained, but these irregularities are very much less than would be the case if meteoric matter existed in quantity adequate to the sustentation of the sun. Astronomers, then, believe that though meteors may provide a rate in aid of the sun's current expenditure, yet that the greater portion of that expenditure must be defrayed from other resources.

It is one of the achievements of modern science to have effected the solution of the problem—to have shown how it is that, notwithstanding the stupendous radiation, the sun still maintains its temperature. The question is not free from difficulty in its exposition, but the matter is one of such very great importance that we are compelled to make the attempt.

Let us imagine a vast globe of heated gas in space. This is not an entirely gratuitous supposition, inasmuch as there are globes apparently of this character; they have been already alluded to as planetary nebulæ. This globe will radiate heat, and we shall suppose that it emits more heat than it receives from the radiation of other bodies. The globe will accordingly lose heat, or what is equivalent thereto, but it will be incorrect to assume that the globe will necessarily fall in temperature. That the contrary is, indeed, the case is a result almost paradoxical at the first glance; but yet it can be shown to be a necessary consequence of the laws of heat and of gases.

Let us fix our attention on a portion of the gas lying on the surface of the globe. This is, of course, attracted by all the rest of the globe, and thus tends in towards the centre of the globe. If equilibrium subsists, this tendency must be neutralised by the pressure of the gas beneath; so that the greater the gravitation, the greater is the pressure. When the globe of gas loses heat by radiation, let us suppose that it grows colder—that its temperature accordingly falls; then, since the pressure of a gas decreases when the temperature falls, the pressure beneath the superficial layer of the gas will decrease, while the gravitation is unaltered. The consequence will inevitably be that the gravitation will now conquer the pressure, and the globe of gas will accordingly contract. There is, however, another way in which we can look at the matter. We know that heat is equivalent to energy, so that when the globe radiates forth heat, it must expend energy. A part of the energy of the globe will be due to its temperature; but another, and in some respects a more important, part is that due to the separation of its particles. If we allow the particles to come closer together we shall diminish the energy due to separation, and the energy thus set free can take the form of heat. But this drawing in of the particles necessarily involves a shrinking of the globe.

And now for the remarkable consequence, which seems to have a very important application in astronomy. As the globe contracts, a part of its energy of separation is changed into heat; that heat is partly radiated away, but not so rapidly as it is produced by the contraction. The consequence is, that although the globe is really losing heat and really contracting, yet that its temperature is actually rising.[43] A simple case will suffice to demonstrate this result, paradoxical as it may at first seem. Let us suppose that by contraction of the sphere it had diminished to one-half its diameter; and let us fix our attention on a cubic inch of the gaseous matter in any point of the mass. After the contraction has taken place each edge of the cube would be reduced to half an inch, and the volume would therefore be reduced to one-eighth part of its original amount. The law of gases tells us that if the temperature be unaltered the pressure varies inversely as the volume, and consequently the internal pressure in the cube would in that case be increased eightfold. As, however, in the case before us, the distance between every two particles is reduced to one-half, it will follow that the gravitation between every two particles is increased fourfold, and as the area is also reduced to one-fourth, it will follow that the pressure inside the reduced cube is increased sixteenfold; but we have already seen that with a constant temperature it only increases eightfold, and hence the temperature cannot be constant, but must rise with the contraction.

We thus have the somewhat astonishing result that a gaseous globe in space radiating heat, and thereby growing smaller, is all the time actually increasing in temperature. But, it may be said, surely this cannot go on for ever. Are we to suppose that the gaseous mass will go on contracting and contracting with a temperature ever fiercer and fiercer, and actually radiating out more and more heat the more it loses? Where lies the limit to such a prospect? As the body contracts, its density must increase, until it either becomes a liquid, or a solid, or, at any rate, until it ceases to obey the laws of a purely gaseous body which we have supposed. Once these laws cease to be observed the argument disappears; the loss of heat may then really be attended with a loss of temperature, until in the course of time the body has sunk to the temperature of space itself.

It is not assumed that this reasoning can be applied in all its completeness to the present state of the sun. The sun's density is now so great that the laws of gases cannot be there strictly followed. There is, however, good reason to believe that the sun was once more gaseous than at present; possibly at one time he may have been quite gaseous enough to admit of this reasoning in all its fulness. At present the sun appears to be in some intermediate stage of its progress from the gaseous condition to the solid condition. We cannot, therefore, say that the temperature of the sun is now increasing in correspondence with the process of contraction. This may be true or it may not be true; we have no means of deciding the point. We may, however, feel certain that the sun is still sufficiently gaseous to experience in some degree the rise of temperature associated with the contraction. That rise in temperature may be partly or wholly obscured by the fall in temperature which would be the more obvious consequence of the radiation of heat from the partially solid body. It will, however, be manifest that the cooling of the sun may be enormously protracted if the fall of temperature from the one cause be nearly compensated by the rise of temperature from the other. It can hardly be doubted that in this we find the real explanation of the fact that we have no historical evidence of any appreciable alteration in the radiation of heat from the sun.

This question is one of such interest that it may be worth while to look at it from a slightly different point of view. The sun contains a certain store of energy, part of which is continually disappearing in the form of radiant heat. The energy remaining in the sun is partly transformed in character; some of it is transformed into heat, which goes wholly or partly to supply the loss by radiation. The total energy of the sun must, however, be decreasing; and hence it would seem the sun must at some time or other have its energy exhausted, and cease to be a source of light and of heat. It is true that the rate at which the sun contracts is very slow. We are, indeed, not able to measure with certainty the decrease in the sun's bulk. It is a quantity so minute, that the contraction since the birth of accurate astronomy is not large enough to be perceptible in our telescopes. It is, however, possible to compute what the contraction of the sun's bulk must be, on the supposition that the energy lost by that contraction just suffices to supply the daily radiation of heat. The change is very small when we consider the present size of the sun. At the present time the sun's diameter is about 860,000 miles. If each year this diameter decreases by about 300 feet, sufficient energy will be yielded to account for the entire radiation. This gradual decrease is always in progress.

These considerations are of considerable interest when we apply them retrospectively. If it be true that the sun is at this moment shrinking, then in past times his globe must have been greater than it is at present. Assuming the figures already given, it follows that one hundred years ago the diameter of the sun must have been nearly six miles greater than it is now; one thousand years ago the diameter was fifty-seven miles greater; ten thousand years ago the diameter of the sun was five hundred and seventy miles greater than it is to-day. When man first trod this earth it would seem that the sun must have been many hundreds, perhaps many thousands, of miles greater than it is at this time.

We must not, however, over-estimate the significance of this statement. The diameter of the sun is so great, that a diminution of 10,000 miles would be but little more than the hundredth part of its diameter. If it were suddenly to shrink to the extent of 10,000 miles, the change would not be appreciable to ordinary observation, though a much smaller change would not elude delicate astronomical measurement. It does not necessarily follow that the climates on our earth in these early times must have been very different from those which we find at this day, for the question of climate depends upon other matters besides sunbeams.

