Kitabı oku: «Time and Tide: A Romance of the Moon»

View of the Moon two days after first quarter.

From a photograph by Mr. Lewis M. Rutherford.

Frontispiece.

TO

The Members of the London Institution

I DEDICATE

THIS LITTLE BOOK

PREFACE

Having been honoured once again with a request that I should lecture before the London Institution, I chose for my subject the Theory of Tidal Evolution. The kind reception which these lectures received has led to their publication in the present volume. I have taken the opportunity to supplement the lectures as actually delivered by the insertion of some additional matter. I am indebted to my friends Mr. Close and Mr. Rambaut for their kindness in reading the proofs.

Robert S. Ball.

Observatory, Co. Dublin,

April 26, 1889.

LECTURE I

It is my privilege to address you this afternoon on a subject in which science and poetry are blended in a happy conjunction. If there be a peculiar fascination about the earlier chapters of any branch of history, how great must be the interest which attaches to that most primeval of all terrestrial histories which relates to the actual beginnings of this globe on which we stand.

In our efforts to grope into the dim recesses of this awful past, we want the aid of some steadfast light which shall illumine the dark places without the treachery of the will-o'-the-wisp. In the absence of that steadfast light, vague conjectures as to the beginning of things could never be entitled to any more respect than was due to mere matters of speculation.

Of late, however, the required light has been to some considerable extent forthcoming, and the attempt has been made, with no little success, to elucidate a most interesting and wonderful chapter of an exceedingly remote history. To chronicle this history is the object of the present lectures before this Institution.

First, let us be fully aware of the extraordinary remoteness of that period of which our history treats. To attempt to define that period chronologically would be utterly futile. When we have stated that it is more ancient than almost any other period which we can discuss, we have expressed all that we are really entitled to say. Yet this conveys not a little. It directs us to look back through all the ages of modern human history, through the great days of ancient Greece and Rome, back through the times when Egypt and Assyria were names of renown, through the days when Nineveh and Babylon were mighty and populous cities in the zenith of their glory. Back earlier still to those more ancient nations of which we know hardly anything, and still earlier to the prehistoric man, of whom we know less; back, finally, to the days when man first trod on this planet, untold ages ago. Here is indeed a portentous retrospect from most points of view, but it is only the commencement of that which our subject suggests.

For man is but the final product of the long anterior ages during which the development of life seems to have undergone an exceedingly gradual elevation. Our retrospect now takes its way along the vistas opened up by the geologists. We look through the protracted tertiary ages, when mighty animals, now generally extinct, roamed over the continents. Back still earlier through those wondrous secondary periods, where swamps or oceans often covered what is now dry land, and where mighty reptiles of uncouth forms stalked and crawled and swam through the old world and the new. Back still earlier through those vitally significant ages when the sunbeams were being garnered and laid aside for man's use in the great forests, which were afterwards preserved by being transformed into seams of coal. Back still earlier through endless thousands of years, when lustrous fishes abounded in the oceans; back again to those periods characterized by the lower types of life; and still earlier to that incredibly remote epoch when life itself began to dawn on our awakening globe. Even here the epoch of our present history can hardly be said to have been reached. We have to look through a long succession of ages still antecedent. The geologist, who has hitherto guided our view, cannot render us much further assistance; but the physicist is at hand—he teaches us that the warm globe on which life is beginning has passed in its previous stages through every phase of warmth, of fervour, of glowing heat, of incandescence, and of actual fusion; and thus at last our retrospect reaches to that particular period of our earth's past history which is specially illustrated by the modern doctrine of Time and Tide.

The present is the clue to the past. It is the steady application of this principle which has led to such epoch-making labours as those by which Lyell disclosed the origin of the earth's crust, Darwin the origin of species, Max Müller the origin of language. In our present subject the course is equally clear. Study exactly what is going on at present, and then have the courage to apply consistently and rigorously what we have learned from the present to the interpretation of the past.

Thus we begin with the ripple of the tide on the sea-beach which we see to-day. The ebb and the flow of the tide are the present manifestations of an agent which has been constantly at work. Let that present teach us what tides must have done in the indefinite past.

It has been known from the very earliest times that the moon and the tides were connected together—connected, I say, for a great advance had to be made in human knowledge before it would have been possible to understand the true relation between the tides and the moon. Indeed, that relation is so far from being of an obvious character, that I think I have read of a race who felt some doubt as to whether the moon was the cause of the tides, or the tides the cause of the moon. I should, however, say that the moon is not the sole agent engaged in producing this periodic movement of our waters. The sun also arouses a tide, but the solar tide is so small in comparison with that produced by the moon, that for our present purpose we may leave it out of consideration. We must, however, refer to the solar tide at a later period of our discourses, for it will be found to have played a very splendid part at the initial stage of the Earth-Moon History, while in the remote future it will again rise into prominence.

It will be well to set forth a few preliminary figures which shall explain how it comes to pass that the efficiency of the sun as a tide-producing agent is so greatly inferior to that of the moon. Indeed, considering that the sun has a mass so stupendous, that it controls the entire planetary system, how is it that a body so insignificant as the moon can raise a bigger tide on the ocean than can the sun, of which the mass is 26,000,000 times as great as that of our satellite?

