Read Ebook: Time and Tide: A Romance of the Moon by Ball Robert S Robert Stawell

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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.

The art 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has now been found possible to call in the aid of ingenious mechanism, by which the labours of computation are entirely superseded.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.

The computations necessary for this purpose are no doubt simple, so far as their principle is concerned; but they are exceedingly tedious, and any process must be welcomed which affords mitigation of a task so laborious. The entire theory of the tides owes much to Sir William Thomson in the methods of observation and in the methods of reduction. He has now completed the practical parts of the subject by inventing and constructing the famous tide-predicting engine.

The principle of this engine is comparatively simple. There is a chain which at one end is fixed, and at the other end carries the pencil which is pressed against the revolving drum on which the prediction is to be inscribed. Between its two ends the chain passes up and down over pulleys. Each pulley corresponds to one of the "tides," and there are about a dozen altogether, some of which exercise but little effect. Of course if the centres of the pulleys were all fixed the pen could not move, but the centre of each pulley describes a circle with a radius proportional to the amplitude of the corresponding tide, and in a time proportional to the period of that tide. When these pulleys are all set so as to start at the proper phases, the motion is produced by turning round a handle which makes the drum rotate, and sets all the pulleys in motion. The tide curve is thus rapidly drawn out; and so expeditious is the machine, that the tides of a port for an entire year can be completely worked out in a couple of hours.

While the student or the philosopher who seeks to render any account of the tide on dynamical grounds is greatly embarrassed by the difficulties introduced by friction, we, for our present purpose in the study of the great romance of modern science opened up to us by the theory of the tides, have to welcome friction as the agent which gives to the tides their significance from our point of view.

There is the greatest difference between the height of the rise and fall of the tide at different localities. Out in mid-ocean, for instance, an island like St. Helena is washed by a tide only about three feet in range; an enclosed sea like the Caspian is subject to no appreciable tides whatever, while the Mediterranean, notwithstanding its connection with the Atlantic, is still only subject to very inconsiderable tides, varying from one foot to a few feet. The statement that water always finds its own level must be received, like many another proposition in nature, with a considerable degree of qualification. Long ere one tide could have found its way through the Straits of Gibraltar in sufficient volume to have appreciably affected the level of the great inland sea, its effects would have been obliterated by succeeding tides. On the other hand, there are certain localities which expose a funnel-shape opening to the sea; into these the great tidal wave rushes, and as it passes onwards towards the narrow part, the waters become piled up so as to produce tidal phenomena of abnormal proportions. Thus, in our own islands, we have in the Bristol Channel a wide mouth into which a great tide enters, and as it hurries up the Severn it produces the extraordinary phenomenon of the Bore. The Bristol Channel also concentrates the great wave which gives Chepstow and Cardiff a tidal range of thirty-seven or thirty-eight feet at springs, and forces the sea up the river Avon so as to give Bristol a wonderful tide. There is hardly any more interesting spot in our islands for the observation of tides than is found on Clifton Suspension Bridge. From that beautiful structure you look down on a poor and not very attractive stream, which two hours later becomes transformed into a river of ample volume, down which great ships are navigated. But of all places in the world, the most colossal tidal phenomena are those in the Bay of Fundy. Here the Atlantic passes into a long channel whose sides gradually converge. When the great pulse of the tide rushes up this channel, it is gradually accumulated into a mighty volume at the upper end, the ebb and flow of which at spring tides extends through the astonishing range of not less than fifty feet.

These discrepancies between the tides at different places are chiefly due to the local formations of the coasts and the sea-beds. Indeed, it seems that if the whole earth were covered with an uniform and deep ocean of water, the tides would be excessively feeble. On no other supposition can we reasonably account for the fact that our barometric records fail to afford us any very distinct evidence as to the existence of tides in the atmosphere. For you will, of course, remember that our atmosphere may be regarded as a deep and vast ocean of air, which embraces the whole earth, extending far above the loftiest summits of the mountains.

It is one of the profoundest of nature's laws that wherever friction takes place, energy has to be consumed. Perhaps I ought rather to say transformed, for of course it is now well known that consumption of energy in the sense of absolute loss is impossible. Thus, when energy is expended in moving a body in opposition to the force of friction, or in agitating a liquid, the energy which disappears in its mechanical form reappears in the form of heat. The agitation of water by paddles moving through it warms the water, and the accession of heat thus acquired measures the energy which has been expended in making the paddles rotate. The motion of a liquid of which the particles move among each other with friction, can only be sustained by the incessant degradation of energy from the mechanical form into the lower form of diffused heat. Thus the very fact that the tides are ebbing and flowing, and that there is consequently incessant friction going on among all the particles of water in the ocean, shows us that there must be some great store of energy constantly available to supply the incessant draughts made upon it by the daily oscillation of the tides. In addition to the mere friction between the particles of water, there are also many other ways in which the tides proclaim to us that there is some great hoard of energy which is continually accessible to their wants. Stand on the bank of an estuary or river up and down which a great tidal current ebbs and flows; you will see the water copiously charged with sediment which the tide is bearing along. Engineers are well aware of the potency of the tide as a vehicle for transporting stupendous quantities of sand or mud. A sand-bank impedes the navigation of a river; the removal of that sand-bank would be a task, perhaps, conceivably possible by the use of steam dredges and other appliances, whereby vast quantities of sand could be raised and transported to another locality where they would be innocuous. It is sometimes possible to effect the desired end by applying the power of the tide. A sea-wall judiciously thrown out will sometimes concentrate the tide into a much narrower channel. Its daily oscillations will be accomplished with greater vehemence, and as the tide rushes furiously backwards and forwards over the obstacle, the incessant action will gradually remove it, and the impediment to navigation may be cleared away. Here we actually see the tides performing a piece of definite and very laborious work, to accomplish which by the more ordinary agents would be a stupendous task.