Yet we need not abruptly stop our retrospect at any epoch, however remote. We may go back earlier and earlier, through the long ages which geologists claim for the deposition of the stratified rocks; and back again still further, to those very earliest epochs when life began to dawn on the earth. Still we can find no reason to suppose that the law of the sun's decreasing heat is not maintained; and thus we would seem bound by our present knowledge to suppose that the sun grows larger and larger the further our retrospect extends. We cannot assume that the rate of that growth is always the same. No such assumption is required; it is sufficient for our purpose that we find the sun growing larger and larger the further we peer back into the remote abyss of time past. If the present order of things in our universe has lasted long enough, then it would seem that there was a time when the sun must have been twice as large as it is at present; it must once have been ten times as large. How long ago that was no one can venture to say. But we cannot stop at the stage when the sun was even ten times as large as it is at present; the arguments will still apply in earlier ages. We see the sun swelling and swelling, with a corresponding decrease in its density, until at length we find, instead of our sun as we know it, a mighty nebula filling a gigantic region of space.

Such is, in fact, the doctrine of the origin of our system which has been advanced in that celebrated speculation known as the nebular theory of Laplace. Nor can it be ever more than a speculation; it cannot be established by observation, nor can it be proved by calculation. It is merely a conjecture, more or less plausible, but perhaps in some degree necessarily true, if our present laws of heat, as we understand them, admit of the extreme application here required, and if also the present order of things has reigned for sufficient time without the intervention of any influence at present unknown to us. This nebular theory is not confined to the history of our sun. Precisely similar reasoning may be extended to the individual planets: the farther we look back, the hotter and the hotter does the whole system become. It has been thought that if we could look far enough back, we should see the earth too hot for life; back further still, we should find the earth and all the planets red-hot; and back further still, to an exceedingly remote epoch, when the planets would be heated just as much as our sun is now. In a still earlier stage the whole solar system is thought to have been one vast mass of glowing gas, from which the present forms of the sun, with the planets and their satellites, have been gradually evolved. We cannot be sure that the course of events has been what is here indicated; but there are sufficient grounds for thinking that this doctrine substantially represents what has actually occurred.

Many of the features in the solar system harmonise with the supposition that the origin of the system has been that suggested by the nebular theory. We have already had occasion in an earlier chapter to allude to the fact that all the planets perform their revolutions around the sun in the same direction. It is also to be observed that the rotation of the planets on their axes, as well as the movements of the satellites around their primaries, all follow the same law, with two slight exceptions in the case of the Uranian and Neptunian systems. A coincidence so remarkable naturally suggests the necessity for some physical explanation. Such an explanation is offered by the nebular theory. Suppose that countless ages ago a mighty nebula was slowly rotating and slowly contracting. In the process of contraction, portions of the condensed matter of the nebula would be left behind. These portions would still revolve around the central mass, and each portion would rotate on its axis in the same direction. As the process of contraction proceeded, it would follow from dynamical principles that the velocity of rotation would increase; and thus at length these portions would consolidate into planets, while the central mass would gradually contract to form the sun. By a similar process on a smaller scale the systems of satellites were evolved from the contracting primary. These satellites would also revolve in the same direction, and thus the characteristic features of the solar system could be accounted for.

The nebular origin of the solar system receives considerable countenance from the study of the sidereal heavens. We have already dwelt upon the resemblance between the sun and the stars. If, then, our sun has passed through such changes as the nebular theory requires, may we not anticipate that similar phenomena should be met with in other stars? If this be so, it is reasonable to suppose that the evolution of some of the stars may not have progressed so far as has that of the sun, and thus we may be able actually to witness stars in the earlier phases of their development. Let us see how far the telescope responds to these anticipations.

The field of view of a large telescope usually discloses a number of stars scattered over a black background of sky; but the blackness of the background is not uniform: the practised eye of the skilled observer will detect in some parts of the heavens a faint luminosity. This will sometimes be visible over the whole extent of the field, or it may even occupy several fields. Years may pass on, and still there is no perceptible change. There can be no illusion, and the conclusion is irresistible that the object is a stupendous mass of faintly luminous glowing gas or vapour. This is the simplest type of nebula; it is characterised by extreme faintness, and seems composed of matter of the utmost tenuity. On the other hand we are occasionally presented with the beautiful and striking phenomenon of a definite and brilliant star surrounded by a luminous atmosphere. Between these two extreme types of a faint diffused mass on the one hand, and a bright star with a nebula surrounding it on the other, a graduated series of various other nebulæ can be arranged. We thus have a series of links passing by imperceptible gradations from the most faintly diffused nebulæ on the one side, into stars on the other.

The nebulæ seemed to Herschel to be vast masses of phosphorescent vapour. This vapour gradually cools down, and ultimately condenses into a star, or a cluster of stars. When the varied forms of nebulæ were classified, it almost seemed as if the different links in the process could be actually witnessed. In the vast faint nebulæ the process of condensation had just begun; in the smaller and brighter nebulæ the condensation had advanced farther; while in others, the star, or stars, arising from the condensation had already become visible.

But, it may be asked, how did Herschel know this? what is his evidence? Let us answer this question by an illustration. Go into a forest, and look at a noble old oak which has weathered the storm for centuries; have we any doubt that the oak-tree was once a young small plant, and that it grew stage by stage until it reached maturity? Yet no one has ever followed an oak-tree through its various stages; the brief span of human life has not been long enough to do so. The reason why we believe the oak-tree to have passed through all these stages is, because we are familiar with oak-trees of every gradation in size, from the seedling up to the noble veteran. Having seen this gradation in a vast multitude of trees, we are convinced that each individual passes through all these stages.

It was by a similar train of reasoning that Herschel was led to adopt the view of the origin of the stars which we have endeavoured to describe. The astronomer's life is not long enough, the life of the human race might not be long enough, to watch the process by which a nebula condenses down so as to form a solid body. But by looking at one nebula after another, the astronomer thinks he is able to detect the various stages which connect the nebula in its original form with the final form. He is thus led to believe that each of the nebulæ passes, in the course of ages, through these stages. And thus Herschel adopted the opinion that stars—some, many, or all—have each originated from what was once a glowing nebula.

Such a speculation may captivate the imagination, but it must be carefully distinguished from the truths of astronomy, properly so called. Remote posterity may perhaps obtain evidence on the subject which to us is inaccessible: our knowledge of nebulæ is too recent. There has not yet been time enough to detect any appreciable changes: for the study of nebulæ can only be said to date from Messier's Catalogue in 1771.

Since Herschel's time, no doubt, many careful drawings and observations of the nebulæ have been obtained; but still the interval has been much too short, and the earlier observations are too imperfect, to enable any changes in the nebulæ to be investigated with sufficient accuracy. If the human race lasts for very many centuries, and if our present observations are preserved during that time for comparison, then Herschel's theory may perhaps be satisfactorily tested.

A hundred years have passed since Laplace, with some diffidence, set forth his hypothesis as to the mode of formation of the solar system. On the whole it must be said that this "nebular hypothesis" has stood the test of advancing science well, though some slight modifications have become necessary in the light of more recent discoveries. Laplace (and Herschel also) seems to have considered a primitive nebula to consist of a "fiery mist" or glowing gas at a very high temperature. But this is by no means necessary, as we have seen that the gradual contraction of the vast mass supplies energy which may be converted into heat, and the spectroscopic evidence seems also to point to the existence of a moderate temperature in the gaseous nebulæ, which must be considered to be representatives of the hypothetical primitive chaos out of which our sun and planets have been evolved. Another point which has been reconsidered is the formation of the various planets. It was formerly thought that the rotation of the original mass had by degrees caused a number of rings of different dimensions to be separated from the central part, the material of which rings in time collected into single planets. The ring of Saturn was held to be a proof of this process, since we here have a ring, the condensation of which into one or more satellites has somehow been arrested. But while it is not impossible that matter in the shape of rings may have been left behind during the contraction of the nebulous mass (indeed, the minor planets between Mars and Jupiter have perhaps originated in this way), it seems likely that the larger planets were formed from the agglomeration of matter at a point on the equator of the rotating nebula.

