Without pretending to any numerical accuracy, we can at all events give a qualitative if not a quantitative analysis of the past history of our earth, in so far as its changes of temperature are concerned. A million years ago our earth doubtless contained appreciably more heat than it does at present. I speak not now, of course, of mere solar heat—of the heat which gives us the vicissitudes of seasons; I am only referring to the original hoard of internal heat which is gradually waning. As therefore our retrospect extends through millions and millions of past ages, we see our earth ever growing warmer and warmer the further and further we look back. There was a time when those heated strata which we have now to go deep down in mines to find were considerably nearer the surface. At present, were it not for the sun, the heat of the earth where we stand would hardly be appreciably above the temperature of infinite space—perhaps some 200 or 300 degrees below zero. But there must have been a time when there was sufficient internal heat to maintain the exterior at a warm and indeed at a very hot temperature. Nor is there any bound to our retrospect arising from the operation or intervention of any other agent, so far as we know; consequently the hotter and the hotter grows the surface the further and the further we look back. Nor can we stop until, at an antiquity so great that I do not venture on any estimate of the date, we discover that this earth must have consisted of glowing hot material. Further and further we can look back, and we see the rocks—or whatever other term we choose to apply to the then ingredients of the earth's crust—in a white-hot and even in a molten condition. Thus our argument has led us to the belief that time was when this now solid globe of ours was a ball of white-hot fluid.
On the argument which I have here used there are just two remarks which I particularly wish to make. Note in the first place, that our reasoning is founded on the fact that the earth is at present, to some extent, heated. It matters not whether this heat be much or little; our argument would have been equally valid had the earth only contained a single particle of its mass at a somewhat higher temperature than the temperature of space. I am, of course, not alluding in this to heat which can be generated by combustion. The other point to which I refer is to remove an objection which may possibly be urged against this line of reasoning. I have argued that because the temperature is continually increasing as we look backwards, that therefore a very great temperature must once have prevailed. Without some explanation this argument is not logically complete. There is, it is well known, the old paradox of the geometric series; you may add a farthing to a halfpenny, and then a half-farthing, and then a quarter-farthing, and then the eighth of a farthing, followed by the sixteenth, and thirty-second, and so on, halving the contribution each time. Now no matter how long you continue this process, even if you went on with it for ever, and thus made an infinite number of contributions, you would never accomplish the task of raising the original halfpenny to the dignity of a penny. An infinite number of quantities may therefore, as this illustration shows, never succeed in attaining any considerable dimensions. Our argument, however, with regard to the increase of heat as we look back is the very opposite of this. It is the essence of a cooling body to lose heat more rapidly in proportion as its temperature is greater. Thus though the one-thousandth of a degree may be all the fall of temperature that our earth now experiences in a twelvemonth, yet in those glowing days when the surface was heated to incandescence, the loss of heat per annum must have been immensely greater than it is now. It therefore follows that the rate of gain of the earth's heat as we look back must be of a different character to that of the geometric series which I have just illustrated; for each addition to the earth's heat, as we look back from year to year, must grow greater and greater, and therefore there is here no shelter for a fallacy in the argument on which the existence of high temperature of primeval times is founded.
The reasoning that I have applied to our earth may be applied in almost similar words to the moon. It is true that we have not any knowledge of the internal nature of the moon at present, nor are we able to point to any active volcanic phenomena at present in progress there in support of the contention that the moon either has now internal heat, or did once possess it. It is, however, impossible to deny the evidence which the lunar craters afford as to the past existence of volcanic activity on our satellite. Heat, therefore, there was once in the moon; and accordingly we are enabled to conclude that, on a retrospect through illimitable periods of time, we must find the moon transformed from that cold and inert body she now seems to a glowing and incandescent mass of molten material. The earth therefore and the moon in some remote ages—not alone anterior to the existence of life, but anterior even to the earliest periods of which geologists have cognizance—must have been both globes of molten materials which have consolidated into the rocks of the present epoch.
