action and reaction are always equal and opposite--it cannot do that
without itself getting pulled forward. The pull of the earth on the moon
will therefore not be quite central, but will be a little in advance of
its centre; hence, by Kepler's second law, the rate of description of
areas by its radius vector cannot be constant, but must increase (p.
208). And the way it increases will be for the radius vector to
lengthen, so as to sweep out a bigger area. Or, to put it another way,
the extra speed tending to be gained by the moon will fling it further
away by extra centrifugal force. This last is not so good a way of
regarding the matter; though it serves well enough for the case of a
ball whirled at the end of an elastic string. After having got up the
whirl, the hand holding the string may remain almost fixed at the centre
of the circle, and the motion will continue steadily; but if the hand be
moved so as always to pull the string a little in advance of the centre,
the speed of whirl will increase, the elastic will be more and more
stretched, until the whirling ball is describing a much larger circle.
But in this case it will likewise be going faster--distance and speed
increase together. This is because it obeys a different law from
gravitation--the force is not inversely as the square, or any other
single power, of the distance. It does not obey any of Kepler's laws,
and so it does not obey the one which now concerns us, viz. the third;
which practically states that the further a planet is from the centre
the slower it goes; its velocity varies inversely with the square root
of its distance (p. 74).
If, instead of a ball held by elastic, it were a satellite held by
gravity, an increase in distance must be accompanied by a diminution in
speed. The time of revolution varies as the square of the cube root of
the distance (Kepler's third law). Hence, the tidal reaction on the
moon, having as its primary effect, as we have seen, the pulling the
moon a little forward, has also the secondary or indirect effect of
making it move slower and go further off. It may seem strange that an
accelerating pull, directed in front of the centre, and therefore always
pulling the moon the way it is going, should retard it; and that a
retarding force like friction, if such a force acted, should hasten it,
and make it complete its orbit sooner; but so it precisely is.
Gradually, but very slowly, the moon is receding from us, and the month
is becoming longer. The tides of the earth are pushing it away. This is
not a periodic disturbance, like the temporary acceleration of its
motion discovered by Laplace, which in a few centuries, more or less,
will be reversed; it is a disturbance which always acts one way, and
which is therefore cumulative. It is superposed upon all periodic
changes, and, though it seems smaller than they, it is more inexorable.
In a thousand years it makes scarcely an appreciable change, but in a
million years its persistence tells very distinctly; and so, in the long
run, the month is getting longer and the moon further off. Working
backwards also, we see that in past ages the moon must have been nearer
to us than it is now, and the month shorter.
Now just note what the effect of the increased nearness of the moon was
upon our tides. Remember that the tide-generating force varies inversely
as the cube of distance, wherefore a small change of distance will
produce a great difference in the tide-force.
The moon's present distance is 240 thousand miles. At a time when it was
only 190 thousand miles, the earth's tides would have been twice as high
as they are now. The pushing away action was then a good deal more
violent, and so the process went on quicker. The moon must at some time
have been just half its present distance, and the tides would then have
risen, not 20 or 30 feet, but 160 or 200 feet. A little further back
still, we have the moon at one-third of its present distance from the
earth, and the tides 600 feet high. Now just contemplate the effect of a
600-foot tide. We are here only about 150 feet above the level of the
sea; hence, the tide would sweep right over us and rush far away inland.
At high tide we should have some 200 feet of blue water over our heads.
There would be nothing to stop such a tide as that in this neighbourhood
till it reached the high lands of Derbyshire. Manchester would be a
seaport then with a vengeance!
The day was shorter then, and so the interval between tide and tide was
more like ten than twelve hours. Accordingly, in about five hours, all
that mass of water would have swept back again, and great tracts of sand
between here and Ireland would be left dry. Another five hours, and the
water would come tearing and driving over the country, applying its
furious waves and currents to the work of denudation, which would
proceed apace. These high tides of enormously distant past ages
constitute the denuding agent which the geologist required. They are
very ancient--more ancient than the Carboniferous period, for instance,
for no trees could stand the furious storms that must have been
prevalent at this time. It is doubtful whether any but the very lowest
forms of life then existed. It is the strata at the bottom of the
geological scale that are of the most portentous thickness, and the only
organism suspected in them is the doubtful _Eozoon Canadense_. Sir
Robert Ball believes, and several geologists agree with him, that the
mighty tides we are contemplating may have been coaeval with this ancient
Laurentian formation, and others of like nature with it.