The actual steps of the process by which the primeval nebula became transformed into the solar system seem to lie beyond reach of discovery.

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CHAPTER XXVII.

THE TIDES.[44]

Mathematical Astronomy—Lagrange's Theories: how far they are really True—The Solar System not Made of Rigid Bodies—Kepler's Laws True to Observation, but not Absolutely True when the Bodies are not Rigid—The Errors of Observation—The Tides—How the Tides were Observed—Discovery of the Connection between the Tides and the Moon—Solar and Lunar Tides—Work done by the Tides—Whence do the Tides obtain the Power to do the Work?—Tides are Increasing the Length of the Day—Limit to the Shortness of the Day—Early History of the Earth-Moon System—Unstable Equilibrium—Ratio of the Month to the Day—The Future Course of the System—Equality of the Month and the Day—The Future Critical Epoch—The Constant Face of the Moon accounted for—The other Side of the Moon—The Satellites of Mars—Their Remarkable Motions—Have the Tides Possessed Influence in Moulding the Solar System generally?—Moment of Momentum—Tides have had little or no Appreciable Effect on the Orbit of Jupiter—Conclusion.

That the great discoveries of Lagrange on the stability of the planetary system are correct is in one sense strictly true. No one has ever ventured to impugn the mathematics of Lagrange. Given the planetary system in the form which Lagrange assumed and the stability of that system is assured for all time. There is, however, one assumption which Lagrange makes, and on which his whole theory was founded: his assumption is that the planets are rigid bodies.

No doubt our earth seems a rigid body. What can be more solid and unyielding than the mass of rocks and metals which form the earth, so far as it is accessible to us? In the wide realms of space the earth is but as a particle; it surely was a natural and a legitimate assumption to suppose that that particle was a rigid body. If the earth were absolutely rigid—if every particle of the earth were absolutely at a fixed distance from every other particle—if under no stress of forces, and in no conceivable circumstance, the earth experienced even the minutest change of form—if the same could be said of the sun and of all the other planets—then Lagrange's prediction of the eternal duration of our system must be fulfilled.

But what are the facts of the case? Is the earth really rigid? We know from experiment that a rigid body in the mathematical sense of the word does not exist. Rocks are not rigid; steel is not rigid; even a diamond is not perfectly rigid. The whole earth is far from being rigid even on the surface, while part of the interior is still, perhaps, more or less fluid. The earth cannot be called a perfectly rigid body; still less can the larger bodies of our system be called rigid. Jupiter and Saturn are perhaps hardly even what could be called solid bodies. The solar system of Lagrange consisted of a rigid sun and a number of minute rigid planets; the actual solar system consists of a sun which is in no sense rigid, and planets which are only partially so.

The question then arises as to whether the discoveries of the great mathematicians of the last century will apply, not only to the ideal solar system which they conceived, but to the actual solar system in which our lot has been cast. There can be no doubt that these discoveries are approximately true: they are, indeed, so near the absolute truth, that observation has not yet satisfactorily shown any departure from them.

But in the present state of science we can no longer overlook the important questions which arise when we deal with bodies not rigid in the mathematical sense of the word. Let us, for instance, take the simplest of the laws to which we have referred, the great law of Kepler, which asserts that a planet will revolve for ever in an elliptic path of which the sun is one focus. This is seen to be verified by actual observation; indeed, it was established by observation before any theoretical explanation of that movement was propounded. If, however, we state the matter with a little more precision, we shall find that what Newton really demonstrated was, that if two rigid particles attract each other by a law of force which varies with the inverse square of the distance between the particles, then each of the particles will describe an ellipse with the common centre of gravity in the focus. The earth is, to some extent, rigid, and hence it was natural to suppose that the relative behaviour of the earth and the sun would, to a corresponding extent, observe the simple elliptic law of Kepler; as a matter of fact, they do observe it with such fidelity that, if we make allowance for other causes of disturbance, we cannot, even by most careful observation, detect the slightest variation in the motion of the earth arising from its want of rigidity.

There is, however, a subtlety in the investigations of mathematics which, in this instance at all events, transcends the most delicate observations which our instruments enable us to make. The principles of mathematics tell us that though Kepler's laws may be true for bodies which are absolutely and mathematically rigid, yet that if the sun or the planets be either wholly, or even in their minutest part, devoid of perfect rigidity, then Kepler's laws can be no longer true. Do we not seem here to be in the presence of a contradiction? Observation tells us that Kepler's laws are true in the planetary system; theory tells us that these laws cannot be true in the planetary system, because the bodies in that system are not perfectly rigid. How is this discrepancy to be removed? Or is there really a discrepancy at all? There is not. When we say that Kepler's laws have been proved to be true by observation, we must reflect on the nature of the proofs which are attainable. We observe the places of the planets with the instruments in our observatories; these places are measured by the help of our clocks and of the graduated circles on the instruments. These observations are no doubt wonderfully accurate; but they do not, they cannot, possess absolute accuracy in the mathematical sense of the word. We can, for instance, determine the place of a planet with such precision that it is certainly not one second of arc wrong; and one second is an extremely small quantity. A foot-rule placed at a distance of about forty miles subtends an angle of a second, and it is surely a delicate achievement to measure the place of a planet, and feel confident that no error greater than this can have intruded into our result.

When we compare the results of observation with the calculations conducted on the assumption of the truth of Kepler's laws, and when we pronounce on the agreement of the observations with the calculations, there is always a reference, more or less explicit, to the inevitable errors of the observations. If the calculations and observations agree so closely that the differences between the two are minute enough to have arisen in the errors inseparable from the observations, then we are satisfied with the accordance; for, in fact, no closer agreement is attainable, or even conceivable. The influence which the want of rigidity exercises on the fulfilment of the laws of Kepler can be estimated by calculation; it is found, as might be expected, to be extremely small—so small, in fact, as to be contained within that slender margin of error by which observations are liable to be affected. We are thus not able to discriminate by actual measurement the effects due to the absence of rigidity; they are inextricably hid among the small errors of observation.

The argument on which we are to base our researches is really founded on a very familiar phenomenon. There is no one who has ever visited the sea-side who is not familiar with that rise and fall of the sea which we call the tide. Twice every twenty-four hours the sea advances on the beach to produce high tide; twice every day the sea again retreats to produce low tide. These tides are not merely confined to the coasts; they penetrate for miles up the courses of rivers; they periodically inundate great estuaries. In a maritime country the tides are of the most profound practical importance; they also possess a significance of a far less obvious character, which it is our object now to investigate.

