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Title: Time and Tide
A Romance of the Moon
Author: Robert S. (Robert Stawell) Ball
Release Date: January 10, 2008 [eBook #24236]
Language: English
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The Romance of Science.
TIME AND TIDE,
A Romance of the Moon.
Being Two Lectures Delivered
in the Theatre of the London Institution,
on the Afternoons of November 19 and 26, 1888.
by
SIR ROBERT S. BALL, LL.D., F.R.S.,
Royal Astronomer of Ireland.
Published under the Direction of the Committee of
General Literature and Education Appointed
by the Society for Promoting Christian
Knowledge.
London:
Society for Promoting Christian Knowledge,
Northumberland Avenue, W.C.; 43, Queen Victoria Street, E.C.
Brighton: 135, North Street.
New York: E. & J. B. Young & Co.
[Illustration: VIEW OF THE MOON TWO DAYS AFTER FIRST QUARTER. From a
photograph by Mr. LEWIS M. RUTHERFORD.
_Frontispiece._]
TO
The Members of the London Institution
I DEDICATE
THIS LITTLE BOOK.
PREFACE.
Having been honoured once again with a request that I should lecture
before the London Institution, I chose for my subject the Theory of
Tidal Evolution. The kind reception which these lectures received has
led to their publication in the present volume. I have taken the
opportunity to supplement the lectures as actually delivered by the
insertion of some additional matter. I am indebted to my friends Mr.
Close and Mr. Rambaut for their kindness in reading the proofs.
ROBERT S. BALL.
_Observatory_, CO. DUBLIN,
_April_ 26, 1889.
TIME AND TIDE.
LECTURE I.
It is my privilege to address you this afternoon on a subject in which
science and poetry are blended in a happy conjunction. If there be a
peculiar fascination about the earlier chapters of any branch of
history, how great must be the interest which attaches to that most
primeval of all terrestrial histories which relates to the actual
beginnings of this globe on which we stand.
In our efforts to grope into the dim recesses of this awful past, we
want the aid of some steadfast light which shall illumine the dark
places without the treachery of the will-o'-the-wisp. In the absence
of that steadfast light, vague conjectures as to the beginning of
things could never be entitled to any more respect than was due to
mere matters of speculation.
Of late, however, the required light has been to some considerable
extent forthcoming, and the attempt has been made, with no little
success, to elucidate a most interesting and wonderful chapter of an
exceedingly remote history. To chronicle this history is the object of
the present lectures before this Institution.
First, let us be fully aware of the extraordinary remoteness of that
period of which our history treats. To attempt to define that period
chronologically would be utterly futile. When we have stated that it
is more ancient than almost any other period which we can discuss, we
have expressed all that we are really entitled to say. Yet this
conveys not a little. It directs us to look back through all the ages
of modern human history, through the great days of ancient Greece and
Rome, back through the times when Egypt and Assyria were names of
renown, through the days when Nineveh and Babylon were mighty and
populous cities in the zenith of their glory. Back earlier still to
those more ancient nations of which we know hardly anything, and
still earlier to the prehistoric man, of whom we know less; back,
finally, to the days when man first trod on this planet, untold ages
ago. Here is indeed a portentous retrospect from most points of view,
but it is only the commencement of that which our subject suggests.
For man is but the final product of the long anterior ages during
which the development of life seems to have undergone an exceedingly
gradual elevation. Our retrospect now takes its way along the vistas
opened up by the geologists. We look through the protracted tertiary
ages, when mighty animals, now generally extinct, roamed over the
continents. Back still earlier through those wondrous secondary
periods, where swamps or oceans often covered what is now dry land,
and where mighty reptiles of uncouth forms stalked and crawled and
swam through the old world and the new. Back still earlier through
those vitally significant ages when the sunbeams were being garnered
and laid aside for man's use in the great forests, which were
afterwards preserved by being transformed into seams of coal. Back
still earlier through endless thousands of years, when lustrous
fishes abounded in the oceans; back again to those periods
characterized by the lower types of life; and still earlier to that
incredibly remote epoch when life itself began to dawn on our
awakening globe. Even here the epoch of our present history can hardly
be said to have been reached. We have to look through a long
succession of ages still antecedent. The geologist, who has hitherto
guided our view, cannot render us much further assistance; but the
physicist is at hand--he teaches us that the warm globe on which life
is beginning has passed in its previous stages through every phase of
warmth, of fervour, of glowing heat, of incandescence, and of actual
fusion; and thus at last our retrospect reaches to that particular
period of our earth's past history which is specially illustrated by
the modern doctrine of Time and Tide.
The present is the clue to the past. It is the steady application of
this principle which has led to such epoch-making labours as those by
which Lyell disclosed the origin of the earth's crust, Darwin the
origin of species, Max Müller the origin of language. In our present
subject the course is equally clear. Study exactly what is going on at
present, and then have the courage to apply consistently and
rigorously what we have learned from the present to the interpretation
of the past.
Thus we begin with the ripple of the tide on the sea-beach which we
see to-day. The ebb and the flow of the tide are the present
manifestations of an agent which has been constantly at work. Let that
present teach us what tides must have done in the indefinite past.
It has been known from the very earliest times that the moon and the
tides were connected together--connected, I say, for a great advance
had to be made in human knowledge before it would have been possible
to understand the true relation between the tides and the moon.
Indeed, that relation is so far from being of an obvious character,
that I think I have read of a race who felt some doubt as to whether
the moon was the cause of the tides, or the tides the cause of the
moon. I should, however, say that the moon is not the sole agent
engaged in producing this periodic movement of our waters. The sun
also arouses a tide, but the solar tide is so small in comparison with
that produced by the moon, that for our present purpose we may leave
it out of consideration. We must, however, refer to the solar tide at
a later period of our discourses, for it will be found to have played
a very splendid part at the initial stage of the Earth-Moon History,
while in the remote future it will again rise into prominence.
It will be well to set forth a few preliminary figures which shall
explain how it comes to pass that the efficiency of the sun as a
tide-producing agent is so greatly inferior to that of the moon.
Indeed, considering that the sun has a mass so stupendous, that it
controls the entire planetary system, how is it that a body so
insignificant as the moon can raise a bigger tide on the ocean than
can the sun, of which the mass is 26,000,000 times as great as that of
our satellite?
This apparent paradox will disappear when we enunciate the law
according to which the efficiency of a tide-producing agent is to be
estimated. This law is somewhat different from the familiar form in
which the law of gravitation is expressed. The gravitation between two
distant masses is to be measured by multiplying these masses together,
and dividing the product by the _square_ of the distance. The law for
expressing the efficiency of a tide-producing agent varies not
according to the inverse square, but according to the inverse _cube_
of the distance. This difference in the expression of the law will
suffice to account for the superiority of the moon as a tide-producer
over the sun. The moon's distance on an average is about one 386th
part of that of the sun, and thus it is easy to show that so far as
the mere attraction of gravitation is concerned, the efficiency of the
sun's force on the earth is about one hundred and seventy-five times
as great as the force with which the moon attracts the earth. That is
of course calculated under the law of the inverse square. To determine
the tidal efficiency we have to divide this by three hundred and
eighty-six, and thus we see that the tidal efficiency of the sun is
less than half that of the moon.
When the solar tide and the lunar tide are acting in unison, they
conspire to produce very high tides and very low tides, or, as we
call them, spring tides. On the other hand, when the sun is so placed
as to give us a low tide while the moon is producing a high tide, the
net result that we actually experience is merely the excess of the
lunar tide over the solar tide; these are what we call neap tides. In
fact, by very careful and long-continued observations of the rise and
fall of the tides at a particular port, it becomes possible to
determine with accuracy the relative ranges of spring tides and neap
tides; and as the spring tides are produced by moon plus sun, while
the neap tides are produced by moon minus sun, we obtain a means of
actually weighing the relative masses of the sun and moon. This is one
of the remarkable facts which can be deduced from a prolonged study of
the tides.
The demonstration of the law of the tide-producing force is of a
mathematical character, and I do not intend in these lectures to enter
into mathematical calculations. There is, however, a simple line of
reasoning which, though it falls far short of actual demonstration,
may yet suffice to give a plausible reason for the law.
The tides really owe their origin to the fact that the tide-producing
agent operates more powerfully on those parts of the tide-exhibiting
body which are near to it, than on the more distant portions of the
same. The nearer the two bodies are together, the larger
proportionally will be the differences in the distances of its various
parts from the tide-producing body; and on this account the leverage,
so to speak, of the action by which the tides are produced is
increased. For instance, if the two bodies were brought within half
their original distance of each other, the relative size of each body,
as viewed from the other, will be doubled; and what we have called the
leverage of the tide-producing ability will be increased twofold. The
gravitation also between the two bodies is increased fourfold when the
distance is halved, and consequently, the tide-producing ability is
doubled for one reason, and increased fourfold again by another;
hence, the tides will be increased eightfold when the distance is
reduced to one half. Now, as eight is the cube of two, this
illustration may be taken as a verification of the law, that the
efficiency of a body as a tide-producer varies inversely as the cube
of the distance between it and the body on which the tides are being
raised.
For simplicity we may make the assumption that the whole of the earth
is buried beneath the ocean, and that the moon is placed in the plane
of the equator. We may also entirely neglect for the present the tides
produced by the sun, and we shall also make the further assumption
that friction is absent. What friction is capable of doing we shall,
however, refer to later on. The moon will act on the ocean and deform
it, so that there will be high tide along one meridian, and high tide
also on the opposite meridian. This is indeed one of the paradoxes by
which students are frequently puzzled when they begin to learn about
the tides. That the moon should pull the water up in a heap on one
side seems plausible enough. High tide will of course be there; and
the student might naturally think that the water being drawn in this
way into a heap on one side, there will of course be low tide on the
opposite side of the earth. A natural assumption, perhaps, but
nevertheless a very wrong one. There are at every moment two opposite
parts of the earth in a condition of high water; in fact, this will
be obvious if we remember that every day, or, to speak a little more
accurately, in every twenty-four hours and fifty one minutes, we have
on the average two high tides at each locality. Of course this could
not be if the moon raised only one heap of high water, because, as the
moon only appears to revolve around the earth once a day, or, more
accurately, once in that same average period of twenty-four hours and
fifty-one minutes, it would be impossible for us to have high tides
succeeding each other as they do in periods a little longer than
twelve hours, if only one heap were carried round the earth.
The first question then is, as to how these two opposite heaps of
water are placed in respect to the position of the moon. The most
obvious explanation would seem to be, that the moon should pull the
waters up into a heap directly underneath it, and that therefore there
should be high water underneath the moon. As to the other side, the
presence of a high tide there was, on this theory, to be accounted for
by the fact that the moon pulled the earth away from the waters on the
more remote side, just as it pulled the waters away from the more
remote earth on the side underneath the moon. It is, however,
certainly not the case that the high tide is situated in the simple
position that this law would indicate, and which we have represented
in Fig. 1, where the circular body is the earth, the ocean surrounding
which is distorted by the action of the tides.
[Illustration: FIG. 1.]
We have here taken an oval to represent the shape into which the water
is supposed to be forced or drawn by the tidal action of the
tide-producing body. This may possibly be a correct representation of
what would occur on an ideal globe entirely covered with a
frictionless ocean. But as our earth is not covered entirely by water,
and as the ocean is very far from being frictionless, the ideal tide
is not the tide that we actually know; nor is the ideal tide
represented by this oval even an approximation to the actual tides to
which our oceans are subject. Indeed, the oval does not represent the
facts at all, and of this it is only necessary to adduce a single fact
in demonstration. I take the fundamental issue so often debated, as to
whether in the ocean vibrating with ideal tides the high water or the
low water should be under the moon. Or to put the matter otherwise;
when we represent the displaced water by an oval, is the long axis of
the oval to be turned to the moon, as generally supposed, or is it to
be directed at right angles therefrom? If the ideal tides were in any
degree representative of the actual tides, so fundamental a question
as this could be at once answered by an appeal to the facts of
observation. Even if friction in some degree masked the phenomena,
surely one would think that the state of the actual tides should still
enable us to answer this question.
But a study of the tides at different ports fails to realize this
expectation. At some ports, no doubt, the tide is high when the moon
is on the meridian. In that case, of course, the high water is under
the moon, as apparently ought to be the case invariably, on a
superficial view. But, on the other hand, there are ports where there
is often low water when the moon is crossing the meridian. Yet other
ports might be cited in which every intermediate phase could be
observed. If the theory of the tides was to be the simple one so often
described, then at every port noon should be the hour of high water on
the day of the new moon or of the full moon, because then both
tide-exciting bodies are on the meridian at the same time. Even if the
friction retarded the great tidal wave uniformly, the high tide on the
days of full or change should always occur at fixed hours; but,
unfortunately, there is no such delightful theory of the tides as this
would imply. At Greenock no doubt there is high water at or about noon
on the day of full or change; and if it could be similarly said that
on the day of full or change there was high water everywhere at local
noon, then the equilibrium theory of the tides, as it is called,
would be beautifully simple. But this is not the case. Even around our
own coasts the discrepancies are such as to utterly discredit the
theory as offering any practical guide. At Aberdeen the high tide does
not appear till an hour later than the doctrine would suggest. It is
two hours late at London, three at Tynemouth, four at Tralee, five at
Sligo, and six at Hull. This last port would be indeed the haven of
refuge for those who believe that the low tide ought to be under the
moon. At Hull this is no doubt the case; and if at all other places
the water behaved as it does at Hull, why then, of course, it would
follow that the law of low water under the moon was generally true.
But then this would not tally with the condition of affairs at the
other places I have named; and to complete the cycle I shall add a few
more. At Bristol the high water does not get up until seven hours
after the moon has passed the meridian, at Arklow the delay is eight
hours, at Yarmouth it is nine, at the Needles it is ten hours, while
lastly, the moon has nearly got back to the meridian again ere it has
succeeded in dragging up the tide on which Liverpool's great commerce
so largely depends.
Nor does the result of studying the tides along other coasts beside
our own decide more conclusively on the mooted point. Even ports in
the vast ocean give a very uncertain response. Kerguelen Island and
Santa Cruz might seem to prove that the high tide occurs under the
moon, but unfortunately both Fiji and Ascension seem to present us
with an equally satisfactory demonstration, that beneath the moon is
the invariable home of low water.
I do not mean to say that the study of the tides is in other respects
such a confused subject as the facts I have stated would seem to
indicate. It becomes rather puzzling, no doubt, when we compare the
tides at one port with the tides elsewhere. The law and order are,
then, by no means conspicuous, they are often hardly discernible. But
when we confine our attention to the tides at a single port, the
problem becomes at once a very intelligible one. Indeed, the
investigation of the tides is an easy subject, if we are contented
with a reasonably approximate solution; should, however, it be
necessary to discuss fully the tides at any port, the theory of the
method necessary for doing so is available, and a most interesting and
beautiful theory it certainly is.