But let us leave geology now, and trace the inverted progress of events
as we recede in imagination back through the geological era, beyond,
into the dim vista of the past, when the moon was still closer and
closer to the earth, and was revolving round it quicker and quicker,
before life or water existed on it, and when the rocks were still
molten.
Suppose the moon once touched the earth's surface, it is easy to
calculate, according to the principles of gravitation, and with a
reasonable estimate of its size as then expanded by heat, how fast it
must then have revolved round the earth, so as just to save itself from
falling in. It must have gone round once every three hours. The month
was only three hours long at this initial epoch.
Remember, however, the initial length of the day. We found that it was
just possible for the earth to rotate on its axis in three hours, and
that when it did so, something was liable to separate from it. Here we
find the moon in contact with it, and going round it in this same
three-hour period. Surely the two are connected. Surely the moon was a
part of the earth, and was separating from it.
That is the great discovery--the origin of the moon.
Once, long ages back, at date unknown, but believed to be certainly as
much as fifty million years ago, and quite possibly one hundred million,
there was no moon, only the earth as a molten globe, rapidly spinning on
its axis--spinning in about three hours. Gradually, by reason of some
disturbing causes, a protuberance, a sort of bud, forms at one side, and
the great inchoate mass separates into two--one about eighty times as
big as the other. The bigger one we now call earth, the smaller we now
call moon. Round and round the two bodies went, pulling each other into
tremendously elongated or prolate shapes, and so they might have gone on
for a long time. But they are unstable, and cannot go on thus: they must
either separate or collapse. Some disturbing cause acts again, and the
smaller mass begins to revolve less rapidly. Tides at once
begin--gigantic tides of molten lava hundreds of miles high; tides not
in free ocean, for there was none then, but in the pasty mass of the
entire earth. Immediately the series of changes I have described begins,
the speed of rotation gets slackened, the moon's mass gets pushed
further and further away, and its time of revolution grows rapidly
longer. The changes went on rapidly at first, because the tides were so
gigantic; but gradually, and by slow degrees, the bodies get more
distant, and the rate of change more moderate. Until, after the lapse of
ages, we find the day twenty-four hours long, the moon 240,000 miles
distant, revolving in 27-1/3 days, and the tides only existing in the
water of the ocean, and only a few feet high. This is the era we call
"to-day."
The process does not stop here: still the stately march of events goes
on; and the eye of Science strives to penetrate into the events of the
future with the same clearness as it has been able to descry the events
of the past. And what does it see? It will take too long to go into full
detail: but I will shortly summarize the results. It sees this
first--the day and the month both again equal, but both now about 1,400
hours long. Neither of these bodies rotating with respect to each
other--the two as if joined by a bar--and total cessation of
tide-generating action between them.
The date of this period is one hundred and fifty millions of years
hence, but unless some unforeseen catastrophe intervenes, it must
assuredly come. Yet neither will even this be the final stage; for the
system is disturbed by the tide-generating force of the sun. It is a
small effect, but it is cumulative; and gradually, by much slower
degrees than anything we have yet contemplated, we are presented with a
picture of the month getting gradually shorter than the day, the moon
gradually approaching instead of receding, and so, incalculable myriads
of ages hence, precipitating itself upon the surface of the earth whence
it arose.
Such a catastrophe is already imminent in a neighbouring planet--Mars.
Mars' principal moon circulates round him at an absurd pace, completing
a revolution in 7-1/2 hours, and it is now only 4,000 miles from his
surface. The planet rotates in twenty-four hours as we do; but its tides
are following its moon more quickly than it rotates after them; they are
therefore tending to increase its rate of spin, and to retard the
revolution of the moon. Mars is therefore slowly but surely pulling its
moon down on to itself, by a reverse action to that which separated our
moon. The day shorter than the month forces a moon further away; the
month shorter than the day tends to draw a satellite nearer.
This moon of Mars is not a large body: it is only twenty or thirty miles
in diameter, but it weighs some forty billion tons, and will ultimately
crash along the surface with a velocity of 8,000 miles an hour. Such a
blow must produce the most astounding effects when it occurs, but I am
unable to tell you its probable date.