These daily pulses of the ocean have long ceased to be a mystery. It was in the earliest times perceived that there was a connection between the tides and the moon. Ancient writers, such as Pliny and Aristotle, have referred to the alliance between the times of high water and the age of the moon. I think we sometimes do not give the ancient astronomers as much credit as their shrewdness really entitles them to. We have all read—we have all been taught—that the moon and the tides are connected together; but how many of us are in a position to say that we have actually noticed that connection by direct personal observation? The first man who studied this matter with sufficient attention to convince himself and to convince others of its reality must have been a great philosopher. We know not his name, we know not his nation, we know not the age in which he lived; but our admiration of his discovery must be increased by the reflection that he had not the theory of gravitation to guide him. A philosopher of the present day who had never seen the sea could still predict the necessity of tides as a consequence of the law of universal gravitation; but the primitive astronomer, who knew not of the invisible bond by which all bodies in the universe are drawn together, made a splendid—indeed, a typical—inductive discovery, when he ascertained the relation between the moon and the tides.

We can surmise that this discovery, in all probability, first arose from the observations of experienced navigators. In all matters of entering port or of leaving port, the state of the tide is of the utmost concern to the sailor. Even in the open sea he has sometimes to shape his course in accordance with the currents produced by the tides; or, in guiding his course by taking soundings, he has always to bear in mind that the depth varies with the tide. All matters relating to the tide would thus come under his daily observation. His daily work, the success of his occupation, the security of his life, depend often on the tides; and hence he would be solicitous to learn from his observation all that would be useful to him in the future. To the coasting sailor the question of the day is the time of high water. That time varies from day to day; it is an hour or more later to-morrow than to-day, and there is no very simple rule which can be enunciated. The sailor would therefore welcome gladly any rule which would guide him in a matter of such importance. We can make a conjecture as to the manner in which such a rule was first discovered. Let us suppose that a sailor at Calais, for example, is making for harbour. He has a beautiful night—the moon is full; it guides him on his way; he gets safely into harbour; and the next morning he finds the tide high between 11 and 12.[45] He often repeats the same voyage, but he finds sometimes a low and inconvenient tide in the morning. At length, however, it occurs to him that when he has a moonlight night he has a high tide at 11. This occurs once or twice: he thinks it but a chance coincidence. It occurs again and again. At length he finds it always occurs. He tells the rule to other sailors; they try it too. It is invariably found that when the moon is full, the high tide always recurs at the same hour at the same place. The connection between the moon and the tide is thus established, and the intelligent sailor will naturally compare other phases of the moon with the times of high water. He finds, for example, that the moon at the first quarter always gives high water at the same hour of the day; and finally, he obtains a practical rule, by which, from the state of the moon, he can at once tell the time when the tide will be high at the port where his occupation lies. A diligent observer will trace a still further connection between the moon and the tides; he will observe that some high tides rise higher than others, that some low tides fall lower than others. This is a matter of much practical importance. When a dangerous bar has to be crossed, the sailor will feel much additional security in knowing that he is carried over it on the top of a spring tide; or if he has to contend against tidal currents, which in some places have enormous force, he will naturally prefer for his voyage the neap tides, in which the strength of these currents is less than usual. The spring tides and the neap tides will become familiar to him, and he will perceive that the spring tides occur when the moon is full or new—or, at all events, that the spring tides are within a certain constant number of days of the full or new moon. It was, no doubt, by reasoning such as this, that in primitive times the connection between the moon and the tides came to be perceived.

It was not, however, until the great discovery of Newton had disclosed the law of universal gravitation that it became possible to give a physical explanation of the tides. It was then seen how the moon attracts the whole earth and every particle of the earth. It was seen how the fluid particles which form the oceans on the earth were enabled to obey the attraction in a way that the solid parts could not. When the moon is overhead it tends to draw the water up, as it were, into a heap underneath, and thus to give rise to the high tide. The water on the opposite side of the earth is also affected in a way that might not be at first anticipated. The moon attracts the solid body of the earth with greater intensity than it attracts the water at the other side which lies more distant from it. The earth is thus drawn away from the water, and there is therefore a tendency to a high tide as well on the side of the earth away from the moon as on that towards the moon. The low tides occupy the intermediate positions.

The sun also excites tides on the earth; but owing to the great distance of the sun, the difference between its attraction on the sea and on the solid interior of the earth is not so appreciable. The solar tides are thus smaller than the lunar tides. When the two conspire, they cause a spring tide; when the solar and lunar tides are opposed, we have the neap tide.

There are, however, a multitude of circumstances to be taken into account when we attempt to apply this general reasoning to the conditions of a particular case. Owing to local peculiarities the tides vary enormously at the different parts of the coast. In a confined area like the Mediterranean Sea, the tides have only a comparatively small range, varying at different places from one foot to a few feet. In mid-ocean also the tidal rise and fall is not large, amounting, for instance, to a range of three feet at St. Helena. Near the great continental masses the tides become very much modified by the coasts. We find at London a tide of eighteen or nineteen feet; but the most remarkable tides in the British Islands are those in the Bristol Channel, where, at Chepstow or Cardiff, there is a rise and fall during spring tides to the height of thirty-seven or thirty-eight feet, and at neap tides to a height of twenty-eight or twenty-nine. These tides are surpassed in magnitude at other parts of the world. The greatest of all tides are those in the Bay of Fundy, at some parts of which the rise and fall at spring tides is not less than fifty feet.

The rising and falling of the tide is necessarily attended with the formation of currents. Such currents are, indeed, well known, and in some of our great rivers they are of the utmost consequence. These currents of water can, like water-streams of any other kind, be made to do useful work. We can, for instance, impound the rising water in a reservoir, and as the tide falls we can compel the enclosed water to work a water-wheel before it returns to the sea. We have, indeed, here a source of actual power; but it is only in very unusual circumstances that it is found to be economical to use the tides for this purpose. The question can be submitted to calculation, and the area of the reservoir can be computed which would retain sufficient water to work a water-wheel of given horse-power. It can be shown that the area of the reservoir necessary to impound water enough to produce 100 horse-power would be 40 acres. The whole question is then reduced to the simple one of expense: would the construction and the maintenance of this reservoir be more or less costly than the erection and the maintenance of a steam-engine of equivalent power? In most cases it would seem that the latter would be by far the cheaper; at all events, we do not practically find tidal engines in use, so that the power of the tides is now running to waste. The economical aspects of the case may, however, be very profoundly altered at some remote epoch, when our stores of fuel, now so lavishly expended, give appreciable signs of approaching exhaustion.

The tides are, however, doing work of one kind or another. A tide in a river estuary will sometimes scour away a bank and carry its materials elsewhere. We have here work done and energy consumed, just as much as if the same task had been accomplished by engineers directing the powerful arms of navvies. We know that work cannot be done without the consumption of energy in some of its forms; whence, then, comes the energy which supplies the power of the tides? At a first glance the answer to this question seems a very obvious one. Have we not said that the tides are caused by the moon? and must not the energy, therefore, be derived from the moon? This seems plain enough, but, unfortunately, it is not true. It is one of those cases by no means infrequent in Dynamics, where the truth is widely different from that which seems to be the case. An illustration will perhaps make the matter clearer. When a rifle is fired, it is the finger of the rifleman that pulls the trigger; but are we, then, to say that the energy by which the bullet has been driven off has been supplied by the rifleman? Certainly not; the energy is, of course, due to the gunpowder, and all the rifleman did was to provide the means by which the energy stored up in the powder could be liberated. To a certain extent we may compare this with the tidal problem; the tides raised by the moon are the originating cause whereby a certain store of energy is drawn upon and applied to do such work as the tides are competent to perform. This store of energy, strange to say, does not lie in the moon; it is in the earth itself. Indeed, it is extremely remarkable that the moon actually gains energy from the tides by itself absorbing some of the store which exists in the earth. This is not put forward as an obvious result; it depends upon a refined dynamical theorem.