Let us then speak for a few moments about the methods by which we can
study the tides at a particular port. The principle on which it is
based is a very simple one.
It is the month of August, the 18th, we shall suppose, and we are
going to enjoy a delicious swim in the sea. We desire, of course, to
secure a high tide for the purpose of doing so, and we call an almanac
to help us. I refer to the Thom's Dublin Directory, where I find the
tide to be high at 10h. 14m. on the morning of the 18th of August.
That will then be the time to go down to the baths at Howth or
Kingstown.
But what I am now going to discourse to you about is not the delights
of sea-bathing, it is rather a different inquiry. I want to ask, How
did the people who prepared that almanac know years beforehand, that
on that particular day the tide would be high at that particular hour?
How do they predict for every day the hour of high water? and how
comes it to pass that these predictions are invariably correct?
We first refer to that wonderful book, the _Nautical Almanac_. In that
volume the movements of the moon are set forth with full detail; and
among other particulars we can learn on page iv of every month the
mean time of the moon's meridian passage. It appears that on the day
in question the moon crossed the meridian at 11h. 23m. Thus we see
there was high water at Dublin at 10h. 14m., and 1h. 9m. later, that
is, at 11h. 23m., the moon crossed the meridian.
Let us take another instance. There is a high tide at 3.40 P.M. on the
25th August, and again the infallible _Nautical Almanac_ tells us that
the moon crossed the meridian at 5h. 44m., that is, at 2h. 4m. after
the high water.
In the first case the moon followed the tide in about an hour, and in
the second case the moon followed in about two hours. Now if we are to
be satisfied with a very rough tide rule for Dublin, we may say
generally that there is always a high tide an hour and a half before
the moon crosses the meridian. This would not be a very accurate
rule, but I can assure you of this, that if you go by it you will
never fail of finding a good tide to enable you to enjoy your swim. I
do not say this rule would enable you to construct a respectable
tide-table. A ship-owner who has to creep up the river, and to whom
often the inches of water are material, will require far more accurate
tables than this simple rule could give. But we enter into rather
complicated matters when we attempt to give any really accurate
methods of computation. On these we shall say a few words presently.
What I first want to do, is to impress upon you in a simple way the
fact of the relation between the tide and the moon.
To give another illustration, let us see how the tides at London
Bridge are related to the moon. On Jan. 1st, 1887, it appeared that
the tide was high at 6h. 26m. P.M., and that the moon had crossed the
meridian 56m. previously; on the 8th Jan. the tide was high at 0h.
43m. P.M., and the moon had crossed the meridian 2h. 1m. previously.
Therefore we would have at London Bridge high water following the
moon's transit in somewhere about an hour and a half.
I choose a day at random, for example--the 12th April. The moon
crosses the upper meridian at 3h. 39m. A.M., and the lower meridian at
4h 6m. P.M. Adding an hour and a half to each would give the high
tides at 5h 9m. A.M. and 5h. 36m. P.M.; as a matter of fact, they are
4h. 58m. A.M. and 5h. 20m. P.M.
But these illustrations are sufficient. We find that at London, in a
general way, high water appears at London Bridge about an hour and a
half after the moon has passed the meridian of London. It so happens
that the interval at Dublin is about the same, _i.e._ an hour and a
half; only that in the latter case the high water precedes the moon by
that interval instead of following it. We may employ the same simple
process at other places. Choose two days about a week distant; find on
each occasion the interval between the transit of the moon and the
time of high water, then the mean of these two differences will always
give some notion of the interval between high water and the moon's
transit. If then we take from the _Nautical Almanac_ the time of the
moon's transit, and apply to it the correction proper for the port,
we shall always have a sufficiently good tide-table to guide us in
choosing a suitable time for taking our swim or our walk by the
sea-side; though if you be the captain of a vessel, you will not be so
imprudent as to enter port without taking counsel of the accurate
tide-tables, for which we are indebted to the Admiralty.
Every one who visits the sea-side, or who lives at a sea-port, should
know this constant for the tides, which affect him and his movements
so materially. If he will discover it from his own experience, so much
the better.
The first point to be ascertained is the time of high water. Do not
take this from any local table; you ought to observe it for yourself.
You will go to the pier head, or, better still, to some place where
the rise and fall of the mere waves of the sea will not embarrass you
in your work. You must note by your watch the time when the tide is
highest. An accurate way of doing this will be to have a scale on
which you can measure the height at intervals of five minutes about
the time of high water. You will then be able to conclude the time at
which the tide was actually at its highest point; but even if no
great accuracy be obtainable, you can still get much interesting
information, for you will without much difficulty be right within ten
minutes or a quarter of an hour.
The correction for the port is properly called the "establishment,"
this being the average time of high water on the days of full and
change of the moon at the particular port in question.
We can considerably amend the elementary notion of the tides which the
former method has given us, if we adopt the plan described by Dr.
Whewell in the first four editions of the _Admiralty Manual of
Scientific Inquiry_. We speak of the interval between the transit of
the moon and the time of high water as the luni-tidal interval. Of
course at full and change this is the same thing as the establishment,
but for other phases of the moon the establishment must receive a
correction before being used as the luni-tidal interval. The
correction is given by the following table--
Hour of Moon's transit after Sun:
0| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
-+----+----+----+----+----+----+----+----+----+----+----
0|-20m|-30m|-50m|-60m|-60m|-60m|-40m|-10m|+10m|+20m|+10m
Correction of establishment to find luni-tidal interval:
Thus at a port where the establishment was 3h. 25m., let us suppose
that the transit of the moon took place at 6 P.M.; then we correct the
establishment by -60m., and find the luni-tidal interval to be 2h.
25m., and accordingly the high water takes place at 8h. 25m. P.M.
But even this method is only an approximation. The study of the tides
is based on accurate observation of their rise and fall on different
places round the earth. To show how these observations are to be made,
and how they are to be discussed and reduced when they have been made,
I may refer to the last edition of the _Admiralty Manual of Scientific
Inquiry_, 1886. For a complete study of the tides at any port a
self-registering tide-gauge should be erected, on which not alone the
heights and times of high and low water should be depicted, but also
the continuous curve which shows at any time the height of the water.
In fact, the whole subject of the practical observation and discussion
and prediction of tides is full of valuable instruction, and may be
cited as one of the most complete examples of the modern scientific
methods.
In the first place, the tide-gauge itself is a delicate instrument;
it is actuated by a float which rises and falls with the water, due
provision being made that the mere influence of waves shall not make
it to oscillate inconveniently. The motion of the float when suitably
reduced by mechanism serves to guide a pencil, which, acting on the
paper round a revolving drum, gives a faithful and unintermitting
record of the height of the water.
Thus what the tide-gauge does is to present to us a long curved line
of which the summits correspond to the heights of high water, while
the depressions are the corresponding points of low water. The long
undulations of this curve are, however, very irregular. At spring
tides, when the sun and the moon conspire, the elevations rise much
higher and the depressions sink much lower than they do at neap tides,
when the high water raised by the moon is reduced by the action of the
sun. There are also many minor irregularities which show the tides to
be not nearly such simple phenomena as might be at first supposed. But
what we might hastily think of as irregularities are, in truth, the
most interesting parts of the whole phenomena. Just as in the
observations of the planets the study of the perturbations has led us
to results of the widest interest and instruction, so it is these
minor phenomena of the tides which seem most pregnant with scientific
interest.
The tide-gauge gives us an elaborate curve. How are we to interpret
that curve? Here indeed a most beautiful mathematical theorem comes to
our aid. Just as ordinary sounds consist of a number of undulations
blended together, so the tidal wave consists of a number of distinct
undulations superposed. Of these the ordinary lunar tide and the
ordinary solar tide are the two principal; but there are also minor
undulations, harmonics, so to speak, some originating from the moon,
some originating from the sun, and some from both bodies acting in
concert.
In the study of sound we can employ an acoustic apparatus for the
purpose of decomposing any proposed note, and finding not only the
main undulation itself, but the several superposed harmonics which
give to the note its timbre. So also we can analyze the undulation of
the tide, and show the component parts. The decomposition is effected
by the process known as harmonic analysis. The principle of the
method may be very simply described. Let us fix our attention on any
particular "tide," for so the various elements are denoted. We can
always determine beforehand, with as much accuracy as we may require,
what the period of that tide will be. For instance, the period of the
lunar semi-diurnal tide will of course be half the average time
occupied by the moon to travel round from the meridian of any place
until it regains the same meridian; the period of the lunar diurnal
tide will be double as great; and there are fortnightly tides, and
others of periods still greater. The essential point to notice is,
that the periods of these tides are given by purely astronomical
considerations from the periods of the motions theory, and do not
depend upon the actual observations.
We measure off on the curve the height of the tide at intervals of an
hour. The larger the number of such measures that are available the
better; but even if there be only three hundred and sixty or seven
hundred and twenty consecutive hours, then, as shown by Professor G.
H. Darwin in the _Admiralty Manual_ already referred to, it will
still be possible to obtain a very competent knowledge of the tides in
the particular port where the gauge has been placed.
The art (for such indeed it may be described) of harmonic analysis
consists in deducing from the hourly observations the facts with
regard to each of the constituent tides. This art has been carried to
such perfection, that it has been reduced to a very simple series of
arithmetical operations. Indeed it has now been found possible to call
in the aid of ingenious mechanism, by which the labours of computation
are entirely superseded. The pointer of the harmonic analyzer has
merely to be traced over the curve which the tide-gauge has drawn, and
it is the function of the machine to decompose the composite
undulation into its parts, and to exhibit the several constituent
tides whose confluence gives the total result.
As if nothing should be left to complete the perfection of a process
which, both from its theoretical and its practical sides, is of such
importance, a machine for predicting tides has been designed,
constructed, and is now in ordinary use. When by the aid of the
harmonic analysis the effectiveness of the several constituent tides
affecting a port have become fully determined, it is of course
possible to predict the tides for that port. Each "tide" is a simple
periodic rise and fall, and we can compute for any future time the
height of each were it acting alone. These heights can all be added
together, and thus the height of the water is obtained. In this way a
tide-table is formed, and such a table when complete will express not
alone the hours and heights of high water on every day, but the height
of the water at any intervening hour.
The computations necessary for this purpose are no doubt simple, so
far as their principle is concerned; but they are exceedingly tedious,
and any process must be welcomed which affords mitigation of a task so
laborious. The entire theory of the tides owes much to Sir William
Thomson in the methods of observation and in the methods of reduction.
He has now completed the practical parts of the subject by inventing
and constructing the famous tide-predicting engine.
The principle of this engine is comparatively simple. There is a chain
which at one end is fixed, and at the other end carries the pencil
which is pressed against the revolving drum on which the prediction is
to be inscribed. Between its two ends the chain passes up and down
over pulleys. Each pulley corresponds to one of the "tides," and there
are about a dozen altogether, some of which exercise but little
effect. Of course if the centres of the pulleys were all fixed the pen
could not move, but the centre of each pulley describes a circle with
a radius proportional to the amplitude of the corresponding tide, and
in a time proportional to the period of that tide. When these pulleys
are all set so as to start at the proper phases, the motion is
produced by turning round a handle which makes the drum rotate, and
sets all the pulleys in motion. The tide curve is thus rapidly drawn
out; and so expeditious is the machine, that the tides of a port for
an entire year can be completely worked out in a couple of hours.
While the student or the philosopher who seeks to render any account
of the tide on dynamical grounds is greatly embarrassed by the
difficulties introduced by friction, we, for our present purpose in
the study of the great romance of modern science opened up to us by
the theory of the tides, have to welcome friction as the agent which
gives to the tides their significance from our point of view.
There is the greatest difference between the height of the rise and
fall of the tide at different localities. Out in mid-ocean, for
instance, an island like St. Helena is washed by a tide only about
three feet in range; an enclosed sea like the Caspian is subject to no
appreciable tides whatever, while the Mediterranean, notwithstanding
its connection with the Atlantic, is still only subject to very
inconsiderable tides, varying from one foot to a few feet. The
statement that water always finds its own level must be received, like
many another proposition in nature, with a considerable degree of
qualification. Long ere one tide could have found its way through the
Straits of Gibraltar in sufficient volume to have appreciably affected
the level of the great inland sea, its effects would have been
obliterated by succeeding tides. On the other hand, there are certain
localities which expose a funnel-shape opening to the sea; into these
the great tidal wave rushes, and as it passes onwards towards the
narrow part, the waters become piled up so as to produce tidal
phenomena of abnormal proportions. Thus, in our own islands, we have
in the Bristol Channel a wide mouth into which a great tide enters,
and as it hurries up the Severn it produces the extraordinary
phenomenon of the Bore. The Bristol Channel also concentrates the
great wave which gives Chepstow and Cardiff a tidal range of
thirty-seven or thirty-eight feet at springs, and forces the sea up
the river Avon so as to give Bristol a wonderful tide. There is hardly
any more interesting spot in our islands for the observation of tides
than is found on Clifton Suspension Bridge. From that beautiful
structure you look down on a poor and not very attractive stream,
which two hours later becomes transformed into a river of ample
volume, down which great ships are navigated. But of all places in the
world, the most colossal tidal phenomena are those in the Bay of
Fundy. Here the Atlantic passes into a long channel whose sides
gradually converge. When the great pulse of the tide rushes up this
channel, it is gradually accumulated into a mighty volume at the
upper end, the ebb and flow of which at spring tides extends through
the astonishing range of not less than fifty feet.
These discrepancies between the tides at different places are chiefly
due to the local formations of the coasts and the sea-beds. Indeed, it
seems that if the whole earth were covered with an uniform and deep
ocean of water, the tides would be excessively feeble. On no other
supposition can we reasonably account for the fact that our barometric
records fail to afford us any very distinct evidence as to the
existence of tides in the atmosphere. For you will, of course,
remember that our atmosphere may be regarded as a deep and vast ocean
of air, which embraces the whole earth, extending far above the
loftiest summits of the mountains.