In some places the tides are actually harnessed so as to accomplish useful work. I have read that underneath old London Bridge there used formerly to be great water-wheels, which were turned by the tide as it rushed up the river, and turned again, though in the opposite way, by the ebbing tide. These wheels were, I believe, employed to pump up water, though it does not seem obvious for what purposes the water would have been suitable. Indeed in the ebb and flow all round our coasts there is a potential source of energy which has hitherto been allowed to run to waste. The tide could be utilized in various ways. Many of you will remember the floating mills on the Rhine. They are vessels like paddle steamers anchored in the rapid current. The flow of the river makes the paddles rotate, and thus the machinery in the interior is worked. Such craft moored in a rapid tide-way could also be made to convey the power of the tides into the mechanism of the mill. Or there is still another method which has been employed, and which will perhaps have a future before it in those approaching times when the coal-cellars of England shall be exhausted. Imagine on the sea-coast a large flat extent which is inundated twice every day by the tide. Let us build a stout wall round this area, and provide it with a sluice-gate. Open the gate as the tide rises, and the great pond will be filled; then at the moment of high water close the sluice, and the pond-full will be impounded. If at low tide the sluice be opened the water will rush tumultuously out. Now suppose that a water-wheel be provided, so that the rapid rush of water from the exit shall fall upon its blades; then a source of power is obviously the result.

At present, however, such a contrivance would naturally find no advocates, for of course the commercial aspect of the question is that which will decide whether the scheme is practicable and economical. The issue indeed can be very simply stated. Suppose that a given quantity of power be required--let us say that of one hundred horse. Then we have to consider the conditions under which a contrivance of the kind we have sketched shall yield a power of this amount. Sir William Thomson, in a very interesting address to the British Association at York in 1881, discussed this question, and I shall here make use of the facts he brought forward on that occasion. He showed that to obtain as much power as could be produced by a steam-engine of one hundred horse power, a very large reservoir would be required. It is doubtful indeed whether there would be many localities on the earth which would be suitable for the purpose. Suppose, however, an estuary could be found which had an area of forty acres; then if a wall were thrown across the mouth so that the tide could be impounded, the total amount of power that could be yielded by a water-wheel worked by the incessant influx and efflux of the tide would be equal to that yielded by the one hundred horse engine, running continuously from one end of the year to the other.

There are many drawbacks to a tide-mill of this description. In the first place, its situation would naturally be far removed from other conveniences necessary for manufacturing purposes. Then too there is the great irregularity in the way in which the power is rendered available. At certain periods during the twenty-four hours the mill would stop running, and the hours when this happened would be constantly changing. The inconvenience from the manufacturer's point of view of a deficiency of power during neap tides might not be compensated by the fact that he had an excessive supply of power at spring-tides. Before tide-mills could be suitable for manufacturing purposes, some means must be found for storing away the energy when it is redundant, and applying it when its presence is required. We should want in fact for great sources of energy some contrivance which shall fulfil the same purpose as the accumulators do in an electrical installation.

Even then, however, the financial consideration remains, as to whether the cost of building the dam and maintaining the tide-mill in good order will not on the whole exceed the original price and the charges for the maintenance of a hundred horse power steam-engine. There cannot be a doubt that in this epoch of the earth's history, so long as the price of coal is only a few shillings a ton, the tide-mill, even though we seem to get its power without current expense, is vastly more expensive than a steam-engine. Indeed, Sir William Thomson remarks, that wherever a suitable tidal basin could be found, it would be nearly as easy to reclaim the land altogether from the sea. And if this were in any locality where manufactures were possible, the commercial value of forty acres of reclaimed land would greatly exceed all the expenses attending the steam-engine. But when the time comes, as come it apparently will, that the price of coal shall have risen to several pounds a ton, the economical aspect of steam as compared with other prime movers will be greatly altered; it will then no doubt be found advantageous to utilize great sources of energy, such as Niagara and the tides, which it is now more prudent to let run to waste.

