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.
[Illustration: 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.
You will now see the bearing which the movement of the inner satellite
of Mars has on the doctrine of tidal evolution. As a legitimate
consequence of that doctrine, we came to the conclusion that our
earth-moon system must ultimately attain a condition in which the day
is longer than the month. But this conclusion stood unsupported by any
analogous facts in the more anciently-known truths of astronomy. The
movement of the satellite of Mars, however, affords the precise
illustration we want; and this fact, I think, adds an additional
significance to the interest and the beauty of Professor Hall's
discovery.
It is of particular interest to investigate the possible connection
which the phenomena of tidal evolution may have had in connection with
the geological phenomena of the earth. We have already pointed out the
greater closeness of the moon to us in times past. The tides raised by
the moon on the earth must therefore have been greater in past ages
than they are now, for of course the nearer the moon the bigger the
tide. As soon as the earth and the moon had separated to a
considerable distance we may say that the height of the tide will vary
inversely as the cube of the moon's distance; it will therefore
happen, that when the moon was at half its present distance from us,
his tide-producing capacity was not alone twice as much or four times
as much, but even eight times as much as it is at present; and a much
greater rate of tidal rise and fall indicates, of course, a
preponderance in every other manifestation of tidal activity. The
tidal currents, for instance, must have been much greater in volume
and in speed; even now there are places in which the tidal currents
flow at four or more miles per hour. We can imagine, therefore, the
vehemence of the tidal currents which must have flowed in those days
when the moon was a much smaller distance from us. It is interesting
to view these considerations in their possible bearings on geological
phenomena. It is true that we have here many elements of uncertainty,
but there is, however, a certain general outline of facts which may
be laid down, and which appears to be instructive, with reference to
the past history of our earth.
I have all through these lectures indicated a mighty system of
chronology for the earth-moon system. It is true that we cannot give
our chronology any accurate expression in years. The various stages of
this history are to be represented by the successive distances between
the earth and the moon. Each successive epoch, for instance, may be
marked by the number of thousands of miles which separate the moon
from the earth.
But we have another system of chronology derived from a wholly
different system of ideas; it too relates to periods of vast duration,
and, like our great tidal periods, extends to times anterior to human
history, or even to the duration of human life on this globe. The
facts of geology open up to us a majestic chronology, the epochs of
which are familiar to us by the succession of strata forming the crust
of the earth, and by the succession of living beings whose remains
these strata have preserved. From the present or recent age our
retrospect over geological chronology leads us to look through a
vista embracing periods of time overwhelming in their duration, until
at last our view becomes lost, and our imagination is baffled in the
effort to comprehend the formation of those vast stratified rocks, a
dozen miles or more in thickness, which seem to lie at the very base
of the stratified system on the earth, and in which it would appear
that the dawnings of life on this globe may be almost discerned. We
have thus the two systems of chronology to compare--one, the
astronomical chronology measured by the successive stages in the
gradual retreat of the moon; the other, the geological chronology
measured by the successive strata constituting the earth's crust.
Never was a more noble problem proposed in the physical history of our
earth than that which is implied in the attempt to correlate these two
systems of chronology. What we would especially desire to know is the
moon's distance which corresponds to each of the successive strata on
the earth. How far off, for instance, was that moon which looked down
on the coal forests in the time of their greatest luxuriance? or what
was the apparent size of the full moon at which the ichthyosaurus
could have peeped when he turned that wonderful eye of his to the sky
on a fine evening? But interesting as this great problem is, it lies,
alas! outside the possibility of exact solution. Indeed we shall not
make any attempt which must necessarily be futile to correlate these
chronologies; all we can do is to state the one fact which is
absolutely undeniable in the matter.