It is one of the profoundest of nature's laws that wherever friction
takes place, energy has to be consumed. Perhaps I ought rather to say
transformed, for of course it is now well known that consumption of
energy in the sense of absolute loss is impossible. Thus, when energy
is expended in moving a body in opposition to the force of friction,
or in agitating a liquid, the energy which disappears in its
mechanical form reappears in the form of heat. The agitation of water
by paddles moving through it warms the water, and the accession of
heat thus acquired measures the energy which has been expended in
making the paddles rotate. The motion of a liquid of which the
particles move among each other with friction, can only be sustained
by the incessant degradation of energy from the mechanical form into
the lower form of diffused heat. Thus the very fact that the tides are
ebbing and flowing, and that there is consequently incessant friction
going on among all the particles of water in the ocean, shows us that
there must be some great store of energy constantly available to
supply the incessant draughts made upon it by the daily oscillation of
the tides. In addition to the mere friction between the particles of
water, there are also many other ways in which the tides proclaim to
us that there is some great hoard of energy which is continually
accessible to their wants. Stand on the bank of an estuary or river up
and down which a great tidal current ebbs and flows; you will see the
water copiously charged with sediment which the tide is bearing along.
Engineers are well aware of the potency of the tide as a vehicle for
transporting stupendous quantities of sand or mud. A sand-bank impedes
the navigation of a river; the removal of that sand-bank would be a
task, perhaps, conceivably possible by the use of steam dredges and
other appliances, whereby vast quantities of sand could be raised and
transported to another locality where they would be innocuous. It is
sometimes possible to effect the desired end by applying the power of
the tide. A sea-wall judiciously thrown out will sometimes concentrate
the tide into a much narrower channel. Its daily oscillations will be
accomplished with greater vehemence, and as the tide rushes furiously
backwards and forwards over the obstacle, the incessant action will
gradually remove it, and the impediment to navigation may be cleared
away. Here we actually see the tides performing a piece of definite
and very laborious work, to accomplish which by the more ordinary
agents would be a stupendous task.
In some places the tides are actually harnessed so as to accomplish
useful work. I have read that underneath old London Bridge there used
formerly to be great water-wheels, which were turned by the tide as it
rushed up the river, and turned again, though in the opposite way, by
the ebbing tide. These wheels were, I believe, employed to pump up
water, though it does not seem obvious for what purposes the water
would have been suitable. Indeed in the ebb and flow all round our
coasts there is a potential source of energy which has hitherto been
allowed to run to waste. The tide could be utilized in various ways.
Many of you will remember the floating mills on the Rhine. They are
vessels like paddle steamers anchored in the rapid current. The flow
of the river makes the paddles rotate, and thus the machinery in the
interior is worked. Such craft moored in a rapid tide-way could also
be made to convey the power of the tides into the mechanism of the
mill. Or there is still another method which has been employed, and
which will perhaps have a future before it in those approaching times
when the coal-cellars of England shall be exhausted. Imagine on the
sea-coast a large flat extent which is inundated twice every day by
the tide. Let us build a stout wall round this area, and provide it
with a sluice-gate. Open the gate as the tide rises, and the great
pond will be filled; then at the moment of high water close the
sluice, and the pond-full will be impounded. If at low tide the sluice
be opened the water will rush tumultuously out. Now suppose that a
water-wheel be provided, so that the rapid rush of water from the exit
shall fall upon its blades; then a source of power is obviously the
result.
At present, however, such a contrivance would naturally find no
advocates, for of course the commercial aspect of the question is that
which will decide whether the scheme is practicable and economical.
The issue indeed can be very simply stated. Suppose that a given
quantity of power be required--let us say that of one hundred horse.
Then we have to consider the conditions under which a contrivance of
the kind we have sketched shall yield a power of this amount. Sir
William Thomson, in a very interesting address to the British
Association at York in 1881, discussed this question, and I shall here
make use of the facts he brought forward on that occasion. He showed
that to obtain as much power as could be produced by a steam-engine
of one hundred horse power, a very large reservoir would be required.
It is doubtful indeed whether there would be many localities on the
earth which would be suitable for the purpose. Suppose, however, an
estuary could be found which had an area of forty acres; then if a
wall were thrown across the mouth so that the tide could be impounded,
the total amount of power that could be yielded by a water-wheel
worked by the incessant influx and efflux of the tide would be equal
to that yielded by the one hundred horse engine, running continuously
from one end of the year to the other.
There are many drawbacks to a tide-mill of this description. In the
first place, its situation would naturally be far removed from other
conveniences necessary for manufacturing purposes. Then too there is
the great irregularity in the way in which the power is rendered
available. At certain periods during the twenty-four hours the mill
would stop running, and the hours when this happened would be
constantly changing. The inconvenience from the manufacturer's point
of view of a deficiency of power during neap tides might not be
compensated by the fact that he had an excessive supply of power at
spring-tides. Before tide-mills could be suitable for manufacturing
purposes, some means must be found for storing away the energy when it
is redundant, and applying it when its presence is required. We should
want in fact for great sources of energy some contrivance which shall
fulfil the same purpose as the accumulators do in an electrical
installation.
Even then, however, the financial consideration remains, as to whether
the cost of building the dam and maintaining the tide-mill in good
order will not on the whole exceed the original price and the charges
for the maintenance of a hundred horse power steam-engine. There
cannot be a doubt that in this epoch of the earth's history, so long
as the price of coal is only a few shillings a ton, the tide-mill,
even though we seem to get its power without current expense, is
vastly more expensive than a steam-engine. Indeed, Sir William Thomson
remarks, that wherever a suitable tidal basin could be found, it would
be nearly as easy to reclaim the land altogether from the sea. And if
this were in any locality where manufactures were possible, the
commercial value of forty acres of reclaimed land would greatly exceed
all the expenses attending the steam-engine. But when the time comes,
as come it apparently will, that the price of coal shall have risen to
several pounds a ton, the economical aspect of steam as compared with
other prime movers will be greatly altered; it will then no doubt be
found advantageous to utilize great sources of energy, such as Niagara
and the tides, which it is now more prudent to let run to waste.
For my argument, however, it matters little that the tides are not
constrained to do much useful work. They are always doing work of some
kind, whether that be merely heating the particles of water by
friction, or vaguely transporting sand from one part of the ocean to
the other. Useful work or useless work are alike for the purpose of my
argument. We know that work can never be done unless by the
consumption or transformation of energy. For each unit of work that is
done--whether by any machine or contrivance, by the muscles of man or
any other animal, by the winds, the waves, or the tides, or in any
other way whatever--a certain equivalent quantity of energy must have
been expended. When, therefore, we see any work being performed, we
may always look for the source of energy to which the machine owes its
efficiency. In fact, it is the old story illustrated, that perpetual
motion is impossible. A mechanical device, however ingenious may be
the construction, or however accurate the workmanship, can never
possess what is called perpetual motion. It is needless to enter into
details of any proposed contrivance of wheels, of pumps, of pulleys;
it is sufficient to say that nothing in the shape of mechanism can
work without friction, that friction produces heat, that heat is a
form of energy, and that to replace the energy consumed in producing
the heat there must be some source from which the machine is
replenished if its motion is to be continued indefinitely.
Hence, as the tides may be regarded as a machine doing work, we have
to ascertain the origin of that energy which they are continually
expending. It is at this point that we first begin to feel the
difficulties inherent in the theory of tidal evolution. I do not mean
difficulties in the sense of doubts, for up to the present I have
mentioned no doubtful point. When I come to such I shall give due
warning. By difficulties I now mean points which it is not easy to
understand without a little dynamical theory; but we must face these
difficulties, and endeavour to elucidate them as well as we can.
Let us first see what the sources of energy can possibly be on which
the tides are permitted to draw. Our course is simplified by the fact
that the energy of which we have to speak is of a mechanical
description, that is to say, not involving heat or other more obscure
forms of energy. A simple type of energy is that possessed by a
clock-weight after the clock has been wound. A store of power is thus
laid up which is gradually doled out during the week in small
quantities, second by second, to sustain the motion of the pendulum.
The energy in this case is due to the fact that the weight is
attracted by the earth, and is yielded according as the weight sinks
downwards. In the separation between two mutually attracting bodies, a
store of energy is thus implied. What we learn from an ordinary clock
may be extended to the great bodies of the universe. The moon is a
gigantic globe separated from our earth by a distance of 240,000
miles. The attraction between these two bodies always tends to bring
them together. No doubt the moon is not falling towards the earth as
the descending clock-weight is doing. We may, in fact, consider the
moon, so far as our present object is concerned, to be revolving
almost in a circle, of which the earth is the centre. If the moon,
however, were to be stopped, it would at once commence to rush down
towards the earth, whither it would arrive with an awful crash in the
course of four or five days. It is fortunately true that the moon does
not behave thus; but it has the ability of doing so, and thus the mere
separation between the earth and the moon involves the existence of a
stupendous quantity of energy, capable under certain conditions of
undergoing transformation.
There is also another source of mechanical energy besides that we have
just referred to. A rapidly moving body possesses, in virtue of its
motion, a store of readily available energy, and it is easy to show
that energy of this type is capable of transformation into other
types. Think of a cannon-ball rushing through the air at a speed of a
thousand feet per second; it is capable of wreaking disaster on
anything which it meets, simply because its rapid motion is the
vehicle by which the energy of the gunpowder is transferred from the
gun to where the blow is to be struck. Had the cannon been directed
vertically upwards, then the projectile, leaving the muzzle with the
same initial velocity as before, would soar up and up, with gradually
abating speed, until at last it reached a turning-point, the elevation
of which would depend upon the initial velocity. Poised for a moment
at the summit, the cannon-ball may then be likened to the
clock-weight, for the entire energy which it possessed by its motion
has been transformed into the statical energy of a raised weight. Thus
we see these two forms of energy are mutually interchangeable. The
raised weight if allowed to fall will acquire velocity, or the rapidly
moving weight if directed upwards will acquire altitude.
The quantity of energy which can be conveyed by a rapidly moving body
increases greatly with its speed. For instance, if the speed of the
body be doubled, the energy will be increased fourfold, or, in
general, the energy which a moving body possesses may be said to be
proportional to the square of its speed. Here then we have another
source of the energy present in our earth-moon system; for the moon is
hurrying along in its path with a speed of two-thirds of a mile per
second, or about twice or three times the speed of a cannon-shot.
Hence the fact that the moon is continuously revolving in a circle
shows us that it possesses a store of energy which is nine times as
great as that which a cannon-ball as massive as the moon, and fired
with the ordinary velocity, would receive from the powder which
discharged it.
Thus we see that the moon is endowed with two sources of energy, one
of which is due to its separation from the earth, and the other to the
speed of its motion. Though these are distinct, they are connected
together by a link which it is important for us to comprehend. The
speed with which the moon revolves around the earth is connected with
the moon's distance from the earth. The moon might, for instance,
revolve in a larger circle than that which it actually pursues; but
if it did so, the speed of its motion would have to be appropriately
lessened. The orbit of the moon might have a much smaller radius than
it has at present, provided that the speed was sufficiently increased
to compensate for the increased attraction which the earth would
exercise at the lessened distance. Indeed, I am here only stating what
every one is familiar with under the form of Kepler's Law, that the
square of the periodic time is in proportion to the cube of the mean
distance. To each distance of the moon therefore belongs an
appropriate speed. The energy due to the moon's position and the
energy due to its motion are therefore connected together. One of
these quantities cannot be altered without the other undergoing
change. If the moon's orbit were increased there would be a gain of
energy due to the enlarged distance, and a loss of energy due to the
diminished speed. These would not, however, exactly compensate. On the
whole, we may represent the total energy of the moon as a single
quantity, which increases when the distance of the moon from the earth
increases, and lessens when the distance from the earth to the moon
lessens. For simplicity we may speak of this as moon-energy.
But the most important constituent of the store of energy in the
earth-moon system is that contributed by the earth itself. I do not
now speak of the energy due to the velocity of the earth in its orbit
round the sun. The moon indeed participates in this equally with the
earth, but it does not affect those mutual actions between the earth
and moon with which we are at present concerned. We are, in fact,
discussing the action of that piece of machinery the earth-moon
system; and its action is not affected by the circumstance that the
entire machine is being bodily transported around the sun in a great
annual voyage. This has little more to do with the action of our
present argument than has the fact that a man is walking about to do
with the motions of the works of the watch in his pocket. We shall,
however, have to allude to this subject further on.
The energy of the earth which is significant in the earth-moon theory
is due to the earth's rotation upon its axis. We may here again use as
an illustration the action of machinery; and the special contrivance
that I now refer to is the punching-engine that is used in our
ship-building works. In preparing a plate of iron to be riveted to the
side of a ship, a number of holes have to be made all round the margin
of the plate. These holes must be half an inch or more in diameter,
and the plate is sometimes as much as, or more than, half an inch in
thickness. The holes are produced in the metal by forcing a steel
punch through it; and this is accomplished without even heating the
plate so as to soften the iron. It is needless to say that an intense
force must be applied to the punch. On the other hand, the distance
through which the punch has to be moved is comparatively small. The
punch is attached to the end of a powerful lever, the other end of the
lever is raised by a cam, so as to depress the punch to do its work.
An essential part of the machine is a small but heavy fly-wheel
connected by gearing with the cam.
This fly-wheel when rapidly revolving contains within it, in virtue of
its motion, a large store of energy which has gradually accumulated
during the time that the punch is not actually in action. The energy
is no doubt originally supplied from a steam-engine. What we are
especially concerned with is the action of the rapidly rotating wheel
as a reservoir in which a large store of energy can be conveniently
maintained until such time as it is wanted. In the action of punching,
when the steel die comes down upon the surface of the plate, a large
quantity of energy is suddenly demanded to force the punch against the
intense resistance it experiences; the energy for this purpose is
drawn from the store in the fly-wheel, which experiences no doubt a
check in its velocity, to be regained again from the energy of the
engine during the interval which elapses before the punch is called on
to make the next hole.
Another illustration of the fly-wheel on a splendid scale is seen in
our mighty steel works, where ponderous rails are being manufactured.
A white-hot ingot of steel is presented to a pair of powerful rollers,
which grip the steel, and send it through at the other side both
compressed and elongated. Tremendous power is required to meet the
sudden demand on the machine at the critical moment. To obtain this
power an engine of stupendous proportions is sometimes attached
directly to the rollers, but more frequently an engine of rather less
horse-power will be used, the might of this engine being applied to
giving rapid rotation to an immense fly-wheel, which may thus be
regarded as a reservoir full of energy. The rolling mills then obtain
from this store in the fly-wheel whatever energy is necessary for
their gigantic task.