For my argument, however, it matters little that the tides are not constrained to do much useful work. They are always doing work of some kind, whether that be merely heating the particles of water by friction, or vaguely transporting sand from one part of the ocean to the other. Useful work or useless work are alike for the purpose of my argument. We know that work can never be done unless by the consumption or transformation of energy. For each unit of work that is done--whether by any machine or contrivance, by the muscles of man or any other animal, by the winds, the waves, or the tides, or in any other way whatever--a certain equivalent quantity of energy must have been expended. When, therefore, we see any work being performed, we may always look for the source of energy to which the machine owes its efficiency. In fact, it is the old story illustrated, that perpetual motion is impossible. A mechanical device, however ingenious may be the construction, or however accurate the workmanship, can never possess what is called perpetual motion. It is needless to enter into details of any proposed contrivance of wheels, of pumps, of pulleys; it is sufficient to say that nothing in the shape of mechanism can work without friction, that friction produces heat, that heat is a form of energy, and that to replace the energy consumed in producing the heat there must be some source from which the machine is replenished if its motion is to be continued indefinitely.

Let us first see what the sources of energy can possibly be on which the tides are permitted to draw. Our course is simplified by the fact that the energy of which we have to speak is of a mechanical description, that is to say, not involving heat or other more obscure forms of energy. A simple type of energy is that possessed by a clock-weight after the clock has been wound. A store of power is thus laid up which is gradually doled out during the week in small quantities, second by second, to sustain the motion of the pendulum. The energy in this case is due to the fact that the weight is attracted by the earth, and is yielded according as the weight sinks downwards. In the separation between two mutually attracting bodies, a store of energy is thus implied. What we learn from an ordinary clock may be extended to the great bodies of the universe. The moon is a gigantic globe separated from our earth by a distance of 240,000 miles. The attraction between these two bodies always tends to bring them together. No doubt the moon is not falling towards the earth as the descending clock-weight is doing. We may, in fact, consider the moon, so far as our present object is concerned, to be revolving almost in a circle, of which the earth is the centre. If the moon, however, were to be stopped, it would at once commence to rush down towards the earth, whither it would arrive with an awful crash in the course of four or five days. It is fortunately true that the moon does not behave thus; but it has the ability of doing so, and thus the mere separation between the earth and the moon involves the existence of a stupendous quantity of energy, capable under certain conditions of undergoing transformation.

There is also another source of mechanical energy besides that we have just referred to. A rapidly moving body possesses, in virtue of its motion, a store of readily available energy, and it is easy to show that energy of this type is capable of transformation into other types. Think of a cannon-ball rushing through the air at a speed of a thousand feet per second; it is capable of wreaking disaster on anything which it meets, simply because its rapid motion is the vehicle by which the energy of the gunpowder is transferred from the gun to where the blow is to be struck. Had the cannon been directed vertically upwards, then the projectile, leaving the muzzle with the same initial velocity as before, would soar up and up, with gradually abating speed, until at last it reached a turning-point, the elevation of which would depend upon the initial velocity. Poised for a moment at the summit, the cannon-ball may then be likened to the clock-weight, for the entire energy which it possessed by its motion has been transformed into the statical energy of a raised weight. Thus we see these two forms of energy are mutually interchangeable. The raised weight if allowed to fall will acquire velocity, or the rapidly moving weight if directed upwards will acquire altitude.

The quantity of energy which can be conveyed by a rapidly moving body increases greatly with its speed. For instance, if the speed of the body be doubled, the energy will be increased fourfold, or, in general, the energy which a moving body possesses may be said to be proportional to the square of its speed. Here then we have another source of the energy present in our earth-moon system; for the moon is hurrying along in its path with a speed of two-thirds of a mile per second, or about twice or three times the speed of a cannon-shot. Hence the fact that the moon is continuously revolving in a circle shows us that it possesses a store of energy which is nine times as great as that which a cannon-ball as massive as the moon, and fired with the ordinary velocity, would receive from the powder which discharged it.

Thus we see that the moon is endowed with two sources of energy, one of which is due to its separation from the earth, and the other to the speed of its motion. Though these are distinct, they are connected together by a link which it is important for us to comprehend. The speed with which the moon revolves around the earth is connected with the moon's distance from the earth. The moon might, for instance, revolve in a larger circle than that which it actually pursues; but if it did so, the speed of its motion would have to be appropriately lessened. The orbit of the moon might have a much smaller radius than it has at present, provided that the speed was sufficiently increased to compensate for the increased attraction which the earth would exercise at the lessened distance. Indeed, I am here only stating what every one is familiar with under the form of Kepler's Law, that the square of the periodic time is in proportion to the cube of the mean distance. To each distance of the moon therefore belongs an appropriate speed. The energy due to the moon's position and the energy due to its motion are therefore connected together. One of these quantities cannot be altered without the other undergoing change. If the moon's orbit were increased there would be a gain of energy due to the enlarged distance, and a loss of energy due to the diminished speed. These would not, however, exactly compensate. On the whole, we may represent the total energy of the moon as a single quantity, which increases when the distance of the moon from the earth increases, and lessens when the distance from the earth to the moon lessens. For simplicity we may speak of this as moon-energy.

But the most important constituent of the store of energy in the earth-moon system is that contributed by the earth itself. I do not now speak of the energy due to the velocity of the earth in its orbit round the sun. The moon indeed participates in this equally with the earth, but it does not affect those mutual actions between the earth and moon with which we are at present concerned. We are, in fact, discussing the action of that piece of machinery the earth-moon system; and its action is not affected by the circumstance that the entire machine is being bodily transported around the sun in a great annual voyage. This has little more to do with the action of our present argument than has the fact that a man is walking about to do with the motions of the works of the watch in his pocket. We shall, however, have to allude to this subject further on.