This apparent paradox will disappear when we enunciate the law according to which the efficiency of a tide-producing agent is to be estimated. This law is somewhat different from the familiar form in which the law of gravitation is expressed. The gravitation between two distant masses is to be measured by multiplying these masses together, and dividing the product by the square of the distance. The law for expressing the efficiency of a tide-producing agent varies not according to the inverse square, but according to the inverse cube of the distance. This difference in the expression of the law will suffice to account for the superiority of the moon as a tide-producer over the sun. The moon's distance on an average is about one 386th part of that of the sun, and thus it is easy to show that so far as the mere attraction of gravitation is concerned, the efficiency of the sun's force on the earth is about one hundred and seventy-five times as great as the force with which the moon attracts the earth. That is of course calculated under the law of the inverse square. To determine the tidal efficiency we have to divide this by three hundred and eighty-six, and thus we see that the tidal efficiency of the sun is less than half that of the moon.

When the solar tide and the lunar tide are acting in unison, they conspire to produce very high tides and very low tides, or, as we call them, spring tides. On the other hand, when the sun is so placed as to give us a low tide while the moon is producing a high tide, the net result that we actually experience is merely the excess of the lunar tide over the solar tide; these are what we call neap tides. In fact, by very careful and long-continued observations of the rise and fall of the tides at a particular port, it becomes possible to determine with accuracy the relative ranges of spring tides and neap tides; and as the spring tides are produced by moon plus sun, while the neap tides are produced by moon minus sun, we obtain a means of actually weighing the relative masses of the sun and moon. This is one of the remarkable facts which can be deduced from a prolonged study of the tides.

The demonstration of the law of the tide-producing force is of a mathematical character, and I do not intend in these lectures to enter into mathematical calculations. There is, however, a simple line of reasoning which, though it falls far short of actual demonstration, may yet suffice to give a plausible reason for the law.

The tides really owe their origin to the fact that the tide-producing agent operates more powerfully on those parts of the tide-exhibiting body which are near to it, than on the more distant portions of the same. The nearer the two bodies are together, the larger proportionally will be the differences in the distances of its various parts from the tide-producing body; and on this account the leverage, so to speak, of the action by which the tides are produced is increased. For instance, if the two bodies were brought within half their original distance of each other, the relative size of each body, as viewed from the other, will be doubled; and what we have called the leverage of the tide-producing ability will be increased twofold. The gravitation also between the two bodies is increased fourfold when the distance is halved, and consequently, the tide-producing ability is doubled for one reason, and increased fourfold again by another; hence, the tides will be increased eightfold when the distance is reduced to one half. Now, as eight is the cube of two, this illustration may be taken as a verification of the law, that the efficiency of a body as a tide-producer varies inversely as the cube of the distance between it and the body on which the tides are being raised.

For simplicity we may make the assumption that the whole of the earth is buried beneath the ocean, and that the moon is placed in the plane of the equator. We may also entirely neglect for the present the tides produced by the sun, and we shall also make the further assumption that friction is absent. What friction is capable of doing we shall, however, refer to later on. The moon will act on the ocean and deform it, so that there will be high tide along one meridian, and high tide also on the opposite meridian. This is indeed one of the paradoxes by which students are frequently puzzled when they begin to learn about the tides. That the moon should pull the water up in a heap on one side seems plausible enough. High tide will of course be there; and the student might naturally think that the water being drawn in this way into a heap on one side, there will of course be low tide on the opposite side of the earth. A natural assumption, perhaps, but nevertheless a very wrong one. There are at every moment two opposite parts of the earth in a condition of high water; in fact, this will be obvious if we remember that every day, or, to speak a little more accurately, in every twenty-four hours and fifty one minutes, we have on the average two high tides at each locality. Of course this could not be if the moon raised only one heap of high water, because, as the moon only appears to revolve around the earth once a day, or, more accurately, once in that same average period of twenty-four hours and fifty-one minutes, it would be impossible for us to have high tides succeeding each other as they do in periods a little longer than twelve hours, if only one heap were carried round the earth.

The first question then is, as to how these two opposite heaps of water are placed in respect to the position of the moon. The most obvious explanation would seem to be, that the moon should pull the waters up into a heap directly underneath it, and that therefore there should be high water underneath the moon. As to the other side, the presence of a high tide there was, on this theory, to be accounted for by the fact that the moon pulled the earth away from the waters on the more remote side, just as it pulled the waters away from the more remote earth on the side underneath the moon. It is, however, certainly not the case that the high tide is situated in the simple position that this law would indicate, and which we have represented in Fig. 1, where the circular body is the earth, the ocean surrounding which is distorted by the action of the tides.

Fig. 1

We have here taken an oval to represent the shape into which the water is supposed to be forced or drawn by the tidal action of the tide-producing body. This may possibly be a correct representation of what would occur on an ideal globe entirely covered with a frictionless ocean. But as our earth is not covered entirely by water, and as the ocean is very far from being frictionless, the ideal tide is not the tide that we actually know; nor is the ideal tide represented by this oval even an approximation to the actual tides to which our oceans are subject. Indeed, the oval does not represent the facts at all, and of this it is only necessary to adduce a single fact in demonstration. I take the fundamental issue so often debated, as to whether in the ocean vibrating with ideal tides the high water or the low water should be under the moon. Or to put the matter otherwise; when we represent the displaced water by an oval, is the long axis of the oval to be turned to the moon, as generally supposed, or is it to be directed at right angles therefrom? If the ideal tides were in any degree representative of the actual tides, so fundamental a question as this could be at once answered by an appeal to the facts of observation. Even if friction in some degree masked the phenomena, surely one would think that the state of the actual tides should still enable us to answer this question.

