indicated by A and B. The sun attracts a pound of matter at B more than it
attracts a pound of matter at A, because B is nearer than A, and hence the
total resultant force is in the direction M N rather than O O, through the
centre of the earth's mass. But we know that a force in the direction M N
is equivalent to a force O O parallel to M N, together with a tilting
couple of forces tending to turn the equator edge on to the sun. You will
get the true result as to the tilting tendency by imagining the earth to be
motionless, and the sun's mass to be distributed as a circular ring of
matter 184 millions of miles in diameter, inclined to the equator at
23-1/2deg. Under the influence of the attraction of this ring the earth
would heave like a great ship on a calm sea, rolling very slowly; in fact,
making one complete swing in about three years. But the earth is spinning,
and the tilting couple or torque acts upon it just like the forces which
are always tending to cause this ship-model to stand upright, and hence it
has a precessional motion whose complete period is 26,000 years. When there
is no spin in the ship, its complete oscillation takes place in three
seconds, and {85} when I spin the gyrostat on board the ship, the complete
period of its precession is two minutes. In both cases the effect of the
spin is to convert what would be an oscillation into a very much slower
precession.
There is, however, a great difference between the earth and the gyrostat.
The forces acting on the top are always the same, but the forces acting on
the earth are continually altering. At midwinter and midsummer the tilting
forces are greatest, and at the equinoxes in spring and autumn there are no
such forces. So that the precessional motion changes its rate every quarter
year from a maximum to nothing, or from nothing to a maximum. It is,
however, always in the same direction--the direction opposed to the earth's
spin. When we speak then of the precessional motion of the earth, we
usually think of the mean or average motion, since the motion gets quicker
and slower every quarter year.
Further, the moon is like the sun in its action. It tries to tilt the
equatorial part of the earth into the plane of the moon's orbit. The plane
of the moon's orbit is nearly the same as that of the ecliptic, and hence
the average precession of the earth is of much the same kind as if only one
of the two, the moon or the sun, alone acted. That is, the general
phenomenon of precession of the {86} earth's axis in a conical path in
26,000 years is the effect of the combined tilting actions of the sun and
moon.
You will observe here an instance of the sort of untruth which it is almost
imperative to tell in explaining natural phenomena. Hitherto I had spoken
only of the sun as producing precession of the earth. This was convenient,
because the plane of the ecliptic makes always almost exactly 23-1/2deg
with the earth's equator, and although on the whole the moon's action is
nearly identical with that of the sun, and about twice as great, yet it
varies considerably. The superior tilting action of the moon, just like its
tide-producing action, is due to its being so much nearer us than the sun,
and exists in spite of the very small mass of the moon as compared with
that of the sun.
As the ecliptic makes an angle of 23-1/2deg with the earth's equator, and
the moon's orbit makes an angle 5-1/2deg with the ecliptic, we see that the
moon's orbit sometimes makes an angle of 29deg with the earth's equator,
and sometimes only 18deg, changing from 29deg to 18deg, and back to 29deg
again in about nineteen years. This causes what is called "Nutation," or
the nodding of the earth, for the tilting action due to the sun is greatly
helped and greatly modified by it. The result of the variable nature of the
moon's action is then that the earth's axis {87} rotates in an elliptic
conical path round what might be called its mean position. We have also to
remember that twice in every lunar month the moon's tilting action on the
earth is greater, and twice it is zero, and that it continually varies in
value.
On the whole, then, the moon and sun, and to a small extent the planets,
produce the general effect of a precession, which repeats itself in a
period of about 25,695 years. It is not perfectly uniform, being performed
at a speed which is a maximum in summer and winter; that is, there is a
change of speed whose period is half a year; and there is a change of speed
whose period is half a lunar month, the precession being quicker to-night
than it will be next Saturday, when it will increase for about another
week, and diminish the next. Besides this, because of 5-1/2deg of
angularity of the orbits, we have something like the nodding of our
precessing gyrostat, and the inclination of the earth's axis to the
ecliptic is not constant at 23-1/2deg, but is changing, its periodic time
being nineteen years. Regarding the earth's centre as fixed at O we see
then, as illustrated in this model and in Fig. 40, the axis of the earth
describes almost a perfect circle on the celestial sphere once in 25,866
years, its speed fluctuating every half year and every half month. But it
is not a perfect circle, it is really a wavy {88} line, there being a
complete wave every nineteen years, and there are smaller ripples in it,
corresponding to the half-yearly and fortnightly periods. But the very
cause of the nutation, the nineteen-yearly period of retrogression of the
moon's nodes, as it is called, is itself really produced as the precession
of a gyrostat is produced, that is, by tilting forces acting on a spinning
body.