The energy of the earth which is significant in the earth-moon theory is due to the earth's rotation upon its axis. We may here again use as an illustration the action of machinery; and the special contrivance that I now refer to is the punching-engine that is used in our ship-building works. In preparing a plate of iron to be riveted to the side of a ship, a number of holes have to be made all round the margin of the plate. These holes must be half an inch or more in diameter, and the plate is sometimes as much as, or more than, half an inch in thickness. The holes are produced in the metal by forcing a steel punch through it; and this is accomplished without even heating the plate so as to soften the iron. It is needless to say that an intense force must be applied to the punch. On the other hand, the distance through which the punch has to be moved is comparatively small. The punch is attached to the end of a powerful lever, the other end of the lever is raised by a cam, so as to depress the punch to do its work. An essential part of the machine is a small but heavy fly-wheel connected by gearing with the cam.

This fly-wheel when rapidly revolving contains within it, in virtue of its motion, a large store of energy which has gradually accumulated during the time that the punch is not actually in action. The energy is no doubt originally supplied from a steam-engine. What we are especially concerned with is the action of the rapidly rotating wheel as a reservoir in which a large store of energy can be conveniently maintained until such time as it is wanted. In the action of punching, when the steel die comes down upon the surface of the plate, a large quantity of energy is suddenly demanded to force the punch against the intense resistance it experiences; the energy for this purpose is drawn from the store in the fly-wheel, which experiences no doubt a check in its velocity, to be regained again from the energy of the engine during the interval which elapses before the punch is called on to make the next hole.

Another illustration of the fly-wheel on a splendid scale is seen in our mighty steel works, where ponderous rails are being manufactured. A white-hot ingot of steel is presented to a pair of powerful rollers, which grip the steel, and send it through at the other side both compressed and elongated. Tremendous power is required to meet the sudden demand on the machine at the critical moment. To obtain this power an engine of stupendous proportions is sometimes attached directly to the rollers, but more frequently an engine of rather less horse-power will be used, the might of this engine being applied to giving rapid rotation to an immense fly-wheel, which may thus be regarded as a reservoir full of energy. The rolling mills then obtain from this store in the fly-wheel whatever energy is necessary for their gigantic task.

Reverting now to the earth-moon system, the energy which that system contains consists essentially of two parts--the moon-energy, whose composite character I have already explained, and the earth-energy, which has its origin solely in the rotation of the earth on its axis. It is necessary to observe that these are essentially distinct--there is no necessary relation between the speed of the earth's rotation and the distance of the moon, such as there is between the distance of the moon and the speed with which it revolves in its orbit.

For completeness, it ought to be added that there is also some energy due to the moon's rotation on its axis, but this is very small for two reasons: first, because the moon is small compared with the earth, and second, because the angular velocity of the moon is also very small compared with that of the earth. We may therefore dismiss as insignificant the contributions from this source of energy to the sum total.

In the earth-moon system there is no engine at hand to restore the losses of energy which are inevitable when work has to be done. But we have seen that work is done; we have shown, in fact, that the tides are at present doing work, and have been doing work for as long a period in the past as our imagination can extend to. The energy which this work has necessitated can only have been drawn from the existing store in the system; that energy consists of two parts--the moon-energy and the earth's rotation energy. The problem therefore for us to consider is, which of these two banks the tides have drawn on to meet their constant expenditure. This is not a question that can be decided offhand; in fact, if we attempt to decide it in an offhand manner we shall certainly go wrong. It seems so very plausible to say that as the moon causes the tides, therefore the energy which these tides expend should be contributed by the moon. But this is not the case. It actually happens that though the moon does cause the tides, yet when those tides consume energy they draw it not from the distant moon, but from the vast supply which they find ready to their hand, stored up in the rotation of the earth.

The demonstration of this is not a very simple matter; in fact, it is so far from being simple that many philosophers, including some eminent ones too, while admitting that of course the tides must have drawn their energy from one or other or both of these two sources, yet found themselves unable to assign how the demand was distributed between the two conceivable sources of supply.

Let us look at the earth-moon system. The law of the conservation of moment of momentum may, with sufficient accuracy for our present purpose, be interpreted to mean that the total quantity of spin in the system remains unaltered. In our system the spin is threefold; there is first the rotation of the earth on its axis, there is the rotation of the moon on its axis, and then there is the orbital revolution of the moon around the earth. The law to which we refer asserts that the total quantity of these three spins, each estimated in the proper way, will remain constant. It matters not that tides may ebb and flow, or that the distribution of the spin shall vary, but its total amount remains inflexibly constant. One constituent of the total amount--that is, the rotation of the moon on its axis--is so insignificant, that for our present purposes it may be entirely disregarded. We may therefore assert that the amount of spin in the earth, due to its rotation round its axis, added to the amount of spin in the moon due to its revolution round the earth, remains unalterable. If one of these quantities change by increase or by decrease, the other must correspondingly change by decrease or by increase. If, therefore, from any cause, the earth began to spin a little more quickly round its axis, the moon must do a little less spin; and consequently, it must shorten its distance from the earth. Or suppose that the earth's velocity of rotation is abated, then its contribution to the total amount of spin is lessened; the deficiency must therefore be made up by the moon, but this can only be done by an enlargement of the moon's orbit. I should add, as a caution, that these results are true only on the supposition that the earth-moon system is isolated from all external interference. With this proviso, however, it matters not what may happen to the earth or moon, or what influence one of them may exert upon the other, no matter what tides may be raised, no matter even if the earth fly into fragments, the whole quantity of spin of all those fragments would, if added to the spin of the moon, yield the same unalterable total. We are here in possession of a most valuable dynamical principle. We are not concerned with any special theory as to the action of the tides; it is sufficient for us that in some way or other the tides have been caused by the moon, and that being so, the principle of the conservation of spin will apply.

