predicted is to set up a tide-gauge, or automatic recorder, and keep it
working for a year or two. The tide-gauge is easy enough to understand:
it marks the height of the tide at every instant by an irregular curved
line like a barometer chart (Fig. 117). These observational curves so
obtained have next to be fed into a fearfully complex machine, which it
would take a whole lecture to make even partially intelligible, but Fig.
118 shows its aspect. It consists of ten integrating machines in a row,
coupled up and working together. This is the "harmonic analyzer," and
the result of passing the curve through this machine is to give you all
the constituents of which it is built up, viz. the lunar tide, the solar
tide, and eight of the sub-tides or disturbances. These ten values are
then set off into a third machine, the tide-predicter proper. The
general mode of action of this machine is not difficult to understand.
It consists of a string wound over and under a set of pulleys, which are
each set on an excentric, so as to have an up-and-down motion. These
up-and-down motions are all different, and there are ten of these
movable pulleys, which by their respective excursions represent the
lunar tide, the solar tide, and the eight disturbances already analyzed
out of the tide-gauge curve by the harmonic analyzer. One end of the
string is fixed, the other carries a pencil which writes a trace on a
revolving drum of paper--a trace which represents the combined motion of
all the pulleys, and so predicts the exact height of the tide at the
place, at any future time you like. The machine can be turned quite
quickly, so that a year's tides can be run off with every detail in
about half-an-hour. This is the easiest part of the operation. Nothing
has to be done but to keep it supplied with paper and pencil, and turn a
handle as if it were a coffee-mill instead of a tide-mill. (Figs. 119
and 120.)
[Illustration: FIG. 118.--Harmonic analyzer; for analyzing out the
constituents from a set of observational curves.]
My subject is not half exhausted. I might go on to discuss the question
of tidal energy--whether it can be ever utilized for industrial
purposes; and also the very interesting question whence it comes. Tidal
energy is almost the only terrestrial form of energy that does not
directly or indirectly come from the sun. The energy of tides is now
known to be obtained at the expense of the earth's rotation; and
accordingly our day must be slowly, very slowly, lengthening. The tides
of past ages have destroyed the moon's rotation, and so it always turns
the same face to us. There is every reason to believe that in geologic
ages the moon was nearer to us than it is now, and that accordingly our
tides were then far more violent, rising some hundreds of feet instead
of twenty or thirty, and sweeping every six hours right over the face of
a country, ploughing down hills, denuding rocks, and producing a copious
sedimentary deposit.
[Illustration: FIG. 119.--Tide-predicter, for combining the ascertained
constituents into a tidal curve for the future.]
In thus discovering the probable violent tides of past ages, astronomy
has, within the last few years, presented geology with the most powerful
denuding agent known; and the study of the earth's past history cannot
fail to be greatly affected by the modern study of the intricate and
refined conditions attending prolonged tidal action on incompletely
rigid bodies. [Read on this point the last chapter of Sir R. Ball's
_Story of the Heavens_.]
[Illustration: Fig. 120.--Weekly sheet of curves. Tides for successive
days are predicted on the same sheet of paper, to economise space.]
I might also point out that the magnitude of our terrestrial tides
enables us to answer the question as to the internal fluidity of the
earth. It used to be thought that the earth's crust was comparatively
thin, and that it contained a molten interior. We now know that this is
not the case. The interior of the earth is hot indeed, but it is not
fluid. Or at least, if it be fluid, the amount of fluid is but very
small compared with the thickness of the unyielding crust. All these,
and a number of other most interesting questions, fringe the subject of
the tides; the theoretical study of which, started by Newton, has
developed, and is destined in the future to further develop, into one of
the most gigantic and absorbing investigations--having to do with the
stability or instability of solar systems, and with the construction and
decay of universes.
These theories are the work of pioneers now living, whose biographies it
is therefore unsuitable for us to discuss, nor shall I constantly
mention their names. But Helmholtz, and Thomson, are household words,
and you well know that in them and their disciples the race of Pioneers
maintains its ancient glory.