So far we have dealt mainly with the earth and its moon; but is the
existence of tides limited to these bodies? By no means. No body in the
solar system is rigid, no body in the stellar universe is rigid. All
must be susceptible of some tidal deformation, and hence, in all of
them, agents like those we have traced in the history of the earth and
moon must be at work: the motion of all must be complicated by the
phenomena of tides. It is Prof. George Darwin who has worked out the
astronomical influence of the tides, on the principles of Sir William
Thomson: it is Sir Robert Ball who has extended Mr. Darwin's results to
the past history of our own and other worlds.[32]
Tides are of course produced in the sun by the action of the
planets, for the sun rotates in twenty-five days or thereabouts,
while the planets revolve in much longer periods than that. The
principal tide-generating bodies will be Venus and Jupiter; the
greater nearness of one rather more than compensating for the
greater mass of the other.
It may be interesting to tabulate the relative tide-producing
powers of the planets on the sun. They are as follows, calling that
of the earth 1,000:--
RELATIVE TIDE-PRODUCING POWERS OF THE PLANETS
ON THE SUN.
Mercury 1,121
Venus 2,339
Earth 1,000
Mars 304
Jupiter 2,136
Saturn 1,033
Uranus 21
Neptune 9
The power of all of them is very feeble, and by acting on different
sides they usually partly neutralize each other's action; but
occasionally they get all on one side, and in that case some
perceptible effect may be produced; the probable effect seems
likely to be a gentle heaving tide in the solar surface, with
breaking up of any incipient crust; and such an effect may be
considered as evidenced periodically by the great increase in the
number of solar spots which then break out.
The solar tides are, however, much too small to appreciably push
any planet away, hence we are not to suppose that the planets
originated by budding from the sun, in contradiction of the nebular
hypothesis. Nor is it necessary to assume that the satellites, as a
class, originated in the way ours did; though they may have done
so. They were more probably secondary rings. Our moon differs from
other satellites in being exceptionally large compared with the
size of its primary; it is as big as some of the moons of Jupiter
and Saturn. The earth is the only one of the small planets that has
an appreciable moon, and hence there is nothing forced or unnatural
in supposing that it may have had an exceptional history.
Evidently, however, tidal phenomena must be taken into
consideration in any treatment of the solar system through enormous
length of time, and it will probably play a large part in
determining its future.
When Laplace and Lagrange investigated the question of the stability or
instability of the solar system, they did so on the hypothesis that the
bodies composing it were rigid. They reached a grand conclusion--that
all the mutual perturbations of the solar system were periodic--that
whatever changes were going on would reach a maximum and then begin to
diminish; then increase again, then diminish, and so on. The system was
stable, and its changes were merely like those of a swinging pendulum.
But this conclusion is not final. The hypothesis that the bodies are
rigid is not strictly true: and directly tidal deformation is taken into
consideration it is perceived to be a potent factor, able in the long
run to upset all their calculations. But it is so utterly and
inconceivably minute--it only produces an appreciable effect after
millions of years--whereas the ordinary perturbations go through their
swings in some hundred thousand years or so at the most. Granted it is
small, but it is terribly persistent; and it always acts in one
direction. Never does it cease: never does it begin to act oppositely
and undo what it has done. It is like the perpetual dropping of water.
There may be only one drop in a twelvemonth, but leave it long enough,
and the hardest stone must be worn away at last.
* * * * *
We have been speaking of millions of years somewhat familiarly; but
what, after all, is a million years that we should not speak familiarly
of it? It is longer than our lifetime, it is true. To the ephemeral
insects whose lifetime is an hour, a year might seem an awful period,
the mid-day sun might seem an almost stationary body, the changes of the
seasons would be unknown, everything but the most fleeting and rapid
changes would appear permanent and at rest. Conversely, if our
life-period embraced myriads of aeons, things which now seem permanent
would then appear as in a perpetual state of flux. A continent would be
sometimes dry, sometimes covered with ocean; the stars we now call fixed
would be moving visibly before our eyes; the earth would be humming on
its axis like a top, and the whole of human history might seem as
fleeting as a cloud of breath on a mirror.
