law means, but I must make the attempt, and if you think the
illustration beneath the dignity of the subject, I can only plead the
difficulty of mathematics as an excuse. Let us suppose that a
ball-room is fairly filled with dancers, or those willing to dance,
and that a merry waltz is being played; the couples have formed, and
the floor is occupied with pairs who are whirling round and round in
that delightful amusement. Some couples drop out for a while and
others strike in; the fewer couples there are the wider is the range
around which they can waltz, the more numerous the couples the less
individual range will they possess. I want you to realize that in the
progress of the dance there is a certain total quantity of spin at any
moment in progress; this spin is partly made up of the rotation by
which each dancer revolves round his partner, and partly of the
circular orbit about the room which each couple endeavours to
describe. If there are too many couples on the floor for the general
enjoyment of the dance, then both the orbit and the angular velocity
of each couple will be restricted by the interference with their
neighbours. We may, however, assert that so long as the dance is in
full swing the total quantity of spin, partly rotational and partly
orbital, will remain constant. When there are but few couples the
unimpeded rotation and the large orbits will produce as much spin as
when there is a much larger number of couples, for in the latter case
the diminished freedom will lessen the quantity of spin produced by
each individual pair. It will sometimes happen too that collision will
take place, but the slight diversions thus arising only increase the
general merriment, so that the total quantity of spin may be
sustained, even though one or two couples are placed temporarily _hors
de combat_. I have invoked a ball-room for the purpose of bringing out
what we may call the law of the conservation of spin. No matter how
much the individual performers may change, or no matter what
vicissitudes arise from their collision and other mutual actions, yet
the total quantity of spin remains unchanged.
Let us look at the earth-moon system. The law of the conservation of
moment of momentum may, with sufficient accuracy for our present
purpose, be interpreted to mean that the total quantity of spin in the
system remains unaltered. In our system the spin is threefold; there
is first the rotation of the earth on its axis, there is the rotation
of the moon on its axis, and then there is the orbital revolution of
the moon around the earth. The law to which we refer asserts that the
total quantity of these three spins, each estimated in the proper way,
will remain constant. It matters not that tides may ebb and flow, or
that the distribution of the spin shall vary, but its total amount
remains inflexibly constant. One constituent of the total amount--that
is, the rotation of the moon on its axis--is so insignificant, that
for our present purposes it may be entirely disregarded. We may
therefore assert that the amount of spin in the earth, due to its
rotation round its axis, added to the amount of spin in the moon due
to its revolution round the earth, remains unalterable. If one of
these quantities change by increase or by decrease, the other must
correspondingly change by decrease or by increase. If, therefore, from
any cause, the earth began to spin a little more quickly round its
axis, the moon must do a little less spin; and consequently, it must
shorten its distance from the earth. Or suppose that the earth's
velocity of rotation is abated, then its contribution to the total
amount of spin is lessened; the deficiency must therefore be made up
by the moon, but this can only be done by an enlargement of the moon's
orbit. I should add, as a caution, that these results are true only on
the supposition that the earth-moon system is isolated from all
external interference. With this proviso, however, it matters not what
may happen to the earth or moon, or what influence one of them may
exert upon the other, no matter what tides may be raised, no matter
even if the earth fly into fragments, the whole quantity of spin of
all those fragments would, if added to the spin of the moon, yield the
same unalterable total. We are here in possession of a most valuable
dynamical principle. We are not concerned with any special theory as
to the action of the tides; it is sufficient for us that in some way
or other the tides have been caused by the moon, and that being so,
the principle of the conservation of spin will apply.
Were the earth and the moon both rigid bodies, then there could be of
course no tides on the earth, it being rigid and devoid of ocean. The
rotation of the earth on its axis would therefore be absolutely
without change, and therefore the necessary condition of the
conservation of spin would be very simply attained by the fact that
neither of the constituent parts changed. The earth, however, not
being entirely rigid, and being subject to tides, this simple state of
things cannot continue; there must be some change in progress.
I have already shown that the fact of the ebbing and the flowing of
the tide necessitates an expenditure of energy, and we saw that this
energy must come either from that stored up in the earth by its
rotation, or from that possessed by the moon in virtue of its distance
and revolution. The law of the conservation of spin will enable us to
decide at once as to whence the tides get their energy. Suppose they
took it from the moon, the moon would then lose in energy, and
consequently come nearer the earth. The quantity of spin contributed
by the moon would therefore be lessened, and accordingly the spin to
be made up by the earth would be increased. That means, of course,
that the velocity of the earth rotating on its axis must be increased,
and this again would necessitate an increase in the earth's rotational
energy. It can be shown, too, that to keep the total spin right, the
energy of the earth would have to gain more than the moon would have
lost by revolving in a smaller orbit. Thus we find that the total
quantity of energy in the system would be increased. This would lead
to the absurd result that the action of the tides manufactured energy
in our system. Of course, such a doctrine cannot be true; it would
amount to a perpetual motion! We might as well try to get a
steam-engine which would produce enough heat by friction not only to
supply its own boilers, but to satisfy all the thermal wants of the
whole parish. We must therefore adopt the other alternative. The tides
do not draw their energy from the moon; they draw it from the store
possessed by the earth in virtue of its rotation.
We can now state the end of this rather long discussion in a very
simple and brief manner. Energy can only be yielded by the earth at
the expense of some of the speed of its rotation. The tides must
therefore cause the earth to revolve more slowly; in other words, _the
tides are increasing the length of the day_.
The earth therefore loses some of its velocity of rotation;
consequently it does less than its due share of the total quantity of
spin, and an increased quantity of spin must therefore be accomplished
by the moon; but this can only be done by an enlargement of its orbit.
Thus there are two great consequences of the tides in the earth-moon
system--the days are getting longer, the moon is receding further.
These points are so important that I shall try and illustrate them in
another way, which will show, at all events, that one and both of
these tidal phenomena commend themselves to our common sense. Have we
not shown how the tides in their ebb and flow are incessantly
producing friction, and have we not also likened the earth to a great
wheel? When the driver wants to stop a railway train the brakes are
put on, and the brake is merely a contrivance for applying friction to
the circumference of a wheel for the purpose of checking its motion.