These illustrations will suffice to show how a rapidly rotating body
may contain energy in virtue of its rotation, just as a cannon-ball
contains energy in virtue of its speed of translation, or as a
clock-weight has energy in virtue of the fact that it has some
distance to fall before it reaches the earth. The rotating body need
not necessarily have the shape of a wheel--it may be globular in form;
nor need the axes of rotation be fixed in bearings, like those of the
fly-wheel; nor of course is there any limit to the dimensions which
the rotating body may assume. Our earth is, in fact, a vast rotating
body 8000 miles in diameter, and turning round upon its axis once
every twenty-three hours and fifty-six minutes. Viewed in this way,
the earth is to be regarded as a gigantic fly-wheel containing a
quantity of energy great in correspondence with the earth's mass. The
amount of energy which can be stored by rotation also depends upon the
square of the velocity with which the body turns round; thus if our
earth turned round in half the time which it does at present, that is,
if the day was twelve hours instead of twenty-four hours, the energy
contained in virtue of that rotation would be four times its present
amount.
Reverting now to the earth-moon system, the energy which that system
contains consists essentially of two parts--the moon-energy, whose
composite character I have already explained, and the earth-energy,
which has its origin solely in the rotation of the earth on its axis.
It is necessary to observe that these are essentially distinct--there
is no necessary relation between the speed of the earth's rotation and
the distance of the moon, such as there is between the distance of the
moon and the speed with which it revolves in its orbit.
For completeness, it ought to be added that there is also some energy
due to the moon's rotation on its axis, but this is very small for
two reasons: first, because the moon is small compared with the
earth, and second, because the angular velocity of the moon is also
very small compared with that of the earth. We may therefore dismiss
as insignificant the contributions from this source of energy to the
sum total.
I have frequently used illustrations derived from machinery, but I
want now to emphasize the profound distinction that exists between the
rotation of the earth and the rotation of a fly-wheel in a machine
shop. They are both, no doubt, energy-holders, but it must be borne in
mind, that as the fly-wheel doles out its energy to supply the wants
of the machines with which it is connected, a restitution of its store
is continually going on by the action of the engine, so that on the
whole the speed of the fly-wheel does not slacken. The earth, however,
must be likened to a fly-wheel which has been disconnected with the
engine. If, therefore, the earth have to supply certain demands on its
accumulation of energy, it can only do so by a diminution of its
hoard, and this involves a sacrifice of some of its speed.
In the earth-moon system there is no engine at hand to restore the
losses of energy which are inevitable when work has to be done. But we
have seen that work is done; we have shown, in fact, that the tides
are at present doing work, and have been doing work for as long a
period in the past as our imagination can extend to. The energy which
this work has necessitated can only have been drawn from the existing
store in the system; that energy consists of two parts--the
moon-energy and the earth's rotation energy. The problem therefore for
us to consider is, which of these two banks the tides have drawn on to
meet their constant expenditure. This is not a question that can be
decided offhand; in fact, if we attempt to decide it in an offhand
manner we shall certainly go wrong. It seems so very plausible to say
that as the moon causes the tides, therefore the energy which these
tides expend should be contributed by the moon. But this is not the
case. It actually happens that though the moon does cause the tides,
yet when those tides consume energy they draw it not from the distant
moon, but from the vast supply which they find ready to their hand,
stored up in the rotation of the earth.
The demonstration of this is not a very simple matter; in fact, it is
so far from being simple that many philosophers, including some
eminent ones too, while admitting that of course the tides must have
drawn their energy from one or other or both of these two sources, yet
found themselves unable to assign how the demand was distributed
between the two conceivable sources of supply.
We are indebted to Professor Purser of Belfast for having indicated
the true dynamical principle on which the problem depends. It involves
reasoning based simply on the laws of motion and on elementary
mathematics, but not in the least involving questions of astronomical
observation. It would be impossible for me in a lecture like this to
give any explanation of the mathematical principles referred to. I
shall, however, endeavour by some illustrations to set before you what
this profound principle really is. Were I to give it the old name I
should call it the law of the conservation of areas; the more modern
writers, however, speak of it as the conservation of moment of
momentum, an expression which exhibits the nature of the principle in
a more definite manner.
I do not see how to give any very accurate illustration of what this
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.
LECTURE II.
Starting from that fitting commencement of earth-moon history which
the critical epoch affords, we shall now describe the dynamical
phenomena as the tidal evolution progressed. The moon and the earth
initially moved as a solid body, each bending the same face towards
the other; but as the moon retreated, and as tides began to be raised
on the earth, the length of the day began to increase, as did also the
length of the month. We know, however, that the month increased more
rapidly than the day, so that a time was reached when the month was
twice as long as the day; and still both periods kept on increasing,
but not at equal rates, for in progress of time the month grew so much
more rapidly than the day, that many days had to elapse while the moon
accomplished a single revolution. It is, however, only necessary for
us to note those stages of the mighty progress which correspond to
special events. The first of such stages was attained when the month
assumed its maximum ratio to the day. At this time, the month was
about twenty-nine days, and the epoch appears to have occurred at a
comparatively recent date if we use such standards of time as tidal
evolution requires; though measured by historical standards, the epoch
is of incalculable antiquity. I cannot impress upon you too often the
enormous magnitude of the period of time which these phenomena have
required for their evolution. Professor Darwin's theory affords but
little information on this point, and the utmost we can do is to
assign a minor limit to the period through which tidal evolution has
been in progress. It is certain that the birth of the moon must have
occurred at least fifty million years ago, but probably the true
period is enormously greater than this. If indeed we choose to add a
cipher or two to the figure just printed, I do not think there is
anything which could tell us that we have over-estimated the mark.
Therefore, when I speak of the epoch in which the month possessed the
greatest number of days as a recent one, it must be understood that I
am merely speaking of events in relation to the order of tidal
evolution. Viewed from this standpoint, we can show that the epoch is
a recent one in the following manner. At present the month consists of
a little more than twenty-seven days, but at this maximum period to
which I have referred the month was about twenty-nine days; from that
it began to decline, and the decline cannot have proceeded very far,
for even still there are only two days less in the month than at the
time when the month had the greatest number of days. It thus follows
that the present epoch--the human epoch, as we may call it--in the
history of the earth has fallen at a time when the progress of tidal
evolution is about half-way between the initial and the final stage. I
do not mean half-way in the sense of actual measurement of years;
indeed, from this point it would seem that we cannot yet be nearly
half-way, for, vast as are the periods of time that have elapsed since
the moon first took its departure from the earth, they fall far short
of that awful period of time which will intervene between the present
moment and the hour when the next critical state of earth-moon
history shall have been attained. In that state the day is destined
once again to be equal to the month, just as was the case in the
initial stage. The half-way stage will therefore in one sense be that
in which the proportion of the month to the day culminates. This is
the stage which we have but lately passed; and thus it is that at
present we may be said to be almost half-way through the progress of
tidal evolution.
My narrative of the earth-moon evolution must from this point forward
cease to be retrospective. Having begun at that critical moment when
the month and day were first equal, we have traced the progress of
events to the present hour. What we have now to say is therefore a
forecast of events yet to come. So far as we can tell, no agent is
likely to interfere with the gradual evolution caused by the tides,
which dynamical principles have disclosed to us. As the years roll on,
or perhaps, I should rather say, as thousands of years and millions of
years roll on, the day will continue to elongate, or the earth to
rotate more slowly on its axis. But countless ages must elapse before
another critical stage of the history shall be reached. It is needless
for me to ponder over the tedious process by which this interesting
epoch is reached. I shall rather sketch what the actual condition of
our system will be when that moment shall have arrived. The day will
then have expanded from the present familiar twenty-four hours up to a
day more than twice, more than five, even more than fifty times its
present duration. In round numbers, we may say that this great day
will occupy one thousand four hundred of our ordinary hours. To
realize the critical nature of the situation then arrived at, we must
follow the corresponding evolution through which the moon passes. From
its present distance of two hundred and forty thousand miles, the moon
will describe an ever-enlarging orbit; and as it does so the duration
of the month will also increase, until at last a point will be reached
when the month has become more than double its present length, and has
attained the particular value of one thousand four hundred hours. We
are specially to observe that this one-thousand four-hundred-hour
month will be exactly reached when the day has also expanded to one
thousand four hundred hours; and the essence of this critical
condition, which may be regarded as a significant point of tidal
evolution, is that the day and the month have again become equal. The
day and the month were equal at the beginning, the day and the month
will be equal at the end. Yet how wide is the difference between the
beginning and the end. The day or the month at the end is some
hundreds of times as long as the month or the day at the beginning.
I have already fully explained how, in any stage of the evolutionary
progress in which the day and the month became equal, the energy of
the system attained a maximum or a minimum value. At the beginning the
energy was a maximum; at the end the energy will be a minimum. The
most important consequences follow from this consideration. I have
already shown that a condition of maximum energy corresponded to
dynamic instability. Thus we saw that the earth-moon history could not
have commenced without the intervention of some influence other than
tides at the beginning. Now let us learn what the similar doctrine
has to tell us with regard to the end. The condition then arrived at
is one of dynamical stability; for suppose that the system were to
receive a slight alteration, by which the moon went out a little
further, and thus described a larger orbit, and so performed more than
its share of the moment of spin. Then the earth would have to do a
little less spinning, because, under all circumstances, the total
quantity of spin must be preserved unaltered. But the energy being at
a minimum, such a small displacement must of course produce a state of
things in which the energy would be increased. Or if we conceived the
moon to come in towards the earth, the moon would then contribute less
to the total moment of momentum. It would therefore be incumbent on
the earth to do more; and accordingly the velocity of the earth's
rotation would be augmented. But this arrangement also could only be
produced by the addition of some fresh energy to the system, because
the position from which the system is supposed to have been disturbed
is one of minimum energy.
No disturbance of the system from this final position is therefore
conceivable, unless some energy can be communicated to it. But this
will demonstrate the utter incompetency of the tides to shift the
system by a hair's breadth from this position; for it is of the
essence of the tides to waste energy by friction. And the
transformations of the system which the tides have caused are
invariably characterized by a decline of energy, the movements being
otherwise arranged so that the total moment of momentum shall be
preserved intact. Note, how far we were justified in speaking of this
condition as a final one. It is final so far as the lunar tides are
concerned; and were the system to be screened from all outer
interference, this accommodation between the earth and the moon would
be eternal.
There is indeed another way of demonstrating that a condition of the
system in which the day has assumed equality with the month must
necessarily be one of dynamical equilibrium. We have shown that the
energy which the tides demand is derived not from the mere fact that
there are high tides and low tides, but from the circumstance that
these tides do rise and fall; that in falling and rising they do
produce currents; and it is these currents which generate the friction
by which the earth's velocity is slowly abated, its energy wasted, and
no doubt ultimately dissipated as heat. If therefore we can make the
ebbing and the flowing of the tides to cease, then our argument will
disappear. Thus suppose, for the sake of illustration, that at a
moment when the tides happened to be at high water in the Thames, such
a change took place in the behaviour of the moon that the water always
remained full in the Thames, and at every other spot on the earth
remained fixed at the exact height which it possessed at this
particular moment. There would be no more tidal friction, and
therefore the system would cease to course through that series of
changes which the existence of tidal friction necessitates.
But if the tide is always to be full in the Thames, then the moon must
be always in the same position with respect to the meridian, that is,
the moon must always be fixed in the heavens over London. In fact, the
moon must then revolve around the earth just as fast as London
does--the month must have the same length as the day. The earth must
then show the same face constantly to the moon, just as the moon
always does show the same face towards the earth; the two globes will
in fact revolve as if they were connected with invisible bonds, which
united them into a single rigid body.
We need therefore feel no surprise at the cessation of the progress of
tidal evolution when the month and the day are equal, for then the
movement of moon-raised tides has ceased. No doubt the same may be
said of the state at the beginning of the history, when the day and
the month had the brief and equal duration of a few hours. While the
equality of the two periods lasted there could be no tides, and
therefore no progress in the direction of tidal evolution. There is,
however, the profound difference of stability and instability between
the two cases; the most insignificant disturbance of the system at the
initial stage was sufficient to precipitate the revolving moon from
its condition of dynamical equilibrium, and to start the course of
tidal evolution in full vigour. If, however, any trifling derangement
should take place in the last condition of the system, so that the
month and the day departed slightly from equality, there would
instantly be an ebbing and a flowing of the tides; and the friction
generated by these tides would operate to restore the equality because
this condition is one of dynamical stability.
It will thus be seen with what justice we can look forward to the day
and month each of fourteen hundred hours as a finale to the progress
of the luni-tidal evolution. Throughout the whole of this marvellous
series of changes it is always necessary to remember the one constant
and invariable element--the moment of momentum of the system which
tides cannot alter. Whatever else the friction can have done, however
fearful may have been the loss of energy by the system, the moment of
momentum which the system had at the beginning it preserves unto the
end. This it is which chiefly gives us the numerical data on which we
have to rely for the quantitative features of tidal evolution.
We have made so many demands in the course of these lectures on the
capacity of tidal friction to accomplish startling phenomena in the
evolution of the earth-moon system, that it is well for us to seek for
any evidence that may otherwise be obtainable as to the capacity of
tides for the accomplishment of gigantic operations. I do not say
that there is any doubt which requires to be dispelled by such
evidence, for as to the general outlines of the doctrine of tidal
evolution which has been here sketched out there can be no reasonable
ground for mistrust; but nevertheless it is always desirable to widen
our comprehension of any natural phenomena by observing collateral
facts. Now there is one branch of tidal action to which I have as yet
only in the most incidental way referred. We have been speaking of the
tides in the earth which are made to ebb and flow by the action of the
moon; we have now to consider the tides in the moon, which are there
excited by the action of the earth. For between these two bodies there
is a reciprocity of tidal-making energy--each of them is competent to
raise tides in the other. As the moon is so small in comparison with
the earth, and as the tides on the moon are of but little significance
in the progress of tidal evolution, it has been permissible for us to
omit them from our former discussion. But it is these tides on the
moon which will afford us a striking illustration of the competency of
tides for stupendous tasks. The moon presents a monument to show what
tides are able to accomplish.
[Illustration: Fig. 3.--The Moon.]
I must first, however, explain a difficulty which is almost sure to
suggest itself when we speak of tides on the moon. I shall be told
that the moon contains no water on its surface, and how then, it will
be said, can tides ebb and flow where there is no sea to be disturbed?
There are two answers to this difficulty; it is no doubt true that the
moon seems at present entirely devoid of water in so far as its
surface is exposed to us, but it is by no means certain that the moon
was always in this destitute condition. There are very large features
marked on its map as "seas"; these regions are of a darker hue than
the rest of the moon's surface, they are large objects often many
hundreds of miles in diameter, and they form, in fact, those dark
patches on the brilliant surface which are conspicuous to the unaided
eye, and are represented in Fig. 3. Viewed in a telescope these
so-called seas, while clearly possessing no water at the present time,
are yet widely different from the general aspect of the moon's
surface. It has often been supposed that great oceans once filled
these basins, and a plausible explanation has even been offered as to
how the waters they once contained could have vanished. It has been
thought that as the mineral substances deep in the interior of our
satellite assumed the crystalline form during the progress of cooling,
the demand for water of crystallization required for incorporation
with the minerals was so great that the oceans of the moon became
entirely absorbed. It is, however, unnecessary for our present
argument that this theory should be correct. Even if there never was a
drop of water found on our satellite, the tides in its molten
materials would be quite sufficient for our purpose; anything that
tides could accomplish would be done more speedily by vast tides of
flowing lava than by merely oceanic tides.