[Illustration: FIG. 40.]
Imagine the earth to be stationary, and the sun and moon revolving round
it. It was Gauss who found that the present action is the same as if the
masses of the moon and sun were distributed all {89} round their orbits.
For instance, imagine the moon's mass distributed over her orbit in the
form of a rigid ring of 480,000 miles diameter, and imagine less of it to
exist where the present speed is greater, so that the ring would be thicker
at the moon's apogee, and thinner at the perigee. Such a ring round the
earth would be similar to Saturn's rings, which have also a precession of
nodes, only Saturn's rings are not rigid, else there would be no
equilibrium. Now if we leave out of account the earth and imagine this ring
to exist by itself, and that its centre simply had a motion round the sun
in a year, since it makes an angle of 5-1/2deg with the ecliptic it would
vibrate into the ecliptic till it made the same angle on the other side and
back again. But it revolves once about its centre in twenty-seven solar
days, eight hours, and it will no longer swing like a ship in a
ground-swell, but will get a motion of precession opposed in direction to
its own revolution. As the ring's motion is against the hands of a watch,
looking from the north down on the ecliptic, this retrogression of the
moon's nodes is in the direction of the hands of a watch. It is exactly the
same sort of phenomenon as the precession of the equinoxes, only with a
much shorter period of 6798 days instead of 25,866 years.
I told you how, if we knew the moon's mass or the sun's, we could tell the
amount of the forces, or {90} the torque as it is more properly called,
with which it tries to tilt the earth. We know the rate at which the earth
is spinning, and we have observed the precessional motion. Now when we
follow up the method which I have sketched already, we find that the
precessional velocity of a spinning body ought to be equal to the torque
divided by the spinning velocity and by the moment of inertia[7] of the
body about the polar axis. Hence the greater the tilting forces, and the
less the spin and the less the moment of inertia, the greater is the
precessional speed. Given all of these elements except one, it is easy to
calculate that unknown element. Usually what we aim at in such a
calculation is the determination of the moon's mass, as this phenomenon of
precession and the action of the tides are the only two natural phenomena
which have as yet enabled the moon's mass to be calculated.
I do not mean to apologize to you for the introduction of such terms as
_Moment of Inertia_, nor do I mean to explain them. In this lecture I have
avoided, as much as I could, the introduction of mathematical expressions
and the use of technical terms. But I want you to {91} understand that I am
not afraid to introduce technical terms when giving a popular lecture. If
there is any offence in such a practice, it must, in my opinion, be greatly
aggravated by the addition of explanations of the precise meanings of such
terms. The use of a correct technical term serves several useful purposes.
First, it gives some satisfaction to the lecturer, as it enables him to
state, very concisely, something which satisfies his own weak inclination
to have his reasoning complete, but which he luckily has not time to
trouble his audience with. Second, it corrects the universal belief of all
popular audiences that they know everything now that can be said on the
subject. Third, it teaches everybody, including the lecturer, that there is
nothing lost and often a great deal gained by the adoption of a casual
method of skipping when one is working up a new subject.
Some years ago it was argued that if the earth were a shell filled with
liquid, if this liquid were quite frictionless, then the moment of inertia
of the shell is all that we should have to take into account in considering
precession, and that if it were viscous the precession would very soon
disappear altogether. To illustrate the effect of the moment of inertia, I
have hung up here a number of glasses--one _a_ filled with sand, another
_b_ with treacle, a third _c_ with oil, the fourth _d_ with water, {92}
[Illustration: FIG. 41.]