Or when a great weight is being lowered by a crane, the motion is
checked by a band which applies friction on the circumference of a
wheel, arranged for the special purpose. Need we then be surprised
that the friction of the tides acts like a brake on the earth, and
gradually tends to check its mighty rotation? The progress of
lengthening the day by the tides is thus readily intelligible. It is
not quite so easy to see why the ebbing and the flowing of the tide on
the earth should actually have the effect of making the moon to
retreat; this phenomenon is in deference to a profound law of nature,
which tells us that action and reaction are equal and opposite to each
other. If I might venture on a very homely illustration, I may say
that the moon, like a troublesome fellow, is constantly annoying the
earth by dragging its waters backward and forward by means of tides;
and the earth, to free itself from this irritating interference, tries
to push off the aggressor and to make him move further away.
[Illustration: FIG. 2.]
Another way in which we can illustrate the retreat of the moon as the
inevitable consequence of tidal friction is shown in the adjoining
figure, in which the large body E represents the earth, and the small
body M the moon. We may for simplicity regard the moon as a point, and
as this attracts each particle of the earth, the total effect of the
moon on the earth may be represented by a single force. By the law of
equality of action and reaction, the force of the earth on the moon is
to be represented by an equal and opposite force. If there were no
tides then the moon's force would of course pass through the earth's
centre; but as the effect of the moon is to slacken the earth's
rotation, it follows that the total force does not exactly pass
through the line of the earth's centre, but a little to one side, in
order to pull the opposite way to that in which the earth is turning,
and thus bring down its speed. We may therefore decompose the earth's
total force on the moon into two parts, one of which tends directly
towards the earth's centre, while the other acts tangentially to the
moon's orbit. The central force is of course the main guiding power
which keeps the moon in its path; but the incessant tangential force
constantly tends to send the moon out further and further, and thus
the growth of its orbit can be accounted for.
We therefore conclude finally, that the tides are making the day
longer and sending the moon away further. It is the development of the
consequences of these laws that specially demands our attention in
these lectures. We must have the courage to look at the facts
unflinchingly, and deduce from them all the wondrous consequences they
involve. Their potency arises from a characteristic feature--they are
unintermitting. Most of the great astronomical changes with which we
are ordinarily familiar are really periodic: they gradually increase
in one direction for years, for centuries, or for untold ages; but
then a change comes, and the increase is changed into a decrease, so
that after the lapse of becoming periods the original state of things
is restored. Such periodic phenomena abound in astronomy. There is the
annual fluctuation of the seasons; there is the eighteen or nineteen
year period of the moon; there is the great period of the precession
of the equinoxes, amounting to twenty-six thousand years; and then
there is the stupendous Annus Magnus of hundreds of thousands of
years, during which the earth's orbit itself breathes in and out in
response to the attraction of the planets. But these periodic
phenomena, however important they may be to us mere creatures of a
day, are insignificant in their effects on the grand evolution through
which the celestial bodies are passing. The really potent agents in
fashioning the universe are those which, however slow or feeble they
may seem to be, are still incessant in their action. The effect which
a cause shall be competent to produce depends not alone upon the
intensity of that cause, but also upon the time during which it has
been in operation. From the phenomena of geology, as well as from
those of astronomy, we know that this earth and the system to which it
belongs has endured for ages, not to be counted by scores of thousands
of years, or, as Prof. Tyndall has so well remarked, "Not for six
thousand years, nor for sixty thousand years, nor six hundred thousand
years, but for æons of untold millions." Those slender agents which
have devoted themselves unceasingly to the accomplishment of a single
task may in this long lapse of time have accomplished results of
stupendous magnitude. In famed stalactite caverns we are shown a
colossal figure of crystal extending from floor to roof, and the
formation of that column is accounted for when we see a tiny drop
falling from the roof above to the floor beneath. A lifetime may not
suffice for that falling drop to add an appreciable increase to the
stalactite down which it trickles, or to the growing stalagmite on
which it falls; but when the operation has been in progress for
immense ages, it is capable of the formation of the stately column.
Here we have an illustration of an influence which, though apparently
trivial, acquires colossal significance when adequate time is
afforded. It is phenomena of this kind which the student of nature
should most narrowly watch, for they are the real architects of the
universe.
The tidal consequences which we have already demonstrated are
emphatically of this non-periodic class--the day is always
lengthening, the moon is always retreating. To-day is longer than
yesterday; to-morrow will be longer than to-day. It cannot be said
that the change is a great one; it is indeed too small to be
appreciable even by our most delicate observations. In one thousand
years the alteration in the length of a day is only a small fraction
of a second; but what may be a very small matter in one thousand years
can become a very large one in many millions of years. Thus it is that
when we stretch our view through immense vistas of time past, or when
we look forward through immeasurable ages of time to come, the
alteration in the length of the day will assume the most startling
proportions, and involve the most momentous consequences.
Let us first look back. There was a time when the day, instead of
being the twenty-four hours we now have, must have been only
twenty-three hours, How many millions of years ago that was I do not
pretend to say, nor is the point material for our argument; suffice it
to say, that assuming, as geology assures us we may assume, the
existence of these æons of millions of years, there was once a time
when the day was not only one hour shorter, but was even several hours
less than it is at present. Nor need we stop our retrospect at a day
of even twenty, or fifteen, or ten hours long; we shall at once
project our glance back to an immeasurably remote epoch, at which the
earth was spinning round in a time only one sixth or even less of the
length of the present day. There is here a reason for our retrospect
to halt, for at some eventful period, when the day was about three or
four hours long, the earth must have been in a condition of a very
critical kind.