There can be no doubt that tides raised on the moon by the earth would
be greater than the tides raised on the earth by the moon. The
question is, however, not a very simple one, for it depends on the
masses of both bodies as well as on their relative dimensions. In so
far as the masses are concerned, the earth being more than eighty
times as heavy as the moon, the tides would on this account be vastly
larger on the moon than on the earth. On the other hand, the moon's
diameter being much less than that of the earth, the efficiency of a
tide-producing body in its action on the moon would be less than that
of the same body at the same distance in its action on the earth; but
the diminution of the tides from this cause would be not so great as
their increase from the former cause, and therefore the net result
would be to exhibit much greater tides on the moon than on the earth.
Suppose that the moon had been originally endowed with a rapid
movement of rotation around its axis, the effect of the tides on that
rotation would tend to check its velocity just in the same way as the
tides on the earth have effected a continual elongation of the day.
Only as the tides on the moon were so enormously great, their capacity
to check the moon's speed would have corresponding efficacy. In
addition to this, the mass of the moon being so small, it could only
offer feeble resistance to the unceasing action of the tide, and
therefore our satellite must succumb to whatever the tides desired
ages before our earth would have been affected to a like extent. It
must be noticed that the influence of the tidal friction is not
directed to the total annihilation of the rotation of the two bodies
affected by it; the velocity is only checked down until it attains
such a point that the speed in which each body rotates upon its axis
has become equal to that in which it revolves around the
tide-producer. The practical effect of such an adjustment is to make
the tide-agitated body turn a constant face towards its tormentor.
I may here note a point about which people sometimes find a little
difficulty. The moon constantly turns the same face towards the earth,
and therefore people are sometimes apt to think that the moon performs
no rotation whatever around its own axis. But this is indeed not the
case. The true inference to be drawn from the constant face of the
moon is, that the velocity of rotation about its own axis is equal to
that of its rotation around the earth; in fact, the moon revolves
around the earth in twenty-seven days, and its rotation about its axis
is performed in twenty-seven days also. You may illustrate the
movement of the moon around the earth by walking around a table in a
room, keeping all the time your face turned towards the table; in such
a case as this you not only perform a motion of revolution, but you
also perform a rotation in an equal period. The proof that you do
rotate is to be found in the fact that during the movement your face
is being directed successively to all the points of the compass. There
is no more singular fact in the solar system than the constancy of the
moon's face to the earth. The periods of rotation and revolution are
both alike; if one of these periods exceeded the other by an amount so
small as the hundredth part of a second, the moon would in the lapse
of ages permit us to see that other side which is now so jealously
concealed. The marvellous coincidence between these two periods would
be absolutely inexplicable, unless we were able to assign it to some
physical cause. It must be remembered that in this matter the moon
occupies a unique position among the heavenly host. The sun revolves
around on its axis in a period of twenty-five or twenty-six days--thus
we see one side of the sun as frequently as we see the other. The side
of the sun which is turned towards us to-day is almost entirely
different from that we saw a fortnight ago. Nor is the period of the
sun's rotation to be identified with any other remarkable period in
our system. If it were equal to the length of the year, for instance,
or if it were equal to the period of any of the other planets, then it
could hardly be contended that the phenomenon as presented by the moon
was unique; but the sun's period is not simply related, or indeed
related at all, to any of the other periodic times in the system. Nor
do we find anything like the moon's constancy of face in the behaviour
of the other planets. Jupiter turns now one face to us and then
another. Nor is his rotation related to the sun or related to any
other body, as our moon's motion is related to us. It has indeed been
thought that in the movements of the satellites of Jupiter a somewhat
similar phenomenon may be observed to that in the motion of our own
satellite. If this be so, the causes whereby this phenomenon is
produced are doubtless identical in the two cases.
So remarkable a coincidence as that which the moon's motion shows
could not reasonably be explained as a mere fortuitous circumstance;
nor need we hesitate to admit that a physical explanation is required
when we find a most satisfactory one ready for our acceptance, as was
originally pointed out by Helmholtz.
There can be no doubt whatever that the constancy of the moon's face
is the work of ancient tides, which have long since ceased to act. We
have shown that if the moon's rotation had once been too rapid to
permit of the same face being always directed towards us, the tides
would operate as a check by which the velocity of that rotation would
be abated. On the other hand, if the moon rotated so slowly that its
other face would be exposed to us in the course of the revolution, the
tides would then be dragged violently over its surface in the
direction of its rotation; their tendency would thus be to accelerate
the speed until the angular velocity of rotation was equal to that of
revolution. Thus the tides would act as a controlling agent of the
utmost stringency to hurry the moon round when it was not turning fast
enough, and to arrest the motion when going too fast. Peace there
would be none for the moon until it yielded absolute compliance to the
tyranny of the tides, and adjusted its period of rotation with exact
identity to its period of revolution. Doubtless this adjustment was
made countless ages ago, and since that period the tides have acted so
as to preserve the adjustment, as long as any part of the moon was in
a state sufficiently soft or fluid to respond to tidal impression. The
present state of the moon is a monument to which we may confidently
appeal in support of our contention as to the great power of the tides
during the ages which have passed; it will serve as an illustration of
the future which is reserved for our earth in ages yet to come, when
our globe shall have also succumbed to tidal influence.
It is owing to the smallness of the moon relatively to the earth that
the tidal process has reached a much more advanced stage in the moon
than it has on the earth; but the moon is incessant in its efforts to
bring the earth into the same condition which it has itself been
forced to assume. Thus again we look forward to an epoch in the
inconceivably remote future when tidal thraldom shall be supreme, and
when the earth shall turn the same face to the moon, as the moon now
turns the same face to the earth.
In the critical state of things thus looming in the dim future, the
earth and the moon will continue to perform this adjusted revolution
in a period of about fourteen hundred hours, the two bodies being
held, as it were, by invisible bands. Such an arrangement might be
eternal if there were no intrusion of tidal influence from any other
body; but of course in our system as we actually find it the sun
produces tides as well as the moon; and the solar tides being at
present much less than those originated by the moon, we have neglected
them in the general outlines of the theory. The solar tides, however,
must necessarily have an increasing significance. I do not mean that
they will intrinsically increase, for there seems no reason to
apprehend any growth in their actual amount; it is their relative
importance to the lunar tides that is the augmenting quantity. As the
final state is being approached, and as the velocity of the earth's
rotation is approximating to the angular velocity with which the moon
revolves around it, the ebbing and the flowing of the lunar tides must
become of evanescent importance; and this indeed for a double reason,
partly on account of the moon's greatly augmented distance, and
partly on account of the increasing length of the lunar day, and the
extremely tardy movements of ebb and flow that the lunar tides will
then have. Thus the lunar tides, so far as their dynamical importance
is concerned, will ultimately become zero, while the solar tides
retain all their pristine efficiency.
We have therefore to examine the dynamical effects of solar tides on
the earth and moon in the critical stage to which the present course
of things tends. The earth will then rotate in a period of about
fifty-seven of its present days; and considering that the length of
the day, though so much greater than our present day, is still much
less than the year, it follows that the solar tides must still
continue so as to bring the earth's velocity of rotation to a point
even lower than it has yet attained. In fact, if we could venture to
project our glance sufficiently far into the future, it would seem
that the earth must ultimately have its velocity checked by the
sun-raised tides, until the day itself had become equal to the year.
The dynamical considerations become, however, too complex for us to
follow them, so that I shall be content with merely pointing out that
the influence of the solar tides will prevent the earth and moon from
eternally preserving the relations of bending the same face towards
each other; the earth's motion will, in fact, be so far checked, that
the day will become _longer_ than the month.
Thus the doctrine of tidal evolution has conducted us to a prospect of
a condition of things which will some time be reached, when the moon
will have receded to a distance in which the month shall have become
about fifty-seven days, and when the earth around which this moon
revolves shall actually require a still longer period to accomplish
its rotation on its axis. Here is an odd condition for a planet with
its satellite; indeed, until a dozen years ago it would have been
pronounced inconceivable that a moon should whirl round a planet so
quickly that its journey was accomplished in less than one of the
planet's own days. Arguments might be found to show that this was
impossible, or at least unprecedented. There is our own moon, which
now takes twenty-seven days to go round the earth; there is Jupiter,
with four moons, and the nearest of these to the primary goes round
in forty-two and a half hours. No doubt this is a very rapid motion;
but all those matters are much more lively with Jupiter than they are
here. The giant planet himself does not need ten hours for a single
rotation, so that you see his nearest moon still takes between two and
three Jovian days to accomplish a single revolution. The example of
Saturn might have been cited to show that the quickest revolution that
any satellite could perform must still require at least twice as long
as the day in which the planet performed its rotation. Nor could the
rotation of the planets around the sun afford a case which could be
cited. For even Mercury, the nearest of all the planets to the sun of
which the existence is certainly known, and therefore the most rapid
in its revolution, requires eighty-eight days to get round once; and
in the mean time the sun has had time to accomplish between three and
four rotations. Indeed, the analogies would seem to have shown so
great an improbability in the conclusion towards which tidal evolution
points, that they would have contributed a serious obstacle to the
general acceptance of that theory.
But in 1877 an event took place so interesting in astronomical
history, that we have to look back to the memorable discovery of
Uranus in 1781 before we can find a parallel to it in importance. Mars
had always been looked upon as one of the moonless planets, though
grounds were not wanting for the surmise that probably moons to Mars
really existed. It was under the influence of this belief that an
attempt was made by Professor Asaph Hall at Washington to make a
determined search, and see if Mars might not be attended by satellites
large enough to be discoverable. The circumstances under which this
memorable inquiry was undertaken were eminently favourable for its
success. The orbit of Mars is one which possesses an exceptionally
high eccentricity; it consequently happens that the oppositions during
which the planet is to be observed vary very greatly in the facilities
they afford for a search like that contemplated by Professor Hall. It
is obviously advantageous that the planet should be situated as near
as possible to the earth, and in the opposition in 1877 the distance
was almost at the lowest point it is capable of attaining; but this
was not the only point in which Professor Hall was favoured; he had
the use of a telescope of magnificent proportions and of consummate
optical perfection. His observatory was also placed in Washington, so
that he had the advantage of a pure sky and of a much lower latitude
than any observatory in Great Britain is placed at. But the most
conspicuous advantage of all was the practised skill of the astronomer
himself, without which all these other advantages would have been but
of little avail. Great success rewarded his well-designed efforts; not
alone was one satellite discovered which revolved around the planet in
a period conformable with that of other similar cases, but a second
little satellite was found, which accomplished its revolution in a
wholly unexpected and unprecedented manner. The day of Mars himself,
that is, the period in which he can accomplish a rotation around his
axis, very closely approximates to our own day, being in fact half an
hour longer. This little satellite, the inner and more rapid of the
pair, requires for a single revolution a period of only seven hours
thirty-nine minutes, that is to say, the little body scampers more
than three times round its primary before the primary itself has
finished one of its leisurely rotations. Here was indeed a striking
fact, a unique fact in our system, which riveted the attention of
astronomers on this most beautiful discovery.
You will now see the bearing which the movement of the inner satellite
of Mars has on the doctrine of tidal evolution. As a legitimate
consequence of that doctrine, we came to the conclusion that our
earth-moon system must ultimately attain a condition in which the day
is longer than the month. But this conclusion stood unsupported by any
analogous facts in the more anciently-known truths of astronomy. The
movement of the satellite of Mars, however, affords the precise
illustration we want; and this fact, I think, adds an additional
significance to the interest and the beauty of Professor Hall's
discovery.
It is of particular interest to investigate the possible connection
which the phenomena of tidal evolution may have had in connection with
the geological phenomena of the earth. We have already pointed out the
greater closeness of the moon to us in times past. The tides raised by
the moon on the earth must therefore have been greater in past ages
than they are now, for of course the nearer the moon the bigger the
tide. As soon as the earth and the moon had separated to a
considerable distance we may say that the height of the tide will vary
inversely as the cube of the moon's distance; it will therefore
happen, that when the moon was at half its present distance from us,
his tide-producing capacity was not alone twice as much or four times
as much, but even eight times as much as it is at present; and a much
greater rate of tidal rise and fall indicates, of course, a
preponderance in every other manifestation of tidal activity. The
tidal currents, for instance, must have been much greater in volume
and in speed; even now there are places in which the tidal currents
flow at four or more miles per hour. We can imagine, therefore, the
vehemence of the tidal currents which must have flowed in those days
when the moon was a much smaller distance from us. It is interesting
to view these considerations in their possible bearings on geological
phenomena. It is true that we have here many elements of uncertainty,
but there is, however, a certain general outline of facts which may
be laid down, and which appears to be instructive, with reference to
the past history of our earth.
I have all through these lectures indicated a mighty system of
chronology for the earth-moon system. It is true that we cannot give
our chronology any accurate expression in years. The various stages of
this history are to be represented by the successive distances between
the earth and the moon. Each successive epoch, for instance, may be
marked by the number of thousands of miles which separate the moon
from the earth.
But we have another system of chronology derived from a wholly
different system of ideas; it too relates to periods of vast duration,
and, like our great tidal periods, extends to times anterior to human
history, or even to the duration of human life on this globe. The
facts of geology open up to us a majestic chronology, the epochs of
which are familiar to us by the succession of strata forming the crust
of the earth, and by the succession of living beings whose remains
these strata have preserved. From the present or recent age our
retrospect over geological chronology leads us to look through a
vista embracing periods of time overwhelming in their duration, until
at last our view becomes lost, and our imagination is baffled in the
effort to comprehend the formation of those vast stratified rocks, a
dozen miles or more in thickness, which seem to lie at the very base
of the stratified system on the earth, and in which it would appear
that the dawnings of life on this globe may be almost discerned. We
have thus the two systems of chronology to compare--one, the
astronomical chronology measured by the successive stages in the
gradual retreat of the moon; the other, the geological chronology
measured by the successive strata constituting the earth's crust.