We must now revert to the tidal history of the earth-moon system. Did we not show that there was a time when the earth and the moon—or perhaps, I should say, the ingredients of the earth and moon—were close together, were indeed in actual contact? We have now learned, from a wholly different line of reasoning, that in very early ages both bodies were highly heated. Here as elsewhere in this theory we can make little or no attempt to give any chronology, or to harmonize the different lines along which the course of history has run. No one can form the slightest idea as to what the temperature of the earth and of the moon must have been in those primeval ages when they were in contact. It is impossible, however, to deny that they must both have been in a very highly heated state; and everything we know of the matter inclines us to the belief that the temperature of the earth-moon system must at this critical epoch have been one of glowing incandescence and fusion. It is therefore quite possible that these bodies—the moon especially—may have then been not at all of the form we see them now. It has been supposed, and there are some grounds for the supposition, that at this initial stage of earth-moon history the moon materials did not form a globe, but were disposed in a ring which surrounded the earth, the ring being in a condition of rapid rotation. It was at a subsequent period, according to this view, that the substances in the ring gradually drew together, and then by their mutual attractions formed a globe which ultimately consolidated down into the compact moon as we now see it. I must, however, specially draw your attention to the clearly-marked line which divides the facts which dynamics have taught us from those notions which are to be regarded as more or less conjectural. Interpreting the action of the tides by the principles of dynamics, we are assured that the moon was once—or rather the materials of the moon—in the immediate vicinity of the earth. There, however, dynamics leaves us, and unfortunately withholds its accurate illumination from the events which immediately preceded that state of things.
The theory of tidal evolution which I am describing in these lectures is mainly the work of Professor George H. Darwin of Cambridge. Much of the original parts of the theory of the tides was due to Sir William Thomson, and I have also mentioned how Professor Purser contributed an important element to the dynamical theory. It is, however, Darwin who has persistently deduced from the theory all the various consequences which can be legitimately drawn from it. Darwin, for instance, pointed out that as the moon is receding from us, it must, if we only look far enough back, have been once in practical contact with the earth. It is to Darwin also that we owe many of the other parts of a fascinating theory, either in its mathematical or astronomical aspect; but I must take this opportunity of saying, that I do not propose to make Professor Darwin or any of the other mathematicians I have named responsible for all that I shall say in these lectures. I must be myself accountable for the way in which the subject is being treated, as well as for many of the illustrations used, and some of the deductions I have drawn from the subject.
It is almost unavoidable for us to make a surmise as to the cause by which the moon had come into this remarkable position close to the earth at the most critical epoch of earth-moon history.
With reference to this Professor Darwin has offered an explanation, which seems so exceedingly plausible that it is impossible to resist the notion that it must be correct. I will ask you to think of the earth not as a solid body covered largely with ocean, but as a glowing globe of molten material. In a globe of this kind it is possible for great undulations to be set up. Here is a large vase of water, and by displacing it I can cause the water to undulate with a period which depends on the size of the vessel; undulations can be set up in a bucket of water, the period of these undulations being dependent upon the dimensions of the bucket. Similarly in a vast globe of molten material certain undulations could be set up, and those undulations would have a period depending upon the dimensions of this vibrating mass. We may conjecture a mode in which such vibrations could be originated. Imagine a thin shell of rigid material which just encases the globe; suppose this be divided into four quarters, like the four quarters of an orange, and that two of these opposite quarters be rejected, leaving two quarters on the liquid. Now suppose that these two quarters be suddenly pressed in, and then be as suddenly removed—they will produce depressions, of course, on the two opposite quarters, while the uncompressed quarters will become protuberant. In virtue of the mutual attractions between the different particles of the mass, an effort will be made to restore the globular form, but this will of course rather overshoot the mark; and therefore a series of undulations will be originated by which two opposite quarters of the sphere will alternately shrink in and become protuberant. There will be a particular period to this oscillation. For our globe it would appear to be somewhere about an hour and a half or two hours; but there is necessarily a good deal of uncertainty about this point.