Were the earth and the moon both rigid bodies, then there could be of course no tides on the earth, it being rigid and devoid of ocean. The rotation of the earth on its axis would therefore be absolutely without change, and therefore the necessary condition of the conservation of spin would be very simply attained by the fact that neither of the constituent parts changed. The earth, however, not being entirely rigid, and being subject to tides, this simple state of things cannot continue; there must be some change in progress.

I have already shown that the fact of the ebbing and the flowing of the tide necessitates an expenditure of energy, and we saw that this energy must come either from that stored up in the earth by its rotation, or from that possessed by the moon in virtue of its distance and revolution. The law of the conservation of spin will enable us to decide at once as to whence the tides get their energy. Suppose they took it from the moon, the moon would then lose in energy, and consequently come nearer the earth. The quantity of spin contributed by the moon would therefore be lessened, and accordingly the spin to be made up by the earth would be increased. That means, of course, that the velocity of the earth rotating on its axis must be increased, and this again would necessitate an increase in the earth's rotational energy. It can be shown, too, that to keep the total spin right, the energy of the earth would have to gain more than the moon would have lost by revolving in a smaller orbit. Thus we find that the total quantity of energy in the system would be increased. This would lead to the absurd result that the action of the tides manufactured energy in our system. Of course, such a doctrine cannot be true; it would amount to a perpetual motion! We might as well try to get a steam-engine which would produce enough heat by friction not only to supply its own boilers, but to satisfy all the thermal wants of the whole parish. We must therefore adopt the other alternative. The tides do not draw their energy from the moon; they draw it from the store possessed by the earth in virtue of its rotation.

The earth therefore loses some of its velocity of rotation; consequently it does less than its due share of the total quantity of spin, and an increased quantity of spin must therefore be accomplished by the moon; but this can only be done by an enlargement of its orbit. Thus there are two great consequences of the tides in the earth-moon system--the days are getting longer, the moon is receding further.

These points are so important that I shall try and illustrate them in another way, which will show, at all events, that one and both of these tidal phenomena commend themselves to our common sense. Have we not shown how the tides in their ebb and flow are incessantly producing friction, and have we not also likened the earth to a great wheel? When the driver wants to stop a railway train the brakes are put on, and the brake is merely a contrivance for applying friction to the circumference of a wheel for the purpose of checking its motion. Or when a great weight is being lowered by a crane, the motion is checked by a band which applies friction on the circumference of a wheel, arranged for the special purpose. Need we then be surprised that the friction of the tides acts like a brake on the earth, and gradually tends to check its mighty rotation? The progress of lengthening the day by the tides is thus readily intelligible. It is not quite so easy to see why the ebbing and the flowing of the tide on the earth should actually have the effect of making the moon to retreat; this phenomenon is in deference to a profound law of nature, which tells us that action and reaction are equal and opposite to each other. If I might venture on a very homely illustration, I may say that the moon, like a troublesome fellow, is constantly annoying the earth by dragging its waters backward and forward by means of tides; and the earth, to free itself from this irritating interference, tries to push off the aggressor and to make him move further away.

We therefore conclude finally, that the tides are making the day longer and sending the moon away further. It is the development of the consequences of these laws that specially demands our attention in these lectures. We must have the courage to look at the facts unflinchingly, and deduce from them all the wondrous consequences they involve. Their potency arises from a characteristic feature--they are unintermitting. Most of the great astronomical changes with which we are ordinarily familiar are really periodic: they gradually increase in one direction for years, for centuries, or for untold ages; but then a change comes, and the increase is changed into a decrease, so that after the lapse of becoming periods the original state of things is restored. Such periodic phenomena abound in astronomy. There is the annual fluctuation of the seasons; there is the eighteen or nineteen year period of the moon; there is the great period of the precession of the equinoxes, amounting to twenty-six thousand years; and then there is the stupendous Annus Magnus of hundreds of thousands of years, during which the earth's orbit itself breathes in and out in response to the attraction of the planets. But these periodic phenomena, however important they may be to us mere creatures of a day, are insignificant in their effects on the grand evolution through which the celestial bodies are passing. The really potent agents in fashioning the universe are those which, however slow or feeble they may seem to be, are still incessant in their action. The effect which a cause shall be competent to produce depends not alone upon the intensity of that cause, but also upon the time during which it has been in operation. From the phenomena of geology, as well as from those of astronomy, we know that this earth and the system to which it belongs has endured for ages, not to be counted by scores of thousands of years, or, as Prof. Tyndall has so well remarked, "Not for six thousand years, nor for sixty thousand years, nor six hundred thousand years, but for aeons of untold millions." Those slender agents which have devoted themselves unceasingly to the accomplishment of a single task may in this long lapse of time have accomplished results of stupendous magnitude. In famed stalactite caverns we are shown a colossal figure of crystal extending from floor to roof, and the formation of that column is accounted for when we see a tiny drop falling from the roof above to the floor beneath. A lifetime may not suffice for that falling drop to add an appreciable increase to the stalactite down which it trickles, or to the growing stalagmite on which it falls; but when the operation has been in progress for immense ages, it is capable of the formation of the stately column. Here we have an illustration of an influence which, though apparently trivial, acquires colossal significance when adequate time is afforded. It is phenomena of this kind which the student of nature should most narrowly watch, for they are the real architects of the universe.