Never was a more noble problem proposed in the physical history of our
earth than that which is implied in the attempt to correlate these two
systems of chronology. What we would especially desire to know is the
moon's distance which corresponds to each of the successive strata on
the earth. How far off, for instance, was that moon which looked down
on the coal forests in the time of their greatest luxuriance? or what
was the apparent size of the full moon at which the ichthyosaurus
could have peeped when he turned that wonderful eye of his to the sky
on a fine evening? But interesting as this great problem is, it lies,
alas! outside the possibility of exact solution. Indeed we shall not
make any attempt which must necessarily be futile to correlate these
chronologies; all we can do is to state the one fact which is
absolutely undeniable in the matter.
Let us fix our attention on that specially interesting epoch at the
dawn of geological time, when those mighty Laurentian rocks were
deposited of which the thickness is so astounding, and let us consider
what the distance of the moon must have been at this initial epoch of
the earth's history. All we know for certain is, that the moon must
have been nearer, but what proportion that distance bore to the
present distance is necessarily quite uncertain. Some years ago I
delivered a lecture at Birmingham, entitled "A Glimpse through the
Corridors of Time," and in that lecture I threw out the suggestion
that the moon at this primeval epoch may have only been at a small
fraction of its present distance from us, and that consequently
terrific tides may in these days have ravaged the coast. There was a
good deal of discussion on the subject, and while it was universally
admitted that the tides must have been larger in palæozoic times than
they are at present, yet there was a considerable body of opinion to
the effect that the tides even then may have been only about twice, or
possibly not so much, greater than those tides we have at the present.
What the actual fact may be we have no way of knowing; but it is
interesting to note that even the smallest accession to the tides
would be a valuable factor in the performance of geological work.
For let me recall to your minds a few of the fundamental phenomena of
geology. Those stratified rocks with which we are now concerned have
been chiefly manufactured by deposition of sediment in the ocean.
Rivers, swollen, it may be, by floods, and turbid with a quantity of
material held in suspension, discharge their waters into the sea.
Granting time and quiet, this sediment falls to the bottom; successive
additions are made to its thickness during centuries and thousands of
years, and thus beds are formed which in the course of ages
consolidate into actual rock. In the formation of such beds the tides
will play a part. Into the estuaries at the mouths of rivers the tides
hurry in and hurry out, and especially during spring tides there are
currents which flow with tremendous power; then too, as the waves
batter against the coast they gradually wear away and crumble down the
mightiest cliffs, and waft the sand and mud thus produced to augment
that which has been brought down by the rivers. In this operation also
the tides play a part of conspicuous importance, and where the ebb and
flow is greatest it is obvious that an additional impetus will be
given to the manufacture of stratified rocks. In fact, we may regard
the waters of the globe as a mighty mill, incessantly occupied in
grinding up materials for future strata. The main operating power of
this mill is of course derived from the sun, for it is the sun which
brings up the rains to nourish the rivers, it is the sun which raises
the wind which lashes the waves against the shore. But there is an
auxiliary power to keep the mill in motion, and that auxiliary power
is afforded by the tides. If then we find that by any cause the
efficiency of the tides is increased we shall find that the mill for
the manufacture of strata obtains a corresponding accession to its
capacity. Assuming the estimate of Professor Darwin, that the tide may
have had twice as great a vertical range of ebb and flow within
geological times as it has at present, we find a considerable addition
to the efficiency of the ocean in the manufacture of the ancient
stratified rocks. It must be remembered that the extent of the area
through which the tides will submerge and lay bare the country, will
often be increased more than twofold by a twofold increase of height.
A little illustration may show what I mean. Suppose a cone to be
filled with water up to a certain height, and that the quantity of
water in it be measured; now let the cone be filled until the water is
at double the depth; then the surfaces of the water in the two cases
will be in the ratio of the circles, one of which has double the
diameter of the other. The areas of the two surfaces are thus as four
to one; the volumes of the waters in the two cases will be in the
proportion of two similar solids, the ratios of their dimensions being
as two to one. Of course this means that the water in the one case
would be eight times as much as in the other. This particular
illustration will not often apply exactly to tidal phenomena, but I
may mention one place that I happen to know of, in the vicinity of
Dublin, in which the effect of the rise and fall of the tide would be
somewhat of this description. At Malahide there is a wide shallow
estuary cut off from the sea by a railway embankment, and there is a
viaduct in the embankment through which a great tidal current flows in
and out alternately. At low tide there is but little water in this
estuary, but at high tide it extends for miles inland. We may regard
this inlet with sufficient approximation to the truth as half of a
cone with a very large angle, the railway embankment of course forming
the diameter; hence it follows that if the tide was to be raised to
double its height, so large an area of additional land would be
submerged, and so vast an increase of water would be necessary for the
purpose, that the flow under the railway bridge would have to be much
more considerable than it is at present. In some degree the same
phenomena will be repeated elsewhere around the coast. Simply
multiplying the height of the tide by two would often mean that the
border of land between high and low water would be increased more than
twofold, and that the volume of water alternately poured on the land
and drawn off it would be increased in a still larger proportion. The
velocity of all tidal currents would also be greater than at present,
and as the power of a current of water for transporting solid material
held in suspension increases rapidly with the velocity, so we may
infer that the efficiency of tidal currents as a vehicle for the
transport of comminuted rocks would be greatly increased. It is thus
obvious that tides with a rise and fall double in vertical height of
those which we know at present would add a large increase to their
efficiency as geological agents. Indeed, even were the tides only half
or one-third greater than those we know now, we might reasonably
expect that the manufacture of stratified rocks must have proceeded
more rapidly than at present.
The question then will assume this form. We know that the tides must
have been greater in Cambrian or Laurentian days than they are at
present; so that they were available as a means of assisting other
agents in the stupendous operations of strata manufacture which were
then conducted. This certainly helps us to understand how these
tremendous beds of strata, a dozen miles or more in solid thickness,
were deposited. It seems imperative that for the accomplishment of a
task so mighty, some agents more potent than those with which we are
familiar should be required. The doctrine of tidal evolution has shown
us what those agents were. It only leaves us uninformed as to the
degree in which their mighty capabilities were drawn upon.
It is the property of science as it grows to find its branches more
and more interwoven, and this seems especially true of the two
greatest of all natural sciences--geology and astronomy. With the
beginnings of our earth as a globe in the shape in which we find it
both these sciences are directly concerned. I have here touched upon
another branch in which they illustrate and confirm each other.
As the theory of tidal evolution has shed such a flood of light into
the previously dark history of our earth-moon system, it becomes of
interest to see whether the tidal phenomena may not have a wider
scope; whether they may not, for instance, have determined the
formation of the planets by birth from the sun, just as the moon seems
to have originated by birth from the earth. Our first presumption,
that the cases are analogous, is not however justified when the facts
are carefully inquired into. A principle which I have not hitherto
discussed here assumes prominence, and therefore we shall devote our
attention to it for a few minutes.
Let us understand what we mean by the solar system. There is first the
sun at the centre, which preponderates over all the other bodies so
enormously, as shown in Fig. 4, in which the earth and the sun are
placed side by side for comparison. There is then the retinue of
planets, among the smaller of which our earth takes its place, a view
of the comparative sizes of the planets being shown in Fig. 5.
[Illustration: Fig. 4.--Comparative sizes of Earth and Sun.]
Not to embarrass ourselves with the perplexities of a problem so
complicated as our solar system is in its entirety, we shall for the
sake of clear reasoning assume an ideal system, consisting of a sun
and a large planet--in fact, such as our own system would be if we
could withdraw from it all other bodies, leaving the sun and Jupiter
only remaining. We shall suppose, of course, that the sun is much
larger than the planet, in fact, it will be convenient to keep in mind
the relative masses of the sun and Jupiter, the weight of the planet
being less than one-thousandth part of the sun. We know, of course,
that both of those bodies are rotating upon their axes, and the one
is revolving around the other; and for simplicity we may further
suppose that the axes of rotation are perpendicular to the plane of
revolution. In bodies so constituted tides will be manifested.
Jupiter will raise tides in the sun, the sun will raise tides in
Jupiter. If the rotation of each body be performed in a less period
than that of the revolution (the case which alone concerns us), then
the tides will immediately operate in their habitual manner as a brake
for the checking of rotation. The tides raised by the sun on Jupiter
will tend therefore to lengthen Jupiter's day; the tides raised on the
sun by Jupiter will tend to augment the sun's period of rotation. Both
Jupiter and the sun will therefore lose some moment of momentum. We
cannot, however, repeat too often the dynamical truth that the total
moment of momentum must remain constant, therefore what is lost by the
rotation must be made up in the revolution; the orbit of Jupiter
around the sun must accordingly be swelling. So far the reasoning
appears similar to that which led to such startling consequences in
regard to the moon.
[Illustration: Fig. 5.--Comparative sizes of Planets.]
But now for the fundamental difference between the two cases. The
moon, it will be remembered, always revolves with the same face
towards the earth. The tides have ceased to operate there, and
consequently the moon is not able to contribute any moment of
momentum, to be applied to the enlargement of its distance from the
earth; all the moment of momentum necessary for this purpose is of
course drawn from the single supply in the rotation of the earth on
its axis. But in the case of the system consisting of the sun and
Jupiter the circumstances are quite different--Jupiter does not always
bend the same face to the sun; so far, indeed, is this from being
true, that Jupiter is eminently remarkable for the rapidity of his
rotation, and for the incessantly varying aspect in which he would be
seen from the sun. Jupiter has therefore a store of available moment
of momentum, as has also of course the sun. Thus in the sun and planet
system we have in the rotations two available stores of moment of
momentum on which the tides can make draughts for application to the
enlargement of the revolution. The proportions in which these two
available sources can be drawn upon for contributions is not left
arbitrary. The laws of dynamics provide the shares in which each of
the bodies is to contribute for the joint purpose of driving them
further apart.
Let us see if we cannot form an estimate by elementary considerations
as to the division of the labour. The tides raised on Jupiter by the
sun will be practically proportional to the sun's mass and to the
radius of Jupiter. Owing to the enormous size of the sun, the
efficiency of these tides and the moment of the friction-brake they
produce will be far greater on the planet than will the converse
operation of the planet be on the sun. Hence it follows that the
efficiency of the tides in depriving Jupiter of moment of momentum
will be greatly superior to the efficiency of the tides in depriving
the sun of moment of momentum. Without following the matter into any
close numerical calculation, we may assert that for every one part the
sun contributes to the common object, Jupiter will contribute at least
a thousand parts; and this inequality appears all the more striking,
not to say unjust, when it is remembered that the sun is more
abundantly provided with moment of momentum than is Jupiter--the sun
has, in fact, about twenty thousand times as much.
The case may be illustrated by supposing that a rich man and a poor
man combine together to achieve some common purpose to which both are
to contribute. The ethical notion that Dives shall contribute
largely, according to his large means, and Lazarus according to his
slender means, is quite antagonistic to the scale which dynamics has
imposed. Dynamics declares that the rich man need only give a penny to
every pound that has to be extorted from the poor man. Now this is
precisely the case with regard to the sun and Jupiter, and it involves
a somewhat curious consequence. As long as Jupiter possesses available
moment of momentum, we may be certain that no large contribution of
moment of momentum has been obtained from the sun. For, returning to
our illustration, if we find that Lazarus still has something left in
his pocket, we are of course assured that Dives cannot have expended
much, because, as Lazarus had but little to begin with, and as Dives
only puts in a penny for every pound that Lazarus spends, it is
obvious that no large amount can have been devoted to the common
object. Hence it follows that whatever transfer of moment of momentum
has taken place in the sun-Jupiter system has been almost entirely
obtained at the expense of Jupiter. Now in the solar system at
present, the orbital moment of momentum of Jupiter is nearly fifty
thousand times as great as his present store of rotational moment of
momentum. If, therefore, the departure of Jupiter from the sun had
been the consequence of tidal evolution, it would follow that Jupiter
must once have contained many thousands of times the moment of
momentum that he has at present. This seems utterly incredible, for
even were Jupiter dilated into an enormously large mass of vaporous
matter, spinning round with the utmost conceivable speed, it is
impossible that he should ever have possessed enough moment of
momentum. We are therefore forced to the conclusion that the tides
alone do not provide sufficient explanation for the retreat of Jupiter
from the sun.
There is rather a subtle point in the considerations now brought
forward, on which it will be necessary for us to ponder. In the
illustration of Dives and Lazarus, the contributions of Lazarus of
course ceased when his pockets were exhausted, but those of Dives will
continue, and in the lapse of time may attain any amount within the
utmost limits of Dives' resources. The essential point to notice is,
that so long as Lazarus retains anything in _his_ pocket, we know for
certain that Dives has not given much; if Lazarus, however, has his
pocket absolutely empty, and if we do not know how long they may have
been in that condition, we have no means of knowing how large a
portion of wealth Dives may not have actually expended. The
turning-point of the theory thus involves the fact that Jupiter still
retains available moment of momentum in his rotation; and this was our
sole method of proving that the sun, which in this case was Dives, had
never given much. But our argument must have taken an entirely
different line had it so happened that Jupiter constantly turned the
same face to the sun, and that therefore his pockets were entirely
empty in so far as available moment of momentum is concerned. It would
be apparently impossible for us to say to what extent the resources of
the sun may not have been drawn upon; we can, however, calculate
whether in any case the sun could possibly have supplied enough moment
of momentum to account for the recession of Jupiter. Speaking in round
numbers, the revolutional moment of momentum of Jupiter is about
thirty times as great as the rotational moment of momentum at present
possessed by the sun. I do not know that there is anything impossible
in the supposition that the sun might, by an augmented volume and an
augmented velocity of rotation, contain many times the moment of
momentum that it has at this moment. It therefore follows that if it
had happened that Jupiter constantly bent the same face to the sun,
there would apparently be nothing impossible in the fact that Jupiter
had been born of the sun, just as the moon was born of the earth.
These same considerations should also lead us to observe with still
more special attention the development of the earth-moon system. Let
us restate the matter of the earth and moon in the light which the
argument with respect to Jupiter has given us. At present the
rotational moment of momentum of the earth is about a fifth part of
the revolutional moment of momentum of the moon. Owing to the fact
that the moon keeps the same face to us, she has now no available
moment of momentum, and all the moment of momentum required to account
for her retreat has of late come from the rotation of the earth; but
suppose that the moon still had some liquid on its surface which could
be agitated by tides, suppose further that it did not always bend the
same face towards us, that it therefore had some available moment of
momentum due to its rotation on which the tides could operate, then
see how the argument would have been altered. The gradual increase of
the moon's distance could be provided for by a transfer of moment of
momentum from two sources, due of course to the rotational velocities
of the two bodies. Here again the moon and the earth will contribute
according to that dynamical but very iniquitous principle which
regulated the appropriations from the purses of Dives and Lazarus. The
moon must give not according to her abundance, but in the inverse
ratio thereof--because she has little she must give largely. Nor shall
we make an erroneous estimate if we say that nine-tenths of the whole
moment of momentum necessary for the enlargement of the orbit would
have been exacted from the moon; that means that the moon must once
have had about five or six times as much moment of momentum as the
earth possesses at this moment. Considering the small size of the
moon, this could only have arisen by terrific velocity of rotation,
which it is inconceivable that its materials could ever have
possessed.