It is well known that fearful accidents occasionally happen where
large grindstones are being driven at a high speed. The velocity of
rotation becomes too great for the tenacity of the stone to withstand
the stress; a rupture takes place, the stone flies in pieces, and huge
fragments are hurled around. For each particular grindstone there is
a certain special velocity depending upon its actual materials and
character, at which it would inevitably fly in pieces. I have once
before likened our earth to a wheel; now let me liken it to a
grindstone. There is therefore a certain critical velocity of rotation
for the earth at which it would be on the brink of rupture. We cannot
exactly say, in our ignorance of the internal constitution of the
earth, what length of day would be the shortest possible for our earth
to have consistently with the preservation of its integrity; we may,
however, assume that it will be about three or four hours, or perhaps
a little less than three. The exact amount, however, is not really
very material to us; it would be sufficient for our argument to assert
that there is a certain minimum length of day for which the earth can
hold together. In our retrospect, therefore, through the abyss of time
past our view must be bounded by that state of the earth when it is
revolving in this critical period. With what happened before that we
shall not at present concern ourselves. Thus we look back to a time at
the beginning of the present order of things, when the day was only
some three or four hours long.
Let us now look at the moon, and examine where it must have been
during these past ages. As the moon is gradually getting further and
further from us at present, so, looking back into past time, we find
that the moon was nearer and nearer to the earth the further back our
view extends; in fact, concentrating our attention solely on essential
features, we may say that the path of the moon is a sort of spiral
which winds round and round the earth, gradually getting larger,
though with extreme slowness. Looking back we see this spiral
gradually coiling in and in, until in a retrospect of millions of
years, instead of its distance from the earth being 240,000 miles, it
must have been much less. There was a time when the moon was only
200,000 miles away; there was a time many millions of years ago, when
the moon was only 100,000 miles away. Nor can we here stop our
retrospect; we must look further and further back, and follow the
moon's spiral path as it creeps in and in towards the earth, until at
last it appears actually in contact with that great globe of ours,
from which it is now separated by a quarter of a million of miles.
Surely the tides have thus led us to the knowledge of an astounding
epoch in our earth's past history, when the earth is spinning round in
a few hours, and when the moon is, practically speaking, in contact
with it. Perhaps I should rather say, that the materials of our
present moon were in this situation, for we would hardly be entitled
to assume that the moon then possessed the same globular form in which
we see it now. To form a just apprehension of the true nature of both
bodies at this critical epoch, we must study their concurrent history
as it is disclosed to us by a totally different line of reasoning.
Drop, then, for a moment all thought of tides, and let us bring to our
aid the laws of heat, which will disclose certain facts in the ancient
history of the earth-moon system perhaps as astounding as those to
which the tides have conducted us. In one respect we may compare these
laws of heat with the laws of the tides; they are both alike
non-periodic, their effects are cumulative from age to age, and
imagination can hardly even impose a limit to the magnificence of the
works they can accomplish. Our argument from heat is founded on a
very simple matter. It is quite obvious that a heated body tends to
grow cold. I am not now speaking of fires or of actual combustion
whereby heat is produced; I am speaking merely of such heat as would
be possessed by a red-hot poker after being taken from the fire, or by
an iron casting after the metal has been run into the mould. In such
cases as this the general law holds good, that the heated body tends
to grow cold. The cooling may be retarded no doubt if the passage of
heat from the body is impeded. We can, for instance, retard the
cooling of a teapot by the well-known practice of putting a cosy upon
it; but the law remains that, slowly or quickly, the heated body will
tend to grow colder. It seems almost puerile to insist with any
emphasis on a point so obvious as this, but yet I frequently find that
people do not readily apprehend all the gigantic consequences that can
flow from a principle so simple. It is true that a poker cools when
taken from the fire; we also find that a gigantic casting weighing
many tons will grow gradually cold, though it may require days to do
so. The same principle will extend to any object, no matter how vast
it may happen to be. Were that great casting 2000 miles in diameter,
or were it 8000 miles in diameter, it will still steadily part with
its heat, though no doubt the process of cooling becomes greatly
prolonged with an increase in the dimensions of the heated body. The
earth and the moon cannot escape from the application of these simple
principles.
Let us first speak of the earth. There are multitudes of volcanoes in
action at the present moment in various countries upon this earth. Now
whatever explanation may be given of the approximate cause of the
volcanic phenomena, there can be no doubt that they indicate the
existence of heat in the interior of the earth. It may possibly be, as
some have urged, that the volcanoes are merely vents for comparatively
small masses of subterranean molten matter; it may be, as others more
reasonably, in my opinion, believe, that the whole interior of the
earth is at the temperature of incandescence, and that the eruptions
of volcanoes and the shocks of earthquakes are merely consequences of
the gradual shrinkage of the external crust, as it continually strives
to accommodate itself to the lessening bulk of the fluid interior.
But whichever view we may adopt, it is at least obvious that the earth
is in part, at all events, a heated body, and that the heat is not in
the nature of a combustion, generated and sustained by the progress of
chemical action. No doubt there may be local phenomena of this
description, but by far the larger proportion of the earth's internal
heat seems merely the fervour of incandescence. It is to be likened to
the heat of the molten iron which has been run into the sand, rather
than to the glowing coals in the furnace in which that iron has been
smelted.