NOTES FOR LECTURE XVIII
Tides are due to incomplete rigidity of bodies revolving round each
other under the action of gravitation, and at the same time spinning on
their axes.
Two spheres revolving round each other can only remain spherical if
rigid; if at all plastic they become prolate. If either rotate on its
axis, in the same or nearly the same plane as it revolves, that one is
necessarily subject to tides.
The axial rotation tends to carry the humps with it, but the pull of the
other body keeps them from moving much. Hence the rotation takes place
against a pull, and is therefore more or less checked and retarded. This
is the theory of Von Helmholtz.
The attracting force between two such bodies is no longer _exactly_
towards the centre of revolution, and therefore Kepler's second law is
no longer precisely obeyed: the rate of description of areas is subject
to slight acceleration. The effect of this tangential force acting on
the tide-compelling body is gradually to increase its distance from the
other body.
Applying these statements to the earth and moon, we see that tidal
energy is produced at the expense of the earth's rotation, and that the
length of the day is thereby slowly increasing. Also that the moon's
rotation relative to the earth has been destroyed by past tidal action
in it (the only residue of ancient lunar rotation now being a scarcely
perceptible libration), so that it turns always the same face towards
us. Moreover, that its distance from the earth is steadily increasing.
This last is the theory of Professor G.H. Darwin.
Long ago the moon must therefore have been much nearer the earth, and
the day was much shorter. The tides were then far more violent.
Halving the distance would make them eight times as high; quartering it
would increase them sixty-four-fold. A most powerful geological denuding
agent. Trade winds and storms were also more violent.
If ever the moon were close to the earth, it would have to revolve round
it in about three hours. If the earth rotated on its axis in three
hours, when fluid or pasty, it would be unstable, and begin to separate
a portion of itself as a kind of bud, which might then get detached and
gradually pushed away by the violent tidal action. Hence it is possible
that this is the history of the moon. If so, it is probably an
exceptional history. The planets were not formed from the sun in this
way.
Mars' moons revolve round him more quickly than the planet rotates:
hence with them the process is inverted, and they must be approaching
him and may some day crash along his surface. The inner moon is now
about 4,000 miles away, and revolves in 7-1/2 hours. It appears to be
about 20 miles in diameter, and weighs therefore, if composed of rock,
40 billion tons. Mars rotates in 24-1/2 hours.
A similar fate may _possibly_ await our moon ages hence--by reason of
the action of terrestrial tides produced by the sun.
LECTURE XVIII
THE TIDES, AND PLANETARY EVOLUTION
In the last lecture we considered the local peculiarities of the tides,
the way in which they were formed in open ocean under the action of the
moon and the sun, and also the means by which their heights and times
could be calculated and predicted years beforehand. Towards the end I
stated that the subject was very far from being exhausted, and
enumerated some of the large and interesting questions which had been
left untouched. It is with some of these questions that I propose now to
deal.
I must begin by reminding you of certain well-known facts, a knowledge
of which I may safely assume.
And first we must remind ourselves of the fact that almost all the rocks
which form the accessible crust of the earth were deposited by the
agency of water. Nearly all are arranged in regular strata, and are
composed of pulverized materials--materials ground down from
pre-existing rocks by some denuding and grinding action. They nearly all
contain vestiges of ancient life embedded in them, and these vestiges
are mainly of marine origin. The strata which were once horizontal are
now so no longer--they have been tilted and upheaved, bent and
distorted, in many places. Some of them again have been metamorphosed by
fire, so that their organic remains have been destroyed, and the traces
of their aqueous origin almost obliterated. But still, to the eye of the
geologist, all are of aqueous or sedimentary origin: roughly speaking,
one may say they were all deposited at the bottom of some ancient sea.
The date of their formation no man yet can tell, but that it was vastly
distant is certain. For the geological era is not over. Aqueous action
still goes on: still does frost chip the rocks into fragments; still do
mountain torrents sweep stone and mud and _debris_ down the gulleys and
watercourses; still do rivers erode their channels, and carry mud and
silt far out to sea. And, more powerful than any of these agents of
denudation, the waves and the tides are still at work along every
coast-line, eating away into the cliffs, undermining gradually and
submerging acre after acre, and making with the refuse a shingly, or a
sandy, or a muddy beach--the nucleus of a new geological formation.