Let us fix our attention on that specially interesting epoch at the
dawn of geological time, when those mighty Laurentian rocks were
deposited of which the thickness is so astounding, and let us consider
what the distance of the moon must have been at this initial epoch of
the earth's history. All we know for certain is, that the moon must
have been nearer, but what proportion that distance bore to the
present distance is necessarily quite uncertain. Some years ago I
delivered a lecture at Birmingham, entitled "A Glimpse through the
Corridors of Time," and in that lecture I threw out the suggestion
that the moon at this primeval epoch may have only been at a small
fraction of its present distance from us, and that consequently
terrific tides may in these days have ravaged the coast. There was a
good deal of discussion on the subject, and while it was universally
admitted that the tides must have been larger in palæozoic times than
they are at present, yet there was a considerable body of opinion to
the effect that the tides even then may have been only about twice, or
possibly not so much, greater than those tides we have at the present.
What the actual fact may be we have no way of knowing; but it is
interesting to note that even the smallest accession to the tides
would be a valuable factor in the performance of geological work.
For let me recall to your minds a few of the fundamental phenomena of
geology. Those stratified rocks with which we are now concerned have
been chiefly manufactured by deposition of sediment in the ocean.
Rivers, swollen, it may be, by floods, and turbid with a quantity of
material held in suspension, discharge their waters into the sea.
Granting time and quiet, this sediment falls to the bottom; successive
additions are made to its thickness during centuries and thousands of
years, and thus beds are formed which in the course of ages
consolidate into actual rock. In the formation of such beds the tides
will play a part. Into the estuaries at the mouths of rivers the tides
hurry in and hurry out, and especially during spring tides there are
currents which flow with tremendous power; then too, as the waves
batter against the coast they gradually wear away and crumble down the
mightiest cliffs, and waft the sand and mud thus produced to augment
that which has been brought down by the rivers. In this operation also
the tides play a part of conspicuous importance, and where the ebb and
flow is greatest it is obvious that an additional impetus will be
given to the manufacture of stratified rocks. In fact, we may regard
the waters of the globe as a mighty mill, incessantly occupied in
grinding up materials for future strata. The main operating power of
this mill is of course derived from the sun, for it is the sun which
brings up the rains to nourish the rivers, it is the sun which raises
the wind which lashes the waves against the shore. But there is an
auxiliary power to keep the mill in motion, and that auxiliary power
is afforded by the tides. If then we find that by any cause the
efficiency of the tides is increased we shall find that the mill for
the manufacture of strata obtains a corresponding accession to its
capacity. Assuming the estimate of Professor Darwin, that the tide may
have had twice as great a vertical range of ebb and flow within
geological times as it has at present, we find a considerable addition
to the efficiency of the ocean in the manufacture of the ancient
stratified rocks. It must be remembered that the extent of the area
through which the tides will submerge and lay bare the country, will
often be increased more than twofold by a twofold increase of height.
A little illustration may show what I mean. Suppose a cone to be
filled with water up to a certain height, and that the quantity of
water in it be measured; now let the cone be filled until the water is
at double the depth; then the surfaces of the water in the two cases
will be in the ratio of the circles, one of which has double the
diameter of the other. The areas of the two surfaces are thus as four
to one; the volumes of the waters in the two cases will be in the
proportion of two similar solids, the ratios of their dimensions being
as two to one. Of course this means that the water in the one case
would be eight times as much as in the other. This particular
illustration will not often apply exactly to tidal phenomena, but I
may mention one place that I happen to know of, in the vicinity of
Dublin, in which the effect of the rise and fall of the tide would be
somewhat of this description. At Malahide there is a wide shallow
estuary cut off from the sea by a railway embankment, and there is a
viaduct in the embankment through which a great tidal current flows in
and out alternately. At low tide there is but little water in this
estuary, but at high tide it extends for miles inland. We may regard
this inlet with sufficient approximation to the truth as half of a
cone with a very large angle, the railway embankment of course forming
the diameter; hence it follows that if the tide was to be raised to
double its height, so large an area of additional land would be
submerged, and so vast an increase of water would be necessary for the
purpose, that the flow under the railway bridge would have to be much
more considerable than it is at present. In some degree the same
phenomena will be repeated elsewhere around the coast. Simply
multiplying the height of the tide by two would often mean that the
border of land between high and low water would be increased more than
twofold, and that the volume of water alternately poured on the land
and drawn off it would be increased in a still larger proportion. The
velocity of all tidal currents would also be greater than at present,
and as the power of a current of water for transporting solid material
held in suspension increases rapidly with the velocity, so we may
infer that the efficiency of tidal currents as a vehicle for the
transport of comminuted rocks would be greatly increased. It is thus
obvious that tides with a rise and fall double in vertical height of
those which we know at present would add a large increase to their
efficiency as geological agents. Indeed, even were the tides only half
or one-third greater than those we know now, we might reasonably
expect that the manufacture of stratified rocks must have proceeded
more rapidly than at present.