There is one volcanic outbreak of such exceptional interest in these
modern times that I cannot refrain from alluding to it. Doubtless
every one has heard of that marvellous eruption of Krakatoa, which
occurred on August 26th and 27th, 1883, and gives a unique chapter in
the history of volcanic phenomena. Not alone was the eruption of
Krakatoa alarming in its more ordinary manifestations, but it was
unparalleled both in the vehemence of the shock and in the distance to
which the effects of the great eruption were propagated. I speak not
now of the great waves of ocean that inundated the coasts of Sumatra
and Java, and swept away thirty-six thousand people, nor do I allude
to the intense darkness which spread for one hundred and eighty miles
or more all round. I shall just mention the three most important
phenomena, which demonstrate the energy which still resides in the
interior of our earth. Place a terrestrial globe before you, and fix
your attention on the Straits of Sunda; think also of the great
atmospheric ocean some two or three hundred miles deep which envelopes
our earth. When a pebble is tossed into a pond a beautiful series of
concentric ripples diverge from it; so when Krakatoa burst up in that
mighty catastrophe, a series of gigantic waves were propagated through
the air; they embraced the whole globe, converged to the antipodes of
Krakatoa, thence again diverged, and returned to the seat of the
volcano; a second time the mighty series of atmospheric ripples spread
to the antipodes, and a second time returned. Seven times did that
series of waves course over our globe, and leave their traces on every
self-recording barometer that our earth possesses. Thirty-six hours
were occupied in the journey of the great undulation from Krakatoa to
its antipodes. Perhaps even more striking was the extent of our
earth's surface over which the noise of the great explosion spread. At
Batavia, ninety-four miles away, the concussions were simply
deafening; at Macassar, in Celebes, two steamers were sent out to
investigate the explosions which were heard, little thinking that they
came from Krakatoa, nine hundred and sixty-nine miles away. Alarming
sounds were heard over the island of Timor, one thousand three hundred
and fifty-one miles away from Krakatoa. Diego Garcia in the Chogos
islands is two thousand two hundred and sixty-seven miles from
Krakatoa, but the thunders traversed even this distance, and were
attributed to some ship in distress, for which a search was made. Most
astounding of all, there is undoubted evidence that the sound of the
mighty explosion was propagated across nearly the entire Indian ocean,
and was heard in the island of Rodriguez, almost three thousand miles
away. The immense distance over which this sound journeyed will be
appreciated by the fact, that the noise did not reach Rodriguez until
four hours after it had left Krakatoa. In fact, it would seem that if
Vesuvius were to explode with the same vehemence as Krakatoa did, the
thunders of the explosion might penetrate so far as to be heard in
London.
There is another and more beautiful manifestation of the world-wide
significance of the Krakatoa outbreak. The vast column of smoke and
ashes ascended twenty miles high in the air, and commenced a series of
voyages around the equatorial regions of the earth. In three days it
crossed the Indian ocean, and was traversing equatorial Africa; then
came an Atlantic voyage; and then it coursed over central America,
before a Pacific voyage brought it back to its point of departure
after thirteen days; then the dust started again, and was traced
around another similar circuit, while it was even tracked for a
considerable time in placing the third girdle round the earth. Strange
blue suns and green moons and other mysterious phenomena marked the
progress of this vast volcanic cloud. At last the cloud began to lose
its density, the dust spread more widely over the tropics, became
diffused through the temperate regions, and then the whole earth was
able to participate in the glories of Krakatoa. The marvellous sunsets
in the autumn of 1883 are attributable to this cause; and thus once
again was brought before us the fact that the earth still contains
large stores of thermal energy.
Attempts are sometimes made to explain volcanic phenomena on the
supposition that they are entirely of a local character, and that we
are not entitled to infer the incandescent nature of the earth's
interior from the fact that volcanic outbreaks occasionally happen.
For our present purpose this point is immaterial, though I must say it
appears to me unreasonable to deny that the interior of the earth is
in a most highly heated state. Every test we can apply shows us the
existence of internal heat. Setting aside the more colossal phenomena
of volcanic eruptions, we have innumerable minor manifestations of its
presence. Are there not geysers and hot springs in many parts of the
earth? and have we not all over our globe invariable testimony
confirming the statement, that the deeper we go down beneath its
surface the hotter does the temperature become? Every miner is
familiar with these facts; he knows that the deeper are his shafts the
warmer it is down below, and the greater the necessity for providing
increased ventilation to keep the temperature within a limit that
shall be suitable for the workmen. All these varied classes of
phenomena admit solely of one explanation, and that is, that the
interior of the earth contains vast stores of incandescent heat.
We now apply to our earth the same reasoning which we should employ on
a poker taken from the fire, or on a casting drawn from the foundry.
Such bodies will lose their heat by radiation and conduction. The
earth is therefore losing its heat. No doubt the process is an
extremely slow one. The mighty reservoirs of internal heat are covered
by vast layers of rock, which are such excellent non-conductors that
they offer every possible impediment to the leakage of heat from the
interior to the surface. We coat our steam-pipes over with
non-conducting material, and this can now be done so successfully,
that it is beginning to be found economical to transmit steam for a
very long distance through properly protected pipes. But no
non-conducting material that we can manufacture can be half so
effective as the shell of rock twenty miles or more in thickness,
which secures the heated interior of the earth from rapid loss by
radiation into space. Even were the earth's surface solid copper or
solid silver, both most admirable conductors of heat, the cooling down
of this vast globe would be an extremely tardy process; how much more
tardy must it therefore be when such exceedingly bad conductors as
rocks form the envelope? How imperfectly material of this kind will
transmit heat is strikingly illustrated by the great blast iron
furnaces which are so vitally important in one of England's greatest
manufacturing industries. A glowing mass of coal and iron ore and
limestone is here urged to vivid incandescence by a blast of air
itself heated to an intense temperature. The mighty heat thus
generated--sufficient as it is to detach the iron from its close
alliance with the earthy materials and to render the metal out as a
pure stream rushing white-hot from the vent--is sufficiently confined
by a few feet of brick-work, one side of which is therefore at the
temperature of molten iron, while the other is at a temperature not
much exceeding that of the air. We may liken the brick-work of a blast
furnace to the rocky covering of the earth; in each case an
exceedingly high temperature on one side is compatible with a very
moderate temperature on the other.
Although the drainage of heat away from the earth's interior to its
surface, and its loss there by radiation into space, is an extremely
tardy process, yet it is incessantly going on. We have here again to
note the ability for gigantic effect which a small but continually
operating cause may have, provided it always tends in the same
direction. The earth is incessantly losing heat; and though in a day,
a week, or a year the loss may not be very significant, yet when we
come to deal with periods of time that have to be reckoned by millions
of years, it may well be that the effect of a small loss of heat per
annum can, in the course of these ages, reach unimagined dimensions.