{93} and the fifth _e_ is empty (Fig. 41). You see that if I twist these
suspending wires and release them, a vibratory motion is set up, just like
that of the balance of a watch. Observe that the glass with water vibrates
quickly, its effective moment of inertia being merely that of the glass
itself, and you see that the time of swing is pretty much the same as that
of the empty glass; that is, the water does not seem to move with the
glass. Observe that the vibration goes on for a fairly long time.
The glass with sand vibrates slowly; here there is great moment of inertia,
as the sand and glass behave like one rigid body, and again the vibration
goes on for a long time.
In the oil and treacle, however, there are longer periods of vibration than
in the case of the water or empty glass, and less than would be the case if
the vibrating bodies were all rigid, but the vibrations are stilled more
rapidly because of friction.
Boiled (_f_) and unboiled (_g_) eggs suspended from wires in the same way
will exhibit the same differences in the behaviour of bodies, one of which
is rigid and the other liquid inside; you see how much slower an
oscillation the boiled has than the unboiled.
Even on the table here it is easy to show the difference between boiled and
unboiled eggs. {94} Roll them both; you see that one of them stops much
sooner than the other; it is the unboiled one that stops sooner, because of
its internal friction.
I must ask you to observe carefully the following very distinctive test of
whether an egg is boiled or not. I roll the egg or spin it, and then place
my finger on it just for an instant; long enough to stop the motion of the
shell. You see that the boiled egg had quite finished its motion, but the
unboiled egg's shell alone was stopped; the liquid inside goes on moving,
and now renews the motion of the shell when I take my finger away.
It was argued that if the earth were fluid inside, the effective moment of
inertia of the shell being comparatively small, and having, as we see in
these examples, nothing whatever to do with the moment of inertia of the
liquid, the precessional motion of the earth ought to be enormously quicker
than it is. This was used as an argument against the idea of the earth's
being fluid inside.
We know that the observed half-yearly and half-monthly changes of the
precession of the earth would be much greater than they are if the earth
were a rigid shell containing much liquid, and if the shell were not nearly
infinitely rigid the phenomena of the tides would not occur, but in regard
to the general precession of the earth there is now {95} no doubt that the
old line of argument was wrong. Even if the earth were liquid inside, it
spins so rapidly that it would behave like a rigid body in regard to such a
slow phenomenon as precession of the equinoxes. In fact, in the older line
of argument the important fact was lost sight of, that rapid rotation can
give to even liquids a quasi-rigidity. Now here (Fig. 42 _a_) is a hollow
brass top filled with water. The frame is light, and the water inside has
much more mass than the outside frame, and if you test this carefully you
will find that the top spins in almost exactly the same way as if the water
were quite rigid; in fact, as if the whole top were rigid. Here you see it
spinning and precessing just like any rigid top. This top, I know, is not
filled with water, it is only partially filled; but whether partially or
wholly filled it spins very much like a rigid top.
[Illustration: FIG. 42.]
{96}
This is not the case with a long hollow brass top with water inside. I told
you that all bodies have one axis about which they prefer to rotate. The
outside metal part of a top behaves in a way that is now well known to you;
the friction of its peg on the table compels it to get up on its longer
axis. But the fluid inside a top is not constrained to spin on its longer
axis of figure, and as it prefers its shorter axis like all these bodies I
showed you, it spins in its own way, and by friction and pressure against
the case constrains the case to spin about the shorter axis, annulling
completely the tendency of the outside part to rise or keep up on its long
axis. Hence it is found to be simply impossible to spin a long hollow top
when filled with water.
[Illustration: FIG. 43.]
[Illustration: FIG. 44.]
