Before going any further it would be necessary for me to explain more fully than I have hitherto done the true nature of the method of estimating moment of momentum. The moment of momentum consists of two parts: there is first that due to the revolution of the bodies around the sun; there is secondly the rotation of these bodies on their axes.

Let us first think simply of a single planet revolving in a circular orbit around the sun. The momentum of that planet at any moment may be regarded as the product of its ma.s.s and its velocity; then the moment of momentum of the planet in the case mentioned is found by multiplying the momentum by the radius of the path pursued. In a more general case, where the planet does not revolve in a circle, but pursued an elliptic path, the moment of momentum is to be found by multiplying the planet's velocity and its ma.s.s into the perpendicular from the sun on the direction in which the planet is moving.

These rules provide the methods for estimating all the moments of momentum, so far as the revolutions in our system are concerned. For the rotations somewhat more elaborate processes are required. Let us think of a sphere rotating round a fixed axis. Every particle of that sphere will of course describe a circle around the axis, and all these circles will lie in parallel planes. We may for our present purpose regard each atom of the body as a little planet revolving in a circular orbit, and therefore the moment of momentum of the entire sphere will be found by simply adding together the moments of momentum of all the different atoms of which the sphere is composed. To perform this addition the use of an elaborate mathematical method is required.

I do not propose to enter into the matter any further, except to say that the total moment of momentum is the product of two factors--one the angular velocity with which the sphere is turning round, while the other involves the sphere's ma.s.s and dimensions.

To ill.u.s.trate the principles of the computation we shall take one or two examples. Suppose that two circles be drawn, one of which is double the diameter of the other. Let two planets be taken of equal ma.s.s, and one of these be put to revolve in one circle, and the other to revolve in the other circle, in such a way that the periods of both revolutions shall be equal. It is required to find the moments of momentum in the two cases. In the larger of the two circles it is plain that the planet must be moving twice as rapidly as in the smaller, therefore its momentum is twice as great; and as the radius is also double, it follows that the moment of momentum in the large orbit will be four times that in the small orbit. We thus see that the moment of momentum increases in the proportion of the squares of the radii. If, however, the two planets were revolving about the same sun, one of these orbits being double the other, the periodic times could not be equal, for Kepler's law tells us that the square of the periodic time is proportional to the cube of the mean distance.

Suppose, then, that the distance of the first planet is 1, and that of the second planet is 2, the cubes of those numbers are 1 and 8, and therefore the periodic times of the two bodies will be as 1 to the square root of 8. We can thus see that the velocity of the outer body must be less than that of the inner one, for while the length of the path is only double as large, the time taken to describe that path is the square root of eight times as great; in fact, the velocity of the outer body will be only the square root of twice that of the inner one. As, however, its distance from the sun is twice as great, it follows that the moment of momentum of the outer body will be the square root of twice that of the inner body. We may state this result a little more generally as follows--

In comparing the moments of momentum of the several planets which revolve around the sun, that of each planet is proportional to the product of its ma.s.s with the square root of its distance from the sun.

Let us now compare two spheres together, the diameter of one sphere being double that of the other, while the times of rotation of the two are identical. And let us now compare together the moments of momentum in these two cases. It can be shown by reasoning, into which I need not now enter, that the moment of momentum of the large sphere will be thirty-two times that of the small one. In general we may state that if a sphere of h.o.m.ogeneous material be rotating about an axis, its moment of momentum is to be expressed by the product of its angular velocity by the fifth power of its radius.