Evolution is always a slow process. To evolve such an animal as a
greyhound from its remote ancestors, according to Mr. Darwin, needs
immense tracts of time; and if the evolution of some feeble animal
crawling on the surface of this planet is slow, shall the stately
evolution of the planetary orbs themselves be hurried? It may be that we
are able to trace the history of the solar system for some thousand
million years or so; but for how much longer time must it not have a
history--a history, and also a future--entirely beyond our ken?
Those who study the stars have impressed upon them the existence of the
most immeasurable distances, which yet are swallowed up as nothing in
the infinitude of space. No less are we compelled to recognize the
existence of incalculable aeons of time, and yet to perceive that these
are but as drops in the ocean of eternity.
FOOTNOTES:
[1] The following account of Mars's motion is from the excellent small
manual of astronomy by Dr. Haughton of Trinity College, Dublin:--(P.
151) "Mars's motion is very unequal; when he first appears in the
morning emerging from the rays of the sun, his motion is direct and
rapid; it afterwards becomes slower, and he becomes stationary when at
an elongation of 137 deg. from the sun; then his motion becomes retrograde,
and its velocity increases until he is in opposition to the sun at 180 deg.;
at this time the retrograde motion is most rapid, and afterwards
diminishes until he is 137 deg. distant from the sun on the other side, when
Mars again becomes stationary; his motion then becomes direct, and
increases in velocity until it reaches a maximum, when the planet is
again in conjunction with the sun. The retrograde motion of this planet
lasts for 73 days: and its arc of retrogradation is 16 deg.."
[2] It is not so easy to plot the path of the sun among the stars by
direct observation, as it is to plot the path of a planet; because sun
and stars are not visible together. Hipparchus used the moon as an
intermediary; since sun and moon are visible together, and also moon and
stars.
[3] This is, however, by no means the whole of the matter. The motion is
not a simple circle nor has it a readily specifiable period. There are
several disturbing causes. All that is given here is a first rough
approximation.
[4] The proof is easy, and ought to occur in books on solid geometry. By
a "regular" solid is meant one with all its faces, edges, angles, &c.,
absolutely alike: it is of these perfectly symmetrical bodies that there
are only five. Crystalline forms are practically infinite in number.
[5] Best known to us by his Christian name, as so many others of that
time are known, _e.g._ Raphael Sanzio, Dante Alighieri, Michael Angelo
Buonarotti. The rule is not universal. Tasso and Ariosto are surnames.
[6] It would seem that the fact that all bodies of every material tend
to fall at the same rate is still not clearly known. Confusion is
introduced by the resistance of the air. But a little thought should
make it clear that the effect of the air is a mere disturbance, to be
eliminated as far as possible, since the atmosphere has nothing to do
with gravitation. The old fashioned "guinea and feather experiment"
illustrates that in a vacuum things entirely different in specific
gravity or surface drop at the same pace.
[7] Karl von Gebler (Galileo), p. 13.
[8] It is of course the "silver lining" of clouds that outside observers
see.
[9] L.U.K., _Life of Galileo_, p. 26.
[10] _Note added September, 1892._ News from the Lick Observatory makes
a very small fifth satellite not improbable.
[11] They remained there till this century. In 1835 they were quietly
dropped.
[12] It was invented by van Helmont, a Belgian chemist, who died in
1644. He suggested two names _gas_ and _blas_, and the first has
survived. Blas was, I suppose, from _blasen_, to blow, and gas seems to
be an attempt to get at the Sanskrit root underlying all such words as
_geist_.
[13] Such as this, among many others:--The duration of a flame under
different conditions is well worth determining. A spoonful of warm
spirits of wine burnt 116 pulsations. The same spoonful of spirits of
wine with addition of one-sixth saltpetre burnt 94 pulsations. With
one-sixth common salt, 83; with one-sixth gunpowder, 110; a piece of wax
in the middle of the spirit, 87; a piece of _Kieselstein_, 94; one-sixth
water, 86; and with equal parts water, only 4 pulse-beats. This, says
Liebig, is given as an example of a "_licht-bringende Versuch_."
[14] Draper, _History of Civilization in Europe_, vol. ii. p. 259.
[15] Professor Knight's series of Philosophical Classics.