We must clearly understand the nature of this mighty store of energy from which the tides draw their power, and on which the moon is permitted to make large and incessant drafts. Let us see in what sense the earth is said to possess a store of energy. We know that the earth rotates on its axis once every day. It is this rotation which is the source of the energy. Let us compare the rotation of the earth with the rotation of the fly-wheel belonging to a steam-engine. The rotation of the fly-wheel is really a reservoir, into which the engine pours energy at each stroke of the piston. The various machines in the mill worked by the engine merely draw upon the store of energy accumulated in the fly-wheel. The earth may be likened to a gigantic fly-wheel detached from the engine, though still connected with the machines in the mill. From its stupendous dimensions and from its rapid velocity, that great fly-wheel possesses an enormous store of energy, which must be expended before the fly-wheel comes to rest. Hence it is that, though the tides are caused by the moon, yet the energy they require is obtained by simply appropriating some of the vast supply available from the rotation of the earth.

There is, however, a distinction of a very fundamental character between the earth and the fly-wheel of an engine. As the energy is withdrawn from the fly-wheel and consumed by the various machines in the mill, it is continually replaced by fresh energy, which flows in from the exertions of the steam-engine, and thus the velocity of the fly-wheel is maintained. But the earth is a fly-wheel without the engine. When the tides draw upon the store of energy and expend it in doing work, that energy is not replaced. The consequence is irresistible: the energy in the rotation of the earth must be decreasing. This leads to a consequence of the utmost significance. If the engine be cut off from the fly-wheel, then, as everyone knows, the massive fly-wheel may still give a few rotations, but it will speedily come to rest. A similar inference must be made with regard to the earth; but its store of energy is so enormous, in comparison with the demands which are made upon it, that the earth is able to hold out. Ages of countless duration must elapse before the energy of the earth's rotation can be completely exhausted by such drafts as the tides are capable of making. Nevertheless, it is necessarily true that the energy is decreasing; and if it be decreasing, then the speed of the earth's rotation must be surely, if slowly, abating. Now we have arrived at a consequence of the tides which admits of being stated in the simplest language. If the speed of rotation be abating, then the length of the day must be increasing; and hence we are conducted to the following most important statement: that the tides are increasing the length of the day.

To-day is longer than yesterday—to-morrow will be longer than to-day. The difference is so small that even in the course of ages it can hardly be said to have been distinctly established by observation. We do not pretend to say how many centuries have elapsed since the day was even one second shorter than it is at present; but centuries are not the units which we employ in tidal evolution. A million years ago it is quite probable that the divergence of the length of the day from its present value may have been very considerable. Let us take a glance back into the profound depths of times past, and see what the tides have to tell us. If the present order of things has lasted, the day must have been shorter and shorter the farther we look back into the dim past. The day is now twenty-four hours; it was once twenty hours, once ten hours; it was once six hours. How much farther can we go? Once the six hours is past, we begin to approach a limit which must at some point bound our retrospect. The shorter the day the more is the earth bulged at the equator; the more the earth is bulged at the equator the greater is the strain put upon the materials of the earth by the centrifugal force of its rotation. If the earth were to go too fast it would be unable to cohere together; it would separate into pieces, just as a grindstone driven too rapidly is rent asunder with violence. Here, therefore, we discern in the remote past a barrier which stops the present argument. There is a certain critical velocity which is the greatest that the earth could bear without risk of rupture, but the exact amount of that velocity is a question not very easy to answer. It depends upon the nature of the materials of the earth; it depends upon the temperature; it depends upon the effect of pressure, and on other details not accurately known to us. An estimate of the critical velocity has, however, been made, and it has been shown mathematically that the shortest period of rotation which the earth could have, without flying into pieces, is about three or four hours. The doctrine of tidal evolution has thus conducted us to the conclusion that, at some inconceivably remote epoch, the earth was spinning round its axis in a period approximating to three or four hours.

We thus learn that we are indebted to the moon for the gradual elongation of the day from its primitive value up to twenty-four hours. In obedience to one of the most profound laws of nature, the earth has reacted on the moon, and the reaction of the earth has taken a tangible form. It has simply consisted in gradually driving the moon away from the earth. You may observe that this driving away of the moon resembles a piece of retaliation on the part of the earth. The consequence of the retreat of the moon is sufficiently remarkable. The path in which the moon is revolving has at the present time a radius of 240,000 miles. This radius must be constantly growing larger, in consequence of the tides. Provided with this fact, let us now glance back into the past history of the moon. As the moon's distance is increasing when we look forwards, so we find it decreasing when we look backwards. The moon must have been nearer the earth yesterday than it is to-day; the difference is no doubt inappreciable in years, in centuries, or in thousands of years; but when we come to millions of years, the moon must have been significantly closer than it is at present, until at length we find that its distance, instead of 240,000 miles, has dwindled down to 40,000, to 20,000, to 10,000 miles. Nor need we stop—nor can we stop—until we find the moon actually close to the earth's surface. If the present laws of nature have operated long enough, and if there has been no external interference, then it cannot be doubted that the moon and the earth were once in immediate proximity. We can, indeed, calculate the period in which the moon must have been revolving round the earth. The nearer the moon is to the earth the quicker it must revolve; and at the critical epoch when the satellite was in immediate proximity to our earth it must have completed each revolution in about three or four hours.

This has led to one of the most daring speculations which has ever been made in astronomy. We cannot refrain from enunciating it; but it must be remembered that it is only a speculation, and to be received with corresponding reserve. The speculation is intended to answer the question, What brought the moon into that position, close to the surface of the earth? We will only say that there is the gravest reason to believe that the moon was, at some very early period, fractured off from the earth when the earth was in a soft or plastic condition.