The tidal consequences which we have already demonstrated are emphatically of this non-periodic class--the day is always lengthening, the moon is always retreating. To-day is longer than yesterday; to-morrow will be longer than to-day. It cannot be said that the change is a great one; it is indeed too small to be appreciable even by our most delicate observations. In one thousand years the alteration in the length of a day is only a small fraction of a second; but what may be a very small matter in one thousand years can become a very large one in many millions of years. Thus it is that when we stretch our view through immense vistas of time past, or when we look forward through immeasurable ages of time to come, the alteration in the length of the day will assume the most startling proportions, and involve the most momentous consequences.

Let us first look back. There was a time when the day, instead of being the twenty-four hours we now have, must have been only twenty-three hours, How many millions of years ago that was I do not pretend to say, nor is the point material for our argument; suffice it to say, that assuming, as geology assures us we may assume, the existence of these aeons of millions of years, there was once a time when the day was not only one hour shorter, but was even several hours less than it is at present. Nor need we stop our retrospect at a day of even twenty, or fifteen, or ten hours long; we shall at once project our glance back to an immeasurably remote epoch, at which the earth was spinning round in a time only one sixth or even less of the length of the present day. There is here a reason for our retrospect to halt, for at some eventful period, when the day was about three or four hours long, the earth must have been in a condition of a very critical kind.

It is well known that fearful accidents occasionally happen where large grindstones are being driven at a high speed. The velocity of rotation becomes too great for the tenacity of the stone to withstand the stress; a rupture takes place, the stone flies in pieces, and huge fragments are hurled around. For each particular grindstone there is a certain special velocity depending upon its actual materials and character, at which it would inevitably fly in pieces. I have once before likened our earth to a wheel; now let me liken it to a grindstone. There is therefore a certain critical velocity of rotation for the earth at which it would be on the brink of rupture. We cannot exactly say, in our ignorance of the internal constitution of the earth, what length of day would be the shortest possible for our earth to have consistently with the preservation of its integrity; we may, however, assume that it will be about three or four hours, or perhaps a little less than three. The exact amount, however, is not really very material to us; it would be sufficient for our argument to assert that there is a certain minimum length of day for which the earth can hold together. In our retrospect, therefore, through the abyss of time past our view must be bounded by that state of the earth when it is revolving in this critical period. With what happened before that we shall not at present concern ourselves. Thus we look back to a time at the beginning of the present order of things, when the day was only some three or four hours long.

Let us now look at the moon, and examine where it must have been during these past ages. As the moon is gradually getting further and further from us at present, so, looking back into past time, we find that the moon was nearer and nearer to the earth the further back our view extends; in fact, concentrating our attention solely on essential features, we may say that the path of the moon is a sort of spiral which winds round and round the earth, gradually getting larger, though with extreme slowness. Looking back we see this spiral gradually coiling in and in, until in a retrospect of millions of years, instead of its distance from the earth being 240,000 miles, it must have been much less. There was a time when the moon was only 200,000 miles away; there was a time many millions of years ago, when the moon was only 100,000 miles away. Nor can we here stop our retrospect; we must look further and further back, and follow the moon's spiral path as it creeps in and in towards the earth, until at last it appears actually in contact with that great globe of ours, from which it is now separated by a quarter of a million of miles.

Surely the tides have thus led us to the knowledge of an astounding epoch in our earth's past history, when the earth is spinning round in a few hours, and when the moon is, practically speaking, in contact with it. Perhaps I should rather say, that the materials of our present moon were in this situation, for we would hardly be entitled to assume that the moon then possessed the same globular form in which we see it now. To form a just apprehension of the true nature of both bodies at this critical epoch, we must study their concurrent history as it is disclosed to us by a totally different line of reasoning.