Here, for example, is one (Fig. 42 _b_) that only differs from the last in
being longer. It is filled, or partially filled, with water, and you
observe that if {97} I slowly get up a great spin when it is mounted in
this frame, and I let it out on the table as I did the other one, this one
lies down at once and refuses to spin on its peg. This difference of
behaviour is most remarkable in the two hollow tops you see before you
(Fig. 43). They are both nearly spherical, both filled with water. They
look so nearly alike that few persons among the audience are able to detect
any difference in their shape. But one of them (_a_) is really slightly
oblate like an orange, and the other (_b_) is slightly prolate like a
lemon. I will give them both a gradually increasing rotation in this frame
{98} (Fig. 44) for a time sufficient to insure the rotation of the water
inside. When just about to be set free to move like ordinary tops on the
table, water and brass are moving like the parts of a rigid top. You see
that the orange-shaped one continues to spin and precess, and gets itself
upright when disturbed, like an ordinary rigid top; indeed I have seldom
seen a better behaved top; whereas the lemon-shaped one lies down on its
side at once, and quickly ceases to move in any way.
[Illustration: FIG. 45.]
And now you will be able to appreciate a fourth test of a boiled egg, which
is much more easily seen by a large audience than the last. Here is the
unboiled one (Fig. 45 _b_). I try my best to spin it as it lies on the
table, but you see that I cannot give it much spin, and so there is nothing
of any importance to look at. But you observe that it is quite easy to spin
the boiled {99} egg, and that for reasons now well known to you it behaves
like the stones that Thomson spun on the sea-beach; it gets up on its
longer axis, a very pretty object for our educated eyes to look at (Fig. 45
_a_). You are all aware, from the behaviour of the lemon-shaped top, that
even if, by the use of a whirling table suddenly stopped, or by any other
contrivance, I could get up a spin in this unboiled egg, it would never
make the slightest effort to rise on its end and spin about its longer
axis.
I hope you don't think that I have been speaking too long about
astronomical matters, for there is one other important thing connected with
astronomy that I must speak of. You see, I have had almost nothing
practically to do with astronomy, and hence I have a strong interest in the
subject. It is very curious, but quite true, that men practically engaged
in any pursuit are almost unable to see the romance of it. This is what the
imaginative outsider sees. But the overworked astronomer has a different
point of view. As soon as it becomes one's duty to do a thing, and it is
part of one's every-day work, the thing loses a great deal of its interest.
We have been told by a great American philosopher that the only coachmen
who ever saw the romance of coach-driving are those titled individuals who
pay nowadays so largely for the {100} privilege. In almost any branch of
engineering you will find that if any invention is made it is made by an
outsider; by some one who comes to the study of the subject with a fresh
mind. Who ever heard of an old inhabitant of Japan or Peru writing an
interesting book about those countries? At the end of two years' residence
he sees only the most familiar things when he takes his walks abroad, and
he feels unmitigated contempt for the ingenuous globe-trotter who writes a
book about the country after a month's travel over the most beaten tracks
in it. Now the experienced astronomer has forgotten the difficulties of his
predecessors and the doubts of outsiders. It is a long time since he felt
that awe in gazing at a starry sky that we outsiders feel when we learn of
the sizes and distances apart of the hosts of heaven. He speaks quite
coolly of millions of years, and is nearly as callous when he refers to the
ancient history of humanity on our planet as a weather-beaten geologist.
The reason is obvious. Most of you know that the _Nautical Almanac_ is as a
literary production one of the most uninteresting works of reference in
existence. It is even more disconnected than a dictionary, and I should
think that preparing census-tables must be ever so much more romantic as an
occupation than preparing the tables of the _Nautical Almanac_. And yet
{101} a particular figure, one of millions set down by an overworked
calculator, may have all the tragic importance of life or death to the crew
and passengers of a ship, when it is heading for safety or heading for the
rocks under the mandate of that single printed character.
But this may not be a fair sort of criticism. I so seldom deal with
astronomical matters, I know so little of the wear and tear and monotony of
the every-day life of the astronomer, that I do not even know that the
above facts are specially true about astronomers. I only know that they are
very likely to be true because they are true of other professional men.
I am happy to say that I come in contact with all sorts and conditions of
men, and among others, with some men who deny many of the things taught in
our earliest school-books. For example, that the earth is round, or that
the earth revolves, or that Frenchmen speak a language different from ours.