At the beginning of the history we found the earth and the moon close together. We found that the rate of rotation of the earth was only a few hours, instead of twenty-four hours. We found that the moon completed its journey round the primitive earth in exactly the same time as the primitive earth rotated on its axis, so that the two bodies were then constantly face to face. Such a state of things formed what a mathematician would describe as a case of unstable dynamical equilibrium. It could not last. It may be compared to the case of a needle balanced on its point; the needle must fall to one side or the other. In the same way, the moon could not continue to preserve this position. There were two courses open: the moon must either have fallen back on the earth, and been reabsorbed into the mass of the earth, or it must have commenced its outward journey. Which of these courses was the moon to adopt? We have no means, perhaps, of knowing exactly what it was which determined the moon to one course rather than to another, but as to the course which was actually taken there can be no doubt. The fact that the moon exists shows that it did not return to the earth, but commenced its outward journey. As the moon recedes from the earth it must, in conformity with Kepler's laws, require a longer time to complete its revolution. It has thus happened that, from the original period of only a few hours, the duration has increased until it has reached the present number of 656 hours. The rotation of the earth has, of course, also been modified, in accordance with the retreat of the moon. Once the moon had commenced to recede, the earth was released from the obligation which required it constantly to direct the same face to the moon. When the moon had receded to a certain distance, the earth would complete the rotation in less time than that required by the moon for one revolution. Still the moon gets further and further away, and the duration of the revolution increases to a corresponding extent, until three, four, or more days (or rotations of the earth) are identical with the month (or revolution of the moon). Although the number of days in the month increases, yet we are not to suppose that the rate of the earth's rotation is increasing; indeed, the contrary is the fact. The earth's rotation is getting slower, and so is the revolution of the moon, but the retardation of the moon is greater than that of the earth. Even though the period of rotation of the earth has greatly increased from its primitive value, yet the period of the moon has increased still more, so that it is several times as large as that of the rotation of the earth. As ages roll on the moon recedes further and further, its orbit increases, the duration of the revolution augments, until at length a very noticeable epoch is attained, which is, in one sense, a culminating point in the career of the moon. At this epoch the revolution periods of the moon, when measured in rotation periods of the earth, attain their greatest value. It would seem that the month was then twenty-nine days. It is not, of course, meant that the month and the day at that epoch were the month and the day as our clocks now measure time. Both were shorter then than now. But what we mean is, that at this epoch the earth rotated twenty-nine times on its axis while the moon completed one circuit.

This epoch has now been passed. No attempt can be made at present to evaluate the date of that epoch in our ordinary units of measurement. At the same time, however, no doubt can be entertained as to the immeasurable antiquity of the event, in comparison with all historic records; but whether it is to be reckoned in hundreds of thousands of years, in millions of years, or in tens of millions of years, must be left in great degree to conjecture.

This remarkable epoch once passed, we find that the course of events in the earth-moon system begins to shape itself towards that remarkable final stage which has points of resemblance to the initial stage. The moon still continues to revolve in an orbit with a diameter steadily, though very slowly, growing. The length of the month is accordingly increasing, and the rotation of the earth being still constantly retarded, the length of the day is also continually growing. But the ratio of the length of the month to the length of the day now exhibits a change. That ratio had gradually increased, from unity at the commencement, up to the maximum value of somewhere about twenty-nine at the epoch just referred to. The ratio now begins again to decline, until we find the earth makes only twenty-eight rotations, instead of twenty-nine, in one revolution of the moon. The decrease in the ratio continues until the number twenty-seven expresses the days in the month. Here, again, we have an epoch which it is impossible for us to pass without special comment. In all that has hitherto been said we have been dealing with events in the distant past; and we have at length arrived at the present state of the earth-moon system. The days at this epoch are our well-known days, the month is the well-known period of the revolution of our moon. At the present time the month is about twenty-seven of our days, and this relation has remained sensibly true for thousands of years past. It will continue to remain sensibly true for thousands of years to come, but it will not remain true indefinitely. It is merely a stage in this grand transformation; it may possess the attributes of permanence to our ephemeral view, just as the wings of a gnat seem at rest when illuminated by the electric spark; but when we contemplate the history with time conceptions sufficiently ample for astronomy we realise how the present condition of the earth-moon system can have no greater permanence than any other stage in the history.

Our narrative must, however, now assume a different form. We have been speaking of the past; we have been conducted to the present; can we say anything of the future? Here, again, the tides come to our assistance. If we have rightly comprehended the truth of dynamics (and who is there now that can doubt them?), we shall be enabled to make a forecast of the further changes of the earth-moon system. If there be no interruption from any external source at present unknown to us, we can predict—in outline, at all events—the subsequent career of the moon. We can see how the moon will still follow its outward course. The path in which it revolves will grow with extreme slowness, but yet it will always grow; the progress will not be reversed, at all events, before the final stage of our history has been attained. We shall not now delay to dwell on the intervening stages; we will rather attempt to sketch the ultimate type to which our system tends. In the dim future—countless millions of years to come—this final stage will be approached. The ratio of the month to the day, whose decline we have already referred to, will continue to decline. The period of revolution of the moon will grow longer and longer, but the length of the day will increase much more rapidly than the increase in the duration of the moon's period. From the month of twenty-seven days we shall pass to a month of twenty-six days, and so on, until we shall reach a month of ten days, and, finally, a month of one day.

Let us clearly understand what we mean by a month of one day. We mean that the time in which the moon revolves around the earth will be equal to the time in which the earth rotates around its axis. The length of this day will, of course, be vastly greater than our day. The only element of uncertainty in these enquiries arises when we attempt to give numerical accuracy to the statements. It seems to be as true as the laws of dynamics that a state of the earth-moon system in which the day and the month are equal must be ultimately attained; but when we attempt to state the length of that day we introduce a hazardous element into the enquiry. In giving any estimate of its length, it must be understood that the magnitude is stated with great reserve. It may be erroneous to some extent, though, perhaps, not to any considerable amount. The length of this great day would seem to be about equal to fifty-seven of our days. In other words, at some critical time in the excessively distant future, the earth will take something like 1,400 hours to perform a rotation, while the moon will complete its journey precisely in the same time.

We thus see how, in some respects, the first stage of the earth-moon system and the last stage resemble each other. In each case we have the day equal to the month. In the first case the day and the month were only a small fraction of our day; in the last stage the day and the month are each a large multiple of our day. There is, however, a profound contrast between the first critical epoch and the last. We have already mentioned that the first epoch was one of unstability—it could not last; but this second state is one of dynamical stability. Once that state has been acquired, it would be permanent, and would endure for ever if the earth and the moon could be isolated from all external interference.

There is one special feature which characterises the movement when the month is equal to the day. A little reflection will show that when this is the case the earth must constantly direct the same face towards the moon. If the day be equal to the month, then the earth and moon must revolve together, as if bound by invisible bands; and whatever hemisphere of the earth be directed to the moon when this state of things commences will remain there so long as the day remains equal to the month.

At this point it is hardly possible to escape being reminded of that characteristic feature of the moon's motion which has been observed from all antiquity. We refer, of course, to the fact that the moon at the present time constantly turns the same face to the earth.

It is incumbent upon astronomers to provide a physical explanation of this remarkable fact. The moon revolves around our earth once in a definite number of seconds. If the moon always turns the same face to the earth, then it is demonstrated that the moon rotates on its axis once in the same number of seconds also. Now, this would be a coincidence wildly improbable unless there were some physical cause to account for it. We have not far to seek for a cause: the tides on the moon have produced the phenomenon. We now find the moon has a rugged surface, which testifies to the existence of intense volcanic activity in former times. Those volcanoes are now silent—the internal fires in the moon seem to have become exhausted; but there was a time when the moon must have been a heated and semi-molten mass. There was a time when the materials of the moon were so hot as to be soft and yielding, and in that soft and yielding mass the attraction of our earth excited great tides. We have no historical record of these tides (they were long anterior to the existence of telescopes, they were probably long anterior to the existence of the human race), but we know that these tides once existed by the work they have accomplished, and that work is seen to-day in the constant face which the moon turns towards the earth. The gentle rise and fall of the oceans which form our tides present a picture widely different from the tides by which the moon was once agitated. The tides on the moon were vastly greater than those of the earth. They were greater because the weight of the earth is greater than that of the moon, so that the earth was able to produce much more powerful tides in the moon than the moon has ever been able to raise on the earth.