We can now take stock, as it were, of the const.i.tuents of moments of momentum in our system. We may omit the satellites for the present, while such unsubstantial bodies as comets and such small bodies as meteors need not concern us. The present investment of the moment of momentum of our system is to be found by multiplying the ma.s.s of each planet by the square root of its distance from the sun; these products for all the several planets form the total revolutional moment of momentum. The remainder of the investment is in rotational moment of momentum, the collective amount of which is to be estimated by multiplying the angular velocity of each planet into its density, and the fifth power of its radius if the planet be regarded as h.o.m.ogeneous, or into such other power as may be necessary when the planet is not h.o.m.ogeneous. Indeed, as the denser parts of the planet necessarily lie in its interior, and have therefore neither the velocity nor the radius of the more superficial portions, it seems necessary to admit that the moments of momentum of the planets will be proportional to some lower power of the radius than the fifth. The total moment of momentum of the planets by rotation, when multiplied by a constant factor, and added to the revolutional moment of momentum, will remain absolutely constant.

It may be interesting to note the present disposition of this vast inheritance among the different bodies of our system. The biggest item of all is the moment of momentum of Jupiter, due to its revolution around the sun; in fact, in this single investment nearly sixty per cent. of the total moment of momentum of the solar system is found.

The next heaviest item is the moment of momentum of Saturn's revolution, which is twenty-four per cent. Then come the similar contributions of Ura.n.u.s and Neptune, which are six and eight per cent.

respectively. Only one more item is worth mentioning, as far as magnitude is concerned, and that is the nearly two per cent. that the sun contains in virtue of its _rotation_. In fact, all the other moments of momentum are comparatively insignificant in this method of viewing the subject. Jupiter from his rotation has not the fifty thousandth part of his revolutional moment of momentum, while the earth's rotational share is not one ten thousandth part of that of Jupiter, and therefore is without importance in the general aspect of the system. The revolution of the earth contributes about one eight hundredth part of that of Jupiter.

These facts as here stated will suffice for us to make a forecast of the utmost the tides can effect in the future transformation of our system. We have already explained that the general tendency of tidal friction is to augment revolutional moment of momentum at the expense of rotational. The total, however, of the rotational moment of momentum of the system barely reaches two per cent. of the whole amount; this is of course almost entirely contributed by the sun, for all the planets together have not a thousandth part of the sun's rotational moment. The utmost therefore that tidal evolution can effect in the system is to distribute the two per cent. in augmenting the revolutionary moment of momentum. It does not seem that this can produce much appreciable derangement in the configuration of the system. No doubt if it were all applied to one of the smaller planets it would produce very considerable effect. Our earth, for instance, would have to be driven out to a distance many hundreds of times further than it is at present were the sun's disposable moment of momentum ultimately to be transferred to the earth alone. On the other hand, Jupiter could absorb the whole of the sun's share by quite an insignificant enlargement of its present path. It does not seem likely that the distribution that must ultimately take place can much affect the present configuration of the system.

We thus see that the tides do not appear to have exercised anything like the same influence in the affairs of our solar system generally which they have done in that very small part of the solar system which consists of the earth and moon. This is, as I have endeavoured to show in these lectures, the scene of supremely interesting tidal phenomena; but how small it is in comparison with the whole magnitude of our system may be inferred from the following ill.u.s.tration. I represent the whole moment of momentum of our system by 1,000,000,000, the bulk of which is composed of the revolutional moments of momentum of the great planets, and the rotational moment of momentum of the sun. On this scale the rotational share which has fallen to our earth and moon does not even rise to the dignity of a single pound, it can only be represented by the very modest figure of 19_s._ 5_d._ This is divided into two parts--the earth by its rotation accounts for 3_s._ 4_d._, leaving 16_s._ 1_d._ as the equivalent of the revolution of the moon.

The other inferior planets have still less to show than the earth.

Venus can barely have more than 2_s._ 6_d._; even Mars' two satellites cannot bring his figure up beyond the slender value of 1_d._; while Mercury will be amply represented by the smallest coin known at her Majesty's mint.