Drop, then, for a moment all thought of tides, and let us bring to our aid the laws of heat, which will disclose certain facts in the ancient history of the earth-moon system perhaps as astounding as those to which the tides have conducted us. In one respect we may compare these laws of heat with the laws of the tides; they are both alike non-periodic, their effects are cumulative from age to age, and imagination can hardly even impose a limit to the magnificence of the works they can accomplish. Our argument from heat is founded on a very simple matter. It is quite obvious that a heated body tends to grow cold. I am not now speaking of fires or of actual combustion whereby heat is produced; I am speaking merely of such heat as would be possessed by a red-hot poker after being taken from the fire, or by an iron casting after the metal has been run into the mould. In such cases as this the general law holds good, that the heated body tends to grow cold. The cooling may be retarded no doubt if the passage of heat from the body is impeded. We can, for instance, retard the cooling of a teapot by the well-known practice of putting a cosy upon it; but the law remains that, slowly or quickly, the heated body will tend to grow colder. It seems almost puerile to insist with any emphasis on a point so obvious as this, but yet I frequently find that people do not readily apprehend all the gigantic consequences that can flow from a principle so simple. It is true that a poker cools when taken from the fire; we also find that a gigantic casting weighing many tons will grow gradually cold, though it may require days to do so. The same principle will extend to any object, no matter how vast it may happen to be. Were that great casting 2000 miles in diameter, or were it 8000 miles in diameter, it will still steadily part with its heat, though no doubt the process of cooling becomes greatly prolonged with an increase in the dimensions of the heated body. The earth and the moon cannot escape from the application of these simple principles.

Let us first speak of the earth. There are multitudes of volcanoes in action at the present moment in various countries upon this earth. Now whatever explanation may be given of the approximate cause of the volcanic phenomena, there can be no doubt that they indicate the existence of heat in the interior of the earth. It may possibly be, as some have urged, that the volcanoes are merely vents for comparatively small masses of subterranean molten matter; it may be, as others more reasonably, in my opinion, believe, that the whole interior of the earth is at the temperature of incandescence, and that the eruptions of volcanoes and the shocks of earthquakes are merely consequences of the gradual shrinkage of the external crust, as it continually strives to accommodate itself to the lessening bulk of the fluid interior. But whichever view we may adopt, it is at least obvious that the earth is in part, at all events, a heated body, and that the heat is not in the nature of a combustion, generated and sustained by the progress of chemical action. No doubt there may be local phenomena of this description, but by far the larger proportion of the earth's internal heat seems merely the fervour of incandescence. It is to be likened to the heat of the molten iron which has been run into the sand, rather than to the glowing coals in the furnace in which that iron has been smelted.

There is one volcanic outbreak of such exceptional interest in these modern times that I cannot refrain from alluding to it. Doubtless every one has heard of that marvellous eruption of Krakatoa, which occurred on August 26th and 27th, 1883, and gives a unique chapter in the history of volcanic phenomena. Not alone was the eruption of Krakatoa alarming in its more ordinary manifestations, but it was unparalleled both in the vehemence of the shock and in the distance to which the effects of the great eruption were propagated. I speak not now of the great waves of ocean that inundated the coasts of Sumatra and Java, and swept away thirty-six thousand people, nor do I allude to the intense darkness which spread for one hundred and eighty miles or more all round. I shall just mention the three most important phenomena, which demonstrate the energy which still resides in the interior of our earth. Place a terrestrial globe before you, and fix your attention on the Straits of Sunda; think also of the great atmospheric ocean some two or three hundred miles deep which envelopes our earth. When a pebble is tossed into a pond a beautiful series of concentric ripples diverge from it; so when Krakatoa burst up in that mighty catastrophe, a series of gigantic waves were propagated through the air; they embraced the whole globe, converged to the antipodes of Krakatoa, thence again diverged, and returned to the seat of the volcano; a second time the mighty series of atmospheric ripples spread to the antipodes, and a second time returned. Seven times did that series of waves course over our globe, and leave their traces on every self-recording barometer that our earth possesses. Thirty-six hours were occupied in the journey of the great undulation from Krakatoa to its antipodes. Perhaps even more striking was the extent of our earth's surface over which the noise of the great explosion spread. At Batavia, ninety-four miles away, the concussions were simply deafening; at Macassar, in Celebes, two steamers were sent out to investigate the explosions which were heard, little thinking that they came from Krakatoa, nine hundred and sixty-nine miles away. Alarming sounds were heard over the island of Timor, one thousand three hundred and fifty-one miles away from Krakatoa. Diego Garcia in the Chogos islands is two thousand two hundred and sixty-seven miles from Krakatoa, but the thunders traversed even this distance, and were attributed to some ship in distress, for which a search was made. Most astounding of all, there is undoubted evidence that the sound of the mighty explosion was propagated across nearly the entire Indian ocean, and was heard in the island of Rodriguez, almost three thousand miles away. The immense distance over which this sound journeyed will be appreciated by the fact, that the noise did not reach Rodriguez until four hours after it had left Krakatoa. In fact, it would seem that if Vesuvius were to explode with the same vehemence as Krakatoa did, the thunders of the explosion might penetrate so far as to be heard in London.