But a study of the tides at different ports fails to realize this expectation. At some ports, no doubt, the tide is high when the moon is on the meridian. In that case, of course, the high water is under the moon, as apparently ought to be the case invariably, on a superficial view. But, on the other hand, there are ports where there is often low water when the moon is crossing the meridian. Yet other ports might be cited in which every intermediate phase could be observed. If the theory of the tides was to be the simple one so often described, then at every port noon should be the hour of high water on the day of the new moon or of the full moon, because then both tide-exciting bodies are on the meridian at the same time. Even if the friction retarded the great tidal wave uniformly, the high tide on the days of full or change should always occur at fixed hours; but, unfortunately, there is no such delightful theory of the tides as this would imply. At Greenock no doubt there is high water at or about noon on the day of full or change; and if it could be similarly said that on the day of full or change there was high water everywhere at local noon, then the equilibrium theory of the tides, as it is called, would be beautifully simple. But this is not the case. Even around our own coasts the discrepancies are such as to utterly discredit the theory as offering any practical guide. At Aberdeen the high tide does not appear till an hour later than the doctrine would suggest. It is two hours late at London, three at Tynemouth, four at Tralee, five at Sligo, and six at Hull. This last port would be indeed the haven of refuge for those who believe that the low tide ought to be under the moon. At Hull this is no doubt the case; and if at all other places the water behaved as it does at Hull, why then, of course, it would follow that the law of low water under the moon was generally true. But then this would not tally with the condition of affairs at the other places I have named; and to complete the cycle I shall add a few more. At Bristol the high water does not get up until seven hours after the moon has passed the meridian, at Arklow the delay is eight hours, at Yarmouth it is nine, at the Needles it is ten hours, while lastly, the moon has nearly got back to the meridian again ere it has succeeded in dragging up the tide on which Liverpool's great commerce so largely depends.

Nor does the result of studying the tides along other coasts beside our own decide more conclusively on the mooted point. Even ports in the vast ocean give a very uncertain response. Kerguelen Island and Santa Cruz might seem to prove that the high tide occurs under the moon, but unfortunately both Fiji and Ascension seem to present us with an equally satisfactory demonstration, that beneath the moon is the invariable home of low water.

I do not mean to say that the study of the tides is in other respects such a confused subject as the facts I have stated would seem to indicate. It becomes rather puzzling, no doubt, when we compare the tides at one port with the tides elsewhere. The law and order are, then, by no means conspicuous, they are often hardly discernible. But when we confine our attention to the tides at a single port, the problem becomes at once a very intelligible one. Indeed, the investigation of the tides is an easy subject, if we are contented with a reasonably approximate solution; should, however, it be necessary to discuss fully the tides at any port, the theory of the method necessary for doing so is available, and a most interesting and beautiful theory it certainly is.

Let us then speak for a few moments about the methods by which we can study the tides at a particular port. The principle on which it is based is a very simple one.

It is the month of August, the 18th, we shall suppose, and we are going to enjoy a delicious swim in the sea. We desire, of course, to secure a high tide for the purpose of doing so, and we call an almanac to help us. I refer to the Thom's Dublin Directory, where I find the tide to be high at 10h. 14m. on the morning of the 18th of August. That will then be the time to go down to the baths at Howth or Kingstown.

But what I am now going to discourse to you about is not the delights of sea-bathing, it is rather a different inquiry. I want to ask, How did the people who prepared that almanac know years beforehand, that on that particular day the tide would be high at that particular hour? How do they predict for every day the hour of high water? and how comes it to pass that these predictions are invariably correct?

We first refer to that wonderful book, the Nautical Almanac. In that volume the movements of the moon are set forth with full detail; and among other particulars we can learn on page iv of every month the mean time of the moon's meridian passage. It appears that on the day in question the moon crossed the meridian at 11h. 23m. Thus we see there was high water at Dublin at 10h. 14m., and 1h. 9m. later, that is, at 11h. 23m., the moon crossed the meridian.

Let us take another instance. There is a high tide at 3.40 P.M. on the 25th August, and again the infallible Nautical Almanac tells us that the moon crossed the meridian at 5h. 44m., that is, at 2h. 4m. after the high water.

In the first case the moon followed the tide in about an hour, and in the second case the moon followed in about two hours. Now if we are to be satisfied with a very rough tide rule for Dublin, we may say generally that there is always a high tide an hour and a half before the moon crosses the meridian. This would not be a very accurate rule, but I can assure you of this, that if you go by it you will never fail of finding a good tide to enable you to enjoy your swim. I do not say this rule would enable you to construct a respectable tide-table. A ship-owner who has to creep up the river, and to whom often the inches of water are material, will require far more accurate tables than this simple rule could give. But we enter into rather complicated matters when we attempt to give any really accurate methods of computation. On these we shall say a few words presently. What I first want to do, is to impress upon you in a simple way the fact of the relation between the tide and the moon.

To give another illustration, let us see how the tides at London Bridge are related to the moon. On Jan. 1st, 1887, it appeared that the tide was high at 6h. 26m. P.M., and that the moon had crossed the meridian 56m. previously; on the 8th Jan. the tide was high at 0h. 43m. P.M., and the moon had crossed the meridian 2h. 1m. previously. Therefore we would have at London Bridge high water following the moon's transit in somewhere about an hour and a half.

I choose a day at random, for example—the 12th April. The moon crosses the upper meridian at 3h. 39m. A.M., and the lower meridian at 4h 6m. P.M. Adding an hour and a half to each would give the high tides at 5h 9m. A.M. and 5h. 36m. P.M.; as a matter of fact, they are 4h. 58m. A.M. and 5h. 20m. P.M.