Now no man who has been to sea will deny the roundness of the earth,
however greatly he may wonder at it; and no man who has been to France will
deny that the French language is different from ours; but many men who
learnt about the rotation of the earth in their school-days, and have had a
plentiful opportunity of observing the heavenly bodies, deny the rotation
of the earth. {102} They tell you that the stars and moon are revolving
about the earth, for they see them revolving night after night, and the sun
revolves about the earth, for they see it do so every day. And really if
you think of it, it is not so easy to prove the revolution of the earth. By
the help of good telescopes and the electric telegraph or good
chronometers, it is easy to show from the want of parallax in stars that
they must be very far away; but after all, we only know that either the
earth revolves or else the sky revolves.[8] Of course, it seems infinitely
more likely that the small earth should revolve than that the whole
heavenly host should turn about the earth as a centre, and infinite
likelihood is really absolute proof. Yet there is nobody who does not
welcome an independent kind of proof. The phenomena of the tides, and
nearly every new astronomical fact, may be said to be an addition to the
proof. Still there is the absence of perfect certainty, and when we are
told that these spinning-top phenomena give us a real proof of the rotation
of the earth without our leaving the room, we welcome {103} it, even
although we may sneer at it as unnecessary after we have obtained it.
[Illustration: FIG. 17.]
You know that a gyrostat suspended with perfect freedom about axes, which
all pass through its centre of gravity, maintains a constant direction in
space however its support may be carried. Its axis is not forced to alter
its direction in any way. Now this gyrostat (Fig. 17) has not the perfect
absence of friction at its axes of which I speak, and even the slightest
friction will produce some constraint which is injurious to the experiment
I am about to describe. It must be remembered, that if there were
absolutely no constraint, then, even if the {104} gyrostat were _not_
spinning, its axis would keep a constant direction in space. But the
spinning gyrostat shows its superiority in this, that any constraint due to
friction is less powerful in altering the axis. The greater the spin, then,
the better able are we to disregard effects due to friction. You have seen
for yourselves the effect of carrying this gyrostat about in all sorts of
ways--first, when it is not spinning and friction causes quite a large
departure from constancy of direction of the axis; second, when it is
spinning, and you see that although there is now the same friction as
before, and I try to disturb the instrument more than before, the axis
remains sensibly parallel to itself all the time. Now when this instrument
is supported by the table it is really being carried round by the earth in
its daily rotation. If the axis kept its direction perfectly, and it were
now pointing horizontally due east, six hours after this it will point
towards the north, but inclining downwards, six hours afterwards it will
point due west horizontally, and after one revolution of the earth it will
again point as it does now. Suppose I try the experiment, and I see that it
points due east now in this room, and after a time it points due west, and
yet I know that the gyrostat is constantly pointing in the same direction
in space all the time, surely it is obvious that the room must {105} be
turning round in space. Suppose it points to the pole star now, in six
hours, or twelve, or eighteen, or twenty-four, it will still point to the
pole star.
Now it is not easy to obtain so frictionless a gyrostat that it will
maintain a good spin for such a length of time as will enable the rotation
of the room to be made visible to an audience. But I will describe to you
how forty years ago it was proved in a laboratory that the earth turns on
its axis. This experiment is usually connected with the name of Foucault,
the same philosopher who with Fizeau showed how in a laboratory we can
measure the velocity of light, and therefore measure the distance of the
sun. It was suggested by Mr. Lang of Edinburgh in 1836, although only
carried out in 1852 by Foucault. By these experiments, if you were placed
on a body from which you could see no stars or other outside objects, say
that you were living in underground regions, you could discover--first,
whether there is a motion of rotation, and the amount of it; second, the
meridian line or the direction of the true north; third, your latitude.
Obtain a gyrostat like this (Fig. 46) but much larger, and far more
frictionlessly suspended, so that it is free to move vertically or
horizontally. For the vertical motion your gymbal pivots ought to be hard
steel knife-edges. {106}
[Illustration: FIG. 46.]
As for the horizontal freedom, Foucault used a fine steel wire. Let there
be a fine scale engraved crosswise on the outer gymbal ring, and try to
discover if it moves horizontally by means of a microscope with cross
wires. When this is carefully done we find that there is a motion, {107}
but this is not the motion of the gyrostat, it is the motion of the
microscope. In fact, the microscope and all other objects in the room are
going round the gyrostat frame.