Suppose, for instance, that the earth experienced a fall of
temperature in its interior which amounted to only one-thousandth of a
degree in a year. So minute a quantity as this is imperceptible. Even
in a century, the loss of heat at this rate would be only the tenth of
a degree. There would be no possible way of detecting it; the most
careful thermometer could not be relied on to tell us for a certainty
that the temperature of the hot waters of Bath had declined the tenth
of a degree; and I need hardly say, that the fall of a tenth of a
degree would signify nothing in the lavas of Vesuvius, nor influence
the thunders of Krakatoa by one appreciable note. So far as a human
life or the life of the human race is concerned, the decline of a
tenth of a degree per century in the earth's internal heat is
absolutely void of significance. I cannot, however, impress upon you
too strongly, that the mere few thousands of years with which human
history is cognizant are an inappreciable moment in comparison with
those unmeasured millions of years which geology opens out to us, or
with those far more majestic periods which the astronomer demands for
the events he has to describe.
An annual loss of even one-thousandth of a degree will be capable of
stupendous achievements when supposed to operate during epochs of
geological magnitude. In fact, its effects would be so vast, that it
seems hardly credible that the present loss of heat from the earth
should be so great as to amount to an abatement of one-thousandth of a
degree per annum, for that would mean, that in a thousand years the
earth's temperature would decline by one degree, and in a million
years the decline would amount to a thousand degrees. At all events,
the illustration may suffice to show, that the fact that we are not
able to prove by our instruments that the earth is cooling is no
argument whatever against the inevitable law, that the earth, like
every other heated body, must be tending towards a lower temperature.
Without pretending to any numerical accuracy, we can at all events
give a qualitative if not a quantitative analysis of the past history of
our earth, in so far as its changes of temperature are concerned. A
million years ago our earth doubtless contained appreciably more heat
than it does at present. I speak not now, of course, of mere solar
heat--of the heat which gives us the vicissitudes of seasons; I am
only referring to the original hoard of internal heat which is
gradually waning. As therefore our retrospect extends through
millions and millions of past ages, we see our earth ever growing
warmer and warmer the further and further we look back. There was a
time when those heated strata which we have now to go deep down in
mines to find were considerably nearer the surface. At present, were
it not for the sun, the heat of the earth where we stand would hardly
be appreciably above the temperature of infinite space--perhaps some
200 or 300 degrees below zero. But there must have been a time when
there was sufficient internal heat to maintain the exterior at a warm
and indeed at a very hot temperature. Nor is there any bound to our
retrospect arising from the operation or intervention of any other
agent, so far as we know; consequently the hotter and the hotter grows
the surface the further and the further we look back. Nor can we stop
until, at an antiquity so great that I do not venture on any estimate
of the date, we discover that this earth must have consisted of
glowing hot material. Further and further we can look back, and we see
the rocks--or whatever other term we choose to apply to the then
ingredients of the earth's crust--in a white-hot and even in a molten
condition. Thus our argument has led us to the belief that time was
when this now solid globe of ours was a ball of white-hot fluid.
On the argument which I have here used there are just two remarks
which I particularly wish to make. Note in the first place, that our
reasoning is founded on the fact that the earth is at present, to some
extent, heated. It matters not whether this heat be much or little;
our argument would have been equally valid had the earth only
contained a single particle of its mass at a somewhat higher
temperature than the temperature of space. I am, of course, not
alluding in this to heat which can be generated by combustion. The
other point to which I refer is to remove an objection which may
possibly be urged against this line of reasoning. I have argued that
because the temperature is continually increasing as we look
backwards, that therefore a very great temperature must once have
prevailed. Without some explanation this argument is not logically
complete. There is, it is well known, the old paradox of the geometric
series; you may add a farthing to a halfpenny, and then a
half-farthing, and then a quarter-farthing, and then the eighth of a
farthing, followed by the sixteenth, and thirty-second, and so on,
halving the contribution each time. Now no matter how long you
continue this process, even if you went on with it for ever, and thus
made an infinite number of contributions, you would never accomplish
the task of raising the original halfpenny to the dignity of a penny.
An infinite number of quantities may therefore, as this illustration
shows, never succeed in attaining any considerable dimensions. Our
argument, however, with regard to the increase of heat as we look back
is the very opposite of this. It is the essence of a cooling body to
lose heat more rapidly in proportion as its temperature is greater.
Thus though the one-thousandth of a degree may be all the fall of
temperature that our earth now experiences in a twelvemonth, yet in
those glowing days when the surface was heated to incandescence, the
loss of heat per annum must have been immensely greater than it is
now. It therefore follows that the rate of gain of the earth's heat as
we look back must be of a different character to that of the geometric
series which I have just illustrated; for each addition to the
earth's heat, as we look back from year to year, must grow greater and
greater, and therefore there is here no shelter for a fallacy in the
argument on which the existence of high temperature of primeval times
is founded.
The reasoning that I have applied to our earth may be applied in
almost similar words to the moon. It is true that we have not any
knowledge of the internal nature of the moon at present, nor are we
able to point to any active volcanic phenomena at present in progress
there in support of the contention that the moon either has now
internal heat, or did once possess it. It is, however, impossible to
deny the evidence which the lunar craters afford as to the past
existence of volcanic activity on our satellite. Heat, therefore,
there was once in the moon; and accordingly we are enabled to conclude
that, on a retrospect through illimitable periods of time, we must
find the moon transformed from that cold and inert body she now seems
to a glowing and incandescent mass of molten material. The earth
therefore and the moon in some remote ages--not alone anterior to the
existence of life, but anterior even to the earliest periods of which
geologists have cognizance--must have been both globes of molten
materials which have consolidated into the rocks of the present epoch.