The question then will assume this form. We know that the tides must
have been greater in Cambrian or Laurentian days than they are at
present; so that they were available as a means of assisting other
agents in the stupendous operations of strata manufacture which were
then conducted. This certainly helps us to understand how these
tremendous beds of strata, a dozen miles or more in solid thickness,
were deposited. It seems imperative that for the accomplishment of a
task so mighty, some agents more potent than those with which we are
familiar should be required. The doctrine of tidal evolution has shown
us what those agents were. It only leaves us uninformed as to the
degree in which their mighty capabilities were drawn upon.
It is the property of science as it grows to find its branches more
and more interwoven, and this seems especially true of the two
greatest of all natural sciences--geology and astronomy. With the
beginnings of our earth as a globe in the shape in which we find it
both these sciences are directly concerned. I have here touched upon
another branch in which they illustrate and confirm each other.
As the theory of tidal evolution has shed such a flood of light into
the previously dark history of our earth-moon system, it becomes of
interest to see whether the tidal phenomena may not have a wider
scope; whether they may not, for instance, have determined the
formation of the planets by birth from the sun, just as the moon seems
to have originated by birth from the earth. Our first presumption,
that the cases are analogous, is not however justified when the facts
are carefully inquired into. A principle which I have not hitherto
discussed here assumes prominence, and therefore we shall devote our
attention to it for a few minutes.
Let us understand what we mean by the solar system. There is first the
sun at the centre, which preponderates over all the other bodies so
enormously, as shown in Fig. 4, in which the earth and the sun are
placed side by side for comparison. There is then the retinue of
planets, among the smaller of which our earth takes its place, a view
of the comparative sizes of the planets being shown in Fig. 5.
[Illustration: Fig. 4.--Comparative sizes of Earth and Sun.]
Not to embarrass ourselves with the perplexities of a problem so
complicated as our solar system is in its entirety, we shall for the
sake of clear reasoning assume an ideal system, consisting of a sun
and a large planet--in fact, such as our own system would be if we
could withdraw from it all other bodies, leaving the sun and Jupiter
only remaining. We shall suppose, of course, that the sun is much
larger than the planet, in fact, it will be convenient to keep in mind
the relative masses of the sun and Jupiter, the weight of the planet
being less than one-thousandth part of the sun. We know, of course,
that both of those bodies are rotating upon their axes, and the one
is revolving around the other; and for simplicity we may further
suppose that the axes of rotation are perpendicular to the plane of
revolution. In bodies so constituted tides will be manifested.
Jupiter will raise tides in the sun, the sun will raise tides in
Jupiter. If the rotation of each body be performed in a less period
than that of the revolution (the case which alone concerns us), then
the tides will immediately operate in their habitual manner as a brake
for the checking of rotation. The tides raised by the sun on Jupiter
will tend therefore to lengthen Jupiter's day; the tides raised on the
sun by Jupiter will tend to augment the sun's period of rotation. Both
Jupiter and the sun will therefore lose some moment of momentum. We
cannot, however, repeat too often the dynamical truth that the total
moment of momentum must remain constant, therefore what is lost by the
rotation must be made up in the revolution; the orbit of Jupiter
around the sun must accordingly be swelling. So far the reasoning
appears similar to that which led to such startling consequences in
regard to the moon.