We have seen how in those primitive days the earth was spinning around very rapidly; and I have also stated that the earth might at this very critical epoch of its history be compared with a grindstone which is being driven so rapidly that it is on the very brink of rupture. It is remarkable to note, that a cause tending to precipitate a rupture of the earth was at hand. The sun then raised tides in the earth as it does at present. When the earth revolved in a period of some four hours or thereabouts, the high tides caused by the sun succeeded each other at intervals of about two hours. When I speak of tides in this respect, of course I am not alluding to oceanic tides; these were the days long before ocean existed, at least in the liquid form. The tides I am speaking about were raised in the fluids and materials which then constituted the whole of the glowing earth; those tides rose and fell under the throb produced by the sun, just as truly as tides produced in an ordinary ocean. But now note the significant coincidence between the period of the throb produced by the sun-raised tides, and that natural period of vibration which belonged to our earth as a mass of molten material. It therefore follows, that the impulse given to the earth by the sun harmonized in time with that period in which the earth itself was disposed to oscillate. A well-known dynamical principle here comes into play. You see a heavy weight hanging by a string, and in my hand I hold a little slip of wood no heavier than a common pencil; ordinarily speaking, I might strike that heavy weight with this slip of wood, and no effect is produced; but if I take care to time the little blows that I give so that they shall harmonize with the vibrations which the weight is naturally disposed to make, then the effect of many small blows will be cumulative, so much so, that after a short time the weight begins to respond to my efforts, and now you see it has acquired a swing of very considerable amplitude. In Professor Fitzgerald's address to the British Association at Bath last autumn, he gives an account of those astounding experiments of Hertz, in which well-timed electrical impulses broke down an air resistance, and revealed to us ethereal vibrations which could never have been made manifest except by the principle we are here discussing. The ingenious conjecture has been made, that when the earth was thrown into tidal vibrations in those primeval days, these slight vibrations, harmonizing as they did with the natural period of the earth, gradually acquired amplitude; the result being that the pulse of each successive vibration increased at last to such an extent that the earth separated under the stress, and threw off a portion of those semi-fluid materials of which it was composed. In process of time these rejected portions contracted together, and ultimately formed that moon we now see. Such is the origin of the moon which the modern theory of tidal evolution has presented to our notice.
There are two great epochs in the evolution of the earth-moon system—two critical epochs which possess a unique dynamical significance; one of these periods was early in the beginning, while the other cannot arrive for countless ages yet to come. I am aware that in discussing this matter I am entering somewhat largely into mathematical principles; I must only endeavour to state the matter as succinctly as the subject will admit.
In an earlier part of this lecture I have explained how, during all the development of the earth-moon system, the quantity of moment of momentum remains unaltered. The moment of momentum of the earth's rotation added to the moment of momentum of the moon's revolution remains constant; if one of these quantities increase the other must decrease, and the progress of the evolution will have this result, that energy shall be gradually lost in consequence of the friction produced by the tides. The investigation is one appropriate for mathematical formulæ, such as those that can be found in Professor Darwin's memoirs; but nature has in this instance dealt kindly with us, for she has enabled an abstruse mathematical principle to be dealt with in a singularly clear and concise manner. We want to obtain a definite view of the alteration in the energy of the system which shall correspond to a small change in the velocity of the earth's rotation, the moon of course accommodating itself so that the moment of momentum shall be preserved unaltered. We can use for this purpose an angular velocity which represents the excess of the earth's rotation over the angular revolution of the moon; it is, in fact, the apparent angular velocity with which the moon appears to move round the heavens. If we represent by N the angular velocity of the earth, and by M the angular velocity of the moon in its orbit round the earth, the quantity we desire to express is N—M; we shall call it the relative rotation. The mathematical theorem which tells us what we want can be enunciated in a concise manner as follows. The alteration of the energy of the system may be expressed by multiplying the relative rotation by the change in the earth's angular velocity. This result will explain many points to us in the theory, but just at present I am only going to make a single inference from it.
I must advert for a moment to the familiar conception of a maximum or a minimum. If a magnitude be increasing, that is, gradually growing greater and greater, it has obviously not attained a maximum so long as the growth is in progress. Nor if the object be actually decreasing can it be said to be at a maximum either; for then it was greater a second ago than it is now, and therefore it cannot be at a maximum at present. We may illustrate this by the familiar example of a stone thrown up into the air; at first it gradually rises, being higher at each instant than it was previously, until a culminating point is reached, when just for a moment the stone is poised at the summit of its path ere it commences its return to earth again. In this case the maximum point is obtained when the stone, having ceased to ascend, and not having yet commenced to descend, is momentarily at rest.
The same principles apply to the determination of a minimum. As long as the magnitude is declining the minimum has not been reached; it is only when the decline has ceased, and an increase is on the point of setting in, that the minimum can be said to be touched.
The earth-moon system contains at any moment a certain store of energy, and to every conceivable condition of the earth-moon system a certain quantity of energy is appropriate. It is instructive for us to study the different positions in which the earth and the moon might lie, and to examine the different quantities of energy which the system will contain in each of those varied positions. It is however to be understood that the different cases all presuppose the same total moment of momentum.