We must now revert to the tidal history of the earth-moon system. Did
we not show that there was a time when the earth and the moon--or
perhaps, I should say, the ingredients of the earth and moon--were
close together, were indeed in actual contact? We have now learned,
from a wholly different line of reasoning, that in very early ages
both bodies were highly heated. Here as elsewhere in this theory we
can make little or no attempt to give any chronology, or to harmonize
the different lines along which the course of history has run. No one
can form the slightest idea as to what the temperature of the earth
and of the moon must have been in those primeval ages when they were
in contact. It is impossible, however, to deny that they must both
have been in a very highly heated state; and everything we know of the
matter inclines us to the belief that the temperature of the
earth-moon system must at this critical epoch have been one of
glowing incandescence and fusion. It is therefore quite possible that
these bodies--the moon especially--may have then been not at all of
the form we see them now. It has been supposed, and there are some
grounds for the supposition, that at this initial stage of earth-moon
history the moon materials did not form a globe, but were disposed in
a ring which surrounded the earth, the ring being in a condition of
rapid rotation. It was at a subsequent period, according to this view,
that the substances in the ring gradually drew together, and then by
their mutual attractions formed a globe which ultimately consolidated
down into the compact moon as we now see it. I must, however,
specially draw your attention to the clearly-marked line which divides
the facts which dynamics have taught us from those notions which are
to be regarded as more or less conjectural. Interpreting the action of
the tides by the principles of dynamics, we are assured that the moon
was once--or rather the materials of the moon--in the immediate
vicinity of the earth. There, however, dynamics leaves us, and
unfortunately withholds its accurate illumination from the events
which immediately preceded that state of things.
The theory of tidal evolution which I am describing in these lectures
is mainly the work of Professor George H. Darwin of Cambridge. Much of
the original parts of the theory of the tides was due to Sir William
Thomson, and I have also mentioned how Professor Purser contributed an
important element to the dynamical theory. It is, however, Darwin who
has persistently deduced from the theory all the various consequences
which can be legitimately drawn from it. Darwin, for instance, pointed
out that as the moon is receding from us, it must, if we only look far
enough back, have been once in practical contact with the earth. It is
to Darwin also that we owe many of the other parts of a fascinating
theory, either in its mathematical or astronomical aspect; but I must
take this opportunity of saying, that I do not propose to make
Professor Darwin or any of the other mathematicians I have named
responsible for all that I shall say in these lectures. I must be
myself accountable for the way in which the subject is being treated,
as well as for many of the illustrations used, and some of the
deductions I have drawn from the subject.
It is almost unavoidable for us to make a surmise as to the cause by
which the moon had come into this remarkable position close to the
earth at the most critical epoch of earth-moon history.
With reference to this Professor Darwin has offered an explanation,
which seems so exceedingly plausible that it is impossible to resist
the notion that it must be correct. I will ask you to think of the
earth not as a solid body covered largely with ocean, but as a glowing
globe of molten material. In a globe of this kind it is possible for
great undulations to be set up. Here is a large vase of water, and by
displacing it I can cause the water to undulate with a period which
depends on the size of the vessel; undulations can be set up in a
bucket of water, the period of these undulations being dependent upon
the dimensions of the bucket. Similarly in a vast globe of molten
material certain undulations could be set up, and those undulations
would have a period depending upon the dimensions of this vibrating
mass. We may conjecture a mode in which such vibrations could be
originated. Imagine a thin shell of rigid material which just encases
the globe; suppose this be divided into four quarters, like the four
quarters of an orange, and that two of these opposite quarters be
rejected, leaving two quarters on the liquid. Now suppose that these
two quarters be suddenly pressed in, and then be as suddenly
removed--they will produce depressions, of course, on the two opposite
quarters, while the uncompressed quarters will become protuberant. In
virtue of the mutual attractions between the different particles of
the mass, an effort will be made to restore the globular form, but
this will of course rather overshoot the mark; and therefore a series
of undulations will be originated by which two opposite quarters of
the sphere will alternately shrink in and become protuberant. There
will be a particular period to this oscillation. For our globe it
would appear to be somewhere about an hour and a half or two hours;
but there is necessarily a good deal of uncertainty about this point.
We have seen how in those primitive days the earth was spinning around
very rapidly; and I have also stated that the earth might at this
very critical epoch of its history be compared with a grindstone
which is being driven so rapidly that it is on the very brink of
rupture. It is remarkable to note, that a cause tending to precipitate
a rupture of the earth was at hand. The sun then raised tides in the
earth as it does at present. When the earth revolved in a period of
some four hours or thereabouts, the high tides caused by the sun
succeeded each other at intervals of about two hours. When I speak of
tides in this respect, of course I am not alluding to oceanic tides;
these were the days long before ocean existed, at least in the liquid
form. The tides I am speaking about were raised in the fluids and
materials which then constituted the whole of the glowing earth; those
tides rose and fell under the throb produced by the sun, just as truly
as tides produced in an ordinary ocean. But now note the significant
coincidence between the period of the throb produced by the sun-raised
tides, and that natural period of vibration which belonged to our
earth as a mass of molten material. It therefore follows, that the
impulse given to the earth by the sun harmonized in time with that
period in which the earth itself was disposed to oscillate. A
well-known dynamical principle here comes into play. You see a heavy
weight hanging by a string, and in my hand I hold a little slip of
wood no heavier than a common pencil; ordinarily speaking, I might
strike that heavy weight with this slip of wood, and no effect is
produced; but if I take care to time the little blows that I give so
that they shall harmonize with the vibrations which the weight is
naturally disposed to make, then the effect of many small blows will
be cumulative, so much so, that after a short time the weight begins
to respond to my efforts, and now you see it has acquired a swing of
very considerable amplitude. In Professor Fitzgerald's address to the
British Association at Bath last autumn, he gives an account of those
astounding experiments of Hertz, in which well-timed electrical
impulses broke down an air resistance, and revealed to us ethereal
vibrations which could never have been made manifest except by the
principle we are here discussing. The ingenious conjecture has been
made, that when the earth was thrown into tidal vibrations in those
primeval days, these slight vibrations, harmonizing as they did with
the natural period of the earth, gradually acquired amplitude; the
result being that the pulse of each successive vibration increased at
last to such an extent that the earth separated under the stress, and
threw off a portion of those semi-fluid materials of which it was
composed. In process of time these rejected portions contracted
together, and ultimately formed that moon we now see. Such is the
origin of the moon which the modern theory of tidal evolution has
presented to our notice.