But these illustrations are sufficient. We find that at London, in a general way, high water appears at London Bridge about an hour and a half after the moon has passed the meridian of London. It so happens that the interval at Dublin is about the same, i.e. an hour and a half; only that in the latter case the high water precedes the moon by that interval instead of following it. We may employ the same simple process at other places. Choose two days about a week distant; find on each occasion the interval between the transit of the moon and the time of high water, then the mean of these two differences will always give some notion of the interval between high water and the moon's transit. If then we take from the Nautical Almanac the time of the moon's transit, and apply to it the correction proper for the port, we shall always have a sufficiently good tide-table to guide us in choosing a suitable time for taking our swim or our walk by the sea-side; though if you be the captain of a vessel, you will not be so imprudent as to enter port without taking counsel of the accurate tide-tables, for which we are indebted to the Admiralty.

Every one who visits the sea-side, or who lives at a sea-port, should know this constant for the tides, which affect him and his movements so materially. If he will discover it from his own experience, so much the better.

The first point to be ascertained is the time of high water. Do not take this from any local table; you ought to observe it for yourself. You will go to the pier head, or, better still, to some place where the rise and fall of the mere waves of the sea will not embarrass you in your work. You must note by your watch the time when the tide is highest. An accurate way of doing this will be to have a scale on which you can measure the height at intervals of five minutes about the time of high water. You will then be able to conclude the time at which the tide was actually at its highest point; but even if no great accuracy be obtainable, you can still get much interesting information, for you will without much difficulty be right within ten minutes or a quarter of an hour.

The correction for the port is properly called the “establishment,” this being the average time of high water on the days of full and change of the moon at the particular port in question.

We can considerably amend the elementary notion of the tides which the former method has given us, if we adopt the plan described by Dr. Whewell in the first four editions of the Admiralty Manual of Scientific Inquiry. We speak of the interval between the transit of the moon and the time of high water as the luni-tidal interval. Of course at full and change this is the same thing as the establishment, but for other phases of the moon the establishment must receive a correction before being used as the luni-tidal interval. The correction is given by the following table—

Thus at a port where the establishment was 3h. 25m., let us suppose that the transit of the moon took place at 6 P.M.; then we correct the establishment by -60m., and find the luni-tidal interval to be 2h. 25m., and accordingly the high water takes place at 8h. 25m. P.M.

But even this method is only an approximation. The study of the tides is based on accurate observation of their rise and fall on different places round the earth. To show how these observations are to be made, and how they are to be discussed and reduced when they have been made, I may refer to the last edition of the Admiralty Manual of Scientific Inquiry, 1886. For a complete study of the tides at any port a self-registering tide-gauge should be erected, on which not alone the heights and times of high and low water should be depicted, but also the continuous curve which shows at any time the height of the water. In fact, the whole subject of the practical observation and discussion and prediction of tides is full of valuable instruction, and may be cited as one of the most complete examples of the modern scientific methods.

In the first place, the tide-gauge itself is a delicate instrument; it is actuated by a float which rises and falls with the water, due provision being made that the mere influence of waves shall not make it to oscillate inconveniently. The motion of the float when suitably reduced by mechanism serves to guide a pencil, which, acting on the paper round a revolving drum, gives a faithful and unintermitting record of the height of the water.

Thus what the tide-gauge does is to present to us a long curved line of which the summits correspond to the heights of high water, while the depressions are the corresponding points of low water. The long undulations of this curve are, however, very irregular. At spring tides, when the sun and the moon conspire, the elevations rise much higher and the depressions sink much lower than they do at neap tides, when the high water raised by the moon is reduced by the action of the sun. There are also many minor irregularities which show the tides to be not nearly such simple phenomena as might be at first supposed. But what we might hastily think of as irregularities are, in truth, the most interesting parts of the whole phenomena. Just as in the observations of the planets the study of the perturbations has led us to results of the widest interest and instruction, so it is these minor phenomena of the tides which seem most pregnant with scientific interest.

The tide-gauge gives us an elaborate curve. How are we to interpret that curve? Here indeed a most beautiful mathematical theorem comes to our aid. Just as ordinary sounds consist of a number of undulations blended together, so the tidal wave consists of a number of distinct undulations superposed. Of these the ordinary lunar tide and the ordinary solar tide are the two principal; but there are also minor undulations, harmonics, so to speak, some originating from the moon, some originating from the sun, and some from both bodies acting in concert.

In the study of sound we can employ an acoustic apparatus for the purpose of decomposing any proposed note, and finding not only the main undulation itself, but the several superposed harmonics which give to the note its timbre. So also we can analyze the undulation of the tide, and show the component parts. The decomposition is effected by the process known as harmonic analysis. The principle of the method may be very simply described. Let us fix our attention on any particular “tide,” for so the various elements are denoted. We can always determine beforehand, with as much accuracy as we may require, what the period of that tide will be. For instance, the period of the lunar semi-diurnal tide will of course be half the average time occupied by the moon to travel round from the meridian of any place until it regains the same meridian; the period of the lunar diurnal tide will be double as great; and there are fortnightly tides, and others of periods still greater. The essential point to notice is, that the periods of these tides are given by purely astronomical considerations from the periods of the motions theory, and do not depend upon the actual observations.

We measure off on the curve the height of the tide at intervals of an hour. The larger the number of such measures that are available the better; but even if there be only three hundred and sixty or seven hundred and twenty consecutive hours, then, as shown by Professor G. H. Darwin in the Admiralty Manual already referred to, it will still be possible to obtain a very competent knowledge of the tides in the particular port where the gauge has been placed.

The art (for such indeed it may be described) of harmonic analysis consists in deducing from the hourly observations the facts with regard to each of the constituent tides. This art has been carried to such perfection, that it has been reduced to a very simple series of arithmetical operations. Indeed it has now been found possible to call in the aid of ingenious mechanism, by which the labours of computation are entirely superseded. The pointer of the harmonic analyzer has merely to be traced over the curve which the tide-gauge has drawn, and it is the function of the machine to decompose the composite undulation into its parts, and to exhibit the several constituent tides whose confluence gives the total result.