Of all denuding agents, there can be no doubt that, to the land exposed
to them, the waves of the sea are by far the most powerful. Think how
they beat and tear, and drive and drag, until even the hardest rock,
like basalt, becomes honeycombed into strange galleries and
passages--Fingal's Cave, for instance--and the softer parts are crumbled
away. But the area now exposed to the teeth of the waves is not great.
The fury of a winter storm may dash them a little higher than usual, but
they cannot reach cliffs 100 feet high. They can undermine such cliffs
indeed, and then grind the fragments to powder, but their direct action
is limited. Not so limited, however, as they would be without the tides.
Consider for a moment the denudation import of the tides: how does the
existence of tidal rise and fall affect the geological problem?
The scouring action of the tidal currents themselves is not to be
despised. It is the tidal ebb and flow which keeps open channel in the
Mersey, for instance. But few places are so favourably situated as
Liverpool in this respect, and the direct scouring action of the tides
in general is not very great. Their geological import mainly consists in
this--that they raise and lower the surface waves at regular intervals,
so as to apply them to a considerable stretch of coast. The waves are a
great planing machine attacking the land, and the tides raise and lower
this planing machine, so that its denuding tooth is applied, now twenty
feet vertically above mean level, now twenty feet below.
Making all allowance for the power of winds and waves, currents, tides,
and watercourses, assisted by glacial ice and frost, it must be apparent
how slowly the work of forming the rocks is being carried on. It goes on
steadily, but so slowly that it is estimated to take 6000 years to wear
away one foot of the American continent by all the denuding causes
combined. To erode a stratum 5000 feet thick will require at this rate
thirty million years.
The age of the earth is not at all accurately known, but there are many
grounds for believing it not to be much older than some thirty million
years. That is to say, not greatly more than this period of time has
elapsed since it was in a molten condition. It may be as old as a
hundred million years, but its age is believed by those most competent
to judge to be more likely within this limit than beyond it. But if we
ask what is the thickness of the rocks which in past times have been
formed, and denuded, and re-formed, over and over again, we get an
answer, not in feet, but in miles. The Laurentian and Huronian rocks of
Canada constitute a stratum ten miles thick; and everywhere the rocks at
the base of our stratified system are of the most stupendous volume and
thickness.
It has always been a puzzle how known agents could have formed these
mighty masses, and the only solution offered by geologists was,
unlimited time. Given unlimited time, they could, of course, be formed,
no matter how slowly the process went on. But inasmuch as the time
allowable since the earth was cool enough for water to exist on it
except as steam is not by any means unlimited, it becomes necessary to
look for a far more powerful engine than any now existing; there must
have been some denuding agent in those remote ages--ages far more
distant from us than the Carboniferous period, far older than any forms
of life, fossil or otherwise, ages among the oldest known to geology--a
denuding agent must have then existed, far more powerful than any we now
know.
Such an agent it has been the privilege of astronomy and physics, within
the last ten years, to discover. To this discovery I now proceed to lead
up.
Our fundamental standard of time is the period of the earth's
rotation--the length of the day. The earth is our one standard clock:
all time is expressed in terms of it, and if it began to go wrong, or if
it did not go with perfect uniformity, it would seem a most difficult
thing to discover its error, and a most puzzling piece of knowledge to
utilize when found.
That it does not go much wrong is proved by the fact that we can
calculate back to past astronomical events--ancient eclipses and the
like--and we find that the record of their occurrence, as made by the
old magi of Chaldaea, is in very close accordance with the result of
calculation. One of these famous old eclipses was observed in Babylon
about thirty-six centuries ago, and the Chaldaean astronomers have put on
record the time of its occurrence. Modern astronomers have calculated
back when it should have occurred, and the observed time agrees very
closely with the actual, but not exactly. Why not exactly?