There are two great epochs in the evolution of the earth-moon
system--two critical epochs which possess a unique dynamical
significance; one of these periods was early in the beginning, while
the other cannot arrive for countless ages yet to come. I am aware
that in discussing this matter I am entering somewhat largely into
mathematical principles; I must only endeavour to state the matter as
succinctly as the subject will admit.
In an earlier part of this lecture I have explained how, during all
the development of the earth-moon system, the quantity of moment of
momentum remains unaltered. The moment of momentum of the earth's
rotation added to the moment of momentum of the moon's revolution
remains constant; if one of these quantities increase the other must
decrease, and the progress of the evolution will have this result,
that energy shall be gradually lost in consequence of the friction
produced by the tides. The investigation is one appropriate for
mathematical formulæ, such as those that can be found in Professor
Darwin's memoirs; but nature has in this instance dealt kindly with
us, for she has enabled an abstruse mathematical principle to be dealt
with in a singularly clear and concise manner. We want to obtain a
definite view of the alteration in the energy of the system which
shall correspond to a small change in the velocity of the earth's
rotation, the moon of course accommodating itself so that the moment
of momentum shall be preserved unaltered. We can use for this purpose
an angular velocity which represents the excess of the earth's
rotation over the angular revolution of the moon; it is, in fact, the
apparent angular velocity with which the moon appears to move round
the heavens. If we represent by N the angular velocity of the earth,
and by M the angular velocity of the moon in its orbit round the
earth, the quantity we desire to express is N--M; we shall call it the
relative rotation. The mathematical theorem which tells us what we
want can be enunciated in a concise manner as follows. _The alteration
of the energy of the system may be expressed by multiplying the
relative rotation by the change in the earth's angular velocity._ This
result will explain many points to us in the theory, but just at
present I am only going to make a single inference from it.
I must advert for a moment to the familiar conception of a maximum or
a minimum. If a magnitude be increasing, that is, gradually growing
greater and greater, it has obviously not attained a maximum so long
as the growth is in progress. Nor if the object be actually decreasing
can it be said to be at a maximum either; for then it was greater a
second ago than it is now, and therefore it cannot be at a maximum at
present. We may illustrate this by the familiar example of a stone
thrown up into the air; at first it gradually rises, being higher at
each instant than it was previously, until a culminating point is
reached, when just for a moment the stone is poised at the summit of
its path ere it commences its return to earth again. In this case the
maximum point is obtained when the stone, having ceased to ascend, and
not having yet commenced to descend, is momentarily at rest.
The same principles apply to the determination of a minimum. As long
as the magnitude is declining the minimum has not been reached; it is
only when the decline has ceased, and an increase is on the point of
setting in, that the minimum can be said to be touched.
The earth-moon system contains at any moment a certain store of
energy, and to every conceivable condition of the earth-moon system a
certain quantity of energy is appropriate. It is instructive for us to
study the different positions in which the earth and the moon might
lie, and to examine the different quantities of energy which the
system will contain in each of those varied positions. It is however
to be understood that the different cases all presuppose the same
total moment of momentum.
Among the different cases that can be imagined, those will be of
special interest in which the total quantity of energy in the system
is a maximum or a minimum. We must for this purpose suppose the system
gradually to run through all conceivable changes, with the earth and
moon as near as possible, and as far as possible, and in all
intermediate positions; we must also attribute to the earth every
variety in the velocity of its rotation which is compatible with the
preservation of the moment of momentum. Beginning then with the
earth's velocity of rotation at its lowest, we may suppose it
gradually and continually increased, and, as we have already seen, the
change in the energy of the system is to be expressed by multiplying
the relative rotation into the change of the earth's angular velocity.
It follows from the principles we have already explained, that the
maximum or minimum energy is attained at the moment when the
alteration is zero. It therefore follows, that the critical periods of
the system will arise when the relative rotation is zero, that is,
when the earth's rotation on its axis is performed with a velocity
equal to that with which the moon revolves around the earth. This is
truly a singular condition of the earth-moon system; the moon in such
a case would revolve around the earth as if the two bodies were bound
together by rigid bonds into what was practically a single solid body.
At the present moment no doubt to some extent this condition is
realized, because the moon always turns the same face to the earth (a
point on which we shall have something to say later on); but in the
original condition of the earth-moon system, the earth would also
constantly direct the same face to the moon, a condition of things
which is now very far from being realized.
It can be shown from the mathematical nature of the problem that there
are four states of the earth-moon system in which this condition may
be realized, and which are also compatible with the conservation of
the moment of momentum. We can express what this condition implies in
a somewhat more simple manner. Let us understand by the _day_ the
period of the earth's rotation on its axis, whatever that may be, and
let us understand by the _month_ the period of revolution of the moon
around the earth, whatever value it may have; then the condition of
maximum or minimum energy is attained when the day and the month have
become equal to each other. Of the four occasions mathematically
possible in which the day and the month can be equal, there are only
two which at present need engage our attention--one of these occurred
near the beginning of the earth and moon's history, the other remains
to be approached in the immeasurably remote future. The two remaining
solutions are futile, being what the mathematician would describe as
imaginary.
There is a fundamental difference between the dynamical conditions in
these critical epochs--in one of them the energy of the system has
attained a maximum value, and in the other the energy of the system is
at a minimum value. It is impossible to over-estimate the significance
of these two states of the system.
I may recall the fundamental notion which every one has learned in
mechanics, as to the difference between stable and unstable
equilibrium. The conceivable possibility of making an egg stand on its
end is a practical impossibility, because nature does not like
unstable equilibrium, and a body departs therefrom on the least
disturbance; on the other hand, stable equilibrium is the position in
which nature tends to place everything. A log of wood floating on a
river might conceivably float in a vertical position with its end up
out of the water, but you never could succeed in so balancing it,
because no matter how carefully you adjusted the log, it would almost
instantly turn over when you left it free; on the other hand, when the
log floats naturally on the water it assumes a horizontal position, to
which, when momentarily displaced therefrom, it will return if
permitted to do so. We have here an illustration of the contrast
between stable and unstable equilibrium. It will be found generally
that a body is in equilibrium when its centre of gravity is at its
highest point or at its lowest point; there is, however, this
important difference, that when the centre of gravity is highest the
equilibrium is unstable, and when the centre of gravity is lowest the
equilibrium is stable. The potential energy of an egg poised on its
end in unstable equilibrium is greater than when it lies on its side
in stable equilibrium. In fact, energy must be expended to raise the
egg from the horizontal position to the vertical; while, on the other
hand, work could conceivably be done by the egg when it passes from
the vertical position to the horizontal. Speaking generally, we may
say that the stable position indicates low energy, while a redundancy
of that valuable agent is suggestive of instability.