That the moon should bend the same face to the earth depends immediately upon the condition that the moon shall rotate on its axis in precisely the same period as that which it requires to revolve around the earth. The tides are a regulating power of unremitting efficiency to ensure that this condition shall be observed. If the moon rotated more slowly than it ought, then the great lava tides would drag the moon round faster and faster until it attained the desired velocity; and then, but not till then, they would give the moon peace. Or if the moon were to rotate faster on its axis than in its orbit, again the tides would come furiously into play; but this time they would be engaged in retarding the moon's rotation, until they had reduced the speed of the moon to one rotation for each revolution.

Can the moon ever escape from the thraldom of the tides? This is not very easy to answer, but it seems perhaps not impossible that the moon may, at some future time, be freed from tidal control. It is, indeed, obvious that the tides, even at present, have not the extremely stringent control over the moon which they once exercised. We now see no ocean on the moon, nor do the volcanoes show any trace of molten lava. There can hardly be tides on the moon, but there may be tides in the moon. It may be that the interior of the moon is still hot enough to retain an appreciable degree of fluidity, and if so, the tidal control would still retain the moon in its grip; but the time will probably come, if it have not come already, when the moon will be cold to the centre—cold as the temperature of space. If the materials of the moon were what a mathematician would call absolutely rigid, there can be no doubt that the tides could no longer exist, and the moon would be emancipated from tidal control. It seems impossible to predicate how far the moon can ever conform to the circumstances of an actual rigid body, but it may be conceivable that at some future time the tidal control shall have practically ceased. There would then be no longer any necessary identity between the period of rotation and that of revolution. A gleam of hope is thus projected over the astronomy of the distant future. We know that the time of revolution of the moon is increasing, and so long as the tidal governor could act, the time of rotation must increase sympathetically. We have now surmised a state of things in which the control is absent. There will then be nothing to prevent the rotation remaining as at present, while the period of revolution is increasing. The privilege of seeing the other side of the moon, which has been withheld from all previous astronomers, may thus in the distant future be granted to their successors.

The tides which the moon raises in the earth act as a brake on the rotation of the earth. They now constantly tend to bring the period of rotation of the earth to coincide with the period of revolution of the moon. As the moon revolves once in twenty-seven days, the earth is at present going too fast, and consequently the tidal control at the present moment endeavours to retard the rotation of the earth. The rotation of the moon long since succumbed to tidal control, but that was because the moon was comparatively small and the tidal power of the earth was enormous. But this is the opposite case. The earth is large and more massive than the moon, the tides raised by the moon are but small and weak, and the earth has not yet completely succumbed to the tidal action. But the tides are constant, they never for an instant relax the effort to control, and they are gradually tending to render the day and the month coincident, though the progress is a very slow one.

The theory of the tides leads us to look forward to a remote state of things, in which the moon revolves around the earth in a period equal to the day, so that the two bodies shall constantly bend the same face to each other, provided the tidal control be still able to guide the moon's rotation. So far as the mutual action of the earth and the moon is concerned, such an arrangement possesses all the attributes of permanence. If, however, we venture to project our view to a still more remote future, we can discern an external cause which must prevent this mutual accommodation between the earth and the moon from being eternal. The tides raised by the moon on the earth are so much greater than those raised by the sun, that we have, in the course of our previous reasoning, held little account of the sun-raised tides. This is obviously only an approximate method of dealing with the question. The influence of the solar tide is appreciable, and its importance relatively to the lunar tide will gradually increase as the earth and moon approach the final critical stage. The solar tides will have the effect of constantly applying a further brake to the rotation of the earth. It will therefore follow that, after the day and the month have become equal, a still further retardation awaits the length of the day. We thus see that in the remote future we shall find the moon revolving around the earth in a shorter time than that in which the earth rotates on its axis.

A most instructive corroboration of these views is afforded by the discovery of the satellites of Mars. The planet Mars is one of the smaller members of our system. It has a mass which is only the eighth part of the mass of the earth. A small planet like Mars has much less energy of rotation to be destroyed than a larger one like the earth. It may therefore be expected that the small planet will proceed much more rapidly in its evolution than the large one; we might, therefore, anticipate that Mars and his satellites have attained a more advanced stage of their history than is the case with the earth and her satellite.

When the discovery of the satellites of Mars startled the world, in 1877, there was no feature which created so much amazement as the periodic time of the interior satellite. We have already pointed out in [Chapter X]. how Phobos revolves around Mars in a period of 7 hours 39 minutes. The period of rotation of Mars himself is 24 hours 37 minutes, and hence we have the fact, unparalleled in the solar system, that the satellite is actually revolving three times as rapidly as the planet is rotating. There can hardly be a doubt that the solar tides on Mars have abated its velocity of rotation in the manner just suggested.

It has always seemed to me that the matter just referred to is one of the most interesting and instructive in the whole history of astronomy. We have, first, a very beautiful telescopic discovery of the minute satellites of Mars, and we have a determination of the anomalous movement of one of them. We have then found a satisfactory physical explanation of the cause of this phenomenon, and we have shown it to be a striking instance of tidal evolution. Finally, we have seen that the system of Mars and his satellite is really a forecast of the destiny which, after the lapse of ages, awaits the earth-moon system.

It seems natural to enquire how far the influence of tides can have contributed towards moulding the planetary orbits. The circumstances are here very different from those we have encountered in the earth-moon system. Let us first enunciate the problem in a definite shape. The solar system consists of the sun in the centre, and of the planets revolving around the sun. These planets rotate on their axes; and circulating round some of the planets we have their systems of satellites. For simplicity, we may suppose all the planets and their satellites to revolve in the same plane, and the planets to rotate about axes which are perpendicular to that plane. In the study of the theory of tidal evolution we must be mainly guided by a profound dynamical principle known as the conservation of the "moment of momentum." The proof of this great principle is not here attempted; suffice it to say that it can be strictly deduced from the laws of motion, and is thus only second in certainty to the fundamental truths of ordinary geometry or of algebra. Take, for instance, the giant planet, Jupiter. In one second he moves around the sun through a certain angle. If we multiply the mass of Jupiter by that angle, and if we then multiply the product by the square of the distance from Jupiter to the sun, we obtain a certain definite amount. A mathematician calls this quantity the "orbital" moment of momentum of Jupiter.[46] In the same way, if we multiply the mass of Saturn by the angle through which the planet moves in one second, and this product by the square of the distance between the planet and the sun, then we have the orbital moment of momentum of Saturn. In a similar manner we ascertain the moment of momentum for each of the other planets due to revolution around the sun. We have also to define the moment of momentum of the planets around their axes. In one second Jupiter rotates through a certain angle; we multiply that angle by the mass of Jupiter, and by the square of a certain line which depends on his internal constitution: the product forms the "rotational" moment of momentum. In a similar manner we find the rotational moment of momentum for each of the other planets. Each satellite revolves through a certain angle around its primary in one second; we obtain the moment of momentum of each satellite by multiplying its mass into the angle described in one second, and then multiplying the product into the square of the distance of the satellite from its primary. Finally, we compute the moment of momentum of the sun due to its rotation. This we obtain by multiplying the angle through which the sun turns in one second by the whole mass of the sun, and then multiplying the product by the square of a certain line of prodigious length, which depends upon the details of the sun's internal structure.