Partly because of the acceleration of the moon's mean motion, as
explained in the lecture on Laplace (p. 262). The orbit of the earth was
at that time getting rounder, and so, as a secondary result, the speed
of the moon was slightly increasing. It is of the nature of a
perturbation, and is therefore a periodic not a progressive or
continuous change, and in a sufficiently long time it will be reversed.
Still, for the last few thousand years the moon's motion has been, on
the whole, accelerated (though there seems to be a very slight retarding
force in action too).
Laplace thought that this fact accounted for the whole of the
discrepancy; but recently, in 1853, Professor Adams re-examined the
matter, and made a correction in the details of the theory which
diminishes its effect by about one-half, leaving the other half to be
accounted for in some other way. His calculations have been confirmed by
Professor Cayley. This residual discrepancy, when every known cause has
been allowed for, amounts to about one hour.
The eclipse occurred later than calculation warrants. Now this
would have happened from either of two causes, either an
acceleration of the moon in her orbit, or a retardation of the
earth in her diurnal rotation--a shortening of the month or a
lengthening of the day, or both. The total discrepancy being, say,
two hours, an acceleration of six seconds-per-century per century
will in thirty-six centuries amount to one hour; and this,
according to the corrected Laplacian theory, is what has occurred.
But to account for the other hour some other cause must be sought,
and at present it is considered most probably due to a steady
retardation of the earth's rotation--a slow, very slow, lengthening
of the day.
The statement that a solar eclipse thirty-six centuries ago was an
hour late, means that a place on the earth's surface came into the
shadow one hour behind time--that is, had lagged one twenty-fourth
part of a revolution. The earth, therefore, had lost this amount in
the course of 3600 x 365-1/4 revolutions. The loss per revolution
is exceedingly small, but it accumulates, and at any era the total
loss is the sum of all the losses preceding it. It may be worth
while just to explain this point further.
Suppose the earth loses a small piece of time, which I will call an
instant, per day; a locality on the earth will come up to a given
position one instant late on the first day after an event. On the
next day it would come up two instants late by reason of the
previous loss; but it also loses another instant during the course
of the second day, and so the total lateness by the end of that day
amounts to three instants. The day after, it will be going slower
from the beginning at the rate of two instants a day, it will lose
another instant on the fresh day's own account, and it started
three instants late; hence the aggregate loss by the end of the
third day is 1 + 2 + 3 = 6. By the end of the fourth day the whole
loss will be 1 + 2 + 3 + 4, and so on. Wherefore by merely losing
one instant every day the total loss in _n_ days is (1 + 2 + 3 +
... + _n_) instants, which amounts to 1/2_n_ (_n_ + 1) instants;
or practically, when _n_ is big, to 1/2n^2. Now in thirty-six
centuries there have been 3600 x 365-1/4 days, and the total loss
has amounted to an hour; hence the length of "an instant," the loss
per diem, can be found from the equation 1/2(3600 x 365)^2 instants
= 1 hour; whence one "instant" equals the 240 millionth part of a
second. This minute quantity represents the retardation of the
earth per day. In a year the aggregate loss mounts up to 1/3600th
part of a second, in a century to about three seconds, and in
thirty-six centuries to an hour. But even at the end of the
thirty-six centuries the day is barely any longer; it is only 3600
x 365 instants, that is 1/180th of a second, longer than it was at
the beginning. And even a million years ago, unless the rate of
loss was different (as it probably was), the day would only be
thirty-five minutes shorter, though by that time the aggregate
loss, as measured by the apparent lateness of any perfectly
punctual event reckoned now, would have amounted to nine years.
(These numbers are to be taken as illustrative, not as precisely
representing terrestrial fact.)
What can have caused the slowing down? Swelling of the earth by reason
of accumulation of meteoric dust might do something, but probably very
little. Contraction of the earth as it goes on cooling would act in the
opposite direction, and probably more than counterbalance the dust
effect. The problem is thus not a simple one, for there are several
disturbing causes, and for none of them are the data enough to base a
quantitative estimate upon; but one certain agent in lengthening the
day, and almost certainly the main agent, is to be found in the tides.