[Illustration: Fig. 5.--Comparative sizes of Planets.]
But now for the fundamental difference between the two cases. The
moon, it will be remembered, always revolves with the same face
towards the earth. The tides have ceased to operate there, and
consequently the moon is not able to contribute any moment of
momentum, to be applied to the enlargement of its distance from the
earth; all the moment of momentum necessary for this purpose is of
course drawn from the single supply in the rotation of the earth on
its axis. But in the case of the system consisting of the sun and
Jupiter the circumstances are quite different--Jupiter does not always
bend the same face to the sun; so far, indeed, is this from being
true, that Jupiter is eminently remarkable for the rapidity of his
rotation, and for the incessantly varying aspect in which he would be
seen from the sun. Jupiter has therefore a store of available moment
of momentum, as has also of course the sun. Thus in the sun and planet
system we have in the rotations two available stores of moment of
momentum on which the tides can make draughts for application to the
enlargement of the revolution. The proportions in which these two
available sources can be drawn upon for contributions is not left
arbitrary. The laws of dynamics provide the shares in which each of
the bodies is to contribute for the joint purpose of driving them
further apart.
Let us see if we cannot form an estimate by elementary considerations
as to the division of the labour. The tides raised on Jupiter by the
sun will be practically proportional to the sun's mass and to the
radius of Jupiter. Owing to the enormous size of the sun, the
efficiency of these tides and the moment of the friction-brake they
produce will be far greater on the planet than will the converse
operation of the planet be on the sun. Hence it follows that the
efficiency of the tides in depriving Jupiter of moment of momentum
will be greatly superior to the efficiency of the tides in depriving
the sun of moment of momentum. Without following the matter into any
close numerical calculation, we may assert that for every one part the
sun contributes to the common object, Jupiter will contribute at least
a thousand parts; and this inequality appears all the more striking,
not to say unjust, when it is remembered that the sun is more
abundantly provided with moment of momentum than is Jupiter--the sun
has, in fact, about twenty thousand times as much.
The case may be illustrated by supposing that a rich man and a poor
man combine together to achieve some common purpose to which both are
to contribute. The ethical notion that Dives shall contribute
largely, according to his large means, and Lazarus according to his
slender means, is quite antagonistic to the scale which dynamics has
imposed. Dynamics declares that the rich man need only give a penny to
every pound that has to be extorted from the poor man. Now this is
precisely the case with regard to the sun and Jupiter, and it involves
a somewhat curious consequence. As long as Jupiter possesses available
moment of momentum, we may be certain that no large contribution of
moment of momentum has been obtained from the sun. For, returning to
our illustration, if we find that Lazarus still has something left in
his pocket, we are of course assured that Dives cannot have expended
much, because, as Lazarus had but little to begin with, and as Dives
only puts in a penny for every pound that Lazarus spends, it is
obvious that no large amount can have been devoted to the common
object. Hence it follows that whatever transfer of moment of momentum
has taken place in the sun-Jupiter system has been almost entirely
obtained at the expense of Jupiter. Now in the solar system at
present, the orbital moment of momentum of Jupiter is nearly fifty
thousand times as great as his present store of rotational moment of
momentum. If, therefore, the departure of Jupiter from the sun had
been the consequence of tidal evolution, it would follow that Jupiter
must once have contained many thousands of times the moment of
momentum that he has at present. This seems utterly incredible, for
even were Jupiter dilated into an enormously large mass of vaporous
matter, spinning round with the utmost conceivable speed, it is
impossible that he should ever have possessed enough moment of
momentum. We are therefore forced to the conclusion that the tides
alone do not provide sufficient explanation for the retreat of Jupiter
from the sun.