There is another and more beautiful manifestation of the world-wide significance of the Krakatoa outbreak. The vast column of smoke and ashes ascended twenty miles high in the air, and commenced a series of voyages around the equatorial regions of the earth. In three days it crossed the Indian ocean, and was traversing equatorial Africa; then came an Atlantic voyage; and then it coursed over central America, before a Pacific voyage brought it back to its point of departure after thirteen days; then the dust started again, and was traced around another similar circuit, while it was even tracked for a considerable time in placing the third girdle round the earth. Strange blue suns and green moons and other mysterious phenomena marked the progress of this vast volcanic cloud. At last the cloud began to lose its density, the dust spread more widely over the tropics, became diffused through the temperate regions, and then the whole earth was able to participate in the glories of Krakatoa. The marvellous sunsets in the autumn of 1883 are attributable to this cause; and thus once again was brought before us the fact that the earth still contains large stores of thermal energy.

Attempts are sometimes made to explain volcanic phenomena on the supposition that they are entirely of a local character, and that we are not entitled to infer the incandescent nature of the earth's interior from the fact that volcanic outbreaks occasionally happen. For our present purpose this point is immaterial, though I must say it appears to me unreasonable to deny that the interior of the earth is in a most highly heated state. Every test we can apply shows us the existence of internal heat. Setting aside the more colossal phenomena of volcanic eruptions, we have innumerable minor manifestations of its presence. Are there not geysers and hot springs in many parts of the earth? and have we not all over our globe invariable testimony confirming the statement, that the deeper we go down beneath its surface the hotter does the temperature become? Every miner is familiar with these facts; he knows that the deeper are his shafts the warmer it is down below, and the greater the necessity for providing increased ventilation to keep the temperature within a limit that shall be suitable for the workmen. All these varied classes of phenomena admit solely of one explanation, and that is, that the interior of the earth contains vast stores of incandescent heat.

We now apply to our earth the same reasoning which we should employ on a poker taken from the fire, or on a casting drawn from the foundry. Such bodies will lose their heat by radiation and conduction. The earth is therefore losing its heat. No doubt the process is an extremely slow one. The mighty reservoirs of internal heat are covered by vast layers of rock, which are such excellent non-conductors that they offer every possible impediment to the leakage of heat from the interior to the surface. We coat our steam-pipes over with non-conducting material, and this can now be done so successfully, that it is beginning to be found economical to transmit steam for a very long distance through properly protected pipes. But no non-conducting material that we can manufacture can be half so effective as the shell of rock twenty miles or more in thickness, which secures the heated interior of the earth from rapid loss by radiation into space. Even were the earth's surface solid copper or solid silver, both most admirable conductors of heat, the cooling down of this vast globe would be an extremely tardy process; how much more tardy must it therefore be when such exceedingly bad conductors as rocks form the envelope? How imperfectly material of this kind will transmit heat is strikingly illustrated by the great blast iron furnaces which are so vitally important in one of England's greatest manufacturing industries. A glowing mass of coal and iron ore and limestone is here urged to vivid incandescence by a blast of air itself heated to an intense temperature. The mighty heat thus generated--sufficient as it is to detach the iron from its close alliance with the earthy materials and to render the metal out as a pure stream rushing white-hot from the vent--is sufficiently confined by a few feet of brick-work, one side of which is therefore at the temperature of molten iron, while the other is at a temperature not much exceeding that of the air. We may liken the brick-work of a blast furnace to the rocky covering of the earth; in each case an exceedingly high temperature on one side is compatible with a very moderate temperature on the other.

Although the drainage of heat away from the earth's interior to its surface, and its loss there by radiation into space, is an extremely tardy process, yet it is incessantly going on. We have here again to note the ability for gigantic effect which a small but continually operating cause may have, provided it always tends in the same direction. The earth is incessantly losing heat; and though in a day, a week, or a year the loss may not be very significant, yet when we come to deal with periods of time that have to be reckoned by millions of years, it may well be that the effect of a small loss of heat per annum can, in the course of these ages, reach unimagined dimensions. Suppose, for instance, that the earth experienced a fall of temperature in its interior which amounted to only one-thousandth of a degree in a year. So minute a quantity as this is imperceptible. Even in a century, the loss of heat at this rate would be only the tenth of a degree. There would be no possible way of detecting it; the most careful thermometer could not be relied on to tell us for a certainty that the temperature of the hot waters of Bath had declined the tenth of a degree; and I need hardly say, that the fall of a tenth of a degree would signify nothing in the lavas of Vesuvius, nor influence the thunders of Krakatoa by one appreciable note. So far as a human life or the life of the human race is concerned, the decline of a tenth of a degree per century in the earth's internal heat is absolutely void of significance. I cannot, however, impress upon you too strongly, that the mere few thousands of years with which human history is cognizant are an inappreciable moment in comparison with those unmeasured millions of years which geology opens out to us, or with those far more majestic periods which the astronomer demands for the events he has to describe.

An annual loss of even one-thousandth of a degree will be capable of stupendous achievements when supposed to operate during epochs of geological magnitude. In fact, its effects would be so vast, that it seems hardly credible that the present loss of heat from the earth should be so great as to amount to an abatement of one-thousandth of a degree per annum, for that would mean, that in a thousand years the earth's temperature would decline by one degree, and in a million years the decline would amount to a thousand degrees. At all events, the illustration may suffice to show, that the fact that we are not able to prove by our instruments that the earth is cooling is no argument whatever against the inevitable law, that the earth, like every other heated body, must be tending towards a lower temperature.

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.

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.

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