Remember that the tidal humps were produced as the prolateness of a
sphere whirled round and round a fixed centre, like a football whirled
by a string. These humps are pulled at by the moon, and the earth
rotates on its axis against this pull. Hence it tends to be constantly,
though very slightly, dragged back.
In so far as the tidal wave is allowed to oscillate freely, it will
swing with barely any maintaining force, giving back at one
quarter-swing what it has received at the previous quarter; but in so
far as it encounters friction, which it does in all channels where
there is an actual ebb and flow of the water, it has to receive more
than it gives back, and the balance of energy has to be made up to it,
or the tides would cease. The energy of the tides is, in fact,
continually being dissipated by friction, and all the energy so
dissipated is taken from the rotation of the earth. If tidal energy were
utilized by engineers, the machines driven would be really driven at the
expense of the earth's rotation: it would be a mode of harnessing the
earth and using the moon as fixed point or fulcrum; the moon pulling at
the tidal protuberance, and holding it still as the earth rotates, is
the mechanism whereby the energy is extracted, the handle whereby the
friction brake is applied.
Winds and ocean currents have no such effect (as Mr. Fronde in
_Oceania_ supposes they have), because they are all accompanied by
a precisely equal counter-current somewhere else, and no internal
rearrangement of fluid can affect the motion of a mass as a whole;
but the tides are in different case, being produced, not by
internal inequalities of temperature, but by a straightforward pull
from an external body.
The ultimate effect of tidal friction and dissipation of energy will,
therefore, be to gradually retard the earth till it does not rotate with
reference to the moon, _i.e._ till it rotates once while the moon
revolves once; in other words, to make the day and the month equal. The
same cause must have been in operation, but with eighty-fold greater
intensity, on the moon. It has ceased now, because the rotation has
stopped, but if ever the moon rotated on its axis with respect to the
earth, and if it were either fluid itself or possessed any liquid ocean,
then the tides caused by the pull of the earth must have been
prodigious, and would tend to stop its rotation. Have they not
succeeded? Is it not probable that this is _why_ the moon always now
turns the same face towards us? It is believed to be almost certainly
the cause. If so, there was a time when the moon behaved
differently--when it rotated more quickly than it revolved, and
exhibited to us its whole surface. And at this era, too, the earth
itself must have rotated a little faster, for it has been losing speed
ever since.
We have thus arrived at this fact, that a thousand years ago the day was
a trifle shorter than it is now. A million years ago it was, perhaps, an
hour shorter. Twenty million years ago it must have been much shorter.
Fifty million years ago it may have been only a few hours long. The
earth may have spun round then quite quickly. But there is a limit. If
it spun too fast it would fly to pieces. Attach shot by means of wax to
the whirling earth model, Fig. 110, and at a certain speed the cohesion
of the wax cannot hold them, so they fly off. The earth is held together
not by cohesion but by gravitation; it is not difficult to reckon how
fast the earth must spin for gravity at its surface to be annulled, and
for portions to fly off. We find it about one revolution in three hours.
This is a critical speed. If ever the day was three hours long,
something must have happened. The day can never have been shorter than
that; for if it were, the earth would have a tendency to fly in pieces,
or, at least, to separate into two pieces. Remember this, as a natural
result of a three-hour day, which corresponds to an unstable state of
things; remember also that in some past epoch a three-hour day is a
probability.
If we think of the state of things going on in the earth's
atmosphere, if it had an atmosphere at that remote date, we shall
recognize the existence of the most fearful tornadoes. The trade
winds, which are now peaceful agents of commerce, would then be
perpetual hurricanes, and all the denudation agents of the
geologist would be in a state of feverish activity. So, too, would
the tides: instead of waiting six hours between low and high tide,
we should have to wait only three-quarters of an hour. Every
hour-and-a-half the water would execute a complete swing from high
tide to high again.
Very well, now leave the earth, and think what has been happening to the
moon all this while.