This presents the demonstration of tidal evolution in a fresh light.
If the moon now departed to any considerable extent from showing the
constant face to the earth, it would seem that its retreat could not
have been caused by tides. Some other agent for producing the present
configuration would be necessary, just as we found that some other
agent than the tides has been necessary in the case of Jupiter.
But I must say a few words as to the attitude of this question with
regard to the entire solar system. This system consists of the sun
presiding at the centre, and of the planets and their satellites in
revolution around their respective primaries, and each also animated
by a rotation on its axis. I shall in so far depart from the actual
configuration of the system as to transform it into an ideal system,
whereof the masses, the dimensions, and the velocities shall all be
preserved; but that the several planes of revolution shall be all
flattened into one plane, instead of being inclined at small angles as
they are at present; nor will it be unreasonable for us at the same
time to bring into parallelism all the axes of rotation, and to
arrange that their common directions shall be perpendicular to the
plane of their common orbits. For the purpose of our present research
this ideal system may pass for the real system.
In its original state, whatever that state may have been, a
magnificent endowment was conferred upon the system. Perhaps I may,
without derogation from the dignity of my subject, speak of the
endowment as partly personal and partly entailed. The system had of
course different powers with regard to the disposal of the two
portions; the personal estate could be squandered. It consisted
entirely of what we call energy; and considering how frequently we use
the expression conservation of energy, it may seem strange to say now
that this portion of the endowment has been found capable of
alienation, nay, further, that our system has been squandering it
persistently from the first moment until now. Although the doctrine of
the conservation of energy is, we have every reason to believe, a
fundamental law affecting the whole universe, yet it would be wholly
inaccurate to say that any particular system such as our solar system
shall invariably preserve precisely the same quantity of energy
without alteration. The circumstance that heat is a form of energy
indeed negatives this supposition. For our system possesses energy of
all the different kinds: there is energy due to the motions both of
rotation and of revolution; there is energy due to the fact that the
mutually attracting bodies of our system are separated by distances of
enormous magnitude; and there is also energy in the form of heat; and
the laws of heat permit that this form of energy shall be radiated off
into space, and thus disappear entirely, in so far as our system is
concerned. On the other hand, there may no doubt be some small amount
of energy accruing to our system from the other systems in space,
which like ours are radiating forth energy. Any gain from this source,
however, is necessarily so very small in comparison to the loss to
which we have referred, that it is quite impossible that the one
should balance the other. Though it is undoubtedly true that the
total quantity of energy in the universe is constant, yet the share of
that energy belonging to any particular system such as ours declines
steadily from age to age.
I may indeed remark, that the question as to what becomes of all the
radiant energy which the millions of suns in the universe are daily
discharging offers a problem apparently not easy to solve; but we need
not discuss the matter at present, we are only going to trace out the
vicissitudes of our own system; and whatever other changes that system
may exhibit, the fact is certain that the total quantity of energy it
contains is declining.
Of the two endowments of energy and of moment of momentum originally
conferred on our system the moment of momentum is the entailed estate.
No matter how the bodies may move, no matter how their actions may
interfere with one another, no matter how this body is pulled one way
and the other body that way, the conservation of moment of momentum is
not imperilled, nor, no matter what losses of heat may be experienced
by radiation, could the store of moment of momentum be affected. The
only conceivable way in which the quantity of moment of momentum in
the solar system could be tampered with is by the interference of some
external attracting body. We know, however, that the stars are all
situated at such enormous distances, that the influences they can
exert in the perturbation of the solar system are absolutely
insensible; they are beyond the reach of the most delicate
astronomical measurements. Hence we see how the endowment of the
system with moment of momentum has conferred upon that system a
something which is absolutely inalienable, even to the smallest
portion.
Before going any further it would be necessary for me to explain more
fully than I have hitherto done the true nature of the method of
estimating moment of momentum. The moment of momentum consists of two
parts: there is first that due to the revolution of the bodies around
the sun; there is secondly the rotation of these bodies on their axes.
Let us first think simply of a single planet revolving in a circular
orbit around the sun. The momentum of that planet at any moment may be
regarded as the product of its mass and its velocity; then the moment
of momentum of the planet in the case mentioned is found by
multiplying the momentum by the radius of the path pursued. In a more
general case, where the planet does not revolve in a circle, but
pursued an elliptic path, the moment of momentum is to be found by
multiplying the planet's velocity and its mass into the perpendicular
from the sun on the direction in which the planet is moving.
These rules provide the methods for estimating all the moments of
momentum, so far as the revolutions in our system are concerned. For
the rotations somewhat more elaborate processes are required. Let us
think of a sphere rotating round a fixed axis. Every particle of that
sphere will of course describe a circle around the axis, and all these
circles will lie in parallel planes. We may for our present purpose
regard each atom of the body as a little planet revolving in a
circular orbit, and therefore the moment of momentum of the entire
sphere will be found by simply adding together the moments of momentum
of all the different atoms of which the sphere is composed. To perform
this addition the use of an elaborate mathematical method is required.
I do not propose to enter into the matter any further, except to say
that the total moment of momentum is the product of two factors--one
the angular velocity with which the sphere is turning round, while the
other involves the sphere's mass and dimensions.
To illustrate the principles of the computation we shall take one or
two examples. Suppose that two circles be drawn, one of which is
double the diameter of the other. Let two planets be taken of equal
mass, and one of these be put to revolve in one circle, and the other
to revolve in the other circle, in such a way that the periods of both
revolutions shall be equal. It is required to find the moments of
momentum in the two cases. In the larger of the two circles it is
plain that the planet must be moving twice as rapidly as in the
smaller, therefore its momentum is twice as great; and as the radius
is also double, it follows that the moment of momentum in the large
orbit will be four times that in the small orbit. We thus see that the
moment of momentum increases in the proportion of the squares of the
radii. If, however, the two planets were revolving about the same sun,
one of these orbits being double the other, the periodic times could
not be equal, for Kepler's law tells us that the square of the
periodic time is proportional to the cube of the mean distance.
Suppose, then, that the distance of the first planet is 1, and that of
the second planet is 2, the cubes of those numbers are 1 and 8, and
therefore the periodic times of the two bodies will be as 1 to the
square root of 8. We can thus see that the velocity of the outer body
must be less than that of the inner one, for while the length of the
path is only double as large, the time taken to describe that path is
the square root of eight times as great; in fact, the velocity of the
outer body will be only the square root of twice that of the inner
one. As, however, its distance from the sun is twice as great, it
follows that the moment of momentum of the outer body will be the
square root of twice that of the inner body. We may state this result
a little more generally as follows--
In comparing the moments of momentum of the several planets which
revolve around the sun, that of each planet is proportional to the
product of its mass with the square root of its distance from the
sun.
Let us now compare two spheres together, the diameter of one sphere
being double that of the other, while the times of rotation of the two
are identical. And let us now compare together the moments of momentum
in these two cases. It can be shown by reasoning, into which I need
not now enter, that the moment of momentum of the large sphere will be
thirty-two times that of the small one. In general we may state that
if a sphere of homogeneous material be rotating about an axis, its
moment of momentum is to be expressed by the product of its angular
velocity by the fifth power of its radius.
We can now take stock, as it were, of the constituents of moments of
momentum in our system. We may omit the satellites for the present,
while such unsubstantial bodies as comets and such small bodies as
meteors need not concern us. The present investment of the moment of
momentum of our system is to be found by multiplying the mass of each
planet by the square root of its distance from the sun; these products
for all the several planets form the total revolutional moment of
momentum. The remainder of the investment is in rotational moment of
momentum, the collective amount of which is to be estimated by
multiplying the angular velocity of each planet into its density, and
the fifth power of its radius if the planet be regarded as
homogeneous, or into such other power as may be necessary when the
planet is not homogeneous. Indeed, as the denser parts of the planet
necessarily lie in its interior, and have therefore neither the
velocity nor the radius of the more superficial portions, it seems
necessary to admit that the moments of momentum of the planets will be
proportional to some lower power of the radius than the fifth. The
total moment of momentum of the planets by rotation, when multiplied
by a constant factor, and added to the revolutional moment of
momentum, will remain absolutely constant.
It may be interesting to note the present disposition of this vast
inheritance among the different bodies of our system. The biggest item
of all is the moment of momentum of Jupiter, due to its revolution
around the sun; in fact, in this single investment nearly sixty per
cent. of the total moment of momentum of the solar system is found.
The next heaviest item is the moment of momentum of Saturn's
revolution, which is twenty-four per cent. Then come the similar
contributions of Uranus and Neptune, which are six and eight per cent.
respectively. Only one more item is worth mentioning, as far as
magnitude is concerned, and that is the nearly two per cent. that the
sun contains in virtue of its _rotation_. In fact, all the other
moments of momentum are comparatively insignificant in this method of
viewing the subject. Jupiter from his rotation has not the fifty
thousandth part of his revolutional moment of momentum, while the
earth's rotational share is not one ten thousandth part of that of
Jupiter, and therefore is without importance in the general aspect of
the system. The revolution of the earth contributes about one eight
hundredth part of that of Jupiter.
These facts as here stated will suffice for us to make a forecast of
the utmost the tides can effect in the future transformation of our
system. We have already explained that the general tendency of tidal
friction is to augment revolutional moment of momentum at the expense
of rotational. The total, however, of the rotational moment of
momentum of the system barely reaches two per cent. of the whole
amount; this is of course almost entirely contributed by the sun, for
all the planets together have not a thousandth part of the sun's
rotational moment. The utmost therefore that tidal evolution can
effect in the system is to distribute the two per cent. in augmenting
the revolutionary moment of momentum. It does not seem that this can
produce much appreciable derangement in the configuration of the
system. No doubt if it were all applied to one of the smaller planets
it would produce very considerable effect. Our earth, for instance,
would have to be driven out to a distance many hundreds of times
further than it is at present were the sun's disposable moment of
momentum ultimately to be transferred to the earth alone. On the other
hand, Jupiter could absorb the whole of the sun's share by quite an
insignificant enlargement of its present path. It does not seem likely
that the distribution that must ultimately take place can much affect
the present configuration of the system.
We thus see that the tides do not appear to have exercised anything
like the same influence in the affairs of our solar system generally
which they have done in that very small part of the solar system which
consists of the earth and moon. This is, as I have endeavoured to show
in these lectures, the scene of supremely interesting tidal phenomena;
but how small it is in comparison with the whole magnitude of our
system may be inferred from the following illustration. I represent
the whole moment of momentum of our system by £1,000,000,000, the bulk
of which is composed of the revolutional moments of momentum of the
great planets, and the rotational moment of momentum of the sun. On
this scale the rotational share which has fallen to our earth and moon
does not even rise to the dignity of a single pound, it can only be
represented by the very modest figure of 19_s._ 5_d._ This is divided
into two parts--the earth by its rotation accounts for 3_s._ 4_d._,
leaving 16_s._ 1_d._ as the equivalent of the revolution of the moon.
The other inferior planets have still less to show than the earth.
Venus can barely have more than 2_s._ 6_d._; even Mars' two satellites
cannot bring his figure up beyond the slender value of 1½_d._;
while Mercury will be amply represented by the smallest coin known at
her Majesty's mint.
The same illustration will bring out the contrast between the Jovian
system and our earth system. The rotational share of the former would
be totally represented by a sum of nearly £12,000; of this, however,
Jupiter's satellites only contribute about £89, notwithstanding that
there are four of them. Thus Jupiter's satellites have not one
hundredth part of the moment of momentum which the rotation of Jupiter
exhibits. How wide is the contrast between this state of things and
the earth-moon system, for the earth does not contain in its rotation
one-fifth of the moment of momentum that the moon has in its
revolution; in fact, the moon has gradually robbed the earth, which
originally possessed 19_s._ 5_d._, of which the moon has carried off
all but 3_s._ 4_d._
And this process is still going on, so that ultimately the earth will
be left very poor, though not absolutely penniless, at least if the
retention of a halfpenny can be regarded as justifying that assertion.
Saturn, revolving as it does with great rapidity, and having a very
large mass, possesses about £2700, while Uranus and Neptune taken
together would figure for about the same amount.
In conclusion, let us revert again to the two critical conditions of
the earth-moon system. As to what happened before the first critical
period, the tides tell us nothing, and every other line of reasoning
very little; we can to some extent foresee what may happen after the
second critical epoch is reached, at a time so remote that I do not
venture even to express the number of ciphers which ought to follow
the significant digit in the expression for the number of years. I
mentioned, however, that at this time the sun tides will produce the
effect of applying a still further brake to the rotation of the earth,
so that ultimately the month will have become a shorter period than
the day. It is therefore interesting for us to trace out the tidal
history of a system in which the satellite revolves around the primary
in less time than the primary takes to go round on its own axis--such
a system, in fact, as Mars would present at this moment were the outer
satellite to be abstracted. The effect of the tides on the planet
raised by its satellite would then be to accelerate its rotation; for
as the planet, so to speak, lags behind the tides, friction would now
manifest itself by the continuous endeavour to drag the primary round
faster. The gain of speed, however, thus attained would involve the
primary in performing more than its original share of the moment of
momentum; less moment of momentum would therefore remain to be done by
the satellite, and the only way to accomplish this would be for the
satellite to come inwards and revolve in a smaller orbit.
We might indeed have inferred this from the considerations of energy
alone, for whatever happens in the deformation of the orbit, heat is
produced by the friction, and this heat is lost, and the total energy
of the system must consequently decline. Now if it be a consequence of
the tides that the velocity of the primary is accelerated, the energy
corresponding to that velocity is also increased. Hence the primary
has more energy than it had before; this energy must have been
obtained at the expense of the satellite; the satellite must therefore
draw inwards until it has yielded up enough of energy not alone to
account for the increased energy of the primary, but also for the
absolute loss of energy by which the whole operation is characterized.
It therefore appears that in the excessively remote future the retreat
of the moon will not only be checked, but that the moon may actually
return to a point to be determined by the changes in the earth's
rotation. It is, however, extremely difficult to follow up the study
of a case where the problem of three bodies has become even more
complicated than usual.