Among the different cases that can be imagined, those will be of special interest in which the total quantity of energy in the system is a maximum or a minimum. We must for this purpose suppose the system gradually to run through all conceivable changes, with the earth and moon as near as possible, and as far as possible, and in all intermediate positions; we must also attribute to the earth every variety in the velocity of its rotation which is compatible with the preservation of the moment of momentum. Beginning then with the earth's velocity of rotation at its lowest, we may suppose it gradually and continually increased, and, as we have already seen, the change in the energy of the system is to be expressed by multiplying the relative rotation into the change of the earth's angular velocity. It follows from the principles we have already explained, that the maximum or minimum energy is attained at the moment when the alteration is zero. It therefore follows, that the critical periods of the system will arise when the relative rotation is zero, that is, when the earth's rotation on its axis is performed with a velocity equal to that with which the moon revolves around the earth. This is truly a singular condition of the earth-moon system; the moon in such a case would revolve around the earth as if the two bodies were bound together by rigid bonds into what was practically a single solid body. At the present moment no doubt to some extent this condition is realized, because the moon always turns the same face to the earth (a point on which we shall have something to say later on); but in the original condition of the earth-moon system, the earth would also constantly direct the same face to the moon, a condition of things which is now very far from being realized.
It can be shown from the mathematical nature of the problem that there are four states of the earth-moon system in which this condition may be realized, and which are also compatible with the conservation of the moment of momentum. We can express what this condition implies in a somewhat more simple manner. Let us understand by the day the period of the earth's rotation on its axis, whatever that may be, and let us understand by the month the period of revolution of the moon around the earth, whatever value it may have; then the condition of maximum or minimum energy is attained when the day and the month have become equal to each other. Of the four occasions mathematically possible in which the day and the month can be equal, there are only two which at present need engage our attention—one of these occurred near the beginning of the earth and moon's history, the other remains to be approached in the immeasurably remote future. The two remaining solutions are futile, being what the mathematician would describe as imaginary.
There is a fundamental difference between the dynamical conditions in these critical epochs—in one of them the energy of the system has attained a maximum value, and in the other the energy of the system is at a minimum value. It is impossible to over-estimate the significance of these two states of the system.
I may recall the fundamental notion which every one has learned in mechanics, as to the difference between stable and unstable equilibrium. The conceivable possibility of making an egg stand on its end is a practical impossibility, because nature does not like unstable equilibrium, and a body departs therefrom on the least disturbance; on the other hand, stable equilibrium is the position in which nature tends to place everything. A log of wood floating on a river might conceivably float in a vertical position with its end up out of the water, but you never could succeed in so balancing it, because no matter how carefully you adjusted the log, it would almost instantly turn over when you left it free; on the other hand, when the log floats naturally on the water it assumes a horizontal position, to which, when momentarily displaced therefrom, it will return if permitted to do so. We have here an illustration of the contrast between stable and unstable equilibrium. It will be found generally that a body is in equilibrium when its centre of gravity is at its highest point or at its lowest point; there is, however, this important difference, that when the centre of gravity is highest the equilibrium is unstable, and when the centre of gravity is lowest the equilibrium is stable. The potential energy of an egg poised on its end in unstable equilibrium is greater than when it lies on its side in stable equilibrium. In fact, energy must be expended to raise the egg from the horizontal position to the vertical; while, on the other hand, work could conceivably be done by the egg when it passes from the vertical position to the horizontal. Speaking generally, we may say that the stable position indicates low energy, while a redundancy of that valuable agent is suggestive of instability.
We may apply similar principles to the consideration of the earth-moon system. It is true that we have here a series of dynamical phenomena, while the illustrations I have given of stable and unstable equilibrium relate only to statical problems; but we can have dynamical stability and dynamical instability, just as we can have stable and unstable equilibrium. Dynamical instability corresponds with the maximum of energy, and dynamical stability to the minimum of energy.
At that primitive epoch, when the energy of the earth-moon system was a maximum, the condition was one of dynamical instability; it was impossible that it should last. But now mark how truly critical an occurrence this must have been in the history of the earth-moon system, for have I not already explained that it is a necessary condition of the progress of tidal evolution that the energy of the system should be always declining? But here our retrospect has conducted us back to a most eventful crisis, in which the energy was a maximum, and therefore cannot have been immediately preceded by a state in which the energy was greater still; it is therefore impossible for the tidal evolution to have produced this state of things; some other influence must have been in operation at this beginning of the earth-moon system.