The same ill.u.s.tration will bring out the contrast between the Jovian system and our earth system. The rotational share of the former would be totally represented by a sum of nearly 12,000; of this, however, Jupiter's satellites only contribute about 89, notwithstanding that there are four of them. Thus Jupiter's satellites have not one hundredth part of the moment of momentum which the rotation of Jupiter exhibits. How wide is the contrast between this state of things and the earth-moon system, for the earth does not contain in its rotation one-fifth of the moment of momentum that the moon has in its revolution; in fact, the moon has gradually robbed the earth, which originally possessed 19_s._ 5_d._, of which the moon has carried off all but 3_s._ 4_d._

And this process is still going on, so that ultimately the earth will be left very poor, though not absolutely penniless, at least if the retention of a halfpenny can be regarded as justifying that a.s.sertion.

Saturn, revolving as it does with great rapidity, and having a very large ma.s.s, possesses about 2700, while Ura.n.u.s and Neptune taken together would figure for about the same amount.

In conclusion, let us revert again to the two critical conditions of the earth-moon system. As to what happened before the first critical period, the tides tell us nothing, and every other line of reasoning very little; we can to some extent foresee what may happen after the second critical epoch is reached, at a time so remote that I do not venture even to express the number of ciphers which ought to follow the significant digit in the expression for the number of years. I mentioned, however, that at this time the sun tides will produce the effect of applying a still further brake to the rotation of the earth, so that ultimately the month will have become a shorter period than the day. It is therefore interesting for us to trace out the tidal history of a system in which the satellite revolves around the primary in less time than the primary takes to go round on its own axis--such a system, in fact, as Mars would present at this moment were the outer satellite to be abstracted. The effect of the tides on the planet raised by its satellite would then be to accelerate its rotation; for as the planet, so to speak, lags behind the tides, friction would now manifest itself by the continuous endeavour to drag the primary round faster. The gain of speed, however, thus attained would involve the primary in performing more than its original share of the moment of momentum; less moment of momentum would therefore remain to be done by the satellite, and the only way to accomplish this would be for the satellite to come inwards and revolve in a smaller orbit.

We might indeed have inferred this from the considerations of energy alone, for whatever happens in the deformation of the orbit, heat is produced by the friction, and this heat is lost, and the total energy of the system must consequently decline. Now if it be a consequence of the tides that the velocity of the primary is accelerated, the energy corresponding to that velocity is also increased. Hence the primary has more energy than it had before; this energy must have been obtained at the expense of the satellite; the satellite must therefore draw inwards until it has yielded up enough of energy not alone to account for the increased energy of the primary, but also for the absolute loss of energy by which the whole operation is characterized.

It therefore appears that in the excessively remote future the retreat of the moon will not only be checked, but that the moon may actually return to a point to be determined by the changes in the earth's rotation. It is, however, extremely difficult to follow up the study of a case where the problem of three bodies has become even more complicated than usual.

The importance of tidal evolution in our solar system has also to be viewed in connection with the celebrated nebular hypothesis of the origin of the solar system. Of course it would be understood that tidal evolution is in no sense a rival doctrine to that of the nebular theory. The nebular origin of the sun and the planets sculptured out the main features of our system; tidal evolution has merely come into play as a subsidiary agent, by which a detail here or a feature there has been chiselled into perfect form. In the nebular theory it is believed that the planets and the sun have all originated from the cooling and the contraction of a mighty heated ma.s.s of vapours. Of late years this theory, in its main outlines at all events, has strengthened its hold on the belief of those who try to interpret nature in the past by what we see in the present. The fact that our system at present contains some heat in other bodies as well as in the sun, and the fact that the laws of heat require continual loss by radiation, demonstrate that our system, if we look back far enough, and if the present laws have acted, must have had in part, at all events, an origin like that which the nebular theory would suppose.