The importance of tidal evolution in our solar system has also to be
viewed in connection with the celebrated nebular hypothesis of the
origin of the solar system. Of course it would be understood that
tidal evolution is in no sense a rival doctrine to that of the nebular
theory. The nebular origin of the sun and the planets sculptured out
the main features of our system; tidal evolution has merely come into
play as a subsidiary agent, by which a detail here or a feature there
has been chiselled into perfect form. In the nebular theory it is
believed that the planets and the sun have all originated from the
cooling and the contraction of a mighty heated mass of vapours. Of
late years this theory, in its main outlines at all events, has
strengthened its hold on the belief of those who try to interpret
nature in the past by what we see in the present. The fact that our
system at present contains some heat in other bodies as well as in the
sun, and the fact that the laws of heat require continual loss by
radiation, demonstrate that our system, if we look back far enough,
and if the present laws have acted, must have had in part, at all
events, an origin like that which the nebular theory would suppose.
I feel that I have in the progress of these two lectures been only
able to give the merest outline of the theory of tidal evolution in
its application to the earth-moon system. Indeed I have been obliged,
by the nature of the subject, to omit almost entirely any reference to
a large body of the parts of the theory. I cannot bring myself to
close these lectures without just alluding to this omission, and
without giving expression to the fact, that I feel it is impossible
for me to have rendered adequate justice to the strength of the
argument on which we claim that tidal evolution is the most rational
mode of accounting for the present condition in which we find the
earth-moon system. Of course it will be understood that we have never
contended that the tides offer the only conceivable theory as to the
present condition of things. The argument lies in this wise. A certain
body of facts are patent to our observation. The tides offer an
explanation as to the origin of these facts. The tides are a _vera
causa_, and in the absence of other suggested causes, the tidal theory
holds the field. But much will depend on the volume and the
significance of the group of associated facts of which the doctrine
offers a solution. The facts that it has been in my power to discuss
within the compass of discourses like the present, only give a very
meagre and inadequate notion of the entire phenomena connected with
the moon which the tides will explain. We have not unfrequently, for
the sake of simplicity, spoken of the moon's orbit as circular, and we
have not even alluded to the fact that the plane of that orbit is
inclined to the ecliptic. A comprehensive theory of the moon's origin
should render an account of the eccentricity of the moon's orbit; it
must also involve the obliquity of the ecliptic, the inclination of
the moon's orbit, and the direction of the moon's axis. I have been
perforce compelled to omit the discussion of these attributes of the
earth-moon system, and in doing so I have inflicted what is really an
injustice on the tidal theory. For it is the chief claim of the theory
of tidal evolution, as expounded by Professor Darwin, that it links
together all these various features of the earth-moon system. It
affords a connected explanation, not only of the fact that the moon
always turns the same face to the earth, but also of the eccentricity
of the moon's path around the earth, and the still more difficult
points about the inclinations of the various axes and orbits of the
planets. It is the consideration of these points that forms the
stronghold of the doctrine of tidal evolution. For when we find that a
theory depending upon influences that undoubtedly exist, and are in
ceaseless action around us, can at the same time bring into connection
and offer a common explanation of a number of phenomena which would
otherwise have no common bond of union, it is impossible to refuse to
believe that such a theory does actually correspond to nature.
The greatest of mathematicians have ever found in astronomy problems
which tax, and problems which greatly surpass, the utmost efforts of
which they are capable. The usual way in which the powers of the
mathematician have been awakened into action is by the effort to
remove some glaring discrepancy between an imperfect theory and the
facts of observation. The genius of a Laplace or a Lagrange was
expended, and worthily expended, in efforts to show how one planet
acted on another planet, and produced irregularities in its orbit; the
genius of an Adams and a Leverrier was nobly applied to explain the
irregularities in the motion of Uranus, and to discover a cause of
those irregularities in the unseen Neptune. In all these cases, and in
many others which might be mentioned, the mathematician has been
stimulated by the laudable anxiety to clear away some blemish from the
theory of gravitation throughout the system. The blemish was seen to
exist before its removal was suggested. In that application of
mathematics with which we have been concerned in these lectures the
call for the mathematician has been of quite a different kind. A
certain familiar phenomenon on our sea-coasts has invited attention.
The tidal ripples murmur a secret, but not for every ear. To interpret
that secret fully, the hearer must be a mathematician. Even then the
interpretation can only be won after the profoundest efforts of
thought and attention, but at last the language has been made
intelligible. The labour has been gloriously rewarded, and an
interesting chapter of our earth's history has for the first time been
written.
In the progress of these lectures I have sought to interest you in
those profound investigations which the modern mathematician has made
in his efforts to explore the secrets of nature. He has felt that the
laws of motion, as we understand them, are bounded by no
considerations of space, are limited by no duration of time, and he
has commenced to speculate on the logical consequences of those laws
when time of indefinite duration is assumed to be at his disposal.
From the very nature of the case, observations for confirmation were
impossible. Phenomena that required millions of years for their
development cannot be submitted to the instruments in our
observatories. But this is perhaps one of the special reasons which
make such investigations of peculiar interest, and entitle us to speak
of the revelations of Time and Tide as a romance of modern science.
INDEX.
Aberdeen, tides at, 23
Action and reaction, 69
Adams, discoverer of Neptune, 186
Admiralty Manual of Scientific Inquiry, 30, 31, 34
Admiralty tide tables, 29
Africa, Krakatoa dust over, 84
Analysis, harmonic, of tides, 34
Analysis of tide into its constituents, 33
Ancient tides on moon, 136
Annus Magnus of solar system, 72
Areas, conservation of, 61
Arklow, tides at, 23
Ascension, tides at, 24
Astronomical chronology, 147
Atlantic, Krakatoa dust over, 84
Atlantic, tides in, 38
Atmosphere, tides in, 40
Avon, tides in, at Bristol, 39
Ball-room, illustration, 62
Barometric records of Krakatoa airwave, 82
Batavia, Krakatoa heard at, 83
Bath, hot waters at, 89
Bay of Fundy, 39
Beds of rock, how formed, 149
Birmingham, lecture at, 148
Birth of moon, 118
Blast iron furnace, 87
Blue sun produced by Krakatoa, 84
Bodily tides of moon, 131
Brake illustrating friction, 68
Brickwork as a non-conductor, 87
Bristol Channel, 39
Bucket of water, oscillations in, 98
Cambrian rocks, 153
Cannon-ball, energy of, 51
Cardiff, tides at, 39
Caspian Sea, tides in, 38
Casting, cooling of, 79
Celebes, Krakatoa heard at, 83
Central America, 86
Change and full, tides at, 22
Chemical action in earth, 81
Chepstow, tides at, 39
Chronology, the two systems of, 147
Clifton, tides in Avon at, 39
Clock, illustration of, 49
Coal, 11
Coincidence of moon's rotation and revolution, 134
Combustion, heat of, 92
Cones, volume of, 151
Conservation of spin or areas, 64
Constituent tides, 35
Consumption of energy by tides, 48
Cooling, laws of, 79
Cooling of earth from primitive heat, 87
Crane, brakes on, 69
Craters of moon, 94
Critical epoch in earth's history, 75
Darwin, 12
Darwin, G. H., in Admiralty Manual, 34
Darwin, G. H., on tidal evolution, 97, 118, 122, 151, 185
Day and month equal, 107
Day at present increasing, 68
Day of 1400 hours, 121
Day of 3 or 4 hours, 75
Decline of earth's heat, 90
Diagram showing why moon recedes, 70
Diego Garcia, Krakatoa heard at, 83
Difficulties of tidal evolution, 48
Dives and Lazarus, 161
Dust clouds, 84
Dynamical principle, 65
Dynamical stability, 110
Earth and moon as rigid body, 107
Earth a fly-wheel, 57
Earth, fusion of, 12
Earth, heat of, 80
Earth in highly heated early state, 92
Earth's crust, Lyell on, 12
Earth's history, 12
Earth-moon system originally, 107
Earthquakes, 80
Eccentricity of moon's path explained, 184
Ecliptic, obliquity of, explained, 184
Economic aspects of tides, 45
Egg on end, 108
Elliptic orbit of moon, 113
Endowment of moment of momentum, 167
Energy for tides, whence, 67
Energy lost by tides, 60
Energy of moon's position, 50
Energy of motion, 51
Energy of separation, 50
Energy, sources of, 49
Equality of day and month, 122
Equilibrium, stable and unstable, 108
Equinoxes, precession of, 72
Eruption of volcanoes, 80
Establishment, 30
Estuary, tides in, 152
Explosion of Krakatoa, 83
Extinct season, moon, 129
Fiji, 24
Fishes, periods of fossil, 12
Fitzgerald, Prof., on Hertz' experiments, 101
Floating log, 109
Fortnightly tide, 34
Friction brake, 69
Friction, tidal, 40, 47
Full and change, tides during, 22
Fundy, Bay of, tides in, 39
Gauge for tides, 31
Geological chronology, 146
Geology and tides, 144
Geometric series, 92
Geysers, 85
Gibraltar, Straits of, tides in, 38
Glimpse through the corridors of time, 148
Glorious sunsets from Krakatoa, 85
Greatest length of day, 121
Greatest length of month, 122
Greatest tides, 39
Green moons from Krakatoa dust, 84
Greenock, tides at, 22
Grindstone, rupture of, 76
Gunpowder, energy from, 51
Hall, Prof. A., discovers satellites of Mars, 142
Harmonic analyzer, 35
Heated body, cooling of, 79
Height of tide, how to measure, 29
Helmholtz explains the constant face of the moon, 136
Hertz, undulations of ether, 101
High water, is it under the moon? 21
High water, simple rules for, 26
High water twice a day, 18
Hot springs, 85
Hot waters of Bath, 89
Hour of high water, how found, 26
Howth, 25
Hull, tides at, 23
Ichthyosaurus, 148
Impulse, effect of timed, 101
Incandescence of earth's interior, 85
Incandescence of moon, 94
Indian ocean, 84
Initial condition of earth and moon, 76
Instability, dynamical, 110
Interval, luni-tidal, 30
Iron, machine for punching, 55
Iron smelting, 87
Java, 82
Jupiter, 141, 156
Kepler's Law, 53, 114
Kerguelen Island, 24
Kingstown, 21
Krakatoa, 58
Lagrange, 186
Language, origin of, 12
Laplace, 186
Laurentian rocks, 148
Law of tides, 15
Lazarus and Dives, 161
Leakage of heat, 86
Length of day increasing, 68
Leverrier discovers Neptune, 186
London, tides at, 23, 27
Lunar and solar tides compared, 15
Lunar diurnal tide, 34
Luni-tidal interval, 30
Lyell, 12
Macassar, explosion of Krakatoa heard at, 83
Machine for analyzing tides, 35
Machine for predicting tides, 35
Machine for observing tides, 32
Malahide estuary, 152
Mars, moons of, 142
Mass of moon found from tides, 16
Max Müller, 12
Maximum and minimum, 104
Mediterranean, tides in, 38
Meridian position of moon at high water, 26
Mills, tidal, 43
Mines, heat in, 86
Moment of momentum, how estimated, 170
Month equals day, 122
Month of 1400 hours, 122
Month with greatest number of days, 118
Moon and tides connected, 13
Moon, constant face of, 136
Moon-energy, 54
Moon, volcanic activity on, 94
Moons of Mars, 142
Motion, perpetual, 48
Nautical almanac, 26
Neap tides, 16
Needles, tides at, 23
Niagara, utilization of, 47
Noise from Krakatoa, 83
Numerical data of tidal evolution, 127
Obliquity of ecliptic and tides, 184
Observations of tides, 29
Ocean tides, 38
Orbit of earth, changes in, 72
Orbit of moon explained by tides, 184
Origin of moon, 100
Oscillations of water, 98
Pacific, 84
Paradox of geometric series, 92
Pendulum, motion of, 49
Periodic phenomena, 71
Periods of rotation and revolution of moon equal, 43
Periods of tides, 34
Perpetual motion, 48
Planets and tides, 155
Precession of equinoxes, 72
Prediction of tides, 35
Principal tides, 33
Punching-engine, 55
Purser, Professor, 61
Railway brakes, 68
Reaction and action, 69
Relative rotation, 104
Relative tides on earth and moon, 131
Reptiles, fossil, 11
Retreat of moon explained, 69
Retrospect of moon's history, 77
Rhine, mills on, 43
Ring theory of moon's origin, 96
Rivers, 149
Rocks, formation of, 150
Rodriguez, Krakatoa heard at, 83
Rolling mills, 56
Rotation of moon on its axis, 133
Rupture of earth, 76
Rupture of grindstone, 75
St. Helena, tides at, 38
Santa Cruz, 24
Satellites of Jupiter, 135
Saturn, 141
Seasons, 72
Seas, so called, on moon, 129
Secondary ages, 11
Sligo, tides at, 23
Small tides, 38
Smelting of iron, 87
Solar and lunar tides compared, 15
Solar system, 155
Solar tides, 14
Solar tides, ultimate importance of, 139
Sounds from Krakatoa, 83
Source of tidal energy, 67
Spring tides, 16
Stability, dynamical, 110
Stable equilibrium, 108
Stages of special importance, 117
Stalactites, 73
Steam engine and tides compared, 45
Strata, formation of, aided by tides, 154
Sumatra, 82
Sun, rotation of, 134
Sunbeams stored, 11
Sunda, 82
Sunsets, Krakatoa, 85
Table of luni-tidal corrections, 30
Telescope at Washington, 143
Temperature of space, 91
Tertiary ages, 11
Thames, tides in, 125
Thomson, Sir W., 36, 44
Tidal currents used in making rocks, 153
Tidal efficiency, how estimated, 14
Tidal evolution, to whom due, 97
Tide gauge, 31
Tide mills, 44
Tide predicting engine, 36
Tides and geology, 144
Tides at London bridge, 28
Tides, change and full, 22
Tides, commercial value of, 47
Tides, due to moon, 13
Tides, how to observe, 29
Tides in Jupiter, 138
Tides in moon, 128
Tides, small, 38
Tides, solar and lunar compared, 15
Tides varying inversely as cube of distance, 15
Time and tide, 12
Timor, Krakatoa heard at, 83
Tralee, tides at, 23
Tyndall, 73
Tynemouth, 23
Undulations of a fluid globe, 99
Undulations of air from Krakatoa, 82
Unintermitting phenomena, 71
Unstable equilibrium, 108
Uranus, 142
Vast tides, 39
Volcanoes, 80
Voyage of Krakatoa dust, 84
Washington reflector, 142
Water absent from moon, 129
Water-wheels under London bridge, 43
Waves of air from Krakatoa, 82
Whewell on tides, 30
Work done by tides, 42, 47
Work of tides, 42
Yarmouth, tides at, 23
THE END.
_Richard Clay & Sons, Limited, London & Bungay._
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