As if nothing should be left to complete the perfection of a process which, both from its theoretical and its practical sides, is of such importance, a machine for predicting tides has been designed, constructed, and is now in ordinary use. When by the aid of the harmonic analysis the effectiveness of the several constituent tides affecting a port have become fully determined, it is of course possible to predict the tides for that port. Each “tide” is a simple periodic rise and fall, and we can compute for any future time the height of each were it acting alone. These heights can all be added together, and thus the height of the water is obtained. In this way a tide-table is formed, and such a table when complete will express not alone the hours and heights of high water on every day, but the height of the water at any intervening hour.

LECTURE II

Starting from that fitting commencement of earth-moon history which the critical epoch affords, we shall now describe the dynamical phenomena as the tidal evolution progressed. The moon and the earth initially moved as a solid body, each bending the same face towards the other; but as the moon retreated, and as tides began to be raised on the earth, the length of the day began to increase, as did also the length of the month. We know, however, that the month increased more rapidly than the day, so that a time was reached when the month was twice as long as the day; and still both periods kept on increasing, but not at equal rates, for in progress of time the month grew so much more rapidly than the day, that many days had to elapse while the moon accomplished a single revolution. It is, however, only necessary for us to note those stages of the mighty progress which correspond to special events. The first of such stages was attained when the month assumed its maximum ratio to the day. At this time, the month was about twenty-nine days, and the epoch appears to have occurred at a comparatively recent date if we use such standards of time as tidal evolution requires; though measured by historical standards, the epoch is of incalculable antiquity. I cannot impress upon you too often the enormous magnitude of the period of time which these phenomena have required for their evolution. Professor Darwin's theory affords but little information on this point, and the utmost we can do is to assign a minor limit to the period through which tidal evolution has been in progress. It is certain that the birth of the moon must have occurred at least fifty million years ago, but probably the true period is enormously greater than this. If indeed we choose to add a cipher or two to the figure just printed, I do not think there is anything which could tell us that we have over-estimated the mark. Therefore, when I speak of the epoch in which the month possessed the greatest number of days as a recent one, it must be understood that I am merely speaking of events in relation to the order of tidal evolution. Viewed from this standpoint, we can show that the epoch is a recent one in the following manner. At present the month consists of a little more than twenty-seven days, but at this maximum period to which I have referred the month was about twenty-nine days; from that it began to decline, and the decline cannot have proceeded very far, for even still there are only two days less in the month than at the time when the month had the greatest number of days. It thus follows that the present epoch—the human epoch, as we may call it—in the history of the earth has fallen at a time when the progress of tidal evolution is about half-way between the initial and the final stage. I do not mean half-way in the sense of actual measurement of years; indeed, from this point it would seem that we cannot yet be nearly half-way, for, vast as are the periods of time that have elapsed since the moon first took its departure from the earth, they fall far short of that awful period of time which will intervene between the present moment and the hour when the next critical state of earth-moon history shall have been attained. In that state the day is destined once again to be equal to the month, just as was the case in the initial stage. The half-way stage will therefore in one sense be that in which the proportion of the month to the day culminates. This is the stage which we have but lately passed; and thus it is that at present we may be said to be almost half-way through the progress of tidal evolution.

My narrative of the earth-moon evolution must from this point forward cease to be retrospective. Having begun at that critical moment when the month and day were first equal, we have traced the progress of events to the present hour. What we have now to say is therefore a forecast of events yet to come. So far as we can tell, no agent is likely to interfere with the gradual evolution caused by the tides, which dynamical principles have disclosed to us. As the years roll on, or perhaps, I should rather say, as thousands of years and millions of years roll on, the day will continue to elongate, or the earth to rotate more slowly on its axis. But countless ages must elapse before another critical stage of the history shall be reached. It is needless for me to ponder over the tedious process by which this interesting epoch is reached. I shall rather sketch what the actual condition of our system will be when that moment shall have arrived. The day will then have expanded from the present familiar twenty-four hours up to a day more than twice, more than five, even more than fifty times its present duration. In round numbers, we may say that this great day will occupy one thousand four hundred of our ordinary hours. To realize the critical nature of the situation then arrived at, we must follow the corresponding evolution through which the moon passes. From its present distance of two hundred and forty thousand miles, the moon will describe an ever-enlarging orbit; and as it does so the duration of the month will also increase, until at last a point will be reached when the month has become more than double its present length, and has attained the particular value of one thousand four hundred hours. We are specially to observe that this one-thousand four-hundred-hour month will be exactly reached when the day has also expanded to one thousand four hundred hours; and the essence of this critical condition, which may be regarded as a significant point of tidal evolution, is that the day and the month have again become equal. The day and the month were equal at the beginning, the day and the month will be equal at the end. Yet how wide is the difference between the beginning and the end. The day or the month at the end is some hundreds of times as long as the month or the day at the beginning.

I have already fully explained how, in any stage of the evolutionary progress in which the day and the month became equal, the energy of the system attained a maximum or a minimum value. At the beginning the energy was a maximum; at the end the energy will be a minimum. The most important consequences follow from this consideration. I have already shown that a condition of maximum energy corresponded to dynamic instability. Thus we saw that the earth-moon history could not have commenced without the intervention of some influence other than tides at the beginning. Now let us learn what the similar doctrine has to tell us with regard to the end. The condition then arrived at is one of dynamical stability; for suppose that the system were to receive a slight alteration, by which the moon went out a little further, and thus described a larger orbit, and so performed more than its share of the moment of spin. Then the earth would have to do a little less spinning, because, under all circumstances, the total quantity of spin must be preserved unaltered. But the energy being at a minimum, such a small displacement must of course produce a state of things in which the energy would be increased. Or if we conceived the moon to come in towards the earth, the moon would then contribute less to the total moment of momentum. It would therefore be incumbent on the earth to do more; and accordingly the velocity of the earth's rotation would be augmented. But this arrangement also could only be produced by the addition of some fresh energy to the system, because the position from which the system is supposed to have been disturbed is one of minimum energy.