I feel that I have in the progress of these two lectures been only able to give the merest outline of the theory of tidal evolution in its application to the earth-moon system. Indeed I have been obliged, by the nature of the subject, to omit almost entirely any reference to a large body of the parts of the theory. I cannot bring myself to close these lectures without just alluding to this omission, and without giving expression to the fact, that I feel it is impossible for me to have rendered adequate justice to the strength of the argument on which we claim that tidal evolution is the most rational mode of accounting for the present condition in which we find the earth-moon system. Of course it will be understood that we have never contended that the tides offer the only conceivable theory as to the present condition of things. The argument lies in this wise. A certain body of facts are patent to our observation. The tides offer an explanation as to the origin of these facts. The tides are a _vera causa_, and in the absence of other suggested causes, the tidal theory holds the field. But much will depend on the volume and the significance of the group of a.s.sociated facts of which the doctrine offers a solution. The facts that it has been in my power to discuss within the compa.s.s of discourses like the present, only give a very meagre and inadequate notion of the entire phenomena connected with the moon which the tides will explain. We have not unfrequently, for the sake of simplicity, spoken of the moon's...o...b..t as circular, and we have not even alluded to the fact that the plane of that orbit is inclined to the ecliptic. A comprehensive theory of the moon's origin should render an account of the eccentricity of the moon's...o...b..t; it must also involve the obliquity of the ecliptic, the inclination of the moon's...o...b..t, and the direction of the moon's axis. I have been perforce compelled to omit the discussion of these attributes of the earth-moon system, and in doing so I have inflicted what is really an injustice on the tidal theory. For it is the chief claim of the theory of tidal evolution, as expounded by Professor Darwin, that it links together all these various features of the earth-moon system. It affords a connected explanation, not only of the fact that the moon always turns the same face to the earth, but also of the eccentricity of the moon's path around the earth, and the still more difficult points about the inclinations of the various axes and orbits of the planets. It is the consideration of these points that forms the stronghold of the doctrine of tidal evolution. For when we find that a theory depending upon influences that undoubtedly exist, and are in ceaseless action around us, can at the same time bring into connection and offer a common explanation of a number of phenomena which would otherwise have no common bond of union, it is impossible to refuse to believe that such a theory does actually correspond to nature.

The greatest of mathematicians have ever found in astronomy problems which tax, and problems which greatly surpa.s.s, the utmost efforts of which they are capable. The usual way in which the powers of the mathematician have been awakened into action is by the effort to remove some glaring discrepancy between an imperfect theory and the facts of observation. The genius of a Laplace or a Lagrange was expended, and worthily expended, in efforts to show how one planet acted on another planet, and produced irregularities in its...o...b..t; the genius of an Adams and a Leverrier was n.o.bly applied to explain the irregularities in the motion of Ura.n.u.s, and to discover a cause of those irregularities in the unseen Neptune. In all these cases, and in many others which might be mentioned, the mathematician has been stimulated by the laudable anxiety to clear away some blemish from the theory of gravitation throughout the system. The blemish was seen to exist before its removal was suggested. In that application of mathematics with which we have been concerned in these lectures the call for the mathematician has been of quite a different kind. A certain familiar phenomenon on our sea-coasts has invited attention.

The tidal ripples murmur a secret, but not for every ear. To interpret that secret fully, the hearer must be a mathematician. Even then the interpretation can only be won after the profoundest efforts of thought and attention, but at last the language has been made intelligible. The labour has been gloriously rewarded, and an interesting chapter of our earth's history has for the first time been written.

In the progress of these lectures I have sought to interest you in those profound investigations which the modern mathematician has made in his efforts to explore the secrets of nature. He has felt that the laws of motion, as we understand them, are bounded by no considerations of s.p.a.ce, are limited by no duration of time, and he has commenced to speculate on the logical consequences of those laws when time of indefinite duration is a.s.sumed to be at his disposal.

From the very nature of the case, observations for confirmation were impossible. Phenomena that required millions of years for their development cannot be submitted to the instruments in our observatories. But this is perhaps one of the special reasons which make such investigations of peculiar interest, and ent.i.tle us to speak of the revelations of Time and Tide as a romance of modern science.

THE END.