Thus there can be hardly a doubt that immediately preceding the critical epoch the moon originated from the earth in the way we have described. Note also that this condition, being one of maximum energy, was necessarily of dynamical instability, it could not last; the moon must adopt either of two courses—it must tumble back on the earth, or it must start outwards. Now which course was the moon to adopt? The case is analogous to that of an egg standing on its end—it will inevitably tumble one way or the other. Some infinitesimal cause will produce a tendency towards one side, and to that side accordingly the egg will fall. The earth-moon system was similarly in an unstable state, an infinitesimal cause might conceivably decide the fate of the system. We are necessarily in ignorance of what the determining cause might have been, but the effect it produced is perfectly clear; the moon did not again return to its mother earth, but set out on that mighty career which is in progress to-day.
Let it be noted that these critical epochs in the earth-moon history arise when and only when there is an absolute identity between the length of the month and the length of the day. It may be proper therefore that I should provide a demonstration of the fact, that the identity between these two periods must necessarily have occurred at a very early period in the evolution.
The law of Kepler, which asserts that the square of the periodic time is proportioned to the cube of the mean distance, is in its ordinary application confined to a comparison between the revolutions of the several planets about the sun. The periodic time of each planet is connected with its average distance by this law; but there is another application of Kepler's law which gives us information of the distance and the period of the moon in former stages of the earth-moon history. Although the actual path of the moon is of course an ellipse, yet that ellipse is troubled, as is well known, by many disturbing forces, and from this cause alone the actual path of the moon is far from being any of those simple curves with which we are so well acquainted. Even were the earth and the moon absolutely rigid particles, perturbations would work all sorts of small changes in the pliant curve. The phenomena of tidal evolution impart an additional element of complexity into the actual shape of the moon's path. We now see that the ellipse is not merely subject to incessant deflections of a periodic nature, it also undergoes a gradual contraction as we look back through time past; but we may, with all needful accuracy for our present purpose, think of the path of the moon as a circle, only we must attribute to that circle a continuous contraction of its radius the further and the further we look back. The alteration in the radius will be even so slow, that the moon will accomplish thousands of revolutions around the earth without any appreciable alteration in the average distance of the two bodies. We can therefore think of the moon as revolving at every epoch in a circle of special radius, and as accomplishing that revolution in a special time. With this understanding we can now apply Kepler's law to the several stages of the moon's past history. The periodic time of each revolution, and the mean distance at which that revolution was performed, will be always connected together by the formula of Kepler. Thus to take an instance in the very remote past. Let us suppose that the moon was at one hundred and twenty thousand miles instead of two hundred and forty thousand, that is, at half its present distance. Applying the law of Kepler, we see that the time of revolution must then have been only about ten days instead of the twenty-seven it is now. Still further, let us suppose that the moon revolves in an orbit with one-tenth of the diameter it has at present, then the cube of 10 being 1000, and the square root of 1000 being 31.6, it follows that the month must have been less than the thirty-first part of what it is at present, that is, it must have been considerably less than one of our present days. Thus you see the month is growing shorter and shorter the further we look back, the day is also growing shorter and shorter; but still I think we can show that there must have been a time when the month will have been at least as short as the day. For let us take the most extreme case in which the moon shall have made the closest possible approximation to the earth. Two globes in contact will have a distance between their centres which is equal to the sum of their radii. Take the earth as having a radius of four thousand miles, and the moon a radius of one thousand miles, the two centres must at their shortest distance be five thousand miles apart, that is, the moon must then be at the forty-eighth part of its present distance from the earth. Now the cube of 48 is 110,592, and the square root of 110,592 is nearly 333, therefore the length of the month will be one-three hundred and thirty-third part of the duration of the month at present; in other words, the moon must revolve around the earth in a period of somewhat about two hours. It seems impossible that the day can ever have been as brief as this. We have therefore proved that, in the course of its contracting duration, the moon must have overtaken the contracting day, and that therefore there must have been a time when the moon was in the vicinity of the earth, and having a day and month of equal period. Thus we have shown that the critical condition of dynamical instability must have occurred in the early period of the earth-moon history, if the agents then in operation were those which we now know. The further development of the subject must be postponed until the next lecture.