If we have succeeded in explaining what is meant by the moment of momentum, then the statement of the great law is comparatively simple. We are, in the first place, to observe that the moment of momentum of any planet may alter. It would alter if the distance of the planet from the sun changed, or if the velocity with which the planet rotates upon its axis changed; so, too, the moment of momentum of the sun may change, and so may those of the satellites. In the beginning a certain total quantity of moment of momentum was communicated to our system, and not one particle of that total can the solar system, as a whole, squander or alienate. No matter what be the mutual actions of the various bodies of the system, no matter what perturbations they may undergo—what tides may be produced, or even what mutual collisions may occur—the great law of the conservation of moment of momentum must be obeyed. If some bodies in the solar system be losing moment of momentum, then other bodies in the system must be gaining, so that the total quantity shall remain unaltered. This consideration is one of supreme importance in connection with the tides. The distribution of moment of momentum in the system is being continually altered by the tides; but, however the tides may ebb or flow, the total moment of momentum can never alter so long as influences external to the system are absent.

We must here point out the contrast between the endowment of our system with energy and with moment of momentum. The mutual actions of our system, in so far as they produce heat, tend to squander the energy, a considerable part of which can be thus dissipated and lost; but the mutual actions have no power of dissipating the moment of momentum.

The total moment of momentum of the solar system being taken to be 100, this is at present distributed as follows:—

Orbital moment of momentum of Jupiter	60
Orbital moment of momentum of Saturn	24
Orbital moment of momentum of Uranus	6
Orbital moment of momentum of Neptune	8
Rotational moment of momentum of Sun	2
—
100

The contributions of the other items are excessively minute. The orbital moments of momentum of the few interior planets contain but little more than one thousandth part of the total amount. The rotational contributions of all the planets and of their satellites is very much less, being not more than one sixty-thousandth part of the whole. When, therefore, we are studying the general effects of tides on the planetary orbits these trifling matters may be overlooked. We shall, however, find it desirable to narrow the question still more, and concentrate our attention on one splendid illustration. Let us take the sun and the planet Jupiter, and, supposing all other bodies of our system to be absent, let us discuss the influence of tides produced in Jupiter by the sun, and of tides in the sun by Jupiter.

It might be hastily thought that, just as the moon was born of the earth, so the planets were born of the sun, and have gradually receded by tides into their present condition. We have the means of enquiry into this question by the figures just given, and we shall show that it is impossible that Jupiter, or any of the other planets, can ever have been very much closer to the sun than they are at present. In the case of Jupiter and the sun we have the moment of momentum made up of three items. By far the largest of these items is due to the orbital revolution of Jupiter, the next is due to the sun, the third is due to the rotation of Jupiter on its axis. We may put them in round numbers as follows:—

Orbital moment of momentum of Jupiter	600,000
Rotational moment of momentum of Sun	20,000
Rotational moment of momentum of Jupiter	12

The sun produces tides in Jupiter, those tides retard the rotation of Jupiter. They make Jupiter rotate more and more slowly, therefore the moment of momentum of Jupiter is decreasing, therefore its present value of 12 must be decreasing. Even the mighty sun himself may be distracted by tides. Jupiter raises tides in the sun, those tides retard the motion of the sun, and therefore the moment of momentum of the sun is decreasing, and it follows from both causes that the item of 600,000 must be increasing; in other words, the orbital motion of Jupiter must be increasing, or Jupiter must be receding from the sun. To this extent, therefore, the sun-Jupiter system is analogous to the earth-moon system. As the tides on the earth are driving away the moon, so the tides in Jupiter and the sun are gradually driving the two bodies apart. But there is a profound difference between the two cases. It can be proved that the tides produced in Jupiter by the sun are more effective than those produced in the sun by Jupiter. The contribution of the sun may, therefore, be at present omitted; so that, practically, the augmentations of the orbital moment of momentum of Jupiter are now achieved at the expense of that stored up by Jupiter's rotation. But what is 12 compared with 600,000. Even when the whole of Jupiter's rotational moment of momentum and that of his satellites has become absorbed into the orbital motion, there will hardly be an appreciable difference in the latter. In ancient days we may indeed suppose that Jupiter being hotter was larger than at present, and that he had considerably more rotational moment of momentum. But it is hardly credible that Jupiter can ever have had one hundred times the moment of momentum that he has at present. Yet even if 1,200 units of rotational momentum had been transferred to the orbital motion it would only correspond with the most trivial difference in the distance of Jupiter from the sun. We are hence assured that the tides have not appreciably altered the dimensions of the orbit of Jupiter, or of the other great planets.

The time will, however, come when the rotation of Jupiter on his axis will be gradually abated by the influence of the tides. It will then be found that the moment of momentum of the sun's rotation will be gradually expended in increasing the orbits of the planets, but as this reserve only holds about two per cent. of the whole amount in our system it cannot produce any considerable effect.

The theory of tidal evolution, which in the hands of Professor Darwin has taught us so much with regard to the past history of the systems of satellites in the solar system, will doubtless also, as pointed out by Dr. See, be found to account for the highly eccentric orbits of double star systems. In the earth-moon system we have two bodies exceedingly different in bulk, the mass of the earth being about eighty times as great as that of the moon. But in the case of most double stars we have to do with two bodies not very different as regards mass. It can be demonstrated that the orbit must have been originally of slight eccentricity, but that tidal friction is capable not only of extending, but also of elongating it. The accelerating force is vastly greater at periastron (when the two bodies are nearest each other) than at apastron (when their distance is greatest). At periastron the disturbing force will, therefore, increase the apastron distance by an enormous amount, while at apastron it increases the periastron distance by a very small amount. Thus, while the ellipse is being gradually expanded, the orbit grows more and more eccentric, until the axial rotations have been sufficiently reduced by the transfer of axial to orbital moment of momentum.

And now we must draw this chapter to a close, though there are many other subjects that might be included. The theory of tidal evolution is, indeed, one of quite exceptional interest. The earlier mathematicians expended their labour on the determination of the dynamics of a system which consisted of rigid bodies. We are indebted to contemporary mathematicians for opening up celestial mechanics upon the more real supposition that the bodies are not rigid; in other words, that they are subject to tides. The mathematical difficulties are enormously enhanced, but the problem is more true to nature, and has already led to some of the most remarkable astronomical discoveries made in modern times.

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Our Story of the Heavens has now been told. We commenced this work with some account of the mechanical and optical aids to astronomy; we have ended it with a brief description of an intellectual method of research which reveals some of the celestial phenomena that occurred ages before the human race existed. We have spoken of those objects which are comparatively near to us, and then, step by step, we have advanced to the distant nebulæ and clusters which seem to lie on the confines of the visible universe. Yet how little can we see with even our greatest telescopes, when compared with the whole extent of infinite space! No matter how vast may be the depth which our instruments have sounded, there is yet a beyond of infinite extent. Imagine a mighty globe described in space, a globe of such stupendous dimensions that it shall include the sun and his system, all the stars and nebulæ, and even all the objects which our finite capacities can imagine. Yet, what ratio must the volume of this great globe bear to the whole extent of infinite space? The ratio is infinitely less than that which the water in a single drop of dew bears to the water in the whole Atlantic Ocean.

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APPENDIX.