We may apply similar principles to the consideration of the earth-moon
system. It is true that we have here a series of dynamical phenomena,
while the illustrations I have given of stable and unstable
equilibrium relate only to statical problems; but we can have
dynamical stability and dynamical instability, just as we can have
stable and unstable equilibrium. Dynamical instability corresponds
with the maximum of energy, and dynamical stability to the minimum of
energy.
At that primitive epoch, when the energy of the earth-moon system was
a maximum, the condition was one of dynamical instability; it was
impossible that it should last. But now mark how truly critical an
occurrence this must have been in the history of the earth-moon
system, for have I not already explained that it is a necessary
condition of the progress of tidal evolution that the energy of the
system should be always declining? But here our retrospect has
conducted us back to a most eventful crisis, in which the energy was a
maximum, and therefore cannot have been immediately preceded by a
state in which the energy was greater still; it is therefore
impossible for the tidal evolution to have produced this state of
things; some other influence must have been in operation at this
beginning of the earth-moon system.
Thus there can be hardly a doubt that immediately preceding the
critical epoch the moon originated from the earth in the way we have
described. Note also that this condition, being one of maximum energy,
was necessarily of dynamical instability, it could not last; the moon
must adopt either of two courses--it must tumble back on the earth, or
it must start outwards. Now which course was the moon to adopt? The
case is analogous to that of an egg standing on its end--it will
inevitably tumble one way or the other. Some infinitesimal cause will
produce a tendency towards one side, and to that side accordingly the
egg will fall. The earth-moon system was similarly in an unstable
state, an infinitesimal cause might conceivably decide the fate of the
system. We are necessarily in ignorance of what the determining cause
might have been, but the effect it produced is perfectly clear; the
moon did not again return to its mother earth, but set out on that
mighty career which is in progress to-day.
Let it be noted that these critical epochs in the earth-moon history
arise when and only when there is an absolute identity between the
length of the month and the length of the day. It may be proper
therefore that I should provide a demonstration of the fact, that the
identity between these two periods must necessarily have occurred at a
very early period in the evolution.
The law of Kepler, which asserts that the square of the periodic time
is proportioned to the cube of the mean distance, is in its ordinary
application confined to a comparison between the revolutions of the
several planets about the sun. The periodic time of each planet is
connected with its average distance by this law; but there is another
application of Kepler's law which gives us information of the distance
and the period of the moon in former stages of the earth-moon history.
Although the actual path of the moon is of course an ellipse, yet that
ellipse is troubled, as is well known, by many disturbing forces, and
from this cause alone the actual path of the moon is far from being
any of those simple curves with which we are so well acquainted. Even
were the earth and the moon absolutely rigid particles, perturbations
would work all sorts of small changes in the pliant curve. The
phenomena of tidal evolution impart an additional element of
complexity into the actual shape of the moon's path. We now see that
the ellipse is not merely subject to incessant deflections of a
periodic nature, it also undergoes a gradual contraction as we look
back through time past; but we may, with all needful accuracy for our
present purpose, think of the path of the moon as a circle, only we
must attribute to that circle a continuous contraction of its radius
the further and the further we look back. The alteration in the
radius will be even so slow, that the moon will accomplish thousands
of revolutions around the earth without any appreciable alteration in
the average distance of the two bodies. We can therefore think of the
moon as revolving at every epoch in a circle of special radius, and as
accomplishing that revolution in a special time. With this
understanding we can now apply Kepler's law to the several stages of
the moon's past history. The periodic time of each revolution, and the
mean distance at which that revolution was performed, will be always
connected together by the formula of Kepler. Thus to take an instance
in the very remote past. Let us suppose that the moon was at one
hundred and twenty thousand miles instead of two hundred and forty
thousand, that is, at half its present distance. Applying the law of
Kepler, we see that the time of revolution must then have been only
about ten days instead of the twenty-seven it is now. Still further,
let us suppose that the moon revolves in an orbit with one-tenth of
the diameter it has at present, then the cube of 10 being 1000, and
the square root of 1000 being 31.6, it follows that the month must
have been less than the thirty-first part of what it is at present,
that is, it must have been considerably less than one of our present
days. Thus you see the month is growing shorter and shorter the
further we look back, the day is also growing shorter and shorter; but
still I think we can show that there must have been a time when the
month will have been at least as short as the day. For let us take the
most extreme case in which the moon shall have made the closest
possible approximation to the earth. Two globes in contact will have a
distance between their centres which is equal to the sum of their
radii. Take the earth as having a radius of four thousand miles, and
the moon a radius of one thousand miles, the two centres must at their
shortest distance be five thousand miles apart, that is, the moon must
then be at the forty-eighth part of its present distance from the
earth. Now the cube of 48 is 110,592, and the square root of 110,592
is nearly 333, therefore the length of the month will be one-three
hundred and thirty-third part of the duration of the month at present;
in other words, the moon must revolve around the earth in a period of
somewhat about two hours. It seems impossible that the day can ever
have been as brief as this. We have therefore proved that, in the
course of its contracting duration, the moon must have overtaken the
contracting day, and that therefore there must have been a time when
the moon was in the vicinity of the earth, and having a day and month
of equal period. Thus we have shown that the critical condition of
dynamical instability must have occurred in the early period of the
earth-moon history, if the agents then in operation were those which
we now know. The further development of the subject must be postponed
until the next lecture.