There is rather a subtle point in the considerations now brought
forward, on which it will be necessary for us to ponder. In the
illustration of Dives and Lazarus, the contributions of Lazarus of
course ceased when his pockets were exhausted, but those of Dives will
continue, and in the lapse of time may attain any amount within the
utmost limits of Dives' resources. The essential point to notice is,
that so long as Lazarus retains anything in _his_ pocket, we know for
certain that Dives has not given much; if Lazarus, however, has his
pocket absolutely empty, and if we do not know how long they may have
been in that condition, we have no means of knowing how large a
portion of wealth Dives may not have actually expended. The
turning-point of the theory thus involves the fact that Jupiter still
retains available moment of momentum in his rotation; and this was our
sole method of proving that the sun, which in this case was Dives, had
never given much. But our argument must have taken an entirely
different line had it so happened that Jupiter constantly turned the
same face to the sun, and that therefore his pockets were entirely
empty in so far as available moment of momentum is concerned. It would
be apparently impossible for us to say to what extent the resources of
the sun may not have been drawn upon; we can, however, calculate
whether in any case the sun could possibly have supplied enough moment
of momentum to account for the recession of Jupiter. Speaking in round
numbers, the revolutional moment of momentum of Jupiter is about
thirty times as great as the rotational moment of momentum at present
possessed by the sun. I do not know that there is anything impossible
in the supposition that the sun might, by an augmented volume and an
augmented velocity of rotation, contain many times the moment of
momentum that it has at this moment. It therefore follows that if it
had happened that Jupiter constantly bent the same face to the sun,
there would apparently be nothing impossible in the fact that Jupiter
had been born of the sun, just as the moon was born of the earth.
These same considerations should also lead us to observe with still
more special attention the development of the earth-moon system. Let
us restate the matter of the earth and moon in the light which the
argument with respect to Jupiter has given us. At present the
rotational moment of momentum of the earth is about a fifth part of
the revolutional moment of momentum of the moon. Owing to the fact
that the moon keeps the same face to us, she has now no available
moment of momentum, and all the moment of momentum required to account
for her retreat has of late come from the rotation of the earth; but
suppose that the moon still had some liquid on its surface which could
be agitated by tides, suppose further that it did not always bend the
same face towards us, that it therefore had some available moment of
momentum due to its rotation on which the tides could operate, then
see how the argument would have been altered. The gradual increase of
the moon's distance could be provided for by a transfer of moment of
momentum from two sources, due of course to the rotational velocities
of the two bodies. Here again the moon and the earth will contribute
according to that dynamical but very iniquitous principle which
regulated the appropriations from the purses of Dives and Lazarus. The
moon must give not according to her abundance, but in the inverse
ratio thereof--because she has little she must give largely. Nor shall
we make an erroneous estimate if we say that nine-tenths of the whole
moment of momentum necessary for the enlargement of the orbit would
have been exacted from the moon; that means that the moon must once
have had about five or six times as much moment of momentum as the
earth possesses at this moment. Considering the small size of the
moon, this could only have arisen by terrific velocity of rotation,
which it is inconceivable that its materials could ever have
possessed.
This presents the demonstration of tidal evolution in a fresh light.
If the moon now departed to any considerable extent from showing the
constant face to the earth, it would seem that its retreat could not
have been caused by tides. Some other agent for producing the present
configuration would be necessary, just as we found that some other
agent than the tides has been necessary in the case of Jupiter.
But I must say a few words as to the attitude of this question with
regard to the entire solar system. This system consists of the sun
presiding at the centre, and of the planets and their satellites in
revolution around their respective primaries, and each also animated
by a rotation on its axis. I shall in so far depart from the actual
configuration of the system as to transform it into an ideal system,
whereof the masses, the dimensions, and the velocities shall all be
preserved; but that the several planes of revolution shall be all
flattened into one plane, instead of being inclined at small angles as
they are at present; nor will it be unreasonable for us at the same
time to bring into parallelism all the axes of rotation, and to
arrange that their common directions shall be perpendicular to the
plane of their common orbits. For the purpose of our present research
this ideal system may pass for the real system.