No disturbance of the system from this final position is therefore conceivable, unless some energy can be communicated to it. But this will demonstrate the utter incompetency of the tides to shift the system by a hair's breadth from this position; for it is of the essence of the tides to waste energy by friction. And the transformations of the system which the tides have caused are invariably characterized by a decline of energy, the movements being otherwise arranged so that the total moment of momentum shall be preserved intact. Note, how far we were justified in speaking of this condition as a final one. It is final so far as the lunar tides are concerned; and were the system to be screened from all outer interference, this accommodation between the earth and the moon would be eternal.

There is indeed another way of demonstrating that a condition of the system in which the day has assumed equality with the month must necessarily be one of dynamical equilibrium. We have shown that the energy which the tides demand is derived not from the mere fact that there are high tides and low tides, but from the circumstance that these tides do rise and fall; that in falling and rising they do produce currents; and it is these currents which generate the friction by which the earth's velocity is slowly abated, its energy wasted, and no doubt ultimately dissipated as heat. If therefore we can make the ebbing and the flowing of the tides to cease, then our argument will disappear. Thus suppose, for the sake of illustration, that at a moment when the tides happened to be at high water in the Thames, such a change took place in the behaviour of the moon that the water always remained full in the Thames, and at every other spot on the earth remained fixed at the exact height which it possessed at this particular moment. There would be no more tidal friction, and therefore the system would cease to course through that series of changes which the existence of tidal friction necessitates.

But if the tide is always to be full in the Thames, then the moon must be always in the same position with respect to the meridian, that is, the moon must always be fixed in the heavens over London. In fact, the moon must then revolve around the earth just as fast as London does—the month must have the same length as the day. The earth must then show the same face constantly to the moon, just as the moon always does show the same face towards the earth; the two globes will in fact revolve as if they were connected with invisible bonds, which united them into a single rigid body.

We need therefore feel no surprise at the cessation of the progress of tidal evolution when the month and the day are equal, for then the movement of moon-raised tides has ceased. No doubt the same may be said of the state at the beginning of the history, when the day and the month had the brief and equal duration of a few hours. While the equality of the two periods lasted there could be no tides, and therefore no progress in the direction of tidal evolution. There is, however, the profound difference of stability and instability between the two cases; the most insignificant disturbance of the system at the initial stage was sufficient to precipitate the revolving moon from its condition of dynamical equilibrium, and to start the course of tidal evolution in full vigour. If, however, any trifling derangement should take place in the last condition of the system, so that the month and the day departed slightly from equality, there would instantly be an ebbing and a flowing of the tides; and the friction generated by these tides would operate to restore the equality because this condition is one of dynamical stability.

It will thus be seen with what justice we can look forward to the day and month each of fourteen hundred hours as a finale to the progress of the luni-tidal evolution. Throughout the whole of this marvellous series of changes it is always necessary to remember the one constant and invariable element—the moment of momentum of the system which tides cannot alter. Whatever else the friction can have done, however fearful may have been the loss of energy by the system, the moment of momentum which the system had at the beginning it preserves unto the end. This it is which chiefly gives us the numerical data on which we have to rely for the quantitative features of tidal evolution.

We have made so many demands in the course of these lectures on the capacity of tidal friction to accomplish startling phenomena in the evolution of the earth-moon system, that it is well for us to seek for any evidence that may otherwise be obtainable as to the capacity of tides for the accomplishment of gigantic operations. I do not say that there is any doubt which requires to be dispelled by such evidence, for as to the general outlines of the doctrine of tidal evolution which has been here sketched out there can be no reasonable ground for mistrust; but nevertheless it is always desirable to widen our comprehension of any natural phenomena by observing collateral facts. Now there is one branch of tidal action to which I have as yet only in the most incidental way referred. We have been speaking of the tides in the earth which are made to ebb and flow by the action of the moon; we have now to consider the tides in the moon, which are there excited by the action of the earth. For between these two bodies there is a reciprocity of tidal-making energy—each of them is competent to raise tides in the other. As the moon is so small in comparison with the earth, and as the tides on the moon are of but little significance in the progress of tidal evolution, it has been permissible for us to omit them from our former discussion. But it is these tides on the moon which will afford us a striking illustration of the competency of tides for stupendous tasks. The moon presents a monument to show what tides are able to accomplish.

Fig. 3.—The Moon.

I must first, however, explain a difficulty which is almost sure to suggest itself when we speak of tides on the moon. I shall be told that the moon contains no water on its surface, and how then, it will be said, can tides ebb and flow where there is no sea to be disturbed? There are two answers to this difficulty; it is no doubt true that the moon seems at present entirely devoid of water in so far as its surface is exposed to us, but it is by no means certain that the moon was always in this destitute condition. There are very large features marked on its map as “seas”; these regions are of a darker hue than the rest of the moon's surface, they are large objects often many hundreds of miles in diameter, and they form, in fact, those dark patches on the brilliant surface which are conspicuous to the unaided eye, and are represented in Fig. 3. Viewed in a telescope these so-called seas, while clearly possessing no water at the present time, are yet widely different from the general aspect of the moon's surface. It has often been supposed that great oceans once filled these basins, and a plausible explanation has even been offered as to how the waters they once contained could have vanished. It has been thought that as the mineral substances deep in the interior of our satellite assumed the crystalline form during the progress of cooling, the demand for water of crystallization required for incorporation with the minerals was so great that the oceans of the moon became entirely absorbed. It is, however, unnecessary for our present argument that this theory should be correct. Even if there never was a drop of water found on our satellite, the tides in its molten materials would be quite sufficient for our purpose; anything that tides could accomplish would be done more speedily by vast tides of flowing lava than by merely oceanic tides.