[16] To explain why the entire system, horse and cart together, move
forward, the forces acting on the ground must be attended to.
[17] The distance being proportional to the _square_ of the time, see p.
82.
[18] The following letter, recently unearthed and published in _Nature_,
May 12, 1881, seems to me well worth preserving. The feeling of a
respiratory interval which it describes is familiar to students during
the too few periods of really satisfactory occupation. The early guess
concerning atmospheric electricity is typical of his extraordinary
instinct for guessing right.
"LONDON, _Dec. 15, 1716_.
"DEAR DOCTOR,--He that in ye mine of knowledge deepest diggeth, hath,
like every other miner, ye least breathing time, and must sometimes at
least come to terr. alt. for air.
"In one of these respiratory intervals I now sit down to write to you,
my friend.
"You ask me how, with so much study, I manage to retene my health. Ah,
my dear doctor, you have a better opinion of your lazy friend than he
hath of himself. Morpheous is my last companion; without 8 or 9 hours of
him yr correspondent is not worth one scavenger's peruke. My practices
did at ye first hurt my stomach, but now I eat heartily enou' as y' will
see when I come down beside you.
"I have been much amused at ye singular [Greek: _phenomena_] resulting
from bringing of a needle into contact with a piece of amber or resin
fricated on silke clothe. Ye flame putteth me in mind of sheet lightning
on a small--how very small--scale. But I shall in my epistles abjure
Philosophy whereof when I come down to Sakly I'll give you enou'. I
began to scrawl at 5 mins. from 9 of ye clk. and have in writing consmd.
10 mins. My Ld. Somerset is announced.
"Farewell, Gd. bless you and help yr sincere friend.
"ISAAC NEWTON.
"_To_ DR. LAW, Suffolk."
[19] Kepler's laws may be called respectively, the law of path, the law
of speed, and the relationship law. By the "mass" of a body is meant the
number of pounds or tons in it: the amount of matter it contains. The
idea is involved in the popular word "massive."
[20] The equation we have to verify is
4[pi]^2r^3
gR^2 = -----------,
T^2
with the data that _r_, the moon's distance, is 60 times R, the earth's
radius, which is 3,963 miles; while T, the time taken to complete the
moon's orbit, is 27 days, 13 hours, 18 minutes, 37 seconds. Hence,
suppose we calculate out _g_, the intensity of terrestrial gravity, from
the above equation, we get
4[pi]^2 39.92 x 216000 x 3963 miles
_g_ = ---------- x (60)^3R = -----------------------------
T (27 days, 13 hours, &c.)^2
= 32.57 feet-per-second per second,
which is not far wrong.
[21] The two motions may be roughly compounded into a single motion,
which for a few centuries may without much error be regarded as a
conical revolution about a different axis with a different period; and
Lieutenant-Colonel Drayson writes books emphasizing this simple fact,
under the impression that it is a discovery.
[22] Members of the Accademia dei Lyncei, the famous old scientific
Society established in the time of Cosmo de Medici--older than our own
Royal Society.
[23] Newton suspected that the moon really did so oscillate, and so it
may have done once; but any real or physical libration, if existing at
all, is now extremely minute.
[24] An interesting picture in the New Gallery this year (1891),
attempting to depict "Earth-rise in Moon-land," unfortunately errs in
several particulars. First of all, the earth does not "rise," but is
fixed relatively to each place on the moon; and two-fifths of the moon
never sees it. Next, the earth would not look like a map of the world
with a haze on its edge. Lastly, whatever animal remains the moon may
contain would probably be rather in the form of fossils than of
skeletons. The skeleton is of course intended as an image of death and
desolation. It is a matter of taste: but a skeleton, it seems to me,
speaks too recently of life to be as appallingly weird and desolate as a
blank stone or ice landscape, unshaded by atmosphere or by any trace of
animal or plant life, could be made.
[25] Five of Jupiter's revolutions occupy 21,663 days; two of Saturn's
revolutions occupy 21,526 days.
[26] _Excircularity_ is what is meant by this term. It is called
"excentricity" because the foci (not the centre) of an ellipse are
regarded as the representatives of the centre of a circle. Their
distance from the centre, compared with the radius of the unflattened
circle, is called the excentricity.