In its original state, whatever that state may have been, a
magnificent endowment was conferred upon the system. Perhaps I may,
without derogation from the dignity of my subject, speak of the
endowment as partly personal and partly entailed. The system had of
course different powers with regard to the disposal of the two
portions; the personal estate could be squandered. It consisted
entirely of what we call energy; and considering how frequently we use
the expression conservation of energy, it may seem strange to say now
that this portion of the endowment has been found capable of
alienation, nay, further, that our system has been squandering it
persistently from the first moment until now. Although the doctrine of
the conservation of energy is, we have every reason to believe, a
fundamental law affecting the whole universe, yet it would be wholly
inaccurate to say that any particular system such as our solar system
shall invariably preserve precisely the same quantity of energy
without alteration. The circumstance that heat is a form of energy
indeed negatives this supposition. For our system possesses energy of
all the different kinds: there is energy due to the motions both of
rotation and of revolution; there is energy due to the fact that the
mutually attracting bodies of our system are separated by distances of
enormous magnitude; and there is also energy in the form of heat; and
the laws of heat permit that this form of energy shall be radiated off
into space, and thus disappear entirely, in so far as our system is
concerned. On the other hand, there may no doubt be some small amount
of energy accruing to our system from the other systems in space,
which like ours are radiating forth energy. Any gain from this source,
however, is necessarily so very small in comparison to the loss to
which we have referred, that it is quite impossible that the one
should balance the other. Though it is undoubtedly true that the
total quantity of energy in the universe is constant, yet the share of
that energy belonging to any particular system such as ours declines
steadily from age to age.
I may indeed remark, that the question as to what becomes of all the
radiant energy which the millions of suns in the universe are daily
discharging offers a problem apparently not easy to solve; but we need
not discuss the matter at present, we are only going to trace out the
vicissitudes of our own system; and whatever other changes that system
may exhibit, the fact is certain that the total quantity of energy it
contains is declining.
Of the two endowments of energy and of moment of momentum originally
conferred on our system the moment of momentum is the entailed estate.
No matter how the bodies may move, no matter how their actions may
interfere with one another, no matter how this body is pulled one way
and the other body that way, the conservation of moment of momentum is
not imperilled, nor, no matter what losses of heat may be experienced
by radiation, could the store of moment of momentum be affected. The
only conceivable way in which the quantity of moment of momentum in
the solar system could be tampered with is by the interference of some
external attracting body. We know, however, that the stars are all
situated at such enormous distances, that the influences they can
exert in the perturbation of the solar system are absolutely
insensible; they are beyond the reach of the most delicate
astronomical measurements. Hence we see how the endowment of the
system with moment of momentum has conferred upon that system a
something which is absolutely inalienable, even to the smallest
portion.
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 mass 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 mass 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 mass and dimensions.
To illustrate 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
mass, 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 mass 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 homogeneous 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 constituents 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 mass 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
homogeneous, or into such other power as may be necessary when the
planet is not homogeneous. 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 Uranus 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 illustration. 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 1½_d._;
while Mercury will be amply represented by the smallest coin known at
her Majesty's mint.
The same illustration 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 assertion.
Saturn, revolving as it does with great rapidity, and having a very
large mass, possesses about £2700, while Uranus 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 mass 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 associated facts of which the doctrine
offers a solution. The facts that it has been in my power to discuss
within the compass 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 orbit 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 orbit; it
must also involve the obliquity of the ecliptic, the inclination of
the moon's orbit, 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 surpass, 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 orbit; the
genius of an Adams and a Leverrier was nobly applied to explain the
irregularities in the motion of Uranus, 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 space, 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 assumed 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 entitle us to speak
of the revelations of Time and Tide as a romance of modern science.