There can be no doubt that tides raised on the moon by the earth would be greater than the tides raised on the earth by the moon. The question is, however, not a very simple one, for it depends on the masses of both bodies as well as on their relative dimensions. In so far as the masses are concerned, the earth being more than eighty times as heavy as the moon, the tides would on this account be vastly larger on the moon than on the earth. On the other hand, the moon's diameter being much less than that of the earth, the efficiency of a tide-producing body in its action on the moon would be less than that of the same body at the same distance in its action on the earth; but the diminution of the tides from this cause would be not so great as their increase from the former cause, and therefore the net result would be to exhibit much greater tides on the moon than on the earth.

Suppose that the moon had been originally endowed with a rapid movement of rotation around its axis, the effect of the tides on that rotation would tend to check its velocity just in the same way as the tides on the earth have effected a continual elongation of the day. Only as the tides on the moon were so enormously great, their capacity to check the moon's speed would have corresponding efficacy. In addition to this, the mass of the moon being so small, it could only offer feeble resistance to the unceasing action of the tide, and therefore our satellite must succumb to whatever the tides desired ages before our earth would have been affected to a like extent. It must be noticed that the influence of the tidal friction is not directed to the total annihilation of the rotation of the two bodies affected by it; the velocity is only checked down until it attains such a point that the speed in which each body rotates upon its axis has become equal to that in which it revolves around the tide-producer. The practical effect of such an adjustment is to make the tide-agitated body turn a constant face towards its tormentor.

I may here note a point about which people sometimes find a little difficulty. The moon constantly turns the same face towards the earth, and therefore people are sometimes apt to think that the moon performs no rotation whatever around its own axis. But this is indeed not the case. The true inference to be drawn from the constant face of the moon is, that the velocity of rotation about its own axis is equal to that of its rotation around the earth; in fact, the moon revolves around the earth in twenty-seven days, and its rotation about its axis is performed in twenty-seven days also. You may illustrate the movement of the moon around the earth by walking around a table in a room, keeping all the time your face turned towards the table; in such a case as this you not only perform a motion of revolution, but you also perform a rotation in an equal period. The proof that you do rotate is to be found in the fact that during the movement your face is being directed successively to all the points of the compass. There is no more singular fact in the solar system than the constancy of the moon's face to the earth. The periods of rotation and revolution are both alike; if one of these periods exceeded the other by an amount so small as the hundredth part of a second, the moon would in the lapse of ages permit us to see that other side which is now so jealously concealed. The marvellous coincidence between these two periods would be absolutely inexplicable, unless we were able to assign it to some physical cause. It must be remembered that in this matter the moon occupies a unique position among the heavenly host. The sun revolves around on its axis in a period of twenty-five or twenty-six days—thus we see one side of the sun as frequently as we see the other. The side of the sun which is turned towards us to-day is almost entirely different from that we saw a fortnight ago. Nor is the period of the sun's rotation to be identified with any other remarkable period in our system. If it were equal to the length of the year, for instance, or if it were equal to the period of any of the other planets, then it could hardly be contended that the phenomenon as presented by the moon was unique; but the sun's period is not simply related, or indeed related at all, to any of the other periodic times in the system. Nor do we find anything like the moon's constancy of face in the behaviour of the other planets. Jupiter turns now one face to us and then another. Nor is his rotation related to the sun or related to any other body, as our moon's motion is related to us. It has indeed been thought that in the movements of the satellites of Jupiter a somewhat similar phenomenon may be observed to that in the motion of our own satellite. If this be so, the causes whereby this phenomenon is produced are doubtless identical in the two cases.

So remarkable a coincidence as that which the moon's motion shows could not reasonably be explained as a mere fortuitous circumstance; nor need we hesitate to admit that a physical explanation is required when we find a most satisfactory one ready for our acceptance, as was originally pointed out by Helmholtz.

There can be no doubt whatever that the constancy of the moon's face is the work of ancient tides, which have long since ceased to act. We have shown that if the moon's rotation had once been too rapid to permit of the same face being always directed towards us, the tides would operate as a check by which the velocity of that rotation would be abated. On the other hand, if the moon rotated so slowly that its other face would be exposed to us in the course of the revolution, the tides would then be dragged violently over its surface in the direction of its rotation; their tendency would thus be to accelerate the speed until the angular velocity of rotation was equal to that of revolution. Thus the tides would act as a controlling agent of the utmost stringency to hurry the moon round when it was not turning fast enough, and to arrest the motion when going too fast. Peace there would be none for the moon until it yielded absolute compliance to the tyranny of the tides, and adjusted its period of rotation with exact identity to its period of revolution. Doubtless this adjustment was made countless ages ago, and since that period the tides have acted so as to preserve the adjustment, as long as any part of the moon was in a state sufficiently soft or fluid to respond to tidal impression. The present state of the moon is a monument to which we may confidently appeal in support of our contention as to the great power of the tides during the ages which have passed; it will serve as an illustration of the future which is reserved for our earth in ages yet to come, when our globe shall have also succumbed to tidal influence.

It is owing to the smallness of the moon relatively to the earth that the tidal process has reached a much more advanced stage in the moon than it has on the earth; but the moon is incessant in its efforts to bring the earth into the same condition which it has itself been forced to assume. Thus again we look forward to an epoch in the inconceivably remote future when tidal thraldom shall be supreme, and when the earth shall turn the same face to the moon, as the moon now turns the same face to the earth.

In the critical state of things thus looming in the dim future, the earth and the moon will continue to perform this adjusted revolution in a period of about fourteen hundred hours, the two bodies being held, as it were, by invisible bands. Such an arrangement might be eternal if there were no intrusion of tidal influence from any other body; but of course in our system as we actually find it the sun produces tides as well as the moon; and the solar tides being at present much less than those originated by the moon, we have neglected them in the general outlines of the theory. The solar tides, however, must necessarily have an increasing significance. I do not mean that they will intrinsically increase, for there seems no reason to apprehend any growth in their actual amount; it is their relative importance to the lunar tides that is the augmenting quantity. As the final state is being approached, and as the velocity of the earth's rotation is approximating to the angular velocity with which the moon revolves around it, the ebbing and the flowing of the lunar tides must become of evanescent importance; and this indeed for a double reason, partly on account of the moon's greatly augmented distance, and partly on account of the increasing length of the lunar day, and the extremely tardy movements of ebb and flow that the lunar tides will then have. Thus the lunar tides, so far as their dynamical importance is concerned, will ultimately become zero, while the solar tides retain all their pristine efficiency.

We have therefore to examine the dynamical effects of solar tides on the earth and moon in the critical stage to which the present course of things tends. The earth will then rotate in a period of about fifty-seven of its present days; and considering that the length of the day, though so much greater than our present day, is still much less than the year, it follows that the solar tides must still continue so as to bring the earth's velocity of rotation to a point even lower than it has yet attained. In fact, if we could venture to project our glance sufficiently far into the future, it would seem that the earth must ultimately have its velocity checked by the sun-raised tides, until the day itself had become equal to the year. The dynamical considerations become, however, too complex for us to follow them, so that I shall be content with merely pointing out that the influence of the solar tides will prevent the earth and moon from eternally preserving the relations of bending the same face towards each other; the earth's motion will, in fact, be so far checked, that the day will become longer than the month.

Thus the doctrine of tidal evolution has conducted us to a prospect of a condition of things which will some time be reached, when the moon will have receded to a distance in which the month shall have become about fifty-seven days, and when the earth around which this moon revolves shall actually require a still longer period to accomplish its rotation on its axis. Here is an odd condition for a planet with its satellite; indeed, until a dozen years ago it would have been pronounced inconceivable that a moon should whirl round a planet so quickly that its journey was accomplished in less than one of the planet's own days. Arguments might be found to show that this was impossible, or at least unprecedented. There is our own moon, which now takes twenty-seven days to go round the earth; there is Jupiter, with four moons, and the nearest of these to the primary goes round in forty-two and a half hours. No doubt this is a very rapid motion; but all those matters are much more lively with Jupiter than they are here. The giant planet himself does not need ten hours for a single rotation, so that you see his nearest moon still takes between two and three Jovian days to accomplish a single revolution. The example of Saturn might have been cited to show that the quickest revolution that any satellite could perform must still require at least twice as long as the day in which the planet performed its rotation. Nor could the rotation of the planets around the sun afford a case which could be cited. For even Mercury, the nearest of all the planets to the sun of which the existence is certainly known, and therefore the most rapid in its revolution, requires eighty-eight days to get round once; and in the mean time the sun has had time to accomplish between three and four rotations. Indeed, the analogies would seem to have shown so great an improbability in the conclusion towards which tidal evolution points, that they would have contributed a serious obstacle to the general acceptance of that theory.

But in 1877 an event took place so interesting in astronomical history, that we have to look back to the memorable discovery of Uranus in 1781 before we can find a parallel to it in importance. Mars had always been looked upon as one of the moonless planets, though grounds were not wanting for the surmise that probably moons to Mars really existed. It was under the influence of this belief that an attempt was made by Professor Asaph Hall at Washington to make a determined search, and see if Mars might not be attended by satellites large enough to be discoverable. The circumstances under which this memorable inquiry was undertaken were eminently favourable for its success. The orbit of Mars is one which possesses an exceptionally high eccentricity; it consequently happens that the oppositions during which the planet is to be observed vary very greatly in the facilities they afford for a search like that contemplated by Professor Hall. It is obviously advantageous that the planet should be situated as near as possible to the earth, and in the opposition in 1877 the distance was almost at the lowest point it is capable of attaining; but this was not the only point in which Professor Hall was favoured; he had the use of a telescope of magnificent proportions and of consummate optical perfection. His observatory was also placed in Washington, so that he had the advantage of a pure sky and of a much lower latitude than any observatory in Great Britain is placed at. But the most conspicuous advantage of all was the practised skill of the astronomer himself, without which all these other advantages would have been but of little avail. Great success rewarded his well-designed efforts; not alone was one satellite discovered which revolved around the planet in a period conformable with that of other similar cases, but a second little satellite was found, which accomplished its revolution in a wholly unexpected and unprecedented manner. The day of Mars himself, that is, the period in which he can accomplish a rotation around his axis, very closely approximates to our own day, being in fact half an hour longer. This little satellite, the inner and more rapid of the pair, requires for a single revolution a period of only seven hours thirty-nine minutes, that is to say, the little body scampers more than three times round its primary before the primary itself has finished one of its leisurely rotations. Here was indeed a striking fact, a unique fact in our system, which riveted the attention of astronomers on